![PUBLICATIONS
This thesis is based on the following publications:
R. Dijkgraaf, L. Hollands, P. Sułkowski and C. Vafa,
Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions,
arXiv/0709.4446 [hep-th], JHEP 02 (2008) 106.
L. Hollands, J. Marsano, K. Papadodimas and M. Shigemori,
Nonsupersymmetric Flux Vacua and Perturbed N=2 Systems,
arXiv/0804.4006 [hep-th], JHEP 10 (2008) 102.
R. Dijkgraaf, L. Hollands and P. Sułkowski,
Quantum Curves and D-Modules,
arXiv/0810.4157 [hep-th].
M. Cheng and L. Hollands,
A Geometric Derivation of the Dyon Wall-Crossing Group,
arXiv/0901.1758 [hep-th], JHEP 04 (2009) 067.](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-10-320.jpg)
![1.1. Fermions on Riemann surfaces 5
Figure 1.3: The two-dimensional plane R2
seen as an integrable system.
of the 19th century as a non-linear partial differential equation
∂u
∂t
=
∂3
u
∂x3
+ 6u
∂u
∂x
,
it was only realized later that it is part of a very rich geometric and algebraic
structure. It is impossible to do justice to this beautiful story in this short in-
troduction. Instead, let us just touch on the aspects that are relevant for this
thesis.
Geometrically, a special class of solutions to the KdV differential equation yields
linear flows over a 2g-dimensional torus that is associated to a Riemann surface
Σ. This torus is called the Jacobian. Its complex structure is determined by the
period matrix τij of Σ. The Riemann surface Σ is called the spectral curve of the
KdV hierarchy.
The KdV spectral curve is an elliptic curve
Q2
= P3
− g2P − g3,
where g2 and g3 are the Weierstrass invariants. The coordinates P and Q on this
elliptic curve can best be described as commuting differential operators
[P, Q] = 0,
that arise naturally when the KdV differential equation is written in the form of
a Hamiltonian system.
As a side remark we notice that the KdV system is closely related to the Hitchin
integrable system, which studies certain holomorphic bundles over a complex
curve C. In this integrable system, too, the dynamics can be expressed in terms
of a linear flow on the Jacobian associated to a spectral cover Σ of C.
We go one step beyond the geometrical structure of the KdV system when intro-](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-18-320.jpg)
![1.2. Outline of this thesis 9
shown that the partition function of this theory is the tau-function of an inte-
grable hierarchy.
In the theory of random matrices, λ-corrections correspond to finite N correc-
tions. In this theory, too, it has been found that the partition function behaves
as the tau-function of an integrable hierarchy. However, like in the gauge theory
examples, there is no interpretation of the full partition function as a fermion
determinant. Instead, a new perspective arises, in which the spectral curve is
be replaced by a non-commutative spectral curve. In matrix models that relate
to the KdV integrable system, it is found that the KdV differential operators P
and Q do not commute anymore when the quantum corrections are taken into
account:
[P, Q] = λ.
Similar hints have been found in the study of topological string theory on general
local Calabi-Yau’s modeled on a Riemann surface.
This thesis introduces a new physical perspective to study topological string the-
ory, clarifying the interpretation of the parameter λ as a “non-commutativity
parameter”. We show that λ quantizes the Riemann surface Σ, and that fermions
on the curve are sections of a so-called D-module instead. This attributes Fourier-
like transformation properties to the fermions on Σ:
ψ(y) =
Z
dy exy/λ
ψ(x).
1.2 Outline of this thesis
Chapter 2 starts with an introduction to the geometry of Calabi-Yau threefolds.
They serve as an important class of backgrounds in string theory, that can be
used to make contact with the 4-dimensional world we perceive. Specifically,
we explain the idea of a Calabi-Yau compactification, and introduce certain non-
compact Calabi-Yau backgrounds that appear in many guises in this thesis.
Topological string theory on these non-compact Calabi-Yau backgrounds is of
physical relevance in the description of 4-dimensional supersymmetric gauge
theories and supersymmetric black holes. For example, certain holomorphic
corrections to supersymmetric gauge theories are known to have an elegant de-
scription in terms of an auxiliary Riemann surface Σ. A compactification of string
theory on a local Calabi-Yau that is modeled on this Riemann surface “geometri-
cally engineers” the corresponding gauge theory in four dimensions. Topological
string amplitudes capture the holomorphic corrections.
In Chapter 3 and 4 we elucidate these relations by introducing a web of dualities](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-22-320.jpg)
![1.2. Outline of this thesis 11
as a Krichever solution.
In Chapter 5 we explain how to interpret the topological string partition function
on a local Calabi-Yau XΣ as a quantum deformation of a Krichever solution. We
discover that the curve Σ should be replaced by a quantum curve, defined as a
differential operator P(x) obtained by quantizing y = λ∂/∂x. Mathematically,
we are led to a novel description of topological string theory in terms of D-
modules.
Chapter 6 illustrates this formalism with several examples related to matrix mod-
els and gauge theory. In all these examples there is a canonical way to quantize
the curve as a differential operator. Moreover, we can simply read off the known
partition functions from the associated D-modules. This tests our proposal.
The topological string partition function has several interpretations in terms of
topological invariants on a Calabi-Yau threefold. Moreover, its usual expansion
in the topological string coupling constant λ is only valid in a certain regime of
the background Calabi-Yau parameters. Although the topological invariants stay
invariant under small deformations in these background parameters, there are
so-called walls of marginal stability, where the index of these invariants may
jump. This phenomenon has recently attracted much attention among both
physicists and mathematicians.
Chapter 7 studies wall-crossing in string theory compactifications that preserve
N = 4 supersymmetry. The invariants in this theory that are sensitive to wall-
crossing are called quarter BPS states. They have been studied extensively in the
last years. In Chapter 7 we work out in detail the relation of these supersym-
metric states to a genus 2 surface Σ. Moreover, we find that wall-crossing in this
theory has a simple and elegant interpretation in terms of the genus 2 surface.
Finally, Chapter 8 focuses on supersymmetry breaking in Calabi-Yau compactifi-
cations. Since supersymmetry is not an exact symmetry of nature, we need to
go beyond Calabi-Yau compactifications to find a more accurate description of
nature. One of the possibilities is that supersymmetry is broken at a lower scale
than the compactification scale. In Chapter 8 we study such a supersymmetry
breaking mechanism for N = 2 supersymmetric gauge theories. These gauge
theories are related to type II string compactifications on a local Calabi-Yau XΣ.
By turning on non-standard fluxes (in the form of generalized gauge fields) on
the underlying Riemann surface Σ we find a potential on the moduli space of the
theory. We show that this potential generically has non-supersymmetric minima.
Chapter 3 and 4 explain the ideas of the publication [1], whereas Chapter 5 and 6
contain the results of [2]. The first part of Chapter 7 includes examples from [1],
whereas the second part is based on the article [3]. Finally, Chapter 8 is a slightly
shortened version of the work [4].](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-24-320.jpg)
![2.1. The Strominger-Yau-Zaslow conjecture 17
form k and the holomorphic 3-form Ω determine a Calabi-Yau manifold.
The only non-trivial cohomology of a Calabi-Yau X is contained in H1,1
(X) and
H2,1
(X). These cohomology classes parametrize its moduli. The elements in
H1,1
(X) can change the Kähler structure of the Calabi-Yau infinitesimally, and
are therefore called Kähler moduli. In string theory we usually complexify these
moduli. On the other hand, H2,1
(X) parametrizes the complex structure moduli
of the Calabi-Yau. It is related to the choice of the holomorphic 3-form Ω, since
contracting (2, 1)-forms with Ω determines an isomorphism with the cohomology
class H1
¯
∂
(X), that is well-known to characterize infinitesimal complex structure
deformations.
Celebrated work of E. Calabi and S.-T. Yau shows that the Kähler metric g can
be tuned within its cohomology class to a unique Kähler metric that satisfies
Rmn = 0. Calabi-Yau manifolds can therefore also be characterized by a unique
Ricci-flat Kähler metric.
A tantalizing question is how to visualize a Calabi-Yau threefold. The above
definition in terms of a Kähler form k and a holomorphic 3-form Ω is rather
abstract. Is there a more concrete way to picture a Calabi-Yau manifold?
The SYZ conjecture
From the string theory perspective there are many relations between different
string theory set-ups. These are known as dualities, and relate the different in-
carnations of string theory (type I, type IIA/B, heterotic, M-theory and F-theory)
and different background parameters. One of the famous dualities in string the-
ory connects type IIB string theory compactified on some Calabi-Yau X with type
IIA string theory compactified on another Calabi-Yau X̃, where both Calabi-Yau
manifolds are related by swapping their Kähler and complex structure param-
eters. This duality is called mirror symmetry. It suggests that any Calabi-Yau
threefold X has a mirror X̃ such that
H1,1
(X) ∼
= H2,1
(X̃) and H2,1
(X) ∼
= H1,1
(X̃).
Although the mirror conjecture doesn’t hold exactly as it is stated above, many
mirror pairs have been found and much intuition has been obtained about the
underlying Calabi-Yau geometry.
One of the examples that illustrates this best is known as the Strominger-Yau-
Zaslow (SYZ) conjecture [5]. The starting point for this argument is mirror sym-
metry in type II string theory. Without explaining in detail what type II string
theory is, we just note that an important role in this theory is played by so-called
D-branes. These branes can be considered mathematically as submanifolds of
the Calabi-Yau manifold with certain U(1) bundles on them.](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-30-320.jpg)
![20 Chapter 2. Calabi-Yau Geometry
Figure 2.3: In picture (1) we see a regular torus with complex structure τ: below the torus
itself and above its fundamental domain (that is obtained by removing the A and B-cycle
from the torus). In the large complex structure limit τ is sent to i∞. In picture (2a) this
degeneration is illustrated topologically and in (2b) metrically: when we hold the volume
fixed, the torus collapses to a circle.
Moreover, the singular locus of the fibration is thought to be of codimension two
in the base in that limit, so that it defines a 1-dimensional graph on the base of
the SYZ fibration.
In recent years much progress has been made in the mathematical study of the
SYZ conjecture. Comprehensive reviews can be found in e.g. [6, 7, 8]. Especially
the topological approach of M. Gross and others, with recent connections to
tropical geometry, seems very promising. The main idea in this program is to
characterize the SYZ fibration by a natural affine structure on its base B.
This affine structure is found as follows. Once we pick a zero-section of the (in
particular) Lagrangian fibration, we can contract the inverse symplectic form
with a one-form on the base B to find a vector field along the fiber. An integral
affine structure can then be defined on B by an integral affine function f whose
derivative df defines a time-one flow along the fiber that equals the identity. The
affine structure may be visualized as a lattice on the base B.
Although this affine structure will be trivial in a smooth open region of the fi-
bration, it contains important information near the singular locus. Especially
the monodromy of the T3
-fibers is encoded in it. In the large complex structure
limit all non-trivial symplectic information of the SYZ fibration is thought to be
captured in the monodromy of the T3
-fibers around the discriminant locus of the
fibration [9, 7]. The SYZ conjecture thus leads to a beautiful picture of how a
Calabi-Yau threefold may be visualized as an affine real 3-dimensional space with
an additional integral monodromy structure around an affine 1-dimensional de-
generation locus (see also [10]).
One of the concrete results that have been accomplished is an SYZ description](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-33-320.jpg)
![2.2. The Fermat quintic 21
of elliptic K3-surfaces [11] and Calabi-Yau hypersurfaces in toric varieties (in a
series of papers by W.-D. Ruan starting with [12]). Let us illustrate this with
the prime example of a compact Calabi-Yau threefold, the Fermat quintic. This
is also the first Calabi-Yau for which a mirror pair was found [13]. On a first
reading it is not necessary to go through all of the formulas in this example;
looking at the pictures should be sufficient.
2.2 The Fermat quintic
The Fermat quintic is defined by an equation of degree 5 in projective space P4
:
Xµ :
5
X
k=1
z5
k − 5µ
5
Y
k=1
zk = 0,
where [z] = [z1 : . . . : z5] are projective coordinates on P4
. Here µ denotes the
complex structure of the threefold.
Pulling back the Kähler form of P4
provides the Fermat quintic with a Kähler
structure. Moreover, the so-called adjunction formula shows that Xµ has a trivial
canonical bundle. So the Fermat quintic Xµ is a Calabi-Yau threefold. Note as
well that P4
admits an action of T4
that is parametrized by five angles θk with
P
k θk = 0:
T4
: [z1 : . . . : z5] 7→ [eiθ1
z1 : . . . : eiθ5
z5]
Such a variety is called toric. The quintic is thus embedded as a hypersurface in
a toric variety.
When we take µ → ∞ we reach the large complex structure limit point
X∞ :
5
Y
k=1
zk = 0.
This is just a union of five P3
’s, that each inherit a T3
-action from the above toric
action on P4
. From this observation it simply follows that X∞ can be seen as a
SYZ fibration. We can make this explicit by considering the fibration π : P4
→ R4
given by
π([z]) =
5
X
k=1
|zk|2
P5
l=1 |zl|2
pk,
where the pk are five generic points in R4
. The image of F is a 4-simplex ∆
spanned by the five points pk in R4
and X∞ is naturally fibered over the bound-](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-34-320.jpg)
![22 Chapter 2. Calabi-Yau Geometry
Figure 2.4: The base of the SYZ fibration of the Fermat quintic Xµ is the boundary ∂∆ of
a 4-simplex ∆. To be able to draw this base we have placed the vertex pi at infinity. The
tetrahedron ∆i is the projection of the subset {zi = 0} ⊂ P4
. The boundary ∂∆ consists of
5 such tetrahedrons.
ary ∂∆ with generic fiber being T3
. The base of the SYZ fibration is thus topo-
logically a 3-sphere S3
. Similar to the large complex structure limit of a 2-torus,
the Ricci-flat metric on X∞ is degenerate and the SYZ fibers are very small.
Let us introduce some notation to refer to different patches of P4
and ∆. Call
Di1,...,in
the closed part of P4
where zi1
up to zin
vanish, and denote its projec-
tion to R4
by ∆i1,...,in . The ten faces of the boundary ∂∆ are thus labeled by
∆ij, with 1 ≤ i, j ≤ 5.
In the following it is also useful to introduce a notation for the S1
-cycles that are
fibered over the base ∆. So define the circles
γk
i = { zi = 0, |
zk
zj
| = a1,
zl
zj
= a2,
zm
zj
= a3 },
where the indices {i, j, k, l, m} are a permutation of {1, . . . , 5} and the numbers
a1, a2, a3 are determined by choosing a base point [z] on the circle. Since a
different choice for the index j leaves the circle invariant, it is not included
as a label in the name. Which circle shrinks to zero-size over each cell of the
boundary ∂∆ is summarized in Fig. 2.5. Notice that the only non-vanishing
circle in the fibers over the triple intersection ∆ijk is the circle γl
i (which may
also be denoted as γl
j, γl
k, −γm
i , −γm
j or −γm
k ).
W.-D. Ruan works out how to generate a Lagrangian fibration for any Xµ where
µ is large [12]. His idea is to use a gradient flow that deforms the Lagrangian
fibration of X∞ into a Lagrangian fibration of Xµ. Let us consider the region Di](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-35-320.jpg)
![2.2. The Fermat quintic 23
Figure 2.5: The boundary of each tetrahedron is a union of four triangles. The T3
-fibration
of the Fermat quintic degenerates over these triangles. This figure illustrates which cycle in
the T3
-fiber vanishes over which triangle in ∂∆: γj
i degenerates over the triangle ∆ij, γk
j
over ∆jk, etc.
and choose zj 6= 0 for some j 6= i. In this patch xk = zk/zj are local coordinates.
It turns out that the flow of the vector field
V = Re
P
k6=i,j x5
k + 1
Q
k6=i,j xk
∂
∂xi
produces a Lagrangian fibration of Xµ over ∂∆.
All the smooth points of X∞, i.e. those for which only one of the coordinates
zi vanishes, will be transformed into regular points of the Lagrangian fibration.
Only the points in the intersection of Xµ with the singular locus of X∞ won’t
move with the flow. These can be shown to form the complete singular locus of
the Lagrangian fibration of Xµ. Let us call this singular locus Σ and denote
Σij = Dij ∩ Σ = {[z] | zi = zj = 0, z5
k + z5
l + z5
m = 0 }.
The singular locus Σij is thus a projective curve which has genus 6. It intersects
Dijk at the five points. The image of Σij under the projection π is a deformed
triangle in ∆ij that intersects the boundary lines of ∆ij once. This is illustrated
in Fig. 2.6.
In the neighborhood of an inverse image of the intersection point 1
2 (pl + pm) in
Dij the coordinate zk gets very small. This implies that if we write zk = rkeiφk
,
it is the circle parametrized by φk that wraps the leg of the pair of pants in the
limit rk → 0. In the notation we introduced before this circle is γk
j = γk
i .
The study of the cycles in the fibration reveals the structure of the singular locus.
It is built out of two kinds of 3-vertices. We call a 3-vertex whose center lies at](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-36-320.jpg)
![24 Chapter 2. Calabi-Yau Geometry
Figure 2.6: The shaded area in this figure illustrates the discriminant locus of the La-
grangian fibration of the quintic Xµ, for large µ. In the total space it has the shape of a
genus 6 Riemann surface Σij over each 2-cell ∆ij. The five dots on the Riemann surface
project to a single dot on the base.
an edge Dijk a plus-vertex and a 3-vertex that lies in the interior of some 2-cell
Dij a minus-vertex. The plus-vertex is described by a different degenerating
cycle at each of the three legs. Together they sum up to zero. The minus-vertex
is characterized by a single vanishing cycle. The precise topological picture is
shown in Fig. 2.7.
So in the large complex structure limit µ → ∞ the quintic has an elegant struc-
ture in terms of ten transversely intersecting genus 6 Riemann surfaces or equiv-
alently in terms of 50 plus-vertices and 250 minus-vertices.
Two kinds of vertices
The structure of the singular locus of the Fermat quintic in the above example,
ten genus 6 Riemann surfaces that intersect each other transversely, is very el-
egant. It is remarkable that it may be described by just two types of 3-vertices.
This immediately raises the question whether this is accidental or a generic fea-
ture of SYZ fibrations of Calabi-Yau threefolds. In fact, M. Gross shows that
under reasonable assumptions there are just a few possibilities for the topolog-
ical structure in the neighborhood of the discriminant locus [9] (see also [7]).
Indeed only two types of trivalent vertices can occur. Both are characterized by
the monodromy that acts on three generators γi of the homology H1(F, Z) of](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-37-320.jpg)
![2.3. Local Calabi-Yau threefolds 27
2.3 Local Calabi-Yau threefolds
Topological string theory captures topological aspects of type II string theory
compactifications; Kähler aspects of type IIA compactifications and complex
structure aspects of type IIB compactification. Mirror symmetry relates them.
The topological string partition function Ztop can be written as a generating
function of either symplectic or complex structure invariants of the underlying
Calabi-Yau manifold. It contains for example a series in the number of genus zero
curves that are embedded in the Calabi-Yau threefold. Using mirror symmetry it
is possible to go well beyond the classical computations of these numbers. A fa-
mous result is the computation of the whole series of these genus zero invariants
for the Fermat quintic: 2875 different lines, 609250 conics, 317206375 cubics,
etc. [14].
It is much more difficult to find the complete topological string partition func-
tion, which also contains information about higher genus curves in the Calabi-
Yau threefold. The state of the art for the quintic is the computation of these
invariant up to g = 51 [15]. Although this is an impressive result, it is far
from computing the total partition function. In contrast, the all-genus partition
function has been found for a simpler type of Calabi-Yau manifolds, that are
non-compact. What kind of spaces are these? And why is it so much easier to
compute their partition function?
The simplest Calabi-Yau threefold is plain C3
with complex coordinates zi. It
admits a Kähler form
k =
3
X
i=1
dzi ∧ dz̄i,
and a non-vanishing holomorphic 3-form
Ω = dz1 ∧ dz2 ∧ dz3.
A special Lagrangian T2
× R fibration of C3
over R3
has been known for a long
time [16]. It is defined by the map
(z1, z2, z3) 7→ ( Im z1z2z3, |z1|2
− |z2|2
, |z1|2
− |z3|2
).
Notice that its degeneration locus is a 3-vertex with legs (0, t, 0), (0, 0, t) and
(0, −t, −t), for t ∈ R≥0. Over all these legs some cycle in the T2
-fiber shrinks
to zero-size. We can name these cycles γ1, γ2 and −γ1 − γ2 since they add up
to zero. The degenerate fiber over each leg is a pinched cylinder times S1
. It is
thus the noncompact cycle (∼
= R) in the fiber that picks up monodromy when we
move around one of the toric legs. This 3-vertex clearly has the same topological
structure as the plus-vertex.](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-40-320.jpg)
![28 Chapter 2. Calabi-Yau Geometry
Many more non-compact Calabi-Yau’s can be constructed by gluing C3
-pieces
together. In fact, these constitute all non-compact toric Calabi-Yau threefolds.
Their degeneration locus can be drawn in R2
as a 1-dimensional trivalent graph.
The fact that their singular graph is simply planar, as opposed to actually 3-
dimensional (as for the Fermat quintic), makes the computation of the partition
function on such non-compact toric Calabi-Yau’s much simpler. String theorists
have managed to find the full topological partition function Ztop [17] (using
a duality with a 3-dimensional topological theory, called Chern-Simons theory
[18]). The recipe to compute the partition function involves cutting the graph in
basic 3-vertices. To generalize this for compact Calabi-Yau’s it seems one would
need to find a way to glue the partition function for a plus-vertex with that of a
minus-vertex.
Since the partition function is fully known, topological string theory on these
toric manifolds is the ideal playground to learn more about its underlying struc-
ture. This has revealed many interesting mathematical and physical connec-
tions, for example to several algebraic invariants such as Donaldson-Thomas
invariants [19, 20, 21] and Gopakumar-Vafa invariants [22, 23], to knot theory
[24, 25, 26, 27], and to a duality with crystal melting [28, 29, 30].
To illustrate the last duality, let us write down the plain C3
partition function:
Ztop(C3
) =
Y
n0
1
(1 − qn)n
= 1 + q + 3q2
+ 6q3
+ . . . .
This q-expansion is well-known to be generating function of 3-dimensional par-
titions; it is called the MacMahon function [31]. The 3-dimensional partition
can be visualized as boxes that are stacked in the positive octant of R3
. Three of
the sides of each box must either touch the walls or another box. This is pictured
in Fig. 2.10.
Figure 2.10: Interpretation of the first terms in the expansion of Ztop(C3
) in terms of a
three-dimensional crystal in the positive octant of R3
.
Since q = eλ
, where λ is the coupling constant of topological string theory, the
boxes naturally have length λ. Whereas the regime with λ finite is described by
a discrete quantum structure, in the limit λ → 0 surprisingly a smooth Calabi-
Yau geometry emerges. In a duality with statistical mechanics this corresponds](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-41-320.jpg)
![2.3. Local Calabi-Yau threefolds 29
to the shape of a melting crystal. These observations have led to deep insights
in the quantum description of space and time [32, 33].
Remarkably, it has been shown that the emergent smooth geometry of the crystal
can be identified with the mirror of C3
. How does this limit shape look like?
Using local mirror symmetry the equation for the mirror of C3
was found in
[34]. It is given by
uv − x − y + 1 = 0,
where u, v ∈ C and x, y ∈ C∗
. Remember that the topological structure of the
mirror of a plus-vertex should be that of a minus-vertex. Viewing the mirror as
a (u, v)-fibration over a complex plane spanned by x and y confirms this:
The degeneration locus of the fibration equals the zero-locus x + y − 1 = 0.
Parametrizing this curve by x, it is easily seen that this is a 2-sphere with three
punctures at the points x = 0, 1 and ∞. We can equivalently represent this curve
as a pair of pants, by cutting off a disc at each boundary |x̃| = 1, where x̃ is a
local coordinate that vanishes at the corresponding puncture. This realizes the
mirror of C3
topologically as the minus-vertex in Fig. 2.9.
Mirrors of general non-compact toric manifolds are of the same form
XΣ : uv − H(x, y) = 0,
where the equation H(x, y) = 0 now defines a generic Riemann surface Σ em-
bedded in (C∗
)2
. This surface is a thickening of the 1-dimensional degeneration
graph Γ of its mirror. Its non-vanishing holomorphic 3-form is proportional to
Ω =
du
u
∧ dx ∧ dy.
These geometries allow a Ricci-flat metric that is conical at infinity [35, 36, 37,
38]. We refer to the threefold XΣ as the local Calabi-Yau threefold modeled on Σ.
The curve Σ plays a central role in this thesis. We study several set-ups in string
theory whose common denominator is the relevance of the Riemann surface Σ.
In particular, we study the melting crystal picture from the mirror perspective.
One of our main results is a simple representation of topological string theory in
terms a quantum Riemann surface, that reduces to the smooth Riemann surface
Σ in the limit λ → 0.](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-42-320.jpg)
![Chapter 3
I-brane Perspective on
Vafa-Witten Theory and
WZW Models
In the last decades enormous progress has been made in analyzing 4-dimensional
supersymmetric gauge theories. Partition functions and correlation functions of
some theories have been computed, spectra of BPS operators have been discov-
ered and many other structures have been revealed. Most fascinating to us is
that many exact results can be related to two-dimensional geometries.
Since 4-dimensional supersymmetric gauge theories appear in several contexts
in string theory, much of this progress is strongly influenced by string theory.
Often, string theory tools can be used to compute important quantities in su-
persymmetric gauge theories. Moreover, in many cases string theory provides a
key understanding of new results. For example, when auxiliary structures in the
gauge theory can be realized geometrically in string theory and when symme-
tries in the gauge theory can be understood as stringy dualities.
In this chapter we study a remarkable correspondence between 4-dimensional
gauge theories and 2-dimensional conformal field theories. This correspondence
connects a “twisted” version of supersymmetric Yang-Mills theory to a so-called
Wess-Zumino-Witten model. In particular, generating functions of SU(N) gauge
instantons on the 4-manifold C2
/Zk are related to characters of the affine Kac-
Moody algebra c
su(k) at level N. This connection was originally discovered by
Nakajima [39], and further analyzed by Vafa and Witten [40]. The goal of this
chapter is to make it more transparent. Once again, we find that string theory
offers the right perspective.](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-44-320.jpg)
![34 Chapter 3. I-brane Perspective on Vafa-Witten Theory and WZW Models
So electric-magnetic duality is a strong-weak coupling duality, that sends the
complexified gauge coupling τ 7→ −1/τ. Moreover, this argument suggests an
important role for the modular group Sl(2, Z). This group acts on τ as
τ 7→
aτ + b
cτ + d
, for
a b
c d
∈ Sl(2, Z).
and is generated by
S =
0 1
−1 0
and T =
1 1
0 1
.
Hence, S is the generator of electro-magnetic duality (later also called S-duality)
and T the generator of shifts in the θ-angle. The gauge coupling τ is thus part of
the fundamental domain of Sl(2, Z) in the upper-half plane, as shown in Fig. 3.1.
Figure 3.1: The fundamental domain of the modular group Sl(2, Z) in the upper half plane.
Montonen and Olive [41] where pioneers in conjecturing that electro-magnetic
duality is an exact non-abelian symmetry, that exchanges the opposite roles of
electric and magnetic particles in 4-dimensional gauge theories. This involves
replacing the gauge group G by the dual group Ĝ (whose weight lattice is dual
to that of G). The first important tests of S-duality have been performed in
supersymmetric gauge theories.
For U(1) theories the partition function ZU(1)
can be explicitly computed [42,
43]. The classical contribution to the partition function is given by integral fluxes
p ∈ H2
(M, Z), as in equation (3.5), to the Langrangian (3.3). Decomposing the
flux p into a self-dual and anti-selfdual contribution yields the generalized theta-
function
θΓ(q, q̄) =
X
(p+,p−)∈Γ
q
1
2 p2
+ q̄
1
2 p2
− (3.6)](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-47-320.jpg)
![3.1. Instantons and branes 35
with q = exp(2πiτ), whereas Γ = H2
(M, Z) is the intersection lattice of M and
p2
=
R
M
p ∧ p. The total U(1) partition function is found by adding quantum
corrections to the above result, which are captured by some determinants [42].
Instead of transforming as a modular invariant, ZU(1)
transforms as a modular
form under Sl(2, Z)-transformation of τ
ZU(1)
aτ + b
cτ + d
= (cτ + d)u/2
(cτ + d)v/2
ZU(1)
(τ) ,
a b
c d
∈ Sl(2, Z).
We will come back on this in Section 3.2.
3.1.1 Supersymmetry
In supersymmetric theories quantum corrections are much better under control,
so that much more can be learned about non-perturbative properties of the the-
ory. We will soon discuss such elegant results, but let us first introduce super-
symmetric gauge theories.
The field content of the simplest supersymmetric gauge theories just consists
of a bosonic gauge field A and a fermionic gaugino field λ. Supersymmetry
relates the gauge field A to its superpartner χ. In any supersymmetric theory the
number of physical bosonic degrees of freedom must be the same as the number
of physical fermionic degrees of freedom. This constraints supersymmetric Yang-
Mills theories to dimension d ≤ 10.
The Lagrangian of a minimal supersymmetric gauge theory is
L = −
1
4
Tr(FµνFµν
) +
i
2
χ̄Γµ
Dµχ,
and supersymmetry variations of the fields A and χ are generated by a spinor
δAµ =
i
2
¯
Γµχ, δχ =
1
4
FµνΓµν
. (3.7)
The number of supersymmetries equals the number of components of .
Dimensionally reducing the above minimal N = 1 susy gauge theories to lower-
dimensional space-times yield N = 2, 4 and possibly N = 8 susy gauge theories.
Their supersymmetry variations are determined by an extended supersymme-
try algebra. In four dimensions this is a unique extension of the Poincaré al-
gebra generated by the supercharges QA
α and QAα̇, with A ∈ {1, . . . , N} and
α, α̇ ∈ {1, 2} are indices in the 4-dimensional spin group su(2)L × su(2)R. Non-](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-48-320.jpg)
![36 Chapter 3. I-brane Perspective on Vafa-Witten Theory and WZW Models
vanishing anti-commutation relations are given by
{QA
α , QBβ̇} = 2(σµ
)αβ̇PµδA
B
{QA
α , QB
β } = αβZAB
{QAα̇, QBβ̇} = α̇β̇Z†
AB
where ZAB
and its Hermitean conjugate are the central charges. The auto-
morphism group of this algebra, that acts on the supercharges, is known as the
R-symmetry group.
BPS states
A special role in extended supersymmetric theories is played by supersymmetric
BPS states [44]. They are annihilated by a some of the supersymmetry genera-
tors, e.g. quarter BPS states satisfy
QA
|BPSi = 0,
for 1/4 N indices A ∈ {1, . . . , N}. BPS states saturate the bound M2
≤ |Z|2
and form “small” representations of the above supersymmetry algebra. This
implies that supersymmetry protects them against quantum corrections: a small
deformation won’t just change the dimension of the representation.
Twisting
Supersymmetric Yang-Mills requires a covariantly constant spinor in the rigid
supersymmetry variations (3.7). Since these are impossible to find on a generic
manifold M, the concept of twisting has been invented. Twisting makes use of
the fact that supersymmetric gauge theories are invariant under a non-trivial in-
ternal symmetry, the R-symmetry group. By choosing a homomorphism from the
space-time symmetry group into this internal global symmetry group, the spinor
representations change and often contain a representation that transforms as a
scalar under the new Lorentz group.
Such an odd scalar QT can be argued to obey Q2
T = 0. It is a topological
supercharge that turns the theory into a cohomological quantum field theory.
Observables O can be identified with the cohomology generated by QT , and
correlation functions are independent of continuous deformations of the metric
∂
∂gµν
hO1 · · · Oki = 0 (3.8)
These correlation functions can thus be computed by going to short distances.
This yields techniques to study the dynamics of these theories non-perturbatively.](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-49-320.jpg)
![3.1. Instantons and branes 37
For these topological theories it is sometimes possible to compute the partition
function and other correlators. Witten initiated twisting in the context of N = 2
supersymmetric Yang-Mills [45]. He showed that correlators in the so-called
Donaldson-Witten twist compute the famous Donaldson invariants.
A general theme in cohomological field theories is localization. Unlike in general
physical theories, in these topological theories the saddle point approximation
is actually exact. The path integral only receives contributions from fixed point
locus M of the scalar supercharge QT . Since the kinetic part of the action (that
contains all metric-dependent terms) is QT -exact, the only non-trivial contribu-
tion to the path integral is given by topological terms:
ZcohTFT =
Z
DX exp
−
1
e2
Skin(X) + Stop(X)
→
Z
M
DX exp (Stop(X)) .
Here X represents a general field content. An elegant example in this respect
is 2-dimensional gauge theory [46]. Extensive reviews of localization are [47,
48]. We will encounter localization on quite a few occasions, starting with Vafa-
Witten theory in Section 3.2.
3.1.2 Extended objects
Whereas Yang-Mills theory is formulated in terms of a single gauge potential A,
string theory is equipped with a whole set of higher-form gauge fields. Instead
of coupling to electro-magnetic particles they couple to extended objects, such as
D-branes. This is analogous to the coupling of a particle of electric charge q to
the Maxwell gauge field A
q
Z
W
A = q
Z
W
Aµ
∂xµ
∂t
dt,
where W is the worldline of the particle. Notice that we need to pull-back the
space-time gauge field A in the first term before integrating it over the worldline.
We often don’t write down the pull-back explicitly to simplify notation. D-branes
and other extended objects appear all over this thesis. Let us therefore give a
very brief account of the properties that are relevant for us.
Couplings and branes in type II
Gauge potentials in type II theory either belong to the so-called RR or the NS-
NS sector. The RR potentials couple to D-branes, whereas the NS-NS potential
couples to the fundamental string (which is often denoted by F1) and the NS5-
brane. Let us discuss these sectors in a little more detail.
The only NS-NS gauge field is the 2-form B. The B-field plays a crucial role in
Chapter 5. Aside from the B-field the NS-NS sector contains the dilaton field φ](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-50-320.jpg)
![3.1. Instantons and branes 39
a D(6 − p)-brane that wraps a (7 − p)-dimensional submanifold Σ7−p.
Calibrated cycles
D-branes are solitonic states as their tension Tp is proportional to 1/gs. To be
stable against decay the brane needs to wrap a submanifold that preserves some
supersymmetries. Geometrically, such configurations are defined by a calibration
[49]. A calibration form is a closed form Φ such that Φ ≤ vol at any point of the
background. A submanifold Σ that is calibrated satisfies
Z
Σ
Φ =
Z
Σ
vol,
and minimizes the volume in its holomogy class. On a Kähler manifold a calibra-
tion is given by the Kähler form t, and the calibrated submanifolds are complex
submanifolds. On a Calabi-Yau threefold the holomorphic threeform Ω provides
a calibration, whose calibrated submanifolds are special Lagrangians. Calibrated
submanifolds support covariantly constant spinors, and therefore preserve some
supersymmetry. D-branes wrapping them are supersymmetric BPS states.
Worldvolume theory
D-branes have a perturbative description in terms of open strings that end on
them. The massless modes of these open strings recombine in a Yang-Mills gauge
field A. When the D-brane worldvolume is flat the corresponding field theory
on the p-dimensional brane is a reduction of N = 1 susy Yang-Mills from 10
dimensions to p + 1. The 9 − p scalar fields in this theory correspond to the
transverse D-brane excitations. When N D-branes coincide the wordvolume
theory is a U(N) supersymmetric Yang-Mills theory.
For more general calibrated submanifolds the low energy gauge theory is a
twisted topological gauge theory [50], which we introduced in Section 3.1.1.
Which particular twist is realized, can be argued by determining the normal
bundel to the submanifold. Sections of the normal bundel fix the transverse
bosonic excitations of the gauge theory, and should correspond to the bosonic
field content of the twisted theory.
I-branes and bound states
Branes can intersect each others such that they preserve some amount of super-
symmetry. This is called an I-brane configuration. In such a set-up there are
more degrees of freedom than the ones (we described above) that reside on the
individual branes. These extra degrees of freedom are given by the modes of
open strings that stretch between the branes. In stringy constructions of the
standard model on a set of branes they often provide the chiral fermions.](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-52-320.jpg)
![40 Chapter 3. I-brane Perspective on Vafa-Witten Theory and WZW Models
Chiral fermions are intimately connected with quantum anomalies, and brane
intersections likewise. To cure all possible anomalies in an I-brane system, a
topological Chern-Simons term has to be added to the string action
SCS = Tp
Z
Σp+1
Tr exp
F
2π
∧
X
i
Ci ∧
q
Â(R). (3.11)
This term is derived through a so-called anomaly inflow analysis [51, 52]. The
last piece contains the A-roof genus for the 10-dimensional curvature 2-form R
pulled back to Σp+1. It may be expanded as
Â(R) = 1 −
p1(R)
24
+
7p1(R)2
− 4p2(R)
5760
+ . . . ,
where pk(R) is a Pontryagin class. For example, the Chern-Simons term (3.11)
includes a factor
Tp
Z
Σp+1
Tr
F
2π
∧ Cp−1
when a gauge field F on the worldvolume Σp+1 is turned on. It describes an
induced D(p − 2) brane wrapping the Poincare dual of [F/2π] in Σp+1.
Vice versa, a bound state of a D(p − 2)-brane on a Dp-brane may be interpreted
as turning on a field strength F on the Dp-brane. Analogously, instantons (3.1)
in a 4-dimensional gauge theory, say of rank zero and second Chern class n, have
an interpretation in type IIA theory bound states of n D0-branes on a D4-brane.
More generally, topological excitations in the worldvolume theory of a brane are
often caused by other extended objects that end on it [53].
3.1.3 String dualities
The different appearances of string theory, type I, II, heterotic and M-theory, are
connected through a zoo of dualities. Let us briefly introduce T-duality and S-
duality in type II. There are many more dualities, some of which we will meet
on our way.
T-duality
T-duality originates in the worldsheet description of type II theory in terms of
open and closed strings. T-duality on an S1
in the background interchanges the
Dirichlet and Neumann boundary conditions of the open strings on that circle,
and thereby maps branes that wrap this S1
into branes that don’t wrap it (and
vice versa). It thus interchanges type IIA and type IIB theory.
Similar to electro-magnetic duality (see Section 3.1), T-duality follows from a
path integral argument [54]. The sigma model action for a fundamental string](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-53-320.jpg)
![3.1. Instantons and branes 41
is based on the term
−
1
2πα0
Z
Σ
gmndxm
∧ ∗dxn
,
in equation (3.9) when we forget the B-field for simplicity. Let us suppose that
the metric is diagonal in the coordinate x that parametrizes the T-duality circle
S1
. Then we can add a Lagrange multiplier field dy to the relevant part of the
action
Z
DxDy exp
Z
−
1
2πα0
dx ∧ ∗dx +
i
π
dx ∧ ∗dy
.
On the one hand the Lagrange multiplier field dy forces d(dx) = 0, which locally
says that dx is exact. On the other hand integrating out dx yields
Z
Dy exp
Z
−
α0
2π
dy ∧ ∗dy
So T-duality exchanges
α0
↔
1
α0
and is therefore a strong-weak coupling on the worldsheet. More precisely, one
should also take into account the B-field coupling (3.9), which is related to non-
diagonal terms is the space-time metric. This leads to the well-known Buscher
rules [55]
Since the differential dx may be identified with a component of the gauge field
A on the brane, and dy with a normal 1-form to the brane, the reduction of
supersymmetric Yang-Mills from ten to 10 − d dimensions can be understood as
applying T-duality d times.
S-duality
Since N = 4 supersymmetric Yang-Mills is realized as the low-energy effective
theory on a D3-brane wrapping R4
. Since electro-magnetic duality in this theory
is an exact symmetry, it should have a string theoretic realization in type IIB
theory as well. Indeed it does, and this symmetry is known as S-duality. In type
IIB theory S-duality is a (space-time) strong-weak coupling duality that maps
gs ↔ 1/gs. Analogous to Yang-Mills theory the complete symmetry group is
Sl(2, Z), where the complex coupling constant τ (3.4) is realized as
τ = C0 + ie−φ
.
Since the ratio of the tensions of the fundamental string F1 and the D1-brane
is equal to gs, S-duality exchanges these objects as well as the B-field and the](https://image.slidesharecdn.com/2626155-250609220657-54a98794/85/Topological-Strings-And-Quantum-Curves-Lotte-Hollands-54-320.jpg)
Topological Strings And Quantum Curves Lotte Hollands
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- 5. Lotte Hollands Topological Strings and Quantum Curves UvA Dissertation This thesis presents several new insights on the interface between mathematics and theoretical physics, with a central role for Riemann surfaces. First of all, the duality between Vafa-Witten theory and WZW models is embedded in string theory. Secondly, this model is generalized to a web of dualities connecting topological string theory and N=2 supersymmetric gauge theories to a configuration of D-branes that intersect over a Riemann surface. This description yields a new perspective on topological string theory in terms of a KP integrable system based on a quantum curve. Thirdly, this thesis describes a geometric analysis of wall-crossing in N=4 string theory. And lastly, it offers a novel approach to constuct metastable vacua in type IIB string theory. Lotte Hollands (1981) studied mathematics and theoretical physics at the University of Utrecht. In 2004 she started her PhD research at the Institute for Theoretical Physics of the University of Amsterdam, supervised by Prof. R.H. Dijkgraaf. Topological Strings and Quantum Curves Lotte Hollands
- 6. TOPOLOGICAL STRINGS AND QUANTUM CURVES
- 7. This work has been accomplished at the Institute for Theoretical Physics (ITFA) of the University of Amsterdam (UvA) and is financially supported by a Spinoza grant of the Netherlands Organisation for Scientific Research (NWO). Cover illustration: Lotte Hollands Cover design: The DocWorkers Lay-out: Lotte Hollands, typeset using L A TEX ISBN 978 90 8555 020 4 NUR 924 c L. Hollands / Pallas Publications — Amsterdam University Press, 2009 All rights reserved. Without limiting the rights under copyright reserved above, no part of this book may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without the written permission of both the copyright owner and the author of the book.
- 8. TOPOLOGICAL STRINGS AND QUANTUM CURVES ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof. dr. D.C. van den Boom ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Agnietenkapel op donderdag 3 september, te 14.00 uur door LOTTE HOLLANDS geboren te Maasbree
- 9. PROMOTIECOMMISSIE Promotor prof. dr. R.H. Dijkgraaf Overige leden prof. dr. J. de Boer prof. dr. A.O. Klemm prof. dr. E.M. Opdam dr. H.B. Posthuma dr. S.J.G. Vandoren prof. dr. E.P. Verlinde Faculteit der Natuurwetenschappen, Wiskunde en Informatica
- 10. PUBLICATIONS This thesis is based on the following publications: R. Dijkgraaf, L. Hollands, P. Sułkowski and C. Vafa, Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions, arXiv/0709.4446 [hep-th], JHEP 02 (2008) 106. L. Hollands, J. Marsano, K. Papadodimas and M. Shigemori, Nonsupersymmetric Flux Vacua and Perturbed N=2 Systems, arXiv/0804.4006 [hep-th], JHEP 10 (2008) 102. R. Dijkgraaf, L. Hollands and P. Sułkowski, Quantum Curves and D-Modules, arXiv/0810.4157 [hep-th]. M. Cheng and L. Hollands, A Geometric Derivation of the Dyon Wall-Crossing Group, arXiv/0901.1758 [hep-th], JHEP 04 (2009) 067.
- 12. Contents 1 Introduction 1 1.1 Fermions on Riemann surfaces . . . . . . . . . . . . . . . . . . . 1 1.2 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Calabi-Yau Geometry 15 2.1 The Strominger-Yau-Zaslow conjecture . . . . . . . . . . . . . . . 16 2.2 The Fermat quintic . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Local Calabi-Yau threefolds . . . . . . . . . . . . . . . . . . . . . 27 3 I-brane Perspective on Vafa-Witten Theory and WZW Models 31 3.1 Instantons and branes . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Vafa-Witten twist on ALE spaces . . . . . . . . . . . . . . . . . . . 42 3.3 Free fermion realization . . . . . . . . . . . . . . . . . . . . . . . 51 3.4 Nakajima-Vafa-Witten correspondence . . . . . . . . . . . . . . . 58 4 Topological Strings, Free Fermions and Gauge Theory 69 4.1 Curves in N = 2 gauge theories . . . . . . . . . . . . . . . . . . . 70 4.2 Geometric engineering . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3 Topological invariants of Calabi-Yau threefolds . . . . . . . . . . . 89 4.4 BPS states and free fermions . . . . . . . . . . . . . . . . . . . . . 101 5 Quantum Integrable Hierarchies 113 5.1 Semi-classical integrable hierarchies . . . . . . . . . . . . . . . . 114 5.2 D-modules and quantum curves . . . . . . . . . . . . . . . . . . . 119 5.3 Fermionic states and quantum invariants . . . . . . . . . . . . . . 134
- 13. 6 Quantum Curves in Matrix Models and Gauge Theory 147 6.1 Matrix model geometries . . . . . . . . . . . . . . . . . . . . . . . 149 6.2 Conifold and c = 1 string . . . . . . . . . . . . . . . . . . . . . . . 163 6.3 Seiberg-Witten geometries . . . . . . . . . . . . . . . . . . . . . . 169 6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 7 Dyons and Wall-Crossing 189 7.1 Compact curves of genus 1 and 2 . . . . . . . . . . . . . . . . . . 190 7.2 Quarter BPS dyons . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7.3 Wall-crossing in N = 4 theory . . . . . . . . . . . . . . . . . . . . 215 8 Fluxes and Metastability 235 8.1 Ooguri-Ookouchi-Park formalism . . . . . . . . . . . . . . . . . . 237 8.2 Geometrically engineering the OOP formalism . . . . . . . . . . . 238 8.3 Embedding in a larger Calabi-Yau . . . . . . . . . . . . . . . . . . 256 8.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . 269 A Level-rank duality 273 Bibliography 277 Samenvatting 293 Acknowledgments 299
- 14. Chapter 1 Introduction The twentieth century has seen the birth of two influential theories of physics. On very small scales quantum mechanics is very successful, whereas general relativity rules our universe on large scales. Unfortunately, in regimes at small scale where gravity is nevertheless non-negligible—such as black holes or the big bang—neither of these two descriptions suffice. String theory is the best candidate for a unified theoretical description of nature to date. However, strings live at even smaller scales than those governed by quantum mechanics. Hence string theory necessarily involves levels of energy that are so high that they cannot be simulated in a laboratory. Therefore, we need another way to arrive at valid predictions. Although string theory thus rests on physical arguments, it turns out that this framework carries rich mathematical structure. This means that a broad array of mathematical techniques can be deployed to explore string theory. Vice versa, physics can also benefit mathematics. For ex- ample, different physical perspectives can relate previously unconnected topics in mathematics. In this thesis we are motivated by this fertile interaction. We both use string the- ory to find new directions in mathematics, and employ mathematics to discover novel structures in string theory. Let us illustrate this with an example. 1.1 Fermions on Riemann surfaces This section briefly introduces the main ingredients of this thesis. We will meet all in much greater detail in the following chapters. So-called Riemann surfaces play a prominent role in many of the fruitful in- teractions between mathematics and theoretical physics that have developed in
- 15. 2 Chapter 1. Introduction the second half of the twentieth century. Riemann surfaces are smooth two- dimensional curved surfaces that have a number of holes, which is called their genus g. Fig. 1.1 shows an example. Figure 1.1: A compact Riemann surface with just one hole in it is called a 2-torus T2 . We refer to its two 1-cycles as the A and the B-cycle. Riemann surfaces additionally come equipped with a complex structure, so that any small region of the surface resembles the complex plane C. An illustrative class of Riemann surfaces is defined by equations of the form Σ : F(x, y) = 0, where x, y ∈ C. Here, F can for instance be a polynomial in the complex variables x and y. A simple example is a (hyper)elliptic curve defined by y2 = p(x), where p(x) is a polynomial. The curve is elliptic if p has degree 3 or 4, in which case its genus is g = 1. For higher degrees of p it is hyperelliptic and has genus g > 1. Figure 1.2: A simple example of a non-compact Riemann surface is defined by the equation x2 + y2 = 1 in the complex plane C2 . The complex structure of a compact Riemann surface can be conveniently char- acterized by a period matrix. This is a symmetric square matrix τij of complex numbers, whose rank equals the genus of the surface and whose imaginary part is strictly positive. The complex structure of a 2-torus, illustrated in Fig. 1.1, is
- 16. 1.1. Fermions on Riemann surfaces 3 for example determined by one complex number τ that takes values in the upper half plane. The period matrix encodes the contour integrals of the g independent holomor- phic 1-forms over the 1-cycles on the surface. To this end one picks a canon- ical basis of A-cycles and B-cycles, where the only non-trivial intersection is A1 ∩ Bj = δij. The 1-forms ωi may then be normalized by integrating them over the A-cycles, so that their integrals over the B-cycles determine the period matrix: Z Ai ωj = δij, Z Bi ωj = τij. Conformal field theory Riemann surfaces play a dominant role in the study of conformal field theories (CFT’s). Quantum field theories can be defined on a space-time background M, which is usually a Riemannian manifold with a metric g. The quantum field theory is called conformal when it is invariant under arbitrary rescaling of the metric g. A CFT therefore depends only on the conformal class of the metric g. The simplest quantum field theories study a bosonic scalar field φ on the back- ground M. The contribution to the action for a massless scalar is Sboson = Z M √ g gmn ∂mφ ∂nφ, yielding the Klein-Gordon equation ∂m∂m φ = 0 (for zero mass) as equation of motion. Note that, on the level of the classical action, this theory is clearly conformal in 2 dimensions; this still holds for the full quantum theory. Since a 2-dimensional conformal structure uniquely determines a complex structure, the free boson defines a CFT on any Riemann surface Σ. Let us compactify the scalar field φ on a circle S1 , so that it has winding modes along the 1-cycles of the Riemann surface. The classical part of the CFT partition function is determined by the solutions to the equation of motion. The holomor- phic contribution to the classical partition function is well-known to be encoded in a Riemann theta function θ (τ, ν) = X p∈Zg e2πi(1 2 pt τp+pt ν), where the integers p represent the momenta of φ that flow through the A-cycles of the 2-dimensional geometry. Another basic example of a CFT is generated by a fermionic field ψ on a Rie- mann surface. Let us consider a chiral fermion ψ(z). Mathematically, this field
- 17. 4 Chapter 1. Introduction transforms on the Riemann surface as a (1/2, 0)-form, whence ψ(z) √ dz = ψ(z0 ) √ dz0, where z and z0 are two complex coordinates. Such a chiral fermion contributes to the 2-dimensional action as Sfermion = Z Σ ψ(z) ∂Aψ(z), where ∂A = ∂ + A, and A is a connection 1-form on Σ. This action determines the Dirac equation ∂Aψ(z) = 0 as its equation of motion. The total partition function of the fermion field ψ(z) is computed as the determinant Zfermion = det(∂A). Remarkably, this partition function is also proportional to a Riemann theta func- tion. The fact that this is not just a coincidence, but a reflection of a deeper symmetry between 2-dimensional bosons and chiral fermions, is known as the boson-fermion correspondence. Integrable hierarchy Fermions on Riemann surfaces are also familiar from the perspective of inte- grable systems. Traditionally, integrable systems are Hamiltonian systems ẋi = {xi, H} = ∂H ∂pi , ṗi = {pi, H} = ∂H ∂xi , with coordinates xi, momenta pi and a Hamiltonian H, for which there exists an equal amount of integrals of motion Ii such that ˙ Ii = {Ii, H} = 0, {Ii, Ij} = 0. The left equation requires the integrals Ii to be constants of motion whereas the right one forces them to commute among each other. A simple example is the real two-dimensional plane R2 with polar coordinates r and φ, see Fig. 1.3. Take the Hamiltonian H to be the radius r. Then H = r is itself an integral of motion, so that the system is integrable. Notice that H is constant on the flow generated by the differential ∂/∂φ. A characteristic example of an integrable system that is intimately related to Riemann surfaces and theta-functions is the Korteweg de Vries (KdV) hierarchy. Although this system was already studied by Korteweg and de Vries at the end
- 18. 1.1. Fermions on Riemann surfaces 5 Figure 1.3: The two-dimensional plane R2 seen as an integrable system. of the 19th century as a non-linear partial differential equation ∂u ∂t = ∂3 u ∂x3 + 6u ∂u ∂x , it was only realized later that it is part of a very rich geometric and algebraic structure. It is impossible to do justice to this beautiful story in this short in- troduction. Instead, let us just touch on the aspects that are relevant for this thesis. Geometrically, a special class of solutions to the KdV differential equation yields linear flows over a 2g-dimensional torus that is associated to a Riemann surface Σ. This torus is called the Jacobian. Its complex structure is determined by the period matrix τij of Σ. The Riemann surface Σ is called the spectral curve of the KdV hierarchy. The KdV spectral curve is an elliptic curve Q2 = P3 − g2P − g3, where g2 and g3 are the Weierstrass invariants. The coordinates P and Q on this elliptic curve can best be described as commuting differential operators [P, Q] = 0, that arise naturally when the KdV differential equation is written in the form of a Hamiltonian system. As a side remark we notice that the KdV system is closely related to the Hitchin integrable system, which studies certain holomorphic bundles over a complex curve C. In this integrable system, too, the dynamics can be expressed in terms of a linear flow on the Jacobian associated to a spectral cover Σ of C. We go one step beyond the geometrical structure of the KdV system when intro-
- 19. 6 Chapter 1. Introduction ducing chiral fermions on the spectral curve Σ. Their partition function det(∂) transforms as a section of the determinant line bundle over the moduli space of the integrable hierarchy. It is known as the tau-function. The tau-function is proportional to the theta-function associated to the period matrix τij. Gauge theory and random matrices Similar integrable structures have been found in wide variety of physical the- ories recently. An important ingredient of this thesis is the appearance of an auxiliary Riemann surface in 4-dimensional gauge theories. The basic field in a gauge theory is a gauge field A, which is mathematically a connection 1-form of a principal G-bundle over the 4-manifold M. Let us take G = U(1). In that situ- ation the classical equations of motion reproduce the Maxwell equations. Once more, the holomorphic part of the gauge theory partition function is essentially a theta-function Z DA e R M 1 2 τF+∧F+ ∼ X p+ e 1 2 τp2 + , where τ is the complex coupling constant of the gauge theory, and p+ are the fluxes of the self-dual field strength F+ through the 2-cycles of M. It turns out to be useful to think of τ as the complex structure parameter of an auxiliary elliptic curve. In other gauge theories a similar auxiliary curve is known to play a significant role as well. We will explain this in detail in the main body of this thesis. Another interesting application is the theory of random matrices. This is a 0- dimensional quantum field theory based on a Hermitean N-by-N-matrix X. The simplest matrix model is the so-called Gaussian one, whose action reads Smatrix = − 1 2λ TrX2 , where λ is a coupling constant. Other matrix models are obtained by adding higher order interactions to the matrix model potential W(X) = X2 /2. Such a matrix model can be conveniently evaluated by diagonalizing the matrices X. This reduces the path integral to a standard integral over eigenvalues xi, yet adds the extra term 1 N2 X i,j log |xi − xj| to the matrix model action. Depending on the values of the parameters N and λ the eigenvalues will either localize on the minima of the potential W(X) or, oppositely, spread over a larger interval. When one lets N tend to infinity while
- 20. 1.1. Fermions on Riemann surfaces 7 keeping the ’t Hooft parameter µ = Nλ fixed, the eigenvalues can be seen to form a smooth distribution. For instance, in the Gaussian matrix model, the density of eigenvalues is given by the Wigner-Dyson semi-circle ρ(x) ∼ p 4µ − x2, leading to the algebraic curve x2 + y2 = 4µ. Likewise, the eigenvalue distribution of a more general matrix model takes the form of a hyperelliptic curve in the limit N → ∞ with fixed ’t Hooft parameter. Many properties of the matrix model are captured in this spectral curve. Topological string theory String theory provides a unifying framework to discuss all these models. In particular, topological string theory studies embeddings of a Riemann surface C—which shouldn’t be confused with the spectral curve—into 6-dimensional target manifolds X of a certain kind. These target manifolds are called Calabi- Yau manifolds. The Riemann surface C is the worldvolume swept out in time by a 1-dimensional string. The topological string partition function is a series expansion Ztop(λ) = exp X g λ2g−2 Fg ! , where λ is called the topological string coupling constant. Each Fg contains the contribution of curves C of genus g to the partition function. Calabi-Yau manifolds are not well understood in general, and studying topo- logical string theory is very difficult task. Nevertheless, certain classes of non- compact Calabi-Yau manifolds are much easier to analyse, and contain precisely the relevant backgrounds to study the above integrable structures. Consider an equation of the form XΣ : uv − F(x, y) = 0, where u and v are C-coordinates and Σ : F(x, y) = 0 defines the spectral curve Σ in the complex (x, y)-plane. The 6-dimensional manifold XΣ may be regarded as a C∗ -fibration over the (x, y)-plane that de- generates over the Riemann surface F(x, y) = 0.
- 21. 8 Chapter 1. Introduction Indeed, the fiber over a point (x, y) is defined by uv = F(x, y). Over a generic point in the base this fiber is a hyperboloid. However, when F(x, y) → 0 the hyperboloid degenerates to a cone. This is illustrated in Fig. 1.4. The zero locus of the polynomial F(x, y) thus determines the degeneration locus of the fibration. We therefore refer to the variety XΣ as a non-compact Calabi-Yau threefold modeled on a Riemann surface. Figure 1.4: A local Calabi-Yau threefold defined by an equation of the form uv−F(x, y) = 0 can be viewed as a C∗ -fibration over the base parametrized by x and y. The fibers are hyperboloids over general points in the base, but degenerate when µ = F(x, y) → 0. When the Riemann surface Σ equals the auxiliary Riemann surface that emerges in the study of gauge theories, the topological string partition function is known to capture important properties of the gauge theory. Furthermore, when it equals the spectral curve in the matrix model, it is known to capture the full matrix model partition function. Quantum curves and D-modules However, topological string computations go well beyond computing a fermion determinant on Σ. In fact, it is known that the fermion determinant appears as the genus 1 part F1 of the free energy. We call this the semi-classical part of the free energy, as it remains finite when λ → 0. Similarly, F0 is the classical contribution that becomes very large in this limit, and all higher Fg’s encode the quantum corrections. In other words, we may interpret the loop parameter λ as a quantization of the semi-classical tau-function. But what does this mean in terms of the underlying integrable system? Answering this question is one of the goals in this thesis. In both the gauge theory and the matrix model several hints have been obtained. In the context of gauge theories the higher loop corrections Fg are known to correspond to couplings of the gauge theory to gravity. Mathematically, these appear by making the theory equivariant with respect to an SU(2)-action on the underlying 4-dimensional background. N. Nekrasov and A. Okounkov have
- 22. 1.2. Outline of this thesis 9 shown that the partition function of this theory is the tau-function of an inte- grable hierarchy. In the theory of random matrices, λ-corrections correspond to finite N correc- tions. In this theory, too, it has been found that the partition function behaves as the tau-function of an integrable hierarchy. However, like in the gauge theory examples, there is no interpretation of the full partition function as a fermion determinant. Instead, a new perspective arises, in which the spectral curve is be replaced by a non-commutative spectral curve. In matrix models that relate to the KdV integrable system, it is found that the KdV differential operators P and Q do not commute anymore when the quantum corrections are taken into account: [P, Q] = λ. Similar hints have been found in the study of topological string theory on general local Calabi-Yau’s modeled on a Riemann surface. This thesis introduces a new physical perspective to study topological string the- ory, clarifying the interpretation of the parameter λ as a “non-commutativity parameter”. We show that λ quantizes the Riemann surface Σ, and that fermions on the curve are sections of a so-called D-module instead. This attributes Fourier- like transformation properties to the fermions on Σ: ψ(y) = Z dy exy/λ ψ(x). 1.2 Outline of this thesis Chapter 2 starts with an introduction to the geometry of Calabi-Yau threefolds. They serve as an important class of backgrounds in string theory, that can be used to make contact with the 4-dimensional world we perceive. Specifically, we explain the idea of a Calabi-Yau compactification, and introduce certain non- compact Calabi-Yau backgrounds that appear in many guises in this thesis. Topological string theory on these non-compact Calabi-Yau backgrounds is of physical relevance in the description of 4-dimensional supersymmetric gauge theories and supersymmetric black holes. For example, certain holomorphic corrections to supersymmetric gauge theories are known to have an elegant de- scription in terms of an auxiliary Riemann surface Σ. A compactification of string theory on a local Calabi-Yau that is modeled on this Riemann surface “geometri- cally engineers” the corresponding gauge theory in four dimensions. Topological string amplitudes capture the holomorphic corrections. In Chapter 3 and 4 we elucidate these relations by introducing a web of dualities
- 23. 10 Chapter 1. Introduction in string theory. This web is given in full detail in Fig. 1.6. Most important are its outer boxes, that are illustrated pictorially in Fig. 1.5. They correspond to string theory embeddings of the theories we mentioned above. The upper right box is a string theory embedding of supersymmetric gauge theory, whereas the lowest box is a string theory embedding of topological string theory. The main objective of the duality web is to relate both string frames to the upper left box, that describes a string theory configuration of intersecting D- branes. The most relevant feature of this intersecting brane configuration is that the D-branes intersect over a Riemann surface Σ, which is the same Riemann surface that underlies the gauge theory and appears in topological string theory. We explain that the duality chain gives a dual description of topological string theory in terms of a 2-dimensional quantum field theory of free fermions that live on the 2-dimensional intersecting brane, the so-called I-brane, wrapping Σ. Chapter 3 studies 4-dimensional gauge theories that preserve a maximal amount of supersymmetry. They are called N = 4 supersymmetric Yang-Mills theories. Instead of analyzing this theory on flat R4 , we consider more general ALE back- grounds. These non-compact manifolds are resolved singularities of the type C2 /Γ, where Γ is a finite subgroup of SU(2) that acts linearly on C2 . In the mid- nineties it has been discovered that the gauge theory partition function on an ALE space computes 2-dimensional CFT characters. In Chapter 3 we introduce a string theoretic set-up that explains this duality between 4-dimensional gauge theories and 2-dimensional CFT’s from a higher standpoint. In Chapter 4 we extend the duality between supersymmetric gauge theories and intersecting brane configurations to the full duality web in Fig. 1.6. We start out with 4-dimensional N = 2 gauge theories, that preserve half of the supersym- metry of the above N = 4 theories, and line out their relation to the intersecting brane configuration and to local Calabi-Yau compactifications. Furthermore, we relate several types of objects that play an important role in the duality sequence, and propose a partition function for the intersecting brane configuration. The I-brane configuration emphasizes the key role of the auxiliary Riemann sur- face Σ in 4-dimensional gauge theory and topological string theory. All the rele- vant physical modes are localized on the intersecting brane that wraps Σ. These modes turn out to be free fermions. In Chapter 5 and 6 we describe the resulting 2-dimensional quantum field on Σ in the language of integrable hierarchies. We connect local Calabi-Yau compactifications to the Kadomtsev-Petviashvili, or shortly, KP integrable hierarchy, which is closely related to 2-dimensional free fermion conformal field theories. To any semi-classical free fermion system on a curve Σ it associates a solution of the integrable hierarchy. This is well-known
- 24. 1.2. Outline of this thesis 11 as a Krichever solution. In Chapter 5 we explain how to interpret the topological string partition function on a local Calabi-Yau XΣ as a quantum deformation of a Krichever solution. We discover that the curve Σ should be replaced by a quantum curve, defined as a differential operator P(x) obtained by quantizing y = λ∂/∂x. Mathematically, we are led to a novel description of topological string theory in terms of D- modules. Chapter 6 illustrates this formalism with several examples related to matrix mod- els and gauge theory. In all these examples there is a canonical way to quantize the curve as a differential operator. Moreover, we can simply read off the known partition functions from the associated D-modules. This tests our proposal. The topological string partition function has several interpretations in terms of topological invariants on a Calabi-Yau threefold. Moreover, its usual expansion in the topological string coupling constant λ is only valid in a certain regime of the background Calabi-Yau parameters. Although the topological invariants stay invariant under small deformations in these background parameters, there are so-called walls of marginal stability, where the index of these invariants may jump. This phenomenon has recently attracted much attention among both physicists and mathematicians. Chapter 7 studies wall-crossing in string theory compactifications that preserve N = 4 supersymmetry. The invariants in this theory that are sensitive to wall- crossing are called quarter BPS states. They have been studied extensively in the last years. In Chapter 7 we work out in detail the relation of these supersym- metric states to a genus 2 surface Σ. Moreover, we find that wall-crossing in this theory has a simple and elegant interpretation in terms of the genus 2 surface. Finally, Chapter 8 focuses on supersymmetry breaking in Calabi-Yau compactifi- cations. Since supersymmetry is not an exact symmetry of nature, we need to go beyond Calabi-Yau compactifications to find a more accurate description of nature. One of the possibilities is that supersymmetry is broken at a lower scale than the compactification scale. In Chapter 8 we study such a supersymmetry breaking mechanism for N = 2 supersymmetric gauge theories. These gauge theories are related to type II string compactifications on a local Calabi-Yau XΣ. By turning on non-standard fluxes (in the form of generalized gauge fields) on the underlying Riemann surface Σ we find a potential on the moduli space of the theory. We show that this potential generically has non-supersymmetric minima. Chapter 3 and 4 explain the ideas of the publication [1], whereas Chapter 5 and 6 contain the results of [2]. The first part of Chapter 7 includes examples from [1], whereas the second part is based on the article [3]. Finally, Chapter 8 is a slightly shortened version of the work [4].
- 25. 12 Chapter 1. Introduction Figure 1.5: The web of dualities relates 4-dimensional supersymmetric gauge theory to topological string theory and to an intersecting brane configuration.
- 26. 1.2. Outline of this thesis 13 Figure 1.6: Web of dualities.
- 28. Chapter 2 Calabi-Yau Geometry Although string theory lives on 10-dimensional backgrounds, it is possible to make contact with the 4-dimensional world we perceive by shrinking 6 out of the 10 dimensions to very small scales. Remarkably, this also yields many impor- tant examples of interesting interactions between physics and mathematics. The simplest way to compactify a 10-dimensional string background to 4 dimensions is to start with a compactification of the form R4 × X, where X is a compact 6-dimensional Riemannian manifold and R4 represents Minkowski space-time. Apart from the metric, string theory is based on several more quantum fields. Amongst them are generalized Ramond-Ramond (RR) gauge fields, the ana- logues of the famous Yang-Mills field in four dimensions. In a compactification to four dimensions these fields can be integrated over any cycle of the internal manifold X. This results in e.g. scalars and gauge fields, which are the building blocks of the standard model. This is illustrated in Fig. 2.1. Four-dimensional symmetries, like gauge symmetries, can thus be directly related to geometrical symmetries of the internal space. This geometrization of physics is a main theme in string theory, and we will see many examples in this thesis. Although the topological properties of the compactification manifold X are re- stricted by the model that we try to engineer in four dimensions, other moduli of X, such as its size and shape, are a priori allowed to fluctuate. This leads to the so-called landscape of string theory, that parametrizes all the possible vacua. A way to truncate the possibilities into a discrete number of vacua is to introduce extra fluxes of some higher dimensional gauge fields. We will come back to this at the end of this thesis, in Chapter 8. The best studied compactifications are those that preserve supersymmetry in four dimensions. This yields severe constraints on the internal manifold. Not only should it be provided with a complex structure, but also its metric should
- 29. 16 Chapter 2. Calabi-Yau Geometry Figure 2.1: Compactifying string theory over an internal space X geometrizes 4- dimensional physics. Here we represented the 6-dimensional internal space as a 2-torus. The Yang-Mills gauge field Aµ on R4 is obtained by integrating a 10-dimensional RR 4-form over a 3-cycle C. be of a special form. They are known as Calabi-Yau manifolds. Remarkably, this class of manifolds is also very rich from a mathematical perspective. This chapter is meant to acquaint the reader with Calabi-Yau manifolds. The aim of Section 2.1 is to describe compact real 6-dimensional Calabi-Yau manifolds in terms of real 3-dimensional geometry. Section 2.2 illustrates this with the prime example of a compact Calabi-Yau manifold, the Fermat quintic. We find that its 3-dimensional representation is characterized by a Riemann surface. In Section 2.3 we explain how this places the simpler local or non-compact Calabi- Yau compactifications into context. These are studied in the main body of this thesis. 2.1 The Strominger-Yau-Zaslow conjecture Calabi-Yau manifolds X are complex Riemannian manifolds with some addi- tional structures. These structures can be characterized in a few equivalent ways. One approach is to combine the metric g and the complex structure J into a 2-form k = g ◦ J. For Calabi-Yau manifolds this 2-form needs to be closed dk = 0. It is called the Kähler form and equips the Calabi-Yau with a Käher structure. Moreover, a Calabi-Yau manifold must have a trivializable canonical bundle KX = V3 T∗ X, where T∗ X is the holomorphic cotangent space of X. The canonical bundle is a line bundle over X. When it is trivializable the Calabi-Yau manifold contains a non-vanishing holomorphic 3-form Ω. Together, the Kähler
- 30. 2.1. The Strominger-Yau-Zaslow conjecture 17 form k and the holomorphic 3-form Ω determine a Calabi-Yau manifold. The only non-trivial cohomology of a Calabi-Yau X is contained in H1,1 (X) and H2,1 (X). These cohomology classes parametrize its moduli. The elements in H1,1 (X) can change the Kähler structure of the Calabi-Yau infinitesimally, and are therefore called Kähler moduli. In string theory we usually complexify these moduli. On the other hand, H2,1 (X) parametrizes the complex structure moduli of the Calabi-Yau. It is related to the choice of the holomorphic 3-form Ω, since contracting (2, 1)-forms with Ω determines an isomorphism with the cohomology class H1 ¯ ∂ (X), that is well-known to characterize infinitesimal complex structure deformations. Celebrated work of E. Calabi and S.-T. Yau shows that the Kähler metric g can be tuned within its cohomology class to a unique Kähler metric that satisfies Rmn = 0. Calabi-Yau manifolds can therefore also be characterized by a unique Ricci-flat Kähler metric. A tantalizing question is how to visualize a Calabi-Yau threefold. The above definition in terms of a Kähler form k and a holomorphic 3-form Ω is rather abstract. Is there a more concrete way to picture a Calabi-Yau manifold? The SYZ conjecture From the string theory perspective there are many relations between different string theory set-ups. These are known as dualities, and relate the different in- carnations of string theory (type I, type IIA/B, heterotic, M-theory and F-theory) and different background parameters. One of the famous dualities in string the- ory connects type IIB string theory compactified on some Calabi-Yau X with type IIA string theory compactified on another Calabi-Yau X̃, where both Calabi-Yau manifolds are related by swapping their Kähler and complex structure param- eters. This duality is called mirror symmetry. It suggests that any Calabi-Yau threefold X has a mirror X̃ such that H1,1 (X) ∼ = H2,1 (X̃) and H2,1 (X) ∼ = H1,1 (X̃). Although the mirror conjecture doesn’t hold exactly as it is stated above, many mirror pairs have been found and much intuition has been obtained about the underlying Calabi-Yau geometry. One of the examples that illustrates this best is known as the Strominger-Yau- Zaslow (SYZ) conjecture [5]. The starting point for this argument is mirror sym- metry in type II string theory. Without explaining in detail what type II string theory is, we just note that an important role in this theory is played by so-called D-branes. These branes can be considered mathematically as submanifolds of the Calabi-Yau manifold with certain U(1) bundles on them.
- 31. 18 Chapter 2. Calabi-Yau Geometry The submanifolds are even (odd) dimensional in type IIA (type IIB) string the- ory, and they are restricted by supersymmetry. Even-dimensional cycles need to be holomorphic, whereas 3-dimensional cycles have to be of special Lagrangian type. (Note that these are the only odd-dimensional cycles in case the first Betti number b1 of the Calabi-Yau is zero.) A special Lagrangian 3-cycle C is defined by the requirement that k|C = 0 as well as Im Ω|C = 0. Both the holomorphic and the special Lagrangian cycles minimize the volume in their homology class, and are therefore stable against small deformations. This turns them into supersymmetric cycles. Mirror symmetry not only suggests that the backgrounds X and X̃ are related, but that type IIA string theory compactified on X̃ is equivalent to type IIB string theory on X. This implies in particular that moduli spaces of supersymmetric D-branes should be isomorphic. In type IIB theory in the background R4 × X the relevant D-branes are mathematically characterized by special Lagrangian submanifolds in X with flat U(1)-bundles on them. In type IIA theory there is one particularly simple type of D-branes: D0-branes that just wrap a point in X̃. Their moduli space is simply the space X̃ itself. Since mirror symmetry maps D0-branes into D3-branes that wrap special Lagrangian cycles in X, the moduli space of these D3-branes, i.e. the moduli space of special Lagrangian submanifolds with a flat U(1) bundle on it, should be isomorphic to X̃. Figure 2.2: Schematic illustration of an SYZ fibration of X̃ (on the left) and of X (on the right). Mirror symmetry (MS) acts by mapping each point in X̃ to a special Lagrangian cycle of X. In the SYZ fibration of X̃ this point is part of a fiber F̃b over some b ∈ B. Mirror symmetry sends it to the total fiber Fb, over b ∈ B, of the SYZ fibration of X. Special Lagrangian deformations of a special Lagrangian submanifold C are well-known to be characterized by the first cohomology group H1 (C, R). On the other hand, flat U(1)-connections moduli gauge equivalence are parametrized by H1 (C, R)/H1 (C, Z). The first Betti number of C therefore has to match the complex dimension of X̃, which is 3. So mirror symmetry implies that X̃ must
- 32. 2.1. The Strominger-Yau-Zaslow conjecture 19 admit a 3-dimensional fibration of 3-tori T3 = (S1 )3 over a 3-dimensional base space that parametrizes the special Lagrangian submanifolds in X. Of course, the same argument proves the converse. We thus conclude that both X and X̃ are the total space of a special Lagrangian fibration of 3-tori over some real three-dimensional base B. Such a special Lagrangian fibration is called an SYZ fibration. It is illustrated in Fig. 2.2. Notice that generically this fibration is not smooth over the whole base B. We refer to the locus Γ ⊂ B where the fibers degenerate as the singular, degenerate or discriminant locus of the SYZ fibration. Smooth fibers Fb and F̃b, over some point b ∈ B, can be argued to be dual in the sense that H1 (Fb, R/Z) = F̃b and vice versa, since flat U(1) connections on a torus are parametrized by the dual torus. Large complex structure Mirror symmetry, as well as the SYZ conjecture, is not exactly true in the way as stated above. More precisely, it should be considered as a symmetry only around certain special points in the Kähler and complex structure moduli space. In the complex structure moduli space these are the large complex structure limit points, and in the Kähler moduli space the large Kähler structure limit points. Mirror symmetry will exchange a Calabi-Yau threefold near a large com- plex structure point with its mirror near a large Kähler structure limit point. Moreover, only in these regions the SYZ conjecture may be expected to be valid. Topological aspects of type II string theory compactifications can be fully de- scribed in terms of either the Kähler structure deformations or the complex structure deformations of the internal Calabi-Yau manifold. In the following we will focus solely on describing the Kähler structure of an SYZ fibered Calabi- Yau threefold. Starting with this motivation we can freely go to a large complex structure point on the complex structure moduli space. How should we think of such a large complex structure? Let us illustrate this with a 2-torus T2 . This torus is characterized by one Kähler parameter t, which is the volume of the torus, and one complex structure parameter τ. When perform- ing a mirror symmetry, these two parameters are exchanged. When we send the complex structure to infinity τ 7→ i∞, the torus degenerates into a nodal torus. See Fig. 2.3. However, in this limit the volume also blows up. When we hold this volume fixed, by rescaling the Ricci flat metric ds2 = t τ2 dz ⊗ dz̄, the torus will collapse to a line. This line should be viewed as the base of the SYZ fibration. For a general Calabi-Yau threefold a similar picture is thought to be true. When approaching a large complex structure point, the Calabi-Yau threefold should have a description as a special Lagrangian fibration over a 3-sphere whose T3 - fibers get very small. The metric on these fibers is expected to become flat.
- 33. 20 Chapter 2. Calabi-Yau Geometry Figure 2.3: In picture (1) we see a regular torus with complex structure τ: below the torus itself and above its fundamental domain (that is obtained by removing the A and B-cycle from the torus). In the large complex structure limit τ is sent to i∞. In picture (2a) this degeneration is illustrated topologically and in (2b) metrically: when we hold the volume fixed, the torus collapses to a circle. Moreover, the singular locus of the fibration is thought to be of codimension two in the base in that limit, so that it defines a 1-dimensional graph on the base of the SYZ fibration. In recent years much progress has been made in the mathematical study of the SYZ conjecture. Comprehensive reviews can be found in e.g. [6, 7, 8]. Especially the topological approach of M. Gross and others, with recent connections to tropical geometry, seems very promising. The main idea in this program is to characterize the SYZ fibration by a natural affine structure on its base B. This affine structure is found as follows. Once we pick a zero-section of the (in particular) Lagrangian fibration, we can contract the inverse symplectic form with a one-form on the base B to find a vector field along the fiber. An integral affine structure can then be defined on B by an integral affine function f whose derivative df defines a time-one flow along the fiber that equals the identity. The affine structure may be visualized as a lattice on the base B. Although this affine structure will be trivial in a smooth open region of the fi- bration, it contains important information near the singular locus. Especially the monodromy of the T3 -fibers is encoded in it. In the large complex structure limit all non-trivial symplectic information of the SYZ fibration is thought to be captured in the monodromy of the T3 -fibers around the discriminant locus of the fibration [9, 7]. The SYZ conjecture thus leads to a beautiful picture of how a Calabi-Yau threefold may be visualized as an affine real 3-dimensional space with an additional integral monodromy structure around an affine 1-dimensional de- generation locus (see also [10]). One of the concrete results that have been accomplished is an SYZ description
- 34. 2.2. The Fermat quintic 21 of elliptic K3-surfaces [11] and Calabi-Yau hypersurfaces in toric varieties (in a series of papers by W.-D. Ruan starting with [12]). Let us illustrate this with the prime example of a compact Calabi-Yau threefold, the Fermat quintic. This is also the first Calabi-Yau for which a mirror pair was found [13]. On a first reading it is not necessary to go through all of the formulas in this example; looking at the pictures should be sufficient. 2.2 The Fermat quintic The Fermat quintic is defined by an equation of degree 5 in projective space P4 : Xµ : 5 X k=1 z5 k − 5µ 5 Y k=1 zk = 0, where [z] = [z1 : . . . : z5] are projective coordinates on P4 . Here µ denotes the complex structure of the threefold. Pulling back the Kähler form of P4 provides the Fermat quintic with a Kähler structure. Moreover, the so-called adjunction formula shows that Xµ has a trivial canonical bundle. So the Fermat quintic Xµ is a Calabi-Yau threefold. Note as well that P4 admits an action of T4 that is parametrized by five angles θk with P k θk = 0: T4 : [z1 : . . . : z5] 7→ [eiθ1 z1 : . . . : eiθ5 z5] Such a variety is called toric. The quintic is thus embedded as a hypersurface in a toric variety. When we take µ → ∞ we reach the large complex structure limit point X∞ : 5 Y k=1 zk = 0. This is just a union of five P3 ’s, that each inherit a T3 -action from the above toric action on P4 . From this observation it simply follows that X∞ can be seen as a SYZ fibration. We can make this explicit by considering the fibration π : P4 → R4 given by π([z]) = 5 X k=1 |zk|2 P5 l=1 |zl|2 pk, where the pk are five generic points in R4 . The image of F is a 4-simplex ∆ spanned by the five points pk in R4 and X∞ is naturally fibered over the bound-
- 35. 22 Chapter 2. Calabi-Yau Geometry Figure 2.4: The base of the SYZ fibration of the Fermat quintic Xµ is the boundary ∂∆ of a 4-simplex ∆. To be able to draw this base we have placed the vertex pi at infinity. The tetrahedron ∆i is the projection of the subset {zi = 0} ⊂ P4 . The boundary ∂∆ consists of 5 such tetrahedrons. ary ∂∆ with generic fiber being T3 . The base of the SYZ fibration is thus topo- logically a 3-sphere S3 . Similar to the large complex structure limit of a 2-torus, the Ricci-flat metric on X∞ is degenerate and the SYZ fibers are very small. Let us introduce some notation to refer to different patches of P4 and ∆. Call Di1,...,in the closed part of P4 where zi1 up to zin vanish, and denote its projec- tion to R4 by ∆i1,...,in . The ten faces of the boundary ∂∆ are thus labeled by ∆ij, with 1 ≤ i, j ≤ 5. In the following it is also useful to introduce a notation for the S1 -cycles that are fibered over the base ∆. So define the circles γk i = { zi = 0, | zk zj | = a1, zl zj = a2, zm zj = a3 }, where the indices {i, j, k, l, m} are a permutation of {1, . . . , 5} and the numbers a1, a2, a3 are determined by choosing a base point [z] on the circle. Since a different choice for the index j leaves the circle invariant, it is not included as a label in the name. Which circle shrinks to zero-size over each cell of the boundary ∂∆ is summarized in Fig. 2.5. Notice that the only non-vanishing circle in the fibers over the triple intersection ∆ijk is the circle γl i (which may also be denoted as γl j, γl k, −γm i , −γm j or −γm k ). W.-D. Ruan works out how to generate a Lagrangian fibration for any Xµ where µ is large [12]. His idea is to use a gradient flow that deforms the Lagrangian fibration of X∞ into a Lagrangian fibration of Xµ. Let us consider the region Di
- 36. 2.2. The Fermat quintic 23 Figure 2.5: The boundary of each tetrahedron is a union of four triangles. The T3 -fibration of the Fermat quintic degenerates over these triangles. This figure illustrates which cycle in the T3 -fiber vanishes over which triangle in ∂∆: γj i degenerates over the triangle ∆ij, γk j over ∆jk, etc. and choose zj 6= 0 for some j 6= i. In this patch xk = zk/zj are local coordinates. It turns out that the flow of the vector field V = Re P k6=i,j x5 k + 1 Q k6=i,j xk ∂ ∂xi produces a Lagrangian fibration of Xµ over ∂∆. All the smooth points of X∞, i.e. those for which only one of the coordinates zi vanishes, will be transformed into regular points of the Lagrangian fibration. Only the points in the intersection of Xµ with the singular locus of X∞ won’t move with the flow. These can be shown to form the complete singular locus of the Lagrangian fibration of Xµ. Let us call this singular locus Σ and denote Σij = Dij ∩ Σ = {[z] | zi = zj = 0, z5 k + z5 l + z5 m = 0 }. The singular locus Σij is thus a projective curve which has genus 6. It intersects Dijk at the five points. The image of Σij under the projection π is a deformed triangle in ∆ij that intersects the boundary lines of ∆ij once. This is illustrated in Fig. 2.6. In the neighborhood of an inverse image of the intersection point 1 2 (pl + pm) in Dij the coordinate zk gets very small. This implies that if we write zk = rkeiφk , it is the circle parametrized by φk that wraps the leg of the pair of pants in the limit rk → 0. In the notation we introduced before this circle is γk j = γk i . The study of the cycles in the fibration reveals the structure of the singular locus. It is built out of two kinds of 3-vertices. We call a 3-vertex whose center lies at
- 37. 24 Chapter 2. Calabi-Yau Geometry Figure 2.6: The shaded area in this figure illustrates the discriminant locus of the La- grangian fibration of the quintic Xµ, for large µ. In the total space it has the shape of a genus 6 Riemann surface Σij over each 2-cell ∆ij. The five dots on the Riemann surface project to a single dot on the base. an edge Dijk a plus-vertex and a 3-vertex that lies in the interior of some 2-cell Dij a minus-vertex. The plus-vertex is described by a different degenerating cycle at each of the three legs. Together they sum up to zero. The minus-vertex is characterized by a single vanishing cycle. The precise topological picture is shown in Fig. 2.7. So in the large complex structure limit µ → ∞ the quintic has an elegant struc- ture in terms of ten transversely intersecting genus 6 Riemann surfaces or equiv- alently in terms of 50 plus-vertices and 250 minus-vertices. Two kinds of vertices The structure of the singular locus of the Fermat quintic in the above example, ten genus 6 Riemann surfaces that intersect each other transversely, is very el- egant. It is remarkable that it may be described by just two types of 3-vertices. This immediately raises the question whether this is accidental or a generic fea- ture of SYZ fibrations of Calabi-Yau threefolds. In fact, M. Gross shows that under reasonable assumptions there are just a few possibilities for the topolog- ical structure in the neighborhood of the discriminant locus [9] (see also [7]). Indeed only two types of trivalent vertices can occur. Both are characterized by the monodromy that acts on three generators γi of the homology H1(F, Z) of
- 38. 2.2. The Fermat quintic 25 Figure 2.7: This picture shows one plus-vertex surrounded by three minus-vertices. It illus- trates the discriminant locus in the neighborhood of an inverse image of the center of ∆ijk, marked by a dot in the center of the plus-vertex. At this point the three genus 6 surfaces Σij, Σjk and Σki meet transversely. Notice that e.g. γk j + γl j + γm j = 0 and γj i + γk j + γi k = 0. the T3 -fiber F when we transport them around each single leg of the vertex. Let us summarize this monodromy in three matrices Mα such that γi 7→ γj(Mα)ji when we encircle the αth leg of the vertex. The first type of trivalent vertex is described by the three monodromy matrices M+ : 1 0 1 0 1 0 0 0 1 , 1 0 0 0 1 1 0 0 1 , 1 0 −1 0 1 −1 0 0 1 . This vertex is characterized by a a distinguished T2 ⊂ T3 that is generated by γ1 and γ2 and stays invariant under the monodromy. Only the element γ3 picks up a different cycle at each leg. This latter cycle must therefore be a vanishing cycle at the corresponding leg. The legs of this vertex can thus be labeled by the cycles γ1, γ2 and −γ1 − γ2 respectively, so that the vertex is conveniently represented as in Fig. 2.8. Note that this is precisely the topological structure of the plus-vertex in the Lagrangian fibration of the Fermat quintic.
- 39. 26 Chapter 2. Calabi-Yau Geometry Figure 2.8: The plus-vertex is illustrated as a 1-dimensional graph. Its legs are labeled by the cycle in T2 ⊂ T3 that vanishes there. The cycle γ3 picks up the monodromy. Figure 2.9: The minus-vertex is illustrated as a pair of pants. It is characterized by a single vanishing cycle γ3. Furthermore, its legs are labeled by the cycle that wraps it. The monodromy of the second type of vertex is summarized by the matrices M− : 1 0 0 0 1 0 −1 0 1 , 1 0 0 0 1 0 0 −1 1 , 1 0 0 0 1 0 1 1 1 . In contrast to the plus-vertex these monodromy matrices single out a unique 1-cycle γ3 that degenerates at all three legs of the vertex. Instead of labeling the vertex by this vanishing cycle, it is now more convenient to label the legs with the non-vanishing cycle that does not pick up any monodromy. Like in the example of the Fermat quintic these cycles topologically form a pair of pants. This is illustrated in Fig. 2.9. Notice that both sets of monodromy matrices are related by simple duality (M+)−t = M−. Since the mirror of an SYZ fibered Calabi-Yau may be obtained by dualizing the T3 -fibration, the above vertices must be related by mirror sym- metry as well. Obviously these vertices will become important when we describe string compactifications on Calabi-Yau threefolds.
- 40. 2.3. Local Calabi-Yau threefolds 27 2.3 Local Calabi-Yau threefolds Topological string theory captures topological aspects of type II string theory compactifications; Kähler aspects of type IIA compactifications and complex structure aspects of type IIB compactification. Mirror symmetry relates them. The topological string partition function Ztop can be written as a generating function of either symplectic or complex structure invariants of the underlying Calabi-Yau manifold. It contains for example a series in the number of genus zero curves that are embedded in the Calabi-Yau threefold. Using mirror symmetry it is possible to go well beyond the classical computations of these numbers. A fa- mous result is the computation of the whole series of these genus zero invariants for the Fermat quintic: 2875 different lines, 609250 conics, 317206375 cubics, etc. [14]. It is much more difficult to find the complete topological string partition func- tion, which also contains information about higher genus curves in the Calabi- Yau threefold. The state of the art for the quintic is the computation of these invariant up to g = 51 [15]. Although this is an impressive result, it is far from computing the total partition function. In contrast, the all-genus partition function has been found for a simpler type of Calabi-Yau manifolds, that are non-compact. What kind of spaces are these? And why is it so much easier to compute their partition function? The simplest Calabi-Yau threefold is plain C3 with complex coordinates zi. It admits a Kähler form k = 3 X i=1 dzi ∧ dz̄i, and a non-vanishing holomorphic 3-form Ω = dz1 ∧ dz2 ∧ dz3. A special Lagrangian T2 × R fibration of C3 over R3 has been known for a long time [16]. It is defined by the map (z1, z2, z3) 7→ ( Im z1z2z3, |z1|2 − |z2|2 , |z1|2 − |z3|2 ). Notice that its degeneration locus is a 3-vertex with legs (0, t, 0), (0, 0, t) and (0, −t, −t), for t ∈ R≥0. Over all these legs some cycle in the T2 -fiber shrinks to zero-size. We can name these cycles γ1, γ2 and −γ1 − γ2 since they add up to zero. The degenerate fiber over each leg is a pinched cylinder times S1 . It is thus the noncompact cycle (∼ = R) in the fiber that picks up monodromy when we move around one of the toric legs. This 3-vertex clearly has the same topological structure as the plus-vertex.
- 41. 28 Chapter 2. Calabi-Yau Geometry Many more non-compact Calabi-Yau’s can be constructed by gluing C3 -pieces together. In fact, these constitute all non-compact toric Calabi-Yau threefolds. Their degeneration locus can be drawn in R2 as a 1-dimensional trivalent graph. The fact that their singular graph is simply planar, as opposed to actually 3- dimensional (as for the Fermat quintic), makes the computation of the partition function on such non-compact toric Calabi-Yau’s much simpler. String theorists have managed to find the full topological partition function Ztop [17] (using a duality with a 3-dimensional topological theory, called Chern-Simons theory [18]). The recipe to compute the partition function involves cutting the graph in basic 3-vertices. To generalize this for compact Calabi-Yau’s it seems one would need to find a way to glue the partition function for a plus-vertex with that of a minus-vertex. Since the partition function is fully known, topological string theory on these toric manifolds is the ideal playground to learn more about its underlying struc- ture. This has revealed many interesting mathematical and physical connec- tions, for example to several algebraic invariants such as Donaldson-Thomas invariants [19, 20, 21] and Gopakumar-Vafa invariants [22, 23], to knot theory [24, 25, 26, 27], and to a duality with crystal melting [28, 29, 30]. To illustrate the last duality, let us write down the plain C3 partition function: Ztop(C3 ) = Y n0 1 (1 − qn)n = 1 + q + 3q2 + 6q3 + . . . . This q-expansion is well-known to be generating function of 3-dimensional par- titions; it is called the MacMahon function [31]. The 3-dimensional partition can be visualized as boxes that are stacked in the positive octant of R3 . Three of the sides of each box must either touch the walls or another box. This is pictured in Fig. 2.10. Figure 2.10: Interpretation of the first terms in the expansion of Ztop(C3 ) in terms of a three-dimensional crystal in the positive octant of R3 . Since q = eλ , where λ is the coupling constant of topological string theory, the boxes naturally have length λ. Whereas the regime with λ finite is described by a discrete quantum structure, in the limit λ → 0 surprisingly a smooth Calabi- Yau geometry emerges. In a duality with statistical mechanics this corresponds
- 42. 2.3. Local Calabi-Yau threefolds 29 to the shape of a melting crystal. These observations have led to deep insights in the quantum description of space and time [32, 33]. Remarkably, it has been shown that the emergent smooth geometry of the crystal can be identified with the mirror of C3 . How does this limit shape look like? Using local mirror symmetry the equation for the mirror of C3 was found in [34]. It is given by uv − x − y + 1 = 0, where u, v ∈ C and x, y ∈ C∗ . Remember that the topological structure of the mirror of a plus-vertex should be that of a minus-vertex. Viewing the mirror as a (u, v)-fibration over a complex plane spanned by x and y confirms this: The degeneration locus of the fibration equals the zero-locus x + y − 1 = 0. Parametrizing this curve by x, it is easily seen that this is a 2-sphere with three punctures at the points x = 0, 1 and ∞. We can equivalently represent this curve as a pair of pants, by cutting off a disc at each boundary |x̃| = 1, where x̃ is a local coordinate that vanishes at the corresponding puncture. This realizes the mirror of C3 topologically as the minus-vertex in Fig. 2.9. Mirrors of general non-compact toric manifolds are of the same form XΣ : uv − H(x, y) = 0, where the equation H(x, y) = 0 now defines a generic Riemann surface Σ em- bedded in (C∗ )2 . This surface is a thickening of the 1-dimensional degeneration graph Γ of its mirror. Its non-vanishing holomorphic 3-form is proportional to Ω = du u ∧ dx ∧ dy. These geometries allow a Ricci-flat metric that is conical at infinity [35, 36, 37, 38]. We refer to the threefold XΣ as the local Calabi-Yau threefold modeled on Σ. The curve Σ plays a central role in this thesis. We study several set-ups in string theory whose common denominator is the relevance of the Riemann surface Σ. In particular, we study the melting crystal picture from the mirror perspective. One of our main results is a simple representation of topological string theory in terms a quantum Riemann surface, that reduces to the smooth Riemann surface Σ in the limit λ → 0.
- 44. Chapter 3 I-brane Perspective on Vafa-Witten Theory and WZW Models In the last decades enormous progress has been made in analyzing 4-dimensional supersymmetric gauge theories. Partition functions and correlation functions of some theories have been computed, spectra of BPS operators have been discov- ered and many other structures have been revealed. Most fascinating to us is that many exact results can be related to two-dimensional geometries. Since 4-dimensional supersymmetric gauge theories appear in several contexts in string theory, much of this progress is strongly influenced by string theory. Often, string theory tools can be used to compute important quantities in su- persymmetric gauge theories. Moreover, in many cases string theory provides a key understanding of new results. For example, when auxiliary structures in the gauge theory can be realized geometrically in string theory and when symme- tries in the gauge theory can be understood as stringy dualities. In this chapter we study a remarkable correspondence between 4-dimensional gauge theories and 2-dimensional conformal field theories. This correspondence connects a “twisted” version of supersymmetric Yang-Mills theory to a so-called Wess-Zumino-Witten model. In particular, generating functions of SU(N) gauge instantons on the 4-manifold C2 /Zk are related to characters of the affine Kac- Moody algebra c su(k) at level N. This connection was originally discovered by Nakajima [39], and further analyzed by Vafa and Witten [40]. The goal of this chapter is to make it more transparent. Once again, we find that string theory offers the right perspective.
- 45. 32 Chapter 3. I-brane Perspective on Vafa-Witten Theory and WZW Models We have strived to make this chapter self-contained by starting in Section 3.1 with a short introduction in gauge and string theory. We review how supersym- metric gauge theories show up as low energy world-volume theories on D-branes and how they naturally get twisted. Twisting emphasizes the role of topologi- cal contributions to the theory. Furthermore, we introduce fundamental string dualities as T-duality and S-duality. In Section 3.2 we introduce Vafa-Witten theory as an example of a twisted 4- dimensional gauge theory, and study it on non-compact 4-manifolds that are asymptotically Euclidean. In Section 3.3 we show that Vafa-Witten theory on such a 4-manifold is embedded in string theory as a D4-D6 brane intersection over a torus T2 . We refer to the intersecting brane wrapping T2 as the I-brane. Since the open 4-6 strings introduce chiral fermions on the I-brane, we find a duality between Vafa-Witten theory and a CFT of free fermions on T2 . In Section 3.4 we show that the full I-brane partition function is simply given by a fermionic character, and reduces to the Nakajima-Vafa-Witten results after taking a decoupling limit. The I-brane thus elucidates the Nakajima-Vafa-Witten correspondence from a string theoretic perspective. Moreover, we gain more insights in level-rank duality and the McKay correspondence from this stringy point of view. 3.1 Instantons and branes A four-dimensional gauge theory with gauge group G on a Euclidean 4-manifold M is mathematically formulated in terms of a G-bundle E → M. A gauge field A is part of a local connection D = d + A of this bundle, whose curvature is the electro-magnetic field strength F = dA + A ∧ A. If we denote the electro-magnetic gauge coupling by e and call ∗ the Hodge star operator in four dimensions, the Yang-Mills path integral is Z = Z A/G DA exp − 1 e2 Z M d4 x TrF ∧ ∗F , This path integral, over the moduli space of connections A modulo gauge invari- ance, defines quantum corrections to the classical Yang-Mills equation D∗F = 0. When the gauge group is abelian, G = U(1), the equation of motion plus Bianchi identity combine into the familiar Maxwell equations d ∗ F = 0, dF = 0.
- 46. 3.1. Instantons and branes 33 Topological terms Topologically non-trivial configurations of the gauge field are measured by char- acteristic classes. If G is connected and simply-connected the gauge bundle E is characterized topologically by the instanton charge ch2(F) = Tr F ∧ F 8π2 ∈ H4 (M, Z). (3.1) Instanton configuration are included in the Yang-Mills formalism by adding a topological term to the Yang-Mills Lagrangian L = − 1 e2 F ∧ ∗F + iθ 8π2 F ∧ F (3.2) Note that this doesn’t change the equations of motion. The path integral is invariant under θ → θ + 2π, and the parameter θ is therefore called the θ-angle. The total Yang-Mills Lagrangian can be rewritten as L = iτ 4π F+ ∧ F+ + iτ̄ 4π F− ∧ F−, (3.3) where F± = 1 2 (F ± ∗F) are the (anti-)selfdual field strengths while τ = θ 2π + 4πi e2 (3.4) is the complexified gauge coupling constant . When G is not simply-connected magnetic fluxes on 2-cycles in M are detected by the first Chern class c1(F) = Tr F 2π ∈ H2 (M, Z). (3.5) Electro-magnetic duality The Maxwell equations are clearly invariant under the transformation F ↔ ∗F that exchanges the electric and the magnetic field. To see that this is even a symmetry at the quantum level, we introduce a Lagrange multiplier field AD in the U(1) Yang-Mills path integral that explicitly imposes dF = 0: Z DADAD exp Z M iτ 4π F+ ∧ F+ + iτ̄ 4π F− ∧ F− + 1 2π F ∧ ∗dAD . Integrating out A yields the dual path integral Z DAD exp Z M i 4πτ FD + ∧ FD + + i 4πτ̄ FD − ∧ FD − .
- 47. 34 Chapter 3. I-brane Perspective on Vafa-Witten Theory and WZW Models So electric-magnetic duality is a strong-weak coupling duality, that sends the complexified gauge coupling τ 7→ −1/τ. Moreover, this argument suggests an important role for the modular group Sl(2, Z). This group acts on τ as τ 7→ aτ + b cτ + d , for a b c d ∈ Sl(2, Z). and is generated by S = 0 1 −1 0 and T = 1 1 0 1 . Hence, S is the generator of electro-magnetic duality (later also called S-duality) and T the generator of shifts in the θ-angle. The gauge coupling τ is thus part of the fundamental domain of Sl(2, Z) in the upper-half plane, as shown in Fig. 3.1. Figure 3.1: The fundamental domain of the modular group Sl(2, Z) in the upper half plane. Montonen and Olive [41] where pioneers in conjecturing that electro-magnetic duality is an exact non-abelian symmetry, that exchanges the opposite roles of electric and magnetic particles in 4-dimensional gauge theories. This involves replacing the gauge group G by the dual group Ĝ (whose weight lattice is dual to that of G). The first important tests of S-duality have been performed in supersymmetric gauge theories. For U(1) theories the partition function ZU(1) can be explicitly computed [42, 43]. The classical contribution to the partition function is given by integral fluxes p ∈ H2 (M, Z), as in equation (3.5), to the Langrangian (3.3). Decomposing the flux p into a self-dual and anti-selfdual contribution yields the generalized theta- function θΓ(q, q̄) = X (p+,p−)∈Γ q 1 2 p2 + q̄ 1 2 p2 − (3.6)
- 48. 3.1. Instantons and branes 35 with q = exp(2πiτ), whereas Γ = H2 (M, Z) is the intersection lattice of M and p2 = R M p ∧ p. The total U(1) partition function is found by adding quantum corrections to the above result, which are captured by some determinants [42]. Instead of transforming as a modular invariant, ZU(1) transforms as a modular form under Sl(2, Z)-transformation of τ ZU(1) aτ + b cτ + d = (cτ + d)u/2 (cτ + d)v/2 ZU(1) (τ) , a b c d ∈ Sl(2, Z). We will come back on this in Section 3.2. 3.1.1 Supersymmetry In supersymmetric theories quantum corrections are much better under control, so that much more can be learned about non-perturbative properties of the the- ory. We will soon discuss such elegant results, but let us first introduce super- symmetric gauge theories. The field content of the simplest supersymmetric gauge theories just consists of a bosonic gauge field A and a fermionic gaugino field λ. Supersymmetry relates the gauge field A to its superpartner χ. In any supersymmetric theory the number of physical bosonic degrees of freedom must be the same as the number of physical fermionic degrees of freedom. This constraints supersymmetric Yang- Mills theories to dimension d ≤ 10. The Lagrangian of a minimal supersymmetric gauge theory is L = − 1 4 Tr(FµνFµν ) + i 2 χ̄Γµ Dµχ, and supersymmetry variations of the fields A and χ are generated by a spinor δAµ = i 2 ¯ Γµχ, δχ = 1 4 FµνΓµν . (3.7) The number of supersymmetries equals the number of components of . Dimensionally reducing the above minimal N = 1 susy gauge theories to lower- dimensional space-times yield N = 2, 4 and possibly N = 8 susy gauge theories. Their supersymmetry variations are determined by an extended supersymme- try algebra. In four dimensions this is a unique extension of the Poincaré al- gebra generated by the supercharges QA α and QAα̇, with A ∈ {1, . . . , N} and α, α̇ ∈ {1, 2} are indices in the 4-dimensional spin group su(2)L × su(2)R. Non-
- 49. 36 Chapter 3. I-brane Perspective on Vafa-Witten Theory and WZW Models vanishing anti-commutation relations are given by {QA α , QBβ̇} = 2(σµ )αβ̇PµδA B {QA α , QB β } = αβZAB {QAα̇, QBβ̇} = α̇β̇Z† AB where ZAB and its Hermitean conjugate are the central charges. The auto- morphism group of this algebra, that acts on the supercharges, is known as the R-symmetry group. BPS states A special role in extended supersymmetric theories is played by supersymmetric BPS states [44]. They are annihilated by a some of the supersymmetry genera- tors, e.g. quarter BPS states satisfy QA |BPSi = 0, for 1/4 N indices A ∈ {1, . . . , N}. BPS states saturate the bound M2 ≤ |Z|2 and form “small” representations of the above supersymmetry algebra. This implies that supersymmetry protects them against quantum corrections: a small deformation won’t just change the dimension of the representation. Twisting Supersymmetric Yang-Mills requires a covariantly constant spinor in the rigid supersymmetry variations (3.7). Since these are impossible to find on a generic manifold M, the concept of twisting has been invented. Twisting makes use of the fact that supersymmetric gauge theories are invariant under a non-trivial in- ternal symmetry, the R-symmetry group. By choosing a homomorphism from the space-time symmetry group into this internal global symmetry group, the spinor representations change and often contain a representation that transforms as a scalar under the new Lorentz group. Such an odd scalar QT can be argued to obey Q2 T = 0. It is a topological supercharge that turns the theory into a cohomological quantum field theory. Observables O can be identified with the cohomology generated by QT , and correlation functions are independent of continuous deformations of the metric ∂ ∂gµν hO1 · · · Oki = 0 (3.8) These correlation functions can thus be computed by going to short distances. This yields techniques to study the dynamics of these theories non-perturbatively.
- 50. 3.1. Instantons and branes 37 For these topological theories it is sometimes possible to compute the partition function and other correlators. Witten initiated twisting in the context of N = 2 supersymmetric Yang-Mills [45]. He showed that correlators in the so-called Donaldson-Witten twist compute the famous Donaldson invariants. A general theme in cohomological field theories is localization. Unlike in general physical theories, in these topological theories the saddle point approximation is actually exact. The path integral only receives contributions from fixed point locus M of the scalar supercharge QT . Since the kinetic part of the action (that contains all metric-dependent terms) is QT -exact, the only non-trivial contribu- tion to the path integral is given by topological terms: ZcohTFT = Z DX exp − 1 e2 Skin(X) + Stop(X) → Z M DX exp (Stop(X)) . Here X represents a general field content. An elegant example in this respect is 2-dimensional gauge theory [46]. Extensive reviews of localization are [47, 48]. We will encounter localization on quite a few occasions, starting with Vafa- Witten theory in Section 3.2. 3.1.2 Extended objects Whereas Yang-Mills theory is formulated in terms of a single gauge potential A, string theory is equipped with a whole set of higher-form gauge fields. Instead of coupling to electro-magnetic particles they couple to extended objects, such as D-branes. This is analogous to the coupling of a particle of electric charge q to the Maxwell gauge field A q Z W A = q Z W Aµ ∂xµ ∂t dt, where W is the worldline of the particle. Notice that we need to pull-back the space-time gauge field A in the first term before integrating it over the worldline. We often don’t write down the pull-back explicitly to simplify notation. D-branes and other extended objects appear all over this thesis. Let us therefore give a very brief account of the properties that are relevant for us. Couplings and branes in type II Gauge potentials in type II theory either belong to the so-called RR or the NS- NS sector. The RR potentials couple to D-branes, whereas the NS-NS potential couples to the fundamental string (which is often denoted by F1) and the NS5- brane. Let us discuss these sectors in a little more detail. The only NS-NS gauge field is the 2-form B. The B-field plays a crucial role in Chapter 5. Aside from the B-field the NS-NS sector contains the dilaton field φ
- 51. 38 Chapter 3. I-brane Perspective on Vafa-Witten Theory and WZW Models and the space-time metric gmn. Together these NS-NS fields combine into the sigma model action Sσ-model = − 1 2πα0 Z Σ gmndxm ∧ ∗dxn + iBmndxm ∧ dxn + α0 φR. (3.9) This action describes a string that wraps the Riemann surface Σ and is embedded in a space-time with coordinates xm . In particular, it follows that the B-field couples to a (fundamental) string F1. Remember that the 1-form dxm refers to the pull-back ∂αxm dσα to the world- sheet Σ with coordinates σα . Furthermore, the symbol ∗ stands for the 2- dimensio-nal worldsheet Hodge star operator and R is the worldsheet curvature 2-form. This formula requires some more explanation though. The symbol √ α0 = ls sets the string length, since α0 is inversely proportional to its tension. Since the Ricci scalar of a Riemann surface equals its Euler number, the last term in the action contributes 2g − 2 powers of gs = eφ to a stringy g-loop diagram; gs is therefore called the string coupling constant. The extended object to which the B-field couples magnetically is called the NS5- brane. It can wrap any 6-dimensional geometry in the full 10-dimensional string background, but its presence will deform the transverse geometry. In the trans- verse directions the dilaton field φ is non-constant, and there is a flux H = dB of the B-field through the boundary of the transerve 4-dimensional space. The tension of a NS5-brane is proportional to 1/g2 s so that it is a very heavy object when gs → 0. Moreover, unlike for D-branes open strings cannot end on it. This makes it quite a mysterious object. For the RR-sector it makes a difference whether we are in type IIA or in type IIB theory: type IIA contains all odd-degree RR forms and type IIB the even ones. In particular, the gauge potentials C1 and C4 are known as the graviphoton fields for type IIA and type IIB, respectively, and the RR potential C0 is the axion field. Any RR potential Cp+1 couples electrically to a Dp-brane. This is a p-dimensional extended object that sweeps out a (p+1)-dimensional world-volume Σp+1. Type IIA thus contains Dp-branes with p even, whereas in type IIB p is odd. The electric Dp-brane coupling to Cp+1 introduces the term Tp Z Σp+1 Cp+1 (3.10) in the 10-dimensional string theory action, where 1/Tp = (2π)p √ α0 p+1 gs is the inverse tension of the Dp-brane. Magnetically, the RR-potential Cp+1 couples to
- 52. 3.1. Instantons and branes 39 a D(6 − p)-brane that wraps a (7 − p)-dimensional submanifold Σ7−p. Calibrated cycles D-branes are solitonic states as their tension Tp is proportional to 1/gs. To be stable against decay the brane needs to wrap a submanifold that preserves some supersymmetries. Geometrically, such configurations are defined by a calibration [49]. A calibration form is a closed form Φ such that Φ ≤ vol at any point of the background. A submanifold Σ that is calibrated satisfies Z Σ Φ = Z Σ vol, and minimizes the volume in its holomogy class. On a Kähler manifold a calibra- tion is given by the Kähler form t, and the calibrated submanifolds are complex submanifolds. On a Calabi-Yau threefold the holomorphic threeform Ω provides a calibration, whose calibrated submanifolds are special Lagrangians. Calibrated submanifolds support covariantly constant spinors, and therefore preserve some supersymmetry. D-branes wrapping them are supersymmetric BPS states. Worldvolume theory D-branes have a perturbative description in terms of open strings that end on them. The massless modes of these open strings recombine in a Yang-Mills gauge field A. When the D-brane worldvolume is flat the corresponding field theory on the p-dimensional brane is a reduction of N = 1 susy Yang-Mills from 10 dimensions to p + 1. The 9 − p scalar fields in this theory correspond to the transverse D-brane excitations. When N D-branes coincide the wordvolume theory is a U(N) supersymmetric Yang-Mills theory. For more general calibrated submanifolds the low energy gauge theory is a twisted topological gauge theory [50], which we introduced in Section 3.1.1. Which particular twist is realized, can be argued by determining the normal bundel to the submanifold. Sections of the normal bundel fix the transverse bosonic excitations of the gauge theory, and should correspond to the bosonic field content of the twisted theory. I-branes and bound states Branes can intersect each others such that they preserve some amount of super- symmetry. This is called an I-brane configuration. In such a set-up there are more degrees of freedom than the ones (we described above) that reside on the individual branes. These extra degrees of freedom are given by the modes of open strings that stretch between the branes. In stringy constructions of the standard model on a set of branes they often provide the chiral fermions.
- 53. 40 Chapter 3. I-brane Perspective on Vafa-Witten Theory and WZW Models Chiral fermions are intimately connected with quantum anomalies, and brane intersections likewise. To cure all possible anomalies in an I-brane system, a topological Chern-Simons term has to be added to the string action SCS = Tp Z Σp+1 Tr exp F 2π ∧ X i Ci ∧ q Â(R). (3.11) This term is derived through a so-called anomaly inflow analysis [51, 52]. The last piece contains the A-roof genus for the 10-dimensional curvature 2-form R pulled back to Σp+1. It may be expanded as Â(R) = 1 − p1(R) 24 + 7p1(R)2 − 4p2(R) 5760 + . . . , where pk(R) is a Pontryagin class. For example, the Chern-Simons term (3.11) includes a factor Tp Z Σp+1 Tr F 2π ∧ Cp−1 when a gauge field F on the worldvolume Σp+1 is turned on. It describes an induced D(p − 2) brane wrapping the Poincare dual of [F/2π] in Σp+1. Vice versa, a bound state of a D(p − 2)-brane on a Dp-brane may be interpreted as turning on a field strength F on the Dp-brane. Analogously, instantons (3.1) in a 4-dimensional gauge theory, say of rank zero and second Chern class n, have an interpretation in type IIA theory bound states of n D0-branes on a D4-brane. More generally, topological excitations in the worldvolume theory of a brane are often caused by other extended objects that end on it [53]. 3.1.3 String dualities The different appearances of string theory, type I, II, heterotic and M-theory, are connected through a zoo of dualities. Let us briefly introduce T-duality and S- duality in type II. There are many more dualities, some of which we will meet on our way. T-duality T-duality originates in the worldsheet description of type II theory in terms of open and closed strings. T-duality on an S1 in the background interchanges the Dirichlet and Neumann boundary conditions of the open strings on that circle, and thereby maps branes that wrap this S1 into branes that don’t wrap it (and vice versa). It thus interchanges type IIA and type IIB theory. Similar to electro-magnetic duality (see Section 3.1), T-duality follows from a path integral argument [54]. The sigma model action for a fundamental string
- 54. 3.1. Instantons and branes 41 is based on the term − 1 2πα0 Z Σ gmndxm ∧ ∗dxn , in equation (3.9) when we forget the B-field for simplicity. Let us suppose that the metric is diagonal in the coordinate x that parametrizes the T-duality circle S1 . Then we can add a Lagrange multiplier field dy to the relevant part of the action Z DxDy exp Z − 1 2πα0 dx ∧ ∗dx + i π dx ∧ ∗dy . On the one hand the Lagrange multiplier field dy forces d(dx) = 0, which locally says that dx is exact. On the other hand integrating out dx yields Z Dy exp Z − α0 2π dy ∧ ∗dy So T-duality exchanges α0 ↔ 1 α0 and is therefore a strong-weak coupling on the worldsheet. More precisely, one should also take into account the B-field coupling (3.9), which is related to non- diagonal terms is the space-time metric. This leads to the well-known Buscher rules [55] Since the differential dx may be identified with a component of the gauge field A on the brane, and dy with a normal 1-form to the brane, the reduction of supersymmetric Yang-Mills from ten to 10 − d dimensions can be understood as applying T-duality d times. S-duality Since N = 4 supersymmetric Yang-Mills is realized as the low-energy effective theory on a D3-brane wrapping R4 . Since electro-magnetic duality in this theory is an exact symmetry, it should have a string theoretic realization in type IIB theory as well. Indeed it does, and this symmetry is known as S-duality. In type IIB theory S-duality is a (space-time) strong-weak coupling duality that maps gs ↔ 1/gs. Analogous to Yang-Mills theory the complete symmetry group is Sl(2, Z), where the complex coupling constant τ (3.4) is realized as τ = C0 + ie−φ . Since the ratio of the tensions of the fundamental string F1 and the D1-brane is equal to gs, S-duality exchanges these objects as well as the B-field and the
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- 56. Stood like a tower. His form had not yet lost All its original brightness, nor appeared Less than an Archangel ruined, and the excess Of glory obscured: as when the sun new- risen Looks through the horizontal misty air Shorn of his beams, or, from behind the moon, In dim eclipse, disastrous twilight sheds On half the nations, and with fear of change Perplexes monarchs. Darkened so, yet shone Above them all the Archangel. But his face Deep scars of thunder had intrenched, and care Sat on his faded cheek, but under brows Of dauntless courage, and considerate pride, Waiting revenge.... He now prepared To speak; whereat their doubled ranks they bend From wing to wing, and half enclose him round With all his peers. Attention held them mute. Thrice he assayed and thrice, in spite of scorn, Tears, such as Angels weep, burst forth; at last Words interwove with sighs found out their way:
- 57. O myriads of immortal Spirits! O Powers, Matchless, but with the Almighty!—and that strife Was not inglorious, though the event was dire, As this place testifies, and this dire change, Hateful to utter. But what power of mind, Foreseeing or presaging, from the depth Of knowledge past or present, could have feared How such united force of gods, how such As stood like these, could ever know repulse? He who reigns Monarch in Heaven till then as one secure Sat on his throne, upheld by old repute, Consent, or custom, and his regal state Put forth at full, but still his strength concealed— Which tempted our attempt, and wrought our fall. Henceforth his might we know, and know our own, So as not either to provoke, or dread New war provoked. Our better part remains To work in close design, by fraud or guile, What force effected not; that he no less At length from us may find, Who overcomes By force hath overcome but half his foe. Space may produce more Worlds, whereof so rife
- 58. There went a fame in Heaven that He ere long Intended to create, and therein plant A generation whom his choice regard Should favour equal to the Sons of Heaven. Thither, if but to pry, shall be perhaps Our first eruption—thither, or elsewhere; For this infernal pit shall never hold Celestial Spirits in bondage, nor the Abyss Long under darkness cover. But these thoughts Full counsel must mature. Peace is despaired; For who can think submission? War, then, war Open or understood, must be resolved. He spake; and to confirm his words, out- flew Millions of flaming swords, drawn from the thighs Of mighty Cherubim. The sudden blaze Far round illumined Hell. Highly they raged. Against the Highest, and fierce with grasped arms Clashed on their sounding shields the din of war, Hurling defiance toward the vault of Heaven. The exiled host now led by Mammon, the least erected Spirit that fell from Heaven, proceeded to build Pandemonium, their architect being him whom men called Mulciber, and here
- 59. The great Seraphic Lords and Cherubim In close recess and secret conclave sat A thousand demi-gods on golden seats. II.—The Fiends' Conclave High on a throne of royal state, which far Outshone the wealth of Ormus or of Ind, Or where the gorgeous East with richest hand Showers on her kings barbaric pearl and gold, Satan exalted sat, by merit raised To that bad eminence. Here his compeers gathered round to advise. First Moloch, the strongest and the fiercest Spirit that fought in Heaven, counselled war. Then uprose Belial—a fairer person lost not Heaven—and reasoned that force was futile. The towers of Heaven are filled With armed watch, that render all access Impregnable. Besides, failure might lead to their annihilation, and who wished for that? Who would lose, Though full of pain, this intellectual being, These thoughts that wander through eternity? They were better now than when they were hurled from Heaven, or when they lay chained on the burning lake. Their Supreme Foe might in time remit his anger, and slacken those raging fires.
- 60. Mammon also advised them to keep the peace, and make the best they could of Hell, a policy received with applause; but then Beelzebub, than whom, Satan except, none higher sat, rose, and with a look which drew audience and attention still as night, developed the suggestion previously made by Satan, that they should attack Heaven's High Arbitrator through His new-created Man, waste his creation, and drive as we are driven. This would surpass Common revenge, and interrupt His joy In our confusion, and our joy upraise In His disturbance. This proposal was gleefully received. But then the difficulty arose who should be sent in search of this new world? All sat mute, till Satan declared that he would abroad through all the coasts of dark destruction, a decision hailed with reverent applause. The Council dissolved, the Infernal Peers disperse to their several employments: some to sports, some to warlike feats, some to argument, in wandering mazes lost, some to adventurous discovery; while Satan wings his way to the nine-fold gate of Hell, guarded by Sin, and her abortive offspring, Death; and Sin, opening the gate for him to go out, cannot shut it again. The Fiend stands on the brink, pondering his voyage, while before him appear The secrets of the hoary Deep—on dark Illimitable ocean, without bound, Without dimension; where length, breadth, and highth, And time, and place, are lost; where eldest Night And Chaos, ancestors of Nature, hold Eternal anarchy. At last he spreads his sail-broad vans for flight, and, directed by Chaos and sable-vested Night, comes to where he can see far off The empyreal Heaven, once his native seat, And, fast by, hanging in a golden chain,
- 61. This pendent World. III.—Satan Speeds to Earth An invocation to Light, and a lament for the poet's blindness now preludes a picture of Heaven, and the Almighty Father conferring with the only Son. Hail, holy Light, offspring of Heaven first- born! Bright effluence of bright essence uncreate! Whose fountain who shall tell? Before the Sun, Before the Heavens, thou wert, and at the voice Of God, as with a mantle, didst invest The rising World of waters dark and deep, Won from the void and formless Infinite! ............................ But thou Revisit'st not these eyes, that roll in vain To find thy piercing ray, and find no dawn. ............................ With the year Seasons return; but not to me returns Day, or the sweet approach of even or morn, Or sight of vernal bloom, or summer's rose, Or flocks, or herds, or human face divine; But clouds instead, and ever-during dark Surrounds me, from the cheerful ways of men
- 62. Cut off. God, observing the approach of Satan to the world, foretells the fall of Man to the Son, who listens while In his face Divine compassion visibly appeared, Love without end, and without measure grace. The Father asks where such love can be found as will redeem man by satisfying eternal Justice. He asked, but all the Heavenly Quire stood mute, And silence was in Heaven. Admiration seized all Heaven, and to the ground they cast their crowns in solemn adoration, when the Son replied Account me Man. I for his sake will leave Thy bosom, and this glory next to Thee Freely put off, and for him lastly die Well pleased; on me let Death wreak all his rage. Under his gloomy power I shall not long Lie vanquished. While the immortal quires chanted their praise, Satan drew near, and sighted the World—the sun, earth, moon, and companion planets— As when a scout, Through dark and desert ways with peril gone All night, at last by break of cheerful dawn Obtains the brow of some high-climbing hill,
- 63. Which to his eye discovers unaware The goodly prospect of some foreign land First seen, or some renowned metropolis With glistening spires and pinnacles adorned, Which now the rising Sun gilds with his beams, Such wonder seized, though after Heaven seen, The Spirit malign, but much more envy seized, At sight of all this world beheld so fair. Flying to the Sun, and taking the form of a stripling Cherub, Satan recognises there the Archangel Uriel and accosts him. Brightest Seraph, tell In which of all these shining orbs hath Man His fixed seat. And Uriel, although held to be the sharpest-sighted Spirit of all in Heaven, was deceived, for angels cannot discern hypocrisy. So Uriel, pointing, answers: That place is Earth, the seat of Man.... That spot to which I point is Paradise, Adam's abode; those lofty shades his bower. Thy way thou canst not miss; me mine requires. Thus said, he turned; and Satan, bowing low, As to superior Spirits is wont in Heaven, Where honour due and reverence none neglects,
- 64. Took leave, and toward the coast of Earth beneath, Down from the ecliptic, sped with hoped success, Throws his steep flight in many an aery wheel, Nor stayed till on Niphantes' top he lights. IV.—Of Adam and Eve in Paradise Coming within sight of Paradise Satan's conscience is aroused, and he grieves over the suffering his dire work will entail, exclaiming Me miserable; which way shall I fly Infinite wrath and infinite despair? Which way I fly is Hell; myself am Hell. But he cannot brook submission, and hardens his heart afresh. So farewell hope, and, with hope, farewell fear, Farewell remorse! All good to me is lost; Evil, be thou my Good. As he approaches Paradise more closely, the deliciousness of the place affects even his senses. As when to them who sail Beyond the Cape of Hope, and now are past Mozambic, off at sea north-east winds blow Sabean odours from the spicy shore Of Araby the Blest, with such delay
- 65. Well pleased they slack their course, and many a league Cheered with the grateful smell old Ocean smiles, So entertained those odorous sweets the Fiend. At last, after sighting all kind of living creatures new to sight and strange, he descries Man. Two of far nobler shape, erect and tall, God-like erect, with native honour clad In naked majesty, seemed lords of all, And worthy seemed; for in their looks divine The image of their glorious Maker shone. For contemplation he and valour formed, For softness she and sweet attractive grace; He for God only, she for God in Him. So hand in hand they passed, the loveliest pair That ever since in love's embraces met— Adam the goodliest man of men since born His sons; the fairest of her daughters Eve. At the sight of the gentle pair, Satan again almost relents. Taking the shape of various animals, he approaches to hear them talk and finds from Adam that the only prohibition laid on them is partaking of the Tree of Knowledge. Eve, replying, tells how she found herself alive, saw her form reflected in the water, and thought herself fairer even than Adam until Thy gentle hand Seized mine; I yielded, and from that time see
- 66. How beauty is excelled by manly grace And wisdom, which alone is truly fair. While Satan roams through Paradise, with sly circumspection, Uriel descends on an evening sunbeam to warn Gabriel, chief of the angelic guards, that a suspected Spirit, with looks alien from Heaven, had passed to earth, and Gabriel promises to find him before dawn. Now came still Evening on, and Twilight gray Had in her sober livery all things clad; Silence accompanied; for beast and bird, They to their grassy couch, these to their nests Were slunk, all but the wakeful nightingale. She all night long her amorous descant sung. Silence was pleased. Now glowed the firmament With living sapphires; Hesperus, that led The starry host, rode brightest, till the Moon, Rising in clouded majesty, at length Apparent queen, unveiled her peerless light, And o'er the dark her silver mantle threw. Adam and Eve talk ere they retire to rest—she questioning him Sweet is the breath of Morn, her rising sweet, With charm of earliest birds; pleasant the Sun, When first on this delightful land he spreads
- 67. His orient beams, on herb, tree, fruit, and flower, Glistening with dew; fragrant the fertile Earth After soft showers; and sweet the coming on Of grateful Evening mild; then silent Night With this her solemn bird, and this fair Moon, And these the gems of Heaven, her starry train; But neither breath of Morn, when she ascends With charm of earliest birds; nor rising Sun On this delightful land; nor herb, fruit, flower, Glistening with dew; nor fragrance after showers, Nor grateful Evening mild; nor silent Night, With this her solemn bird; nor walk by moon, Or glittering star-light, without thee is sweet. But wherefore all night long shine these? For whom This glorious sight, when sleep hath shut all eyes? Adam replies: These have their course to finish round the Earth,
- 68. And they, though unbeheld in deep of night, Shine not in vain. Nor think, though men were none, That Heaven would want spectators, God want praise. Millions of spiritual creatures walk the earth Unseen, both when we wake, and when we sleep; All these with ceaseless praise His works behold Both day and night..... Thus talking, hand in hand, alone they passed On to their blissful bower. Gabriel then sends the Cherubim, armed to their night watches, and commands Ithuriel and Zephon to search the Garden, where they find Satan, squat like a toad close to the ear of Eve, seeking to taint her dreams. Him thus intent Ithuriel with his spear Touched lightly; for no falsehood can endure Touch of celestial temper, but returns Of force to its own likeness. Satan therefore starts up in his own person, and is conducted to Gabriel, who sees him coming with them, a third, of regal port, but faded splendour wan. Gabriel and he engage in a heated altercation, and a fight seems imminent between the Fiend and the angelic squadrons that begin to hem him round, when, by a sign in the sky, Satan is reminded of his powerlessness in open fight, and flees, murmuring; and with him fled the shades of Night.
- 69. V.—The Morning Hymn of Praise Adam, waking in the morning, finds Eve flushed and distraught, and she tells him of her troublous dreams. He cheers her, and they pass out to the open field, and, adoring, raise their morning hymn of praise. These are Thy glorious works, Parent of good, Almighty! Thine this universal frame, Thus wondrous fair—Thyself how wondrous then! Unspeakable! Who sittest above these heavens To us invisible, or dimly seen In these Thy lowest works; yet these declare Thy goodness beyond thought, and power divine. Speak, ye who best can tell, ye Sons of Light, Angels—for ye behold Him, and with songs And chloral symphonies, day without night, Circle His throne rejoicing—ye in Heaven; On Earth join, all ye creatures, to extol Him first, Him last, Him midst, and without end. Fairest of Stars, last in the train of Night, If better than belong not to the Dawn, Sure pledge of Day, that crown'st the smiling morn With thy bright circlet, praise Him in thy sphere
- 70. While day arises, that sweet hour of prime. Thou Sun, of this great World both eye and soul, Acknowledge Him thy greater; sound His praise In thy eternal course, both when thou climb'st And when high noon hast gained, and when thou fall'st. Moon, that now meet'st the orient Sun, now fliest, With the fixed Stars, fixed in their orb, that flies; And ye five other wandering Fires, that move In mystic dance, not without song, resound His praise Who out of Darkness called up Light. Ye Mists and Exhalations, that now rise From hill or steaming lake, dusky or gray, Till the Sun paint your fleecy skirts with gold, In honour to the World's great Author rise; Whether to deck with clouds the uncoloured sky, Or wet the thirsty earth with falling showers, Rising or falling, still advance His praise. His praise, ye Winds, that from four quarters blow, Breathe soft or loud; and wave your tops, ye Pines, With every plant in sign of worship wave.
- 71. Fountains, and ye that warble as ye flow, Melodious murmurs, warbling, tune His praise. Join voices, all ye living souls. Ye Birds, That, singing, up to Heaven's gate ascend, Bear on your wings and in your notes His praise. Hail universal Lord! Be bounteous still To give us only good; and, if the night Have gathered aught of evil, or concealed, Disperse it, as now light dispels the dark. So prayed they innocent, and to their thoughts Firm peace recovered soon, and wonted calm. The Almighty now sends Raphael, the sociable Spirit, from Heaven to warn Adam of his danger, and alighting on the eastern cliff of Paradise, the Seraph shakes his plumes and diffuses heavenly fragrance around; then moving through the forest is seen by Adam, who, with Eve, entertains him, and seizes the occasion to ask him of their Being Who dwell in Heaven, and further, what is meant by the angelic caution—If ye be found obedient. Raphael thereupon tells of the disobedience, in Heaven, of Satan, and his fall, from that high state of bliss into what woe. He tells how the Divine decree of obedience to the Only Son was received by Satan with envy, because he felt himself impaired; and how, consulting with Beelzebub, he drew away all the Spirits under their command to the spacious North, and, taunting them with being eclipsed, proposed that they should rebel. Only Abdiel remained faithful, and urged them to cease their impious rage, and seek pardon in time, or they might find that He Who had created them could uncreate them. So spake the Seraph Abdiel, faithful found; Among the faithless faithful only he;
- 72. Among innumerable false unmoved, Unshaken, unseduced, unterrified, His loyalty he kept, his love, his zeal; Nor number nor example with him wrought To swerve from truth, or change his constant mind Though single. VI.—The Story of Satan's Revolt Raphael, continuing, tells Adam how Abdiel flew back to Heaven with the story of the revolt, but found it was known. The Sovran Voice having welcomed the faithful messenger with Servant of God, well done! orders the Archangels Michael and Gabriel to lead forth the celestial armies, while the banded powers of Satan are hastening on to set the Proud Aspirer on the very Mount of God. Long time in even scale the battle hung, but with the dawning of the third day, the Father directed the Messiah to ascend his chariot, and end the strife. Far off his coming shone, and at His presence Heaven his wonted face renewed, and with fresh flowerets hill and valley smiled. But, nearing the foe, His countenance changed into a terror too severe to be beheld. Full soon among them He arrived, in His right hand Grasping ten thousand thunders.... They, astonished, all resistance lost, All courage; down their idle weapons dropt.... .... Headlong themselves they threw Down from the verge of Heaven; eternal wrath Burnt after them to the bottomless pit.
- 73. A like fate, Raphael warns Adam, may befall mankind if they are guilty of disobedience. VII.—The New Creation The affable Archangel, at Adam's request, continues his talk by telling how the world began. Lest Lucifer should take a pride in having dispeopled Heaven, God announces to the Son that he will create another world, and a race to dwell in it who may Open to themselves at length the way Up hither, under long obedience tried, And Earth be changed to Heaven, and Heaven to Earth, This creation is to be the work of the Son, who, girt with omnipotence, prepares to go forth. Heaven opened wide Her ever-daring gates, harmonious sound On golden hinges moving, to let forth The King of Glory, in his powerful Word And Spirit coming to create new worlds. On Heavenly ground they stood, and from the shore They viewed the vast immeasurable Abyss Outrageous as a sea, dark, wasteful, wild, Up from the bottom turned by furious winds And surging waves, as mountains to assault Heaven's highth, and with the centre mix the pole.
- 74. Silence, ye troubled waves, and thou Deep, peace! Said then the omnific Word. Your discord end! Nor stayed; but on the wings of cherubim, Uplifted in paternal glory rode Far into Chaos and the World unborn; For Chaos heard his voice.... And Earth, self-balanced on her centre hung. The six days' creative work is then described in the order of Genesis. VIII.—The Creation of Adam Asked by Adam to tell him about the motions of the heavenly bodies, Raphael adjures him to refrain from thought on matters hid; to serve God and fear; and to be lowly wise. He then asks Adam to tell him of his creation, he having at the time been absent on excursion toward the gates of Hell. Adam complies, and relates how he appealed to God for a companion, and was answered in the fairest of God's gifts. Raphael warns Adam to beware lest passion for Eve sway his judgment, for on him depends the weal or woe, not only of himself, but of all his sons. IX.—The Temptation and the Fall While Raphael was in Paradise, for seven nights, Satan hid himself by circling round in the shadow of the Earth, then, rising as a mist, he crept into Eden undetected, and entered the serpent as the fittest imp of fraud, but not until once more lamenting that the enjoyment of the earth was not for him. In the morning, when the human pair came forth to their pleasant labours, Eve suggested that they should work apart, for when near each other looks intervene and smiles, and casual discourse. Adam replied, defending this
- 75. sweet intercourse of looks and smiles, and saying they had been made not for irksome toil, but for delight. But if much converse perhaps Thee satiate, to short absence I could yield; For solitude sometimes is best society, And short retirement urges sweet return. But other doubt possessed me, lest harm Befall thee.... The wife, where danger or dishonour lurks, Safest and seemliest by her husband stays Who guards her, or the worst with her endures. Eve replies: That such an enemy we have, who seeks Our ruin, both by thee informed I learn, And from the parting Angel overheard, As in a shady nook I stood behind, Just then returned at shut of evening flowers. She, however, repels the suggestion that she can be deceived. Adam replies that he does not wish her to be tempted, and that united they would be stronger and more watchful. Eve responds that if Eden is so exposed that they are not secure apart, how can they be happy? Adams gives way, with the explanation that it is not mistrust but tender love that enjoins him to watch over her, and, as she leaves him, Her long with ardent look his eye pursued Delighted, but desiring more her stay.
- 76. Oft he to her his charge of quick return Repeated; she to him as oft engaged To be returned by noon amid the bower, And all things in best order to invite Noontide repast, or afternoon's repose. O much deceived, much failing, hapless Eve, Of thy presumed return! Event perverse! Thou never from that hour in Paradise Found'st either sweet repast or sound repose. The Fiend, questing through the garden, finds her Veiled in a cloud of fragrance where she stood Half-spied, so thick the roses bushing round About her glowed.... Them she upstays Gently with myrtle band, mindless the while Herself, though fairest unsupported flower, From her best prop so far, and storm so nigh. Seeing her, Satan much the place admired, the person more. As one who, long in populous city pent, Forth issuing on a summer's morn to breathe Among the pleasant villages and farms Adjoined, from each thing met conceives delight— The smell of grain, of tedded grass, of kine,
- 77. Of dairy, each rural sight, each rural sound— If chance with nymph-like step fair virgin pass, What pleasing seemed, for her now pleases more, She most, and in her look seems all delight. Such pleasure took the Serpent to behold This flowery plat, the sweet recess of Eve Thus early, thus alone. The original serpent did not creep on the ground, but was a handsome creature. With burnished neck of verdant gold, erect Amidst his circling spires, that on the grass Floated redundant. Pleasing was his shape And lovely. Appearing before Eve with an air of worshipful admiration, and speaking in human language, the arch-deceiver gains her ear with flattery. Empress of this fair world, resplendent Eve. She asks how it is that man's language is pronounced by tongue of brute. The reply is that the power came through eating the fruit of a certain tree, which gave him reason, and also constrained him to worship her as sovran of creatures. Asked to show her the tree, he leads her swiftly to the Tree of Prohibition, and replying to her scruples and fears, declares— Queen of the Universe! Do not believe Those rigid threats of death. Ye shall not die. How should ye? By the fruit? It gives you life
- 78. To knowledge. By the Threatener? Look on me— Me who have touched and tasted, yet both live And life more perfect have attained than Fate Meant me, by venturing higher than my lot. Shall that be shut to Man which to the Beast Is open? Or will God incense his ire For such a petty trespass?... God therefore cannot hurt ye and be just. Goddess humane, reach, then, and freely taste! He ended; and his words replete with guile Into her heart too easy entrance won. Eve herself then took up the argument and repeated admiringly the Serpent's persuasions. In the day we eat Of this fair fruit our doom is we shall die! How dies the Serpent? He hath eaten and lives, And knows, and speaks, and reasons, and discerns, Irrational till then. For us alone Was death invented? Or to us denied This intellectual food, for beasts reserved? Here grows the care of all, this fruit divine, Fair to the eye, inviting to the taste,
- 79. Of virtue to make wise. What hinders then To reach and feed at once both body and mind? So saying, her rash hand in evil hour Forth-reaching to the fruit, she plucked, she ate. Earth felt the wound, and Nature from her seat, Sighing through all her works, gave signs of woe That all was lost. Back to the thicket slunk The guilty serpent. At first elated by the fruit, Eve presently began to reflect, excuse herself, and wonder what the effect would be on Adam. And I perhaps am secret. Heaven is high — High, and remote to see from thence distinct Each thing on Earth; and other care perhaps May have diverted from continual watch Our great Forbidder, safe with all his spies About him. But to Adam in what sort Shall I appear? Shall I to him make known As yet my change? But what if God have seen And death ensue? Then I shall be no more;
- 80. And Adam, wedded to another Eve, Shall live with her enjoying, I extinct! A death to think! Confirmed then, I resolve Adam shall share with me in bliss or woe, So dear I love him that with him all deaths I could endure, without him live no life. Adam the while Waiting desirous her return, had wove Of choicest flowers a garland, to adorn Her tresses.... Soon as he heard The fatal trespass done by Eve amazed, From his slack hand the garland wreathed for her Down dropt, and all the faded roses shed. Speechless he stood and pale, till thus at length, First to himself he inward silence broke: O fairest of creation, last and best Of all God's works, creature in whom excelled Whatever came to sight or thought be formed, Holy, divine, good, amiable, or sweet, How art thou lost! how on a sudden lost! Some cursed fraud Of enemy hath beguiled thee, yet unknown, And me with thee hath ruined; for with thee Certain my resolution is to die.
- 81. How can I live without thee? How forego Thy sweet converse, and love so dearly joined, To live again in these wild words forlorn?. Then, turning to Eve, he tries to comfort her. Perhaps thou shalt not die ... Nor can I think that God, Creator wise, Though threatening, will in earnest so destroy Us, His prime creatures, dignified so high, Set over all his works.... However, I with thee have fixed my lot, Certain to undergo like doom. If death Consort with thee, death is to me as life. Our state cannot be severed; we are one. So Adam; and thus Eve to him replied: O glorious trial of exceeding love, Illustrious evidence, example high! So saying she embraced him, and for joy Tenderly wept, much won that he his love Had so ennobled as of choice to incur Divine displeasure for her sake, or death. In recompense ... She gave him of that fair enticing fruit With liberal hand. He scrupled not to eat Against his better knowledge, not deceived, But fondly overcome with female charm. The effect of the fruit on them is first to excite lust with guilty shame following, and realising this after the exhilarating vapour bland had spent its force, Adam found utterance for his remorse.
- 82. O Eve, in evil hour thou didst give ear To that false Worm.... ... How shall I behold the face Henceforth of God or Angel, erst with joy And rapture so oft beheld? Those Heavenly shapes Will dazzle now this earthly with their blaze Insufferably bright. Oh, might I here In solitude live savage, in some glade Obscured, where highest winds, impenetrable To star or sunlight, spread their umbrage broad, And brown as evening! Cover me, ye pines! Ye cedars, with innumerable boughs Hide me, where I may never see them more! Then they cower in the woods, and clothe themselves with leaves. Covered, but not at rest or ease of mind They sat them down to weep. But passion also took possession of them, and they began to taunt each other with recriminations. Adam, with estranged look, exclaimed: Would thou hadst hearkened to my words, and stayed With me, as I besought thee, when that strange Desire of wandering, this unhappy morn, I know not whence possessed thee! We had then
- 83. Remained still happy! Eve retorts: Hadst thou been firm and fixed in thy dissent, Neither had I transgressed, nor thou with me. Then Adam: What could I more? I warned thee, I admonished thee, foretold The danger, and the lurking enemy That lay in wait; beyond this had been force. Thus they in mutual accusation spent The fruitless hours, but neither self- condemning; And of their vain contest appeared no end. X.—Sin and Death Triumph The Angels left on guard now slowly return from Paradise to Heaven to report their failure, but are reminded by God that it was ordained; and the Son is sent down to judge the guilty pair, after hearing their excuses, and to punish them with the curses of toil and death. Meantime Sin and Death snuff the smell of mortal change on Earth, and leaving Hell-gate belching outrageous flame, erect a broad road from Hell to Earth through Chaos, and as they come in sight of the World meet Satan steering his way back as an angel, between the Centaur and the Scorpion. He makes Sin and Death his plenipotentiaries on Earth, adjuring them first to make man their thrall, and lastly kill; and as they pass to the evil work the blasted
- 84. stars look wan. The return to Hell is received with loud acclaim, which comes in the form of a hiss, and Satan and all his hosts are turned into grovelling snakes. Adam, now in his repentance, is sternly resentful against Eve, who becomes submissive, and both pass from remorse to sorrow unfeigned and humiliation meek.' XI.—Repentance and the Doom The repentance of the pair is accepted by God, who sends down the Archangel Michael, with a cohort of cherubim, to announce that death will not come until time has been given for repentance, but Paradise can no longer be their home. Whereupon Eve laments. O unexpected stroke, worse than of Death! Must I thus leave thee, Paradise? Thus leave Thee, native soil? These happy walks and shades, Fit haunt of gods, where I had hoped to spend Quiet, though sad, the respite of that day That must be mortal to us both? O flowers, That never will in any other climate grow, My early visitation and my last At even, which I tied up with tender hand From the first opening bud and gave ye names, Who now shall rear ye to the Sun, or rank Your tribes, and water from the ambrosial fount? ... How shall we breathe in other air Less pure, accustomed to immortal fruits?
- 85. The Angel reminds her: Thy going is not lonely; with thee goes Thy husband; him to follow thou art bound. Where he abides think there thy native soil. Michael then ascending a hill with Adam shows him a vision of the world's history, while Eve sleeps. XII.—Paradise Behind, the World Before The history is continued, with its promise of redemption, until Adam exclaims: Full of doubt I stand, Whether I should repent me now of sin By me done and occasioned, or rejoice Much more that much more good thereof shall spring— To God more glory, more good-will to men. Eve awakens from propitious dreams, it having been shown to her that— Though all by me is lost, Such favour I unworthy am vouchsafed. By me the Promised Seed shall all restore. The time, however, has come when they must leave. A flaming sword, fierce as a comet, advances towards them before the bright array of cherubim. Whereat
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