PART XI - M-Theory
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This document discusses M-theory and its relation to string theory. Some key points: 1) M-theory is a theory in 11 dimensions that unifies the five 10-dimensional superstring theories. It reduces to 11-dimensional supergravity at low energies. 2) The document outlines the dimensional reductions of M-theory to the various string theories and their dualities (S-duality, T-duality, U-duality). 3) It presents the field contents and supersymmetries of M-theory compactified on a torus and its connections to the different string theories through dualities.
![From First Principles
June 2016 – R1.0
Maurice R. TREMBLAY
PART XI – M-THEORY
SO(32) ID=10
D=11
D=5-9
IIB IIA E8⊗E8
M
S-duality: φ → −φ
T-duality: ψ → −ψ
U-duality: ψ → ±ψ
U-duality: U ⊃ S ⊗ T
Closed Closed Closed Closed
N=1 SUGRA
⊕E8⊗E8Yang-Mills
N=2A (nonchiral)
SUGRA
N=2B (Chiral)
SUGRA
N=1 SUGRA
⊕SO(32)Yang-Mills
Open & Closed
D=4
(Space-time)
D=3 ⊕ 1
(3-dimensional space plus a separate dimension of time)
AAAA
iX θεδεθδ µµ
Γ== &
HeteroticType Type
T-duality
S-duality
T-duality
S1T0
2 S1 / Z2 T0
2/Z2
∑ ΠΠ=
µ
µ
µ
jijig ∫ ∑∫ =−−=
ji
ji
ji
ji bd
T
SgdTS εσσ 2
2
2
1
2
det andA
i
A
ii iX θθ µµµ
∂Γ−∂=Π
†
θθ µ
Γ=
T-duality
bdhSSS ZW =+= -21 and
∂Γ−∂
∂Γ−∂
∂Γ−∂
=Π
∑ ++
++
)(
)(
)Heterotic(
IIB
IIA
kj
k
i
j
kji
ii
ii
i
iX
iX
iX
θθδ
θθ
θθ
µµ
µµ
µµ
µ
ΓΠ
ΓΠ
ΓΠ
=
∑ +
++
)(
)(
)Heterotic(
-
IIB
IIA
ji
ji
ji
ZW
ddS
dd
dd
h
θθ
θθ
θθ
µ
µ
µ
µ
µ
=
p
E
Pµ
vp
p
F m
td
d
== and
∇∇∇∇hh i
t
iE −→
∂
∂
→ p&
],[ ∇∇∇∇t∂=∂µ
Yang-Mills
),0( 3=== i
X
P µ
δ
δ
µ
µ
&
L
)2,1(0
== aaa
ρψψ
=
−
=
0
0
0
00
i
i
i
i i
ρρ &
ABABBABA g2},{ −=ΓΓ+ΓΓ=ΓΓ )()2( 101 −−±
Γ±Γ=Γ D
1101 −+ ΓΓΓ=Γ DD L
],[2)(2 11 BAABBAAB
ΓΓ=ΓΓ−ΓΓ=Γ −−
θθ µµ
µµ
dd
p
h p
p
LL 1
1
!2
1
ΓΠΠ=
bdh =
0]321[ =ΓΓ ψψψ µ
µ](https://image.slidesharecdn.com/11-mt-160607223641/85/PART-XI-M-Theory-1-320.jpg)
PART XI - M-Theory
- 1. From First Principles June 2016 – R1.0 Maurice R. TREMBLAY PART XI – M-THEORY SO(32) ID=10 D=11 D=5-9 IIB IIA E8⊗E8 M S-duality: φ → −φ T-duality: ψ → −ψ U-duality: ψ → ±ψ U-duality: U ⊃ S ⊗ T Closed Closed Closed Closed N=1 SUGRA ⊕E8⊗E8Yang-Mills N=2A (nonchiral) SUGRA N=2B (Chiral) SUGRA N=1 SUGRA ⊕SO(32)Yang-Mills Open & Closed D=4 (Space-time) D=3 ⊕ 1 (3-dimensional space plus a separate dimension of time) AAAA iX θεδεθδ µµ Γ== & HeteroticType Type T-duality S-duality T-duality S1T0 2 S1 / Z2 T0 2/Z2 ∑ ΠΠ= µ µ µ jijig ∫ ∑∫ =−−= ji ji ji ji bd T SgdTS εσσ 2 2 2 1 2 det andA i A ii iX θθ µµµ ∂Γ−∂=Π † θθ µ Γ= T-duality bdhSSS ZW =+= -21 and ∂Γ−∂ ∂Γ−∂ ∂Γ−∂ =Π ∑ ++ ++ )( )( )Heterotic( IIB IIA kj k i j kji ii ii i iX iX iX θθδ θθ θθ µµ µµ µµ µ ΓΠ ΓΠ ΓΠ = ∑ + ++ )( )( )Heterotic( - IIB IIA ji ji ji ZW ddS dd dd h θθ θθ θθ µ µ µ µ µ = p E Pµ vp p F m td d == and ∇∇∇∇hh i t iE −→ ∂ ∂ → p& ],[ ∇∇∇∇t∂=∂µ Yang-Mills ),0( 3=== i X P µ δ δ µ µ & L )2,1(0 == aaa ρψψ = − = 0 0 0 00 i i i i i ρρ & ABABBABA g2},{ −=ΓΓ+ΓΓ=ΓΓ )()2( 101 −−± Γ±Γ=Γ D 1101 −+ ΓΓΓ=Γ DD L ],[2)(2 11 BAABBAAB ΓΓ=ΓΓ−ΓΓ=Γ −− θθ µµ µµ dd p h p p LL 1 1 !2 1 ΓΠΠ= bdh = 0]321[ =ΓΓ ψψψ µ µ