Connection form

CONNECTION FORM -MOVING FRAME( E=mc 2 ) -DIFFERENTIAL FORM

A connection form is a manner of organizing data of a connection language of moving frame and differential form

Moving Frame Moving frame is a flexible generalization of the notion of a ordered basis of vector space often used to study the exterior differential geometry of smooth manifolds embedded in a homogeneous space.

Differential Form Differential form is a mathematical concept in fields of a multivariative calculus differential topology and tensors.

Moving Frame 1.PROJECTIVE FRAME 2.AFFINE FRAME 3.LINEAR FRAME 4. ORTHONORMAL FRAME 5. EUCLIDEAN FRAME 6. CO-FRAME 7. KLEIN GEO-FRAME 8. FRENET-SERRET FRAME 9.DARBOAX FRAME 10.DESCRETE FRAME/K-THEORY 11.TIME

Projective Frame 1. Sierpinki Space / Homology 2. Pre-regular Space/ Cross Ratio 3. Second Countable Space /Quadrangle Theorem 4. Normal Space /Harmonic Range 5. Lindelof Space/Paskals Theorem 6. Functional Space/ Pappus Theorem 7. Fully Normal Space / Brain Chon 8. Counter Space

Pre–regular Space (SUNDARANAND) X is a pre-regular space if any two topological distinguasible points can be separated by neighborhood.

Homology ( SUNDARANAND) A basic projective transformation is which corresponding slides meet on a fixed line called axis and corresponding points lie on a line through the centre.

Functional Space ( SARBANAND) Functional space is A set of A given kind from A set X to A set Y.

Cross Ratio ( SARBANANDA) Cross ratio of four points is only numerical invariant of projective geometry.

Second Countable Space ( CHANDRASEKHAR) Satisfy the second axiom of count ability. Has a comfortable base. Is separable and lindelof.

Quadrangle Theorem (Chandrashekhar) If two quadrangles have 5 pairs of corresponding sides meeting in collinear points the sixth pair meets on the same line.

Normed Space (AMBER) CONSIST OF T4,T5 &T6

Harmonic Range ( AMBER) Construction of two pairs of points harmonically have cross ratio 1.

Lindelof Space (KAPILAMBER) Is a topological space in which every open cover has a countable sub cover. More commonly used notion of compactness. Strongly lindof / suslin.

Pascal's Theorem ( KAPILAMBAR)

Fully Normal Space ( VAB) Is a topological space in which every open cover admits an open locally finite refinement.

Sierpinki Space (BAMAN) It is a smallest example of a topological space which is neither trivial nor discrete. Is a finite topological space with two points, only one of which is closed.

Affine Frame An affine space is a set with a faithful freely transitive vector space action i,e a tensor for the vector space  :SXS  (a,b)|  (a,b)

Affine Frame 1. QUOTENT SPACE / David Kay’s Axiom 2. FRECHET SPACE (T1) / David Kay’s Axiom ( AS1) 3. HOUSDORF SPACE(T2) / David Kay’s Axiom ( AS2) 4. REGULAR SPACE(T3) / David Kay’s Axiom ( AS3) 5. NORMAL SPACE(T4) / David Kay’s Axiom ( AS4) 6. NORMAL SPACE(T5) / David Kay’s Axiom ( AS5) 7. NORMAL SPACE(T6) / David Kay’s Axiom ( AS6)

Quotient Space (BHUTESH) Identification space is intuitively speaking the result of identifying or “gluing together” certain points of a given space. Equivalence relation

David Kay’s Axim (BHUTESH) David Kay’s description of 3-dimensional affine space is as follows…. “ An affine space is any system of points ,lines and planes which satisfy 6 axioms.”

T1 Space, Frechet Space (SANGBART) It is complete as a uniform space. It is locally convex.

ASI (SANGBARTA) Two distinct points determine a unique line.

T2 Space, Hausdorff Space (BIKRITAKH) A topological space in which points can be separated by neighborhood.

AS2 (BIKRITAKH) Three non-linear points determine a unique plane.

T3 Space (SANGHAR) Normal space

AS3 ( SANGHAR) If two points lie in a plane then the line determined by these points lies in that plane.

T4 Space (DANDAPANI) Normal space

AS4 (DANDAPANI) If two planes meet their intersection is a line.

T5 space (Chakrapani) Normal space

AS5 ( CHAKRAPANI) There exist at least four non-linear points and at least one plane. Each plain contains at least three non-linear points.

AS6 ( KAL) Given any two non co-planer lines, there exist a unique plane through the first line which is parallel to second line.

Linear Frame 1. CO-ODINATE SPACE. 2. SYMMETRIC SPACE/ Standard Basis. 3. CONFORMAL SPACE /Hamel Basis. 4. DUAL SPACE. 5. METRIC SPACE /Orthonormal Base. 6. LUSINS SPACE /Schander Base. 7. ALJEBIC DUAL SPACE. 8. ANTI DE-SITTER SPACE.

Co-ordinate Space ( Kamadiswar) Co- Ordinate space is a proptotypical example of n-dimensional vector space over a field F.

Symmetric Space (AMRITAKH) Same as R0 space

Standard Basis ( Amritakh) Standard basis is a sequence of or the unit vector. Standard basis of a n-dimensional Euclidean space R n is the basis obtained by taking the n-basis vector.  e i : 1  I  n  Where e i is the vector with a 1 in the co-ordinate and 0 elsewhere.

Algebraic Dual Space (KAPALI) Given any vector space V over some field F we define dual space V* to be the set of linear function V , i,e scaler valued linear map on V.

Conformal Space (BOMKESH) Conformal geometry is the study of the set of angle-preserving (conformal) transformation on a Riemann manifold.

Hamel Basis (BOMKESH) Where the number of tensors in the linear combinations A 1 v 1 +……..+a n v n Is always finite.

Metric Space (TRIPURESH) Metric space is a set where a notion of distance (metric) between elements of the set is defined.

Orthonormal Basis (TRIPURESH) Orthonormal basis of an inner product space V is a set of basis vectors whose elements are mutually orthonormal and of magnitude 1.

Dual Space (ESWAR) Defined a finite-dimensional vector space can be used for defined tensors which are studied in tensor space,

Dual Space (ESHAWAR) Given any vector V over some field F we define dual space V* to be the set of linear function V, i,e scaler valued linear map on V.

Lusin Space (NAKULISH) Lusin space is topological space such that some weaken topology makes it into a Polish space.

Schander Basis (NAKULESH) A scander basis is similar to Hamel basis . The difference is that for Hamel basis , linear combination are assured to be finite sums while for schauder basis they may be infinite.

Anti De-sitter Space (KHIROKANTHHA) Anti de-sitter space can be visualized as the lorentizian analogs of a sphere in a space of an additional dimension.

Orthonormal Frame 1. FORCK SPACE / Multilinear Operation. 2. LP space / Bilinear Operation. 3. ORBIT SPACE / Sesquilinear Form.

Fock Space (UNMATT) Fork space is a algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particle. Fork space is a Hilbert space made from direct sum of tensor product of single-particle Hilbert space.

Multilinear Operation (Unmatt) Multilinear operation is a map of type f : V n  k Where V is a vector space over field k, that is separate linear in each its N variable.

Lebesgue Space (LP)( LAMBAKARNA) P-form can be extended to vectors having an infinite number of components; yielding the space Lp.

Bilinear Operation ( Lambakarna) Bilinear is a function which is linear in both of its arguments. Let v , w and be three vector spaces over the same base field F. A linear map B: VXW  X such that for any w in W the map v ם B (v, w) is a linear map from v to x and for any v in V the map w ם B( v , w ) is a linear map for w to x.

Orbit Space (BISHES) Suppose a topological group G acts continuously on a space X. One can form a equivalence relation on X by saying points are equivalent if and only if they lie in the same orbit . The quotient space under this relation is called the Orbit space.

Sesquilinear Form ( BISHES) A Sesquilinear form on a complex vector space v is a map in one argument and anti-linear in other. If  (x+y, z+w) =  (x, y)+  (x, w)+  (y, z)+  (y, w)  (ax, by)= ãb  (x, y) For all x, y, z,w in V AND a ,b in C

Euclidean Frame 1. Baire Space 2.Topological Space. 3. Tychonoff Space.

T 3 ½ , Tychonoff Space (Trisangkeswar) Completely regular space.

Topological Space ( JAGANATH) Topological spaces are mathematical structures that allow the formal definition of concepts:- Convergence Connectedness Continuity

Baire Space ( SAMBRANAND) Baire space is a very large and “enough” points for certain limit process of closed set with empty interior.

Co-frame 1.SUSLIN SPACE/Dual Frame 2.SCHWART SPACE/Co-tangent Bundle 3.RANDOM SPACE/Kronker Delta

Suslin Space ( NIRMISH) Is a separable completely amortizable topological space. I, e a space homogeneous to a complete metric space that has a comfortable dense subset. A Suslin space is the image of a polish space under a continuous mapping.

Dual Frame (NIRMISH) A moving frame determines a dual frame of co-tangent bundle over U, which is sometimes also called Moving Frame. This is a n-tuple of smooth1-form a 1 ,a 2 ,--a n which are linearly independent at each point q in U.

Radon Space (NANDIKESWAR) A Random space is a topological space such that every finite Boral measure is inner regular.

Co-tangent Bundle (NANDIKESHAR) Co-tangent bundle of a smooth manifold is the vector bundle of all the co-tangent space at every point in the manifold.

Schwartz Space (KRODHISH) Schwartz space is the function space of rapidly decreasing function.

Kroncker Delta ( KRODHISH) Kroncker delta is a function of two bundles which is 1 if they are equal, 0 if otherwise.

Darboux Frame 1.CONTOUR SPACE/Adapted Frame 2.LOCALLY CONVEX SPACE/ Maurer – Cartan Frame 3.F SPACE /Push Forward & Pull Back

Contour Space (ABHIRUK) Each lip contour is a point in an 80-dimensional “ contour space”.

Push Backward & Forward (ABHIRUK) Let  :M  N be a smooth map of smooth manifolds. Given some x  M, the Push forward of  at x is a linear map d  x :T x M  T  (x) N from tangent space of M at x to the tangent space of N at  (x). The applicant vector X is sometimes called the Push Forward of x by  .

Locally Convex Space ( BAKRANATH) Locally convex space is defined either in terms of convex set or equivalent in terms of semi norm.

Adapted Frame (BAKRANATH) Let  :M  E n be an embedding of p-dimensional smooth manifold into a euclidean space. The space of adopted frame on M denoted by F  (M) is the collection of tuples (x 1 f 1 ,….f n ) where x  M and f 1 form an ortonormal basis of En such that f 1 ….f q are tangent to  (M) at  (v).

F-space (RAKHASESWAR) Is a vector space V over the real or complex number together with a metric D:VXV  R V is continuous transition invariant.

Maurer-cartan Form (RAKHASHESWAR) Let g = TeG be the tangent space of a Lie group G at identity G acts on itself by left translation L:GXG  G such that g  G We have Lg:G  G where Lg(h)=gh this induces a map of tangent bundle on itself (Lg)*Thg  TghG Aleft invariant vector field is a section x of TG such that (Lg)*X=X  g  G The Mauer-Cartan form w is g-valued one-form on G defined on vectors v  TgG by formula w(v)=(Lg -1 )*v

Klein Geometry Accessible Space 2. Sequential Space 3. De-sitter Space 4. Homogeneous Space 5. Symmetric Space 6. R0 Space

Sequential Space (VADRASEN) Sequential space /Frechet Urysohm space satisfy a very weak axiom of compatibility. A sequential space is a space X satisfying one of the following equivalent conditions. 1. Every sequential open subset of X is open. 2. Every sequential closed subset of X is closed.

De-sitter Space (BHIMLOCHAN) De-Sitter space is Lorentz an analogue of an n-sphere (with its canonical Riemannian metric). De-Sitter space can be defined as a sub manifold of Murkowski space in one higher

Homogeneous Space (BAKRAKUNDA) Traditional spaces are homogeneous space, but not for a uniquely determined group. Changing the group changes the appropriate language.

Regular Space (BAIDYANATH) X is a T3 space if and only if it is both regular and Hausdorff.

R 0 Space (RURU) Let X be a topological space and let x and y be points in X . We say that x and y can be separated if each lies in an open set which does not contain the other point X is a R 0 space if any two topologically distinguishable points in X can be separated.

Accessible Space ( ASITANGA) X is a T1 space if any two distinct points in X can be separated.

K-theory 1.BANACH SPACE/ Representative K-theory 2.ADJUNCTION SPACE/Disjoint Union

Adjunction Space Suppose X is a space and A is a subspace of X . One can identify all points in A to a single equivalence class and leave points outside of A equivalent only to themselves. The resulting quotient space is denoted X/A.

Banach Space (KAPALI) Banach space is a vector space V over the real or complex number with a nom !!.!!.

REPRESENTATIVE K-THEORY ( TRAMBAK)

Franet-serret Frame 1.Affine Space 2.Twistor Space 3.Cauchy Space 4.Euclidean Space

Affine Space (SHUR) An affine space is any system of points , lines and planes which satisfy 6 axioms.

Twister Space (DEBBAHU) Is a 4-dimensional complex space T:=C 4 It has associated to it the double fibration of flag manifold P   F   M Where projective Twister space P:=F1(T)=P3(C)=P( C4)

Cauchy Space (SHIV) A Cauchy space is a set X and a collection of C of proper filters in power set P(X).

Euclidean Space (MAHADAR) An n-dimensional space with a notion of distance and angle that obey Euclidean relationship.

Differential Form Connection Form 2. Curvature Form 3. Torsion Form 4. Solder Form 5. Levi-Civita Connection Form Lie Bracket Form Descrete-1 8. Descrete-2 9. Descrete-3 10. Descrete-4 11. Base States

Connection Form (Shak) 1. Principal Connection 2. Compatible Connection 3. Cartan Connection 4. Ehesmann Connection 5. Associated to Principal 6. Global Connection Form 7. Affine Connection

Principal Connection ( PURAJAB) Let  U  be an open cover of M, along with G-frame on each U, denoted by e U . These one related on the intersection of overlapping open set by e V =e v .h uu for some G-valued function h uu defined in U n V.

Compatible Connection (PABMAN) A connection is compatible with the structure of a G-bundle on E provided that the associated parallel transport maps always sends one G-frame to another.

Cretan Connection (DHRUBHANIK) Cretan Connection are closely related to pseudo-group structure on a manifold. Pseudo group is an extension of group concept , but one that grew out of the geometric approach of Sophus Lie.

Ehesmann Connection ( CHITRAREK) Ehesmann connection is a version of the notion of a connection which is defined on ordinary fiber bundle.

Associated To Principle ( BAHURUP) A Principle G-connection w in a principal G-bundle P  M gives rise to a collection of connection focus on M.

Global Connection Form ( BISHWAVAB) If  Up  is an open covering of M and each Up is equipped with a trivialization ep of E , then it is possible to define a global form in terms of the patches data between the local connection form on the overlap revision.

Affine Connection (MANAJAB) An affine connection is a geometrical object on a smooth manifold which connect near by tangent spaces and so permits tangent vector fields to be differentiable as if they were functions on the manifold with values in a fixed vector space.

Curvature Form (KUSH) 1.Curvilinear Co-ordinate 2.Non-tensor 3.Parallel Transport 4.Geodesics 5.Co-variant Derivative 6.Curvature Tensor 7. Bianchi Identity

Curvilinear Co-ordinate (BASU) Curvilinear co-ordinates are co-ordinate in which angles between axes can changes from point to point.

Nontensors ( HARRUCHI) Nontensor is a tensor like quantity N  that behaves like a tensor in the raising and lowering of indices N  = g  N  N  = g  N  But does not transform like a tensor under a co-ordinate transformation.

Parallel Transport ( NAVIGUPTA)

Geodesics (SUTABRATA) Suppose we have a point x u that moves along a truck in physical space time. Suppose the track is parameterized with the quantity  .The variation of velocity upon parallel displacement along the track can be calculated. If there are no force acting on the point then the velocity is unchanged and we have Geodesics Equation.

Co-variant Derivative ( BIKIKAKT) The partial derivative of a vector w.r.t a space-time co-ordinate is composed of two parts. Normal partial derivative minus the change in the vector due to parallel transport. A  :  =A  1  _A a T  a

Curvature Tensor (BAMDEV) The curvature K of a surface is simply the angle through a vector is turned as we take it around an infinitesimal closed path.

Bianchi Identity (BASUDAN) Following differential relation known as Bianchi identity R   :  +R   :  +R   :  =0

Torsion Form ( KOUNCH) 1. Curvature Tensor 2. Cyclic Sum 3. Curvature Form 4. Affinity Parameterized Geodesic 5. Twisting of Reference Frame 6. Torsion of a Filament 7. Affine Developments

Curvature Tensor (AM) Curvature tensor (  ) is a mapping TM  TM  End (TM) Defined on vector field x, y & z R(X,Y)Z=  x  y Z-  y  x Z-   x, y  Z

Cyclic Sum ( MADHUBRAHA)  (R(X,Y)X):= R(X,Y)Z+R(Y,Z)X+R(Z,X)Y

Curvature Form (MEGHAPRISTA) Curvature form is the gl(n)-valued 2-form.  =Dw = dw +w  w D denotes the exterior covariant derivative 1. D  =  2. D  =0

Affinely Parametrized Geodesic ( SUDHAMA) Suppose y(t) is a curve on M Then  y (t) Y (t)=0 For all time t is the domain of Y.

Twisting Of Reference Frame ( BHAJISTA) Frenet-Serret Formula describe how a particular moving frame twists along a curve.

Torsion Of A Filament ( LOHITAN) Length –maximizing (geodesic) configuration and its energy-minimizing configuration.

Affine Developments (BANASPATY) Suppose that x t is a curve in M.The affine developments of x t is the unique curve ct in Tx 0 M such that Ċ t =  t 0 x t , c 0 =0 Where  t 0 :Tx t M  Tx 0 M is the parallel transport associated to  .

Solder Form (SHALMLI) 1. Smooth Manifold 2. Symplectic Form 3. Complex Manifold 4. Vierbein or Tetrad Theory 5. Signature 6. Palatini Action 7. Interwiners

Smooth Manifold ( SURACHAN) A differentiable manifold for which all the transitions maps are smooth. C 

Symplectic Form ( SOUMANAS) A symplectic form on a manifold M is a non-degenerative closed two form w.

Complex Manifold ( RAMANAK) Complex manifold is a smooth manifold equipped with smooth linear complex structure on each tangent space.

Vierbeib Or Tetrad Theory ( DEVBARS) Special case of application of Cretan connection in four – dimensional manifold.

Signature (PARIVADRA) Signature is the number of positive and negative Eigen values of the vector,

Palatini Action ( APAYAN) In tetrad formulation of general relativity, the action as a function of the co-tetrad e and a connection form A over a differential manifold M is given below where F is the gauge curvature 2-form and  is the anti symmetric inter winer of SO(3,1) normalized by  .

Interwiners (ABIGAN) If X and Y are the representative spaces of two linear representations of G has a linear map f:X  Y is called an interwiner of representations if it commutes with the action of G. Thus an interwiner is an equivalent map in the special case of two linear representations/ action.

Levi-civita Connection (PLAKH) 1. Preserve the metric 2. Derivative along curve 3. Unit sphere in R3 4. Isomorphism 5. Leibniz Rule 6. Torsion Free 7. Weitzenbock Connection

Preserve The Metric ( SHIV) Preserve the metric I,e for any vector field x, y, z we have X (g (Y,Z))=g (  x Y, Z) + g (Y,  x Z) Where X (g (Y,Z)) denotes derivative of a function g (Y,Z) along the vector field X.

Derivative Along Curve (SUVADRA) Derivative along curve D = Pullback Given a smooth curve Y on (M, g) and a vector field V along Y its derivative determined by D t V =  ý (t) V

Unit Sphere In R 3 ( SHANT) Let S2 be the unit sphere in R3 .The tangent space to S2 at a point m is naturally defined with vector sub-space of R3 consisting of all vectors orthogonal to m.

Isomorphism (KHEM) Parallel transport along a curve w.r.t a connection define isomorphism between the tangent space at that point of the curve.

Leibniz RulE (AMRIT) C  (S 2 ) linear is first variable. Consider a map f:S 2  R m  Y ( m ), m  The map f is constant hence differential vanishes dm f (x)=  dm Y (x) m  +  Y (m) ,X (m)  =0  (  x Y) (m), m  =0

Torsion Free ( ABHAY) Torsion free i,e for any vector fields x and y we have  x Y-  y X =  X,Y  Where  X,Y  is the Lie bracket of vector fields X and Y.

Weitzenback Connections ( JABAS) Make the spin connection , non-zero torsion but zero curvature form leads to weitzenback connection. Zero curvature means that there is local moving frame provided the spacetime is simply connected, since the parallel transport of the tetrad is path independent .There is a global moving frame provided the space time is a parallelizable manifold.

L-theory, Lie Bracket ( JAMBU) 1. Derivative of f along the vector field X. 2. Einstein Summation Convention 1-form. 3. Inner product. 4. Isometry. 5. Killing Field. 6. Global Isometry. 7. Linear Isometry. 8. Spin Connection

Derivative Of F Along The Vector Field X ( SWARNAPRASTHA) Given a function f:M  R and a vector field X defined on M. One defines Lie derivative of f at point p  M as £ x f (p) =X p (f) =  x f (p) the usual derivative of f along the vector field X.

Einstein Summation Convention 1-form( CHANDRASHEKHAR) £ x f(p)= d f(p)  X (P)  df = differential of f. Df : M  T * M is the 1- form.

Inner Product (ABARTAN) df(p)  X(P)  Inner product of differential of f ( at point p in M) being taken w.r.t the vector field X ( a point p).

Isometric ( MANDARHAHIN) Distance preserving isomorphism between metric space. Let X and Y be metric spaces with metric dy and dx . A map f: X  Y is called distance preserving if for any x , y  X one has dy( f ( x), f ( y)) = d x (x ,)

Killing Field (PANCHJANN) A vector field X is a killing field if the Lie derivative w.r.t X of the metric g vanish £ x g = 0

Global Isometric ( SINGHAL) Bijective distance preserving maps. Any reflection, translation, rotation is a global isometry on Eucledean space.

Linear Isometric ( LANKA) Given two normed vector spaces V and W a linear isometric is linear map. f :V  W that preserves the nom ||f(v)||=||v|| for all v in V.

Spin Connection (RAMNAK) Spin connection is a connection on a spin or bundle

Descrete-1 (PUSKAR) 1.CONVERGENCE SPACE 2.UNIFORM SPACE

Descrete-2 1.Classica Module Space 2.Quantum Module Space

CLASSICAL MODULLI SPACE (STHANU)

QUANTUM MODULLI SPACE(SARBANANDA)

Descrete-3 1.Product Space 2.Compact Space 3.Locally Compact Space

Descrete-4 (AGNIDH) 1.Satadhenu 2.Swayangbhoj 3.Hyperbola Action 4.Hyperbollic Space

Hyperbolic Motion ( KRITOBARMA) In geometry, a hyperbolic motion is a mapping of a model of hyperbolic geometry that preserves the distance measure in model. Upper half-plane model HP=  X,Y  :Y>0

Hyperbolic Geometry Space ( BIDURATH) Poincare ½-Plane Model. In non-Euclidean Geometry , the Poincare ½ plane model is the upper half, together with a metric , the Poincare metric , that makes it a model of two-dimensional hyperbolic model.

Base States (BHAJAMAN) 1. Path of spin-1 2. Path of spin-1A 3. Path of spin-1B 4. Path of spin-1C 5. Filters in series 6. Filters in series-A 7. 3- Filters in series

Path Of Spin – 1 ( SATADHENU) Stern-Gerlach Apparatus

Path Of Spin-1 A ( KRITABARMA)

Path Of Spin -1 C ( SWANGBHOJ)

Filters In Series-a (BIDURATH)

M-theory 11 th dimension ( ROW & COLUMN). Not a part of moving frames

M-theory (ROW) 1.Type-i String 2.Type-iia String 3.Type-ii B String 4.String With Loose Ends 5. String With Loose Ends 6. Nambu 3-aljebra( Lie 3-aljebra) 7.Non-associative Generalisation Of Lie Aljebra 8. Weakly Interacting 9.Strongly Interacting 10.Prosen 11. Anu

Chiral (NIMLOCHI) A phenomena is said to be chiral if it is not identical to its mirror image. Symmetry translation between the two is called parity . Chirality is same as helicity.

Non-associative Generalization Of Lie Algebra (BRISNI) A binary operation on a set S does not satisfy the associative law is called non-associative. (XxY)xY  Xx(YxZ) For any X,Y,Z  S

Weak Interactions (KINGKIN) It is due to the exchange of heavy W and Z bosons. Most familiar is beta decay and the associated radio activity.

Strongly Interacting (SINI) Strong interaction holds quarks and gluons together to form protons and neutrons.

Type-I String (AYUJIT) Is one of the five consistent super symmetric string theories in 10-dimensions It is the only one whose strings are unorientable and which contains only closed stings and also open strings.

Type-IIA String (SATAJIT) At low energies Type IIA string theory is described by type IIa super gravity in 10 dimensions which is a non-chiral theory (1,1) d=10 super gravity. The fact that the anomalies in this theory does not cancel is therefore trivial. Belong to simplistic topology and algebraic geometry particularly Groove-Witten invariant.

Type –IIB String (SAHASRAJIT) At low energies Type IIB string theory is described by type IIB super gravity in 10 dimensions which is a chiral theory (2,0) d=10 super gravity. The fact that the anomalies in this theory cancel is therefore non- trivial. Belong to algebraic geometry specially the deformation theory of complex structure originally studied by Kunilike Kodaire & Doland C Spencer.

String With Closed End (KAPATOROM) Strings with closed loops are free to move from membrane to membrane.

String With Loose Ends (BILOM) End points of the strings would not be able to move with complete freedom as they were attached or struck within certain region of space.

E8 (PRASEN) E8 is the name of a family of closed related structures. It is the name of some exceptional simple Lie groups. E8 has rank 8 and dimension 248.

U-duality (ANU) U-duality is a symmetry of sting theory or M-theory combining S-duality and T-duality transformation.

M-theory(COLUMN) 1.T-duality 2.S-duality 3.Supergravity 4.Dimensional Reduction To A Line 5.Dimensional Reduction To A Circle 6.Heteriotic String( E8xe8 + So(32) 7.Heteriotic So(32) 8.Joy 9.Judhajit 10.Juldhan

Anamoly (ANAMITRA) An anomaly is an irregularity or a mis proportion on something that is strange or unusual or unique. A chiral anomaly is the anomalous non-conservation of a chiral current.

S& T-duality (SUMITRA) S&T – duality is an equivalence of two quantum fields, string theory or M-theory. S-duality transforms maps the states and vacua with coupling constant g in one theory to states and vacua with coupling constant 1/g in the dual theory. T-duality transformation the radius R of that direction will be changed to 1/R and wrapped string states will be exchanged with high-momentum string states in the dual theory.

Super gravity (KANGBAL) Super gravity is a field theory that combines the principle of super symmetry and general relativity.

Anthropic Explanation (KONI) As string theory presently understood it appears to contain a large number of distance meta-stable vacua perhaps 10 500 or more. Each of these corresponds to a different possible universe with a different collection of particles and forces.

Dimensional Reduction To A Circle ( KONI) Get Type IIA String theory

Holographic Principle ( IBPHALAK) Holographic Principle Which States That The Description Of The Oscillations Of The Surface Of A Black hole Must Also Describe The Space Time Around It.

Dimensional Reduction To A Line Segment ( EBOFALOK) Get Heterotic SO(32) String Theory

Heteriotic (E8XE8+ SO(32))(JUGDHAR) Heteriotic string is a peculiar mixture of bosonic string and superstring. Left moving excitation think that they lie on a bosonic string propagating in D=26 dimension. Right moving excitations think that they belong to a superstring in D=10 dimension.

Heteriotic So (32) (KRUR) (AKRUR) Orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of a matrix multiplication.

Connection form

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