T-duality
Anna Vaughan Supervisor Prof. Peter Bouwknegt Australian National University
12/2/2016
1 Contents
1 Introduction 3
2 The Buscher rules 3
3 Topological T-duality 4
4 Cohomology 7
5 Twisted Cohomology 13
6 T-duality and Twisted Cohomology 17
2 1 Introduction
This report presents an outline of topological T-duality on twisted cohomologies. T-duality is a duality first observed in bosonic string theory with a compactified spacetime dimension which has subsequently been extended to more general settings in pure mathematics. Here we first present a discussion of the Buscher rules for strings moving in background flux, and derive the topological properties of this transformation. T-duality gives an isomorphism on the de Rham cohomology twisted by the H-flux, a result which we attempt to extend to the case of Deligne cohomology. We present a preliminary discussion of twisting the Cech components of this di↵erential, before presenting an outline of how this may be extended to Deligne cohomology. The final section gives an outline of the procedure used to find an isomorphism under T-duality on the Deligne cohomology twisted by the whole Deligne class of H, including a proposed generalization of the Buscher rules.
2 The Buscher rules 2.1 Non-linear sigma model for a bosonic string T-duality may be examined for the case of strings in background flux as described by a nonlinear sigma model. In this setting T-duality is a map between two sigma models which give physically equivalent string theories. The closed string spectrum has three massless fields that a string will naturally couple to, the metric gMN, Kalb ramond field BMN and Dilaton . These are described by a non-linear sigma model
↵ M N ↵ M N S[X]= d d⌧phh gMN(X)@↵X @ X + ✏ BMN(X)@↵X @ X + ↵0phR(h) (X) Z The derivation of this model is outlined in [6]
2.2 Derivation of the Buscher rules Gauging the symmetry this action may be rewritten in terms of the T-dual fields. Following the discussion in [2] consider the case where the sigma model has a U(1) isometry in N of these coordinate directions. The coordinates can therefore be split into isometry (m) and non isometry (µ) directions. For simplicity, the action may be rewritten in complex coordinates, the Dilaton component ignored. BMN and gMN are then combined by defining QMN = gMN + BMN, allowing the action to be rewritten in the form
2 µ ⌫ µ n m ⌫ m n S[X]= d zQµ⌫ @X @¯X + Qµn@X @¯X + Qm⌫ @X @¯X + Qmn@X @¯X Z W emay now gauge this symmetry, i.e introduce auxilary gauge fields and extra degrees to rewrite the action in a form that is invariant under the isometry. To do this introduce U(1)N gauge fields Am, A¯m and couple these to the Xm using the covariant derivatives
@Xm DXm = @Xm + Am ! @¯Xm DX¯ m = @¯Xm + A¯m !
3 Auxillary fields Xˆ M are introduced such that the equations of motion of these fields give vanishing U(1)N curvature. This gives
2 µ ⌫ µ n m ⌫ m n m m S[X]= d z(Qµ⌫ (X)@X @¯X + Qµn(X)@X DX¯ + Qm⌫ (X)DX @¯X + Qmn(X)DX DX¯ + Xˆm(@A¯ @¯A ) Z This gives a new action, which now has the desired U(1) symmetry, and is equivalent to the previous action. Integrating out the auxiliary fields and fixing gives a new action in terms of the auxilary coordinates along the isometry directions coupled to the new combinations of BMN and gMN
1 µ ⌫ 1 µ n 1 m µ 1 m n S[x]= (Q Q (Q )Q @)X @¯X + Q (Q )X @¯Xˆ + (Q Q )Xˆ @¯X + Q Xˆ @¯Xˆ µ⌫ µ⌫ mn n⌫ µm mn mn n⌫ mn Z This action describes a physically equivalent ’T-dual’ theory. The transformation rules for the background are known as the Buscher rules after T. Buscher, and are given by
ˆ ˆ 1 1 Qµ⌫ Qµn Qµ⌫ Qµ⌫ (Qmn )Qn⌫ Qµm(Qmn ) = 1 1 Qˆ Qˆ (Q Qn⌫ ) Q ✓ m⌫ mn◆ ✓ mn mn ◆ Explicitly, in the case of a U(1) isometry 1 gˆ = •• g ••
Bµ gˆµ = • • g •• 1 gˆµ⌫ = gµ⌫ (gµ g⌫ Bµ B⌫ ) g • • • • •• gµ Bˆµ = • • g •• 1 Bˆµ⌫ = Bµ⌫ (gµ B⌫ g⌫ Bµ ) g • • • • •• T-duality therefore ’mixes’ the components of the background fields. Perfroming the same calcula- tion with the Dilaton included gives ˆ = log g •• This justifies not including the dilaton in the discussion of gauging the symmetry, as it is trivially related to the other components.
3 Topological T-duality
Given that gMN and BMN are both locally defined, the Buscher rules are only locally defined expressions. It is however possible to examine what is happening topologically to the space under T-duality. To do this, consider the specific case that the spacetime in question is of the form N Y where Y is a principal circle bundle over a manifold X. In more general cases Y can be considered⇥ to be a principal torus bundle, however for simplicity we restirct this discussion to the case of a principal circle bundle. The action of T-duality on the bundle is then summarized in the following theorem from [2].
4 Figure 1: Correspondence space [2]
Theorem 3.1. (Bouwknegt) Given (Y,[H]) where ⇡ : Y X is a principal circle bundle classified by first Chern class [F] with H-flux [H] the T-dual is given! by (Y,ˆ [Hˆ ]) where⇡ ˆ : Yˆ X is another principal circle bundle classified by Chern class [Fˆ] and Hˆ is the corresponding H flux.! Then these are related by F =ˆ⇡ Hˆ ⇤ Fˆ = ⇡ H ⇤ such that on the correspondence space
p⇤(H)=ˆp⇤(Hˆ ) with p : Y Yˆ Y andp ˆ : Y Yˆ Yˆ ⇤ ⇥X ! ⇤ ⇥X ! We present two proofs of this theorem, the first using the Buscher rules and the second using the dimensionally reduced Gysin sequence. Proof. Proof 1 [2]: Consider a connection on this circle bundle, defined to be
µ A = Aµdx + dx• gMN and BMN may then be decomposed on the base X with respect to the connection to give
µ ⌫ µ 2 gMN =¯gµ⌫ dx dx +(dx• + Aµdx )
1 µ ⌫ µ ⌫ B = B¯ dx dx + B¯ dx (dx• + A¯ dx ) MN 2 µ⌫ ^ µ ⌫ Transforming these expressions under T-duality according to the Buscher rules, T-duality is seen to correspond to the exchange A B µ $ µ This gives a relation between a theory with a principal circle bundle ⇡ : Y X and connection A as defined above, and the T-dual bundle, a principal circle bundle Yˆ over X! with connection
µ Aˆ = Bµdx + dx•
5 Considering just the Kalb - Ramond field, we have on the correspondence space
Bˆ = B A Aˆ + dx• dxˆ• ^ ^ So the H-flux, defined to be the exterior derivative of the B field is
Hˆ = H A Fˆ + F Aˆ ^ ^ Since H and F are globally defined forms, each side must separately equal a form on the base X. The relations are therefore H = M + Fˆ A ^ Hˆ = M + Aˆ F ^ This implies F =ˆ⇡ Hˆ ⇤ Fˆ = ⇡ H ⇤ As required. This only determines H up to cohomology. In order to specify H uniquely, we must impose the initial constraint that the H flux on the base manifold remains constant under T-duality. This is achieved by imposing the extra condition
p⇤(H)=ˆp⇤(Hˆ )
Proof 2 [3]: The topological properties of T-duality are described by the Gysin sequence. Consider the 3 - segment of this sequence for the principal circle bundle ⇡ : E M classified by Euler class F H2 and the T-dual bundle⇡ ˆ : Eˆ M classified by Fˆ H2. ! 2 ! 2 3 ⇡⇤ 3 ⇡ 2 F 4 ... H (M) H (E) ⇤ H (M) ^ H (M)... ! ! ! ! ˆ k ⇡ˆ⇤ 3 ⇡ˆ 2 F 4 ... H (M) H (Eˆ) ⇤ H (M) ^ H (M)... ! ! ! ! This sequence is exact, therefore any H flux in H3(E) on the bundle gives rise to a class ⇡ (H)in H2(M) which can be interpreted as the cohomology class of Fˆ on the T-dual bundle. The⇤ same argument applices for the T-dual bundle. By the definition of an exact sequence, the circle bundle gives [F ] [Fˆ] 0 H4(M) and from the T-dual bundle [Fˆ] [F ] 0 H4(M) We therefore have the relations^ ⌘ 2 ^ ⌘ 2 F =ˆ⇡ Hˆ ⇤ Fˆ = ⇡ H ⇤ As required. As before, since this element can only be determined up to an element in it’s coho- mology class, it is again necessary to impose the condition
p⇤(H)=ˆp⇤(Hˆ )
6 4 Cohomology 4.1 Preliminaries We first present the definition of a presheaf and sheaf, before defining sheaf and de Rham cohomol- ogy following the development in [4]. Definition 4.1. Consider a topological space X. A presheaf F of sets over X consists of the following objects A set F(U) for every open set U of X • For every inclusion V U for open sets, a mapping • ✓ ⇢ : F (U) F (V ) v,u ! This is required to have ⇢U,U as the identity and to obey the transitivity condition for all W V U ✓ ✓ ⇢W,U = ⇢W,V ⇢V,U A sheaf is defined to be a presheaf together with the extra condition For every open set V of X and for every open covering U of V and every family s where • i si F (Ui)with⇢U U ,U (si)=⇢U U ,U (sj)thereisauniques F (V ) such that 2 i\ j i i\ j j 2
⇢Ui,V (s)=si
Cohomology may now be defined using the language of sheaves. Consider first the functor of global sections. This is a functor from the category of sheaves of abelian groups over X to the category of abelian groups (X, ):AB(X) AB,definedby ⇤ ! (X, F) F (X) ⌘ This is used in the definition of sheaf cohomology
Definition 4.2. Let A be a sheaf of abelian groups on a space X and I• an injective resolution of A. Consider the complex ... (X, In) (X, In+1) ... ! ! The sheaf cohomology groups are defined to be the kth cohomology groups of this complex.
4.2 de Rham Cohomology Consider a manifold M with an open covering U. The space of p-forms on M is then given by p p p A (U), with A (U) consisting of a set !i A ( i(Ui U)) where the i are the transition functions 2 \ p satisfying the gluing condition. This gives a sheaf of p-forms AM with AP (U) AP (U) M ⌘ The de Rham complex may then be defined as
A0 d A1 d A2 d A3 d ... M ! M ! M ! M ! Where d is the ordinary exterior di↵erential. We may now define the de Rham cohomology
7 n Definition 4.3. The de Rham cohomology groups of a smooth manifold M, denoted Hd R(M) are the sheaf cohomology groups of the de Rham complex described above. This is equivalent to defining the de Rham cohomology groups as quotient spaces of closed p-forms Zn(M) by exact p-forms Bn(M),
n n n Hd R(M)=Z (M)/B (M)
De Rham cohomology may be viewed as a functor from the category of C1 manifolds to the category of anticommutative graded rings, and as such has applications as a topological invariant of di↵erentiable manifolds [1].
4.3 Cech cohomology The Cech cohomology may be defined for a presheaf A of Abelian groups on a space X with respect to an open covering U. Again following the discussion in [4] let
U U U U ...U i0,i1,i2,...in ⌘ i0 \ i1 \ i2 \ in Define Cp(U, A) A(U ) ⌘ i0,i1,i2,...in i i ...i 0 Y1 n the product of n+1 tuples of elements of the index set of the open cover. Let ↵ be an element of Cp(U, A) We can then define a homomorphism : Cp(U, A) Cp+1(U, A)by ! n+1 j (↵)i0...in+1 = ( 1) (↵i0...ij 1ij+1in+1 Ui0...i(n+1) | j=0 X It is a simple matter to check that delta = 0, therefore we can define a complex ... Cp(U, A) Cp+1(U, A) ! ! ! The Cech cohomology may now be defined as follows. Definition 4.4. The degree p cohomology of the above complex is known as the Cech cohomology of the open cover U with coe cients in the presheaf A. An equivalent way of thinking of Cech cohomology is to consider good open covers and fibred products [2]. Consider an Abelian group G and a fibred product of Y with itself n times over X, [n] denoted as YX . The Cech cochains can then be thought of as smooth collections of maps from the set intersections into the group G, with the maps asymmetric in their indices. Using the di↵erential as defined above, the Abelian group may be taken to be the group of n forms on Y, and consider
C˘q(X, G)=⌦p(Y [q+1])
Theorem 4.1. The spaces of forms and di↵erential discussed above give the following exact complex [2] 0 ⌦p(X) ⌦p(Y ) ⌦p(Y [2]) ⌦p(Y [3])... ! ! ! !
8 Figure 2: Cech de Rham complex [2]
4.4 The Cech de Rham Complex The Cech and de Rham complexes can naturally be combined into a double complex known as the Cech de Rham complex. This complex is seen to arise from the Mayer - Vietoris sequence [1].
Definition 4.5. Let the de Rham complex of forms on X be denoted by ⌦⇤(X). We then have the following exact sequence known as the generalized Mayer - Vietoris sequence.
r [1] [2] [3] 0 ⌦⇤(M) ⌦⇤(Y ) ⌦⇤(Y ) ⌦⇤(Y ) ... ! ! ! ! ! This sequence can be arranged into a double complex with the Cech di↵erential running vertically and the de Rham di↵erential running horizontally, as shown in figure 2. This is known as the Cech de Rham complex, and is an example of a double complex. The double complex can be made into a single graded complex using the di↵erential
D = +( 1)pd Cochains therefore consist of diagonal tuples across the complex. The following theorem relates the Cech and de Rham cohomologies.
9 Theorem 4.2. The Cech cohomology with coe cients in R is isomorphic to the de Rham coho- mology p p H˘ (X, R) H (X) ⌘ dR Proof. For a full proof see [1] and [2].
4.5 Di↵erential Cohomology An ordinary cohomology may be extended using the concept of a di↵erential cohomology. A dif- ferential cohomology is essentially an extension to ordinary cohomology that is able to take more geometric structure into account. We first give a motivating example for di↵erential cohomology Example: Electromagnetism (adapted from [2] and [7])- Consider the information contained in the objects of classical electromagnetism, which is modelled as a principal U(1) bundle ⇡ : Y X. The field strength is given by the curvature, which in this case is also the first Chern class, a 2-form! F on some 4-manifold X. We want to see what other gauge invariant objects are contained in this theory. Firstly, consider a good cover (U↵) of X. Obviously locally on a certain open set U↵
F = dA↵ where A is a connection on the principal bundle. As ⇡ : Y X is a principal bundle, on the double overlaps the locally defined connections must be related by! the transition functions of the bundle, giving d⇤ = A A ↵ ↵ These values of ⇤must take values in the structure group U(1), and hence may be better written as 2⇡i⇤↵ g↵ = e The value is only determined up to some integer however, giving the extra condition
n =⇤ ⇤ +⇤ ↵ ↵ ↵ with the final constraint that
0=n n + n n ↵ ↵ ↵ This is shown diagramatically as a ’staircase’ on the Cech de Rham complex in figure 3. Considering ˘ 1 these relations with a gauge transformation as in [2], we find that g↵, H (X, R/Z) with image 2 ˘ 1 2 2 n↵ breveH (X, Z) under the isomorphism from H (X, R/Z)tobreveH (X, Z). The isomor- 2 2 phism between Cech and de Rham cohomology gives the additional relation that F HdR (X)is 2 ˘ 1 2 2 the image of n↵ breveH (X, Z) under the isomorphism from H (X, R/Z)toHdR (X)Elec- tromagnetism therefore2 contains more gauge invariant information than simply the field strength, there is the extra quantity n↵ known as the holonomy. In summary, this example of electromag- netism has given us a commutative square of cohomologies with one term missing. The cohomology required to complete this square is a di↵erential cohomology. Having given this motivating example, the complete definition of a di↵erential extension is now presented Definition 4.6. [7] A di↵erential extension to a cohomology consists of the following components: Functors • Hˆ M : Manop Abel ! Natural transformations •
10 Figure 3: Electromagnetism on the Cech de Rham complex [2]
Figure 4: Commutative square to be completed by di↵erential cohomology [2]
11 Figure 5: commutative diagram in definition of di↵erential cohomology [7]
n 1 n – a :⌦ (M) Hˆ (M) d ! – I : Hˆ n(M) Hn(M,Z) ! – R : Hˆ n(M) ⌦n (M) ! cl n 1 ˆ n – b : H (M,R/Z) H (M) ! In addition, the commutative diagram shown in figure 5 is required to have an exact row and column.
On closer examination of this diagram, the commutative square from the motivating example of electromagnetism is the square to the bottom right of the complex. There are several examples of this di↵erential cohomology, the most widely used of whcich are Deligne cohomology, Cheeger Simons di↵erential characters and Cheeger Simons cohomology.
4.6 Deligne Cohomology Deligne, or Deligne Bieleson, cohomology is a particular example of a di↵erential cohomology. Deligne gave his original definition as the hypercohomology of the complex of sheaves of forms, as outlined in [4].
Definition 4.7. Let K• be a bounded below complex of sheaves on a space X. Then the hyper- p cohomology group H (X, K•) is defined to be the pth cohomology group of the double complex p,q (X, I )whereI•• is an injective resolution of K•. Deligne cohomology may now be defined as follows
12 Definition 4.8. Consider a subring of R denoted by B(p) and denote the sheaf of p-valued forms p on M as AM . We then have the Deligne complex
B(p) i A0 d A1 d A2 d ... ! M ! M ! M ! The Hypercohomology groups of this complex are known as the Deligne cohomology groups, and denoted Hˆ D
From this definition, Deligne p-cohchains can equivalently be thought of as p-tuples on the graded Cech de Rham complex with di↵erential D [2]. In addition to this the complex is truncated to forms of less than degree p for some value p Z Deligne cohomology is therefore the cohomology of this graded Cech de Rham complex with di↵2erential D, where cochains are of the form of tuples across the diagonals of the complex, explicitly given by
( 1,q) (0,q 1) (q 1,0) ! =(! ,! ,...,! ) in the case that q
5 Twisted Cohomology
Having given an overview of cohomology, we next wish to construct twisted Deligne cohomology. Given that the Deligne di↵erential D = +( 1)qd, in order to do this it is first necessary to understand how to twist Cech and de Rham cohomology. A twisted cohomology may be defined by constructing a twisted di↵erential and grading the cochain complex. We first define the cup product
Definition 5.1. A cup product is an operation defined on cohomology classes,
^: Hp Hq Hp+q ⇥ ! Which for all a Hp and b Hq satisfies the identity 2 2 a^b=( 1)pqb^a A twisted di↵erential on a graded complex may therefore be constructed by taking
d ( )=d( )+K^( ) K ⇤ ⇤ ⇤ Where d is the di↵erential for the untwisted cohomology and K is an arbitrary odd degree cochain.
13 5.1 Twisted de Rham cohomology In de Rham cohomology, the cup product is the familiar wedge product of forms , and we can therefore construct a twisted di↵erential ^
d = d + K K ^ where K is an odd degree form. The next step in constructing a twisted cohomology theory is to grade the complex. Consider the spaces of even and odd forms on a space X, denoted as ⌦0¯(X) and 1¯ 0¯ 1¯ ⌦ (X) respectively. It is clear that dK :⌦ (X) ⌦ (X) The cohomology of this graded complex with the twisted di↵erential is known as the twisted! de Rham cohomology.
5.2 Twisted Cech cohomology We now present an outline of how to construct a twisted Cech cohomology at the level of cochains, which turns out to be considerably more di cult than the case of de Rham cohomology. These following discussions can be attributed in part to conversations with Prof. P. Bouwknegt. Firstly consider the Cech cup product. At the level of cohomology there exists a well defined Cech cup product defined as follows Definition 5.2. Cech cup product for ↵ H˘ p and H˘ q ↵^ is defined to be 2 2 (↵) ( ) H˘ p+q i0,...ip ip...ip+q 2 At the level of cohomology this gives the required relation a^b=( 1)pqb^a. The problem is that at the level of cocycles this no longer holds. Instead,
↵^ +( 1)pq+1 ^↵= (↵ ) ⇤ For the case where we only consider ↵^↵we find
↵^↵= (a a) ⇤ Here this * operation is some relation between the two cocycles. Consider a di↵erential twisted by an odd degree cocycle a defined by
( )= ( )+a^ a ⇤ ⇤ ⇤ The requirement for this to be a di↵erential is that it squares to zero,
2 2 a = +(a^a) ^ In this case however the term a^ais not necessarily zero. Instead, there is the relation
2 = (a a) a ⇤ As a solution to this problem, we give the following proposition Proposition: The Cech cohomology can be twisted by a form ! if and only if it is a Maurer Cartan form, i.e it satisfies the relation ! = !^!Justification: The twisted di↵erential must by definition square to zero. Taking 2 = (a a) it seems as if the di↵erential should be modified to a ⇤ = + a^+(a a) ^ a ⇤
14 We now attempt to square this new di↵erential, which letting a a = a0 gives ⇤ 2 1 1 ( + a^+(a a) ^) ( )= (a^a0 + a0 ^a+ a0 ^a0) ⇤ ⇤ 2 2 The problem continues however in that 1 (a^a0 + a0 ^a+ a0 ^a0) =0 2 6 This iterative approach suggests that the di↵erential should be of the form
a +(e ⇤)
If this is to be a twisted di↵erential, then it would be expected to satisfy the relation
a a a e ⇤ = e ⇤ ^e ⇤ This is just the definition of a Maurer-Cartan form for the Cech cup product. Remark: In general, any cohomology theory may be twisted by a cocycle if and only if it is a Maurer Cartan form. To construct twisted Cech cohomology in general, it is therefore necessary to find an expression for both the * product and the Maurer Cartan form. First consider the * product. In the case of the Deligne class of H, the maximum Cech degree is 3. It is therefore necessary to find an expression for a degree 3 class cupped with another degree three class. We outline a systematic approach to⇤ achieving this below. Initial case: a C˘1, b C˘1. By observation, it is clear that in this case we have 2 2 a b a a ↵0↵1 ⇤ ↵0↵1 ⌘ ↵0↵1 ↵0↵1 We wish to show that 1 a^b= (a b) 2 ⇤ This is easily done by taking the of this expression and then substituting into the LHS using the cocycle conditions a + a a =0 ↵0↵1 ↵1↵2 ↵0↵2 and b + b b =0 ↵0↵1 ↵1↵2 ↵0↵2 This gives an explicit expression for * in the case of 1-cycles. We now wish to exten this to the case of three cycles as this is the important case in the context of T-duality. Unfortunately in these more complex cases it is no longer possible to simply ’guess’ the correct form of * for two arbitrary 3-cycles and check it by direct substitution. Instead, consider the case where we have a three cycle which is the cup product of three one cycles
a = a1 ^a2 ^a3 Where a ,a ,a C˘1 The * product may then be constructed in terms of the 1-cocycle case by 1 2 3 2 use of the following method. Consider a = a1 ^a2 ^a3 and b = b1 ^b2 ^b3.Thentheircup product is given by a^b= a1 ^a2 ^a3 ^b1 ^b2 ^b3
15 The a*b operation may be found in this case through sequentially swapping the cup products. Step one will therefore be a ^a ^a ^b ^b ^b = (a ^a ^b ^a ^b ^b ) (a ^a ^b a ^b ^b ) 1 2 3 1 2 3 1 2 1 3 2 3 1 2 1⇤ 3 2 3 Continuing this process of swapping the one - cycle cup products systematically to go from a^b to b^agives the final expression
a ^a ^a ^b ^b ^b = (b ^b ^b ) 1 2 3 1 2 3 1 2 3 (a ^a ^b a ^b ^b ) 1 2 1 ⇤ 3 2 3 (a ^a b ^a ^b ^b ) 1 2 ⇤ 1 3 2 3 (a b ^a ^a ^b ^b ) 1 ⇤ 1 2 3 2 3 + (b ^a ^a ^b a ^b ) 1 1 2 2 ⇤ 3 3 + (b ^a ^a b ^a ^b ) 1 1 2 ⇤ 2 3 3 + (b ^a b ^a ^a ^b ) 1 1 ⇤ 2 2 3 3 (b ^b ^a ^a ^a b ) 1 2 1 2 3 ⇤ 3 (b ^b ^a ^b a ^b ) 1 2 1 2 ⇤ 2 3 (b ^b ^b a ^a ^a ) 1 2 3 ⇤ 1 2 3 This is therefore an expression for a*b in this delta term in terms of the * product for 1-cocycles which we have already defined. Now rewrite this in terms of a and b rather than their component one cycles ai and bi. To do this use the cocycle condition on any place where the ordering of the indices doesn’t automatically simplify to an expression for the cup product. The expression for three cycles is then given by
(a b) = a b a b + a b ⇤ ↵0↵1↵2↵3↵4↵5 ↵0↵1↵2↵3 ↵0↵3↵4↵5 ↵1↵2↵3↵4 ↵0↵1↵4↵5 ↵2↵3↵4↵5 ↵0↵1↵2↵5 As in the one by one case this may b checked explicitly by substituting into the desired relation and using the cocycle conditions as necessary. Note that although this derivation was performed for the special case where the 3-cycles are composed of cup products of one cycles, the symmetry of the indices means that it will hold for an arbitrary 3-cycle. Having found this operation, it remains to explicitly construct the Maurer - Cartan form. A Maurer cartan form for⇤ a Lie group g naturally has the form 1 ! = g dg Considering the case where a = (b) this leads us to propose the following. Proposition: For a as defined above, the relevant Maurer Cartan form is given by
b b ! = e ^ (e )
Twisting by degree 3-cocycles, the form must therefore be of the form
!MC = !3 + !5 + !7 + ...
16 with individual components in the infinite sum satisfying
(!3)=0 (! )= ! ^! 5 3 3 (! )= ! ^! ! ^! ... 7 3 5 5 3 Considering again the previous expression, the general formula is
! = !n n>0 X where !n is defined to be
1 k n 1 k n 1 k !n n 1( 1) b^ b^a^b^ b ⌘ n! k kX=0 ✓ ◆ It is easily checked that this satisfies the required relations for the maurer Cartan form as described above. We have therefore constructed an explicit method for twisting Cech cohomology.
5.3 Twisted Deligne cohomology Having constructed twisted the Cech and de Rham cohomologies, it is now possible to twist the Deligne cohomology. It is first necessary to construct a cup product on Deligne cochains. This is given explicitly by
(k,l) (k0,l0) kl0 k0l (k,l) (k0,l0) ! ^ ! = ( 1) ! ^/ ! D ^ k+l=p,kX0+l0=q Given that this cup product has both a Cech and a de Rham component, constructing the twisted Deligne di↵erential will have similar di culties to the Cech case. Consider the expression for
!^!
Where ! is an odd degree cocycle. As in the Cech case, this will not in general be equal to zero, rather we will have the expression !^!= D(! !) ⇤ It therefore remains to find an expression for this Deligne equivalent of the Cech cup product.
6 T-duality and Twisted Cohomology
Having examined topological T-duality and twisted cohomology, it now remains to see what e↵ect T-duality has on twisted cohomologies.
17 6.1 Dimensional reduction Dimensional reduction is a technique used to rewrite forms on the total space in terms of forms on the base. Consider an arbitrary principal circle bundle ⇡ : E M and let H be an element of Hk(E) Consider the case in which there is a U(1) isometry, and let! be the corresponding Killing k vector field with invariant forms denoted by ⌦ (E)inv. The Kunneth formula gives a relation for the cohomologies of prouct manifolds
Hn(M N)= (Hp(M) Hq(N)) ⇥ ⌦ p+q=n M This means that locally on the principal bundle (or globally in the trivial case), there exists an isomorphism k k k 1 H (E) H (M) H (M) ' ⇥ This map can in fact be defined globally by use of the connection form A. Considering an arbitrary representative of the cohomology class ⌦ Hk(E) there is then a well defined map 2
A :(!k,!k 1) !k + A !k 1 7! ^ which has a well defined inverse
1 :⌦ (⌦ A ⇡ ⌦,⇡ ⌦) A 7! ^ ⇤ ⇤ Forms on the total space E can therefore be written on the base space by using the connection form [3].
6.2 T-duality on twisted de Rham cohomology It turns out that this dimensional reduction gives an isomorphism under T-duality of the de Rham cohomology twisted by the H flux, as outlined in [3]. Firstly, dimensionally reduce the H flux and an arbitrary form to give H = H3 + A H2 and ⌦=⌦k + A ⌦k 1. The condition for an arbitrary form to define a cohomology class is that^ its twisted di↵erential^ is equal to zero, therefore in this case ⌦is required to satisfy the conditions
d⌦k + H3 ⌦k + F ⌦k 1 =0 ^ ^
d⌦k 1 + H3 ⌦k 1 H2 ⌦k =0 ^ ^ These forms are all defined on M. From section 7, T-duality gives the transformation
((Hˆ , Hˆ ), Fˆ) ((H ,F),H ) 3 2 $ 3 2
It is clear that swapping F and H2 in the conditions for twisted cohomology gives an identical set of conditions under the exchange
(⌦ˆ k, ⌦ˆ k 1) (⌦k 1, ⌦k) $ T-duality therefore gives an isomorphism on the de Rham cohomology twisted by the H flux.
18 6.3 T-duality and twisted Deligne cohomology 6.3.1 Deligne classes of H and F We now wish to look at the Deligne class of the flux H from T-duality. By definition
H = dB↵
On intersections of open sets U↵ the B fields are constrained by the following relation,
dA = B B ↵ ↵ Where the A terms satisfy n = A A + A ↵ ↵ ↵ With n↵ satisfying the relations
0=n n + n n ↵ ↵ ↵ The Deligne class for T-duality therefore consists of the following relations sitting in a ’staircase’ on the Cech de Rham complex.
dB↵ = H↵ dA = B B ↵ ↵ n = A A + A ↵ ↵ ↵ 0=n n + n n ↵ ↵ ↵
These objects H, B↵,A↵ ,n↵ therefore form a class in the Deligne cohomology. Similarly, the Deligne class of F is given by F, A↵0 ,g↵ , satisfying the relations.
dA↵0 = F dg = A0 A0 ↵ ↵ 0=g g + g ↵ ↵ 6.3.2 Buscher rules for the Deligne class In order to examine the isomorphism of T-duality on twisted Deligne cohomology, it is first necessary to find the Buscher rules for the whole Deligne class. These are given by
Hˆ = H F Aˆ + A Fˆ ^ ^ Bˆ = B A Aˆ + dx• dxˆ• ^ ^ Aˆ = A + dx• dxˆ• n =ˆn
Consider now the Deligne class of F. Given that Fˆ and ⇡ H are seen to be interchanged under T-duality, it seems natural to conjecture that the Deligne classes⇤ are related in the following way
19 Hˆ = ⇡ F ⇤ H = ⇡ Fˆ ⇤ Bˆ = ⇡ A0 ⇤ ˆ B = ⇡ A0 ⇤ Aˆ = ⇡ g ⇤ A = ⇡ gˆ ⇤ Note that in order to formally derive these rules it would be necessary to define a new nonlinear sigma model takign the additional terms into account and rederive the Buscher rules using the same techniques as in section 2.
6.3.3 Isomorphism on twisted Deligne cohomology The methods developed in the preceeding sections may now be used to show that T-duality gives an isomorphism on twisted Deligne cohomology. Firstly, construct the relevant Maurer Cartan form to twist the Deligne cohomology. It is then simple to give a twisted di↵erential using the Deligne cup product as defined previously. It would be expected that dimensionally reducing with respect to the form component and substituting in the Buscher rules for the Deligne class T-duality would give an isomorphism on this twisted cohomology.
References
[1] Bott, R and Tu, L (1982) Di↵erential forms in algebraic topology, [2] Bouwknegt, P (2008) Lectures on cohomology, T-duality and generalized geometry, presented at the Summer School New Paths Towards Quantum Gravity, Holbk, Denmark, 10-16 May 2008 [3] Bouwknegt, P. Hannabuss, K. and Mathai, V.(2005), T-duality for principal torus bundles and dimensionally reduced Gysin sequences, Adv. Theor. Math. Phys. 9 749-773, [arXiv:hep- th/0412268] [4] Brylinski, J (1993), Loop groups, Characteristic classes and geometric quantization, Birkhauser, Boston [5] Hatcher, A (2001), Algebraic Topology, Accessed from https://www.math.cornell.edu/ hatcher/AT/AT.pdf [6] Polchinski, J (1998), String theory volume one an introduction to the Bosonic string, Cam- bridge University Press, Cambridge. [7] Upmeier, M (n.d), Notes: Ordinary Di↵erential Cohomology, accessed from http://www.math.uni-augsburg.de/prof/di↵/arbeitsgruppe/upmeier/notes/archiv/vortrag- hannover.pdf [8] Zwiebach, B (2009), A first course in string theory, Cambridge University Press, London
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