Better T-Duality

Better T-Duality

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 Φ. These are described by a non-linear sigma model ↵ M N ↵ M N S[X]= dσd⌧phh gMN(X)@↵X @βX + ✏ BMN(X)@↵X @βX + ↵0phR(h)Φ(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 Φˆ =Φ 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.

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