T-Duality and Differential K-Theory

under construction

This entry is about the article

on the identification of the correct description in differential cohomology (ordinary differential cohomology and differential K-theory) of T-duality or topological T-duality: differential T-duality.

A review is in section 7.4 of

We try to indicate some of the content. There are two main ingredients:

which proposes a refinement of the ingredients of topological T-duality to differential cohomology; and then the

which is the assertion that this setup naturally induces the T-duality operation as an isomorphism on twisted differential K-theory.

Formalization of the setup

Let Λ n\Lambda \subset \mathbb{R}^n be a lattice (an discrete subgroup of the abelian group of real numbers to the nnth cartesian power).


Λ^:=Hom Grp(Λ,) \hat \Lambda := Hom_{Grp}(\Lambda, \mathbb{Z})

for the dual lattice and

(,):Λ×Λ^ (-,-) : \Lambda \times \hat \Lambda \to \mathbb{Z}

for the canonical pairing (the evaluation map). Notice that Λ^\hat\Lambda is canonically identified with the lattice of ( n) *(\mathbb{R}^n)^* consisting of those linear functionals φ: n\varphi:\mathbb{R}^n\to\mathbb{R} such that φ(Λ)\varphi(\Lambda)\subseteq \mathbb{Z}. With this identification, the canonical pairing Λ×Λ^\Lambda \times \hat \Lambda \to \mathbb{Z} can be seen as the restriction to Λ×Λ^\Lambda \times \hat \Lambda of the canonical pairing n( n) *\mathbb{R}^n\otimes(\mathbb{R}^n)^*\to \mathbb{R}.

The quotient n/Λ\mathbb{R}^n / \Lambda is a torus. A n/Λ\mathbb{R}^n/\Lambda-principal bundle is a torus-bundle. Write n/Λ\mathcal{B}\mathbb{R}^n/\Lambda \in Top for the classifying space and B n/Λ\mathbf{B} \mathbb{R}^n / \Lambda for its moduli space: the smooth groupoid delooping the Lie group /Λ\mathbb{R}/\Lambda.

Torus bundles on a smooth manifold XX are classified by H 2(X,Λ)H^2(X, \Lambda). Following the discussion at smooth ∞-groupoid we write here H(X,B n/Λ)\mathbf{H}(X, \mathbf{B}\mathbb{R}^n/\Lambda) for the groupoid of smooth torus bundles and smooth bundle morphisms between. Write H conn(X,B n/Λ)\mathbf{H}_{conn}(X,\mathbf{B} \mathbb{R}^n /\Lambda) for the corresponding differential refinement to bundles with connection.

For AA an abelian group, let A[n]A[n] be the chain complex consisting of AA concentrated in degree nn. Then the tensor product of chain complexes

Λ[2]Λ^[2](ΛΛ^)[4] \Lambda[2]\otimes\hat{\Lambda}[2]\cong (\Lambda\otimes\hat{\Lambda})[4]

together with the map of complexes

(ΛΛ^)[4](,)[4] (\Lambda \otimes \hat \Lambda)[4]\stackrel{(-,-)}{\to} \mathbb{Z}[4]

induces the cup product

H k(X,Λ)×H l(X,Λ^)H k+l(X,ΛΛ^)(,)H k+l(X,). H^k(X, \Lambda) \times H^l(X, \hat \Lambda) \to H^{k+l}(X, \Lambda \otimes \hat \Lambda) \stackrel{(-,-)}{\to} H^{k+l}(X, \mathbb{Z}) \,.

This can be refined to a pairing in differential cohomology

H¯ k(X,Λ)×H¯ l(X,Λ^)H¯ k+l(X,ΛΛ^)(,)H¯ k+l(X,). \bar{H}^k(X, \Lambda) \times \bar{H}^l(X, \hat \Lambda) \to \bar{H}^{k+l}(X, \Lambda \otimes \hat \Lambda) \stackrel{(-,-)}{\to} \bar{H}^{k+l}(X, \mathbb{Z}) \,.

by considering the Deligne complex of sheaves

Λ[2] D :=(ΛC (, n)d ΛΩ 1(, n)), \Lambda[2]^\infty_D:=(\Lambda\hookrightarrow C^\infty(-,\mathbb{R}^n)\xrightarrow{d_\Lambda} \Omega^1(-,\mathbb{R}^n)),

where the differential d Λd_\Lambda is defined as follows: if e 1,e ne_1,\dots e_n is a \mathbb{Z}-basis of Λ\Lambda and e 1,,e ne^1,\dots,e^n are the corresponding projections e i: ne^i:\mathbb{R}^n\to \mathbb{R}, then

d Λ=(de i)e i d_\Lambda=(d\circ e^i)\otimes e_i

(this is independent of the chosen basis). The definition of d Λd_\Lambda is clearly chosen so to have an isomorphism of complexes (Z[2] D ) nΛ[2] D (\mathbf{Z}[2]^\infty_D)^{\otimes n}\cong \Lambda[2]^\infty_D induced by the choice of a \mathbb{Z}-basis of Λ\Lambda.

Write (B n/Λ) conn(\mathbf{B}\mathbb{R}^n/\Lambda)_{conn} for the smooth groupoid associated by the Dold-Kan correspondence to the Deligne complex Λ[2] D \Lambda[2]^\infty_D. Then we have the morphism of smooth groupoids to a morphism

(B n/Λ) conn×(B( n) */Λ^) connB 3U(1) conn (\mathbf{B}\mathbb{R}^n/\Lambda)_{conn}\times (\mathbf{B}(\mathbb{R}^n)^*/\hat\Lambda)_{conn}\to \mathbf{B}^3\mathbf{U}(1)_{conn}

induced by the composition of morphisms of complexes

Λ[2] D Λ^[2] D (ΛΛ^)[4] D [4] D , \Lambda[2]^\infty_D\otimes \hat{\Lambda}[2]^\infty_D \stackrel{\cup}{\to} (\Lambda\otimes\hat\Lambda)[4]^\infty_D \stackrel{}{\to}\mathbb{Z}[4]^\infty_D,

where \cup is the Beilinson-Deligne cup-product.

Notice that this is the one which defines abelian Chern-Simons theories. The higher holonomy of the circle 3-bundle with connection appearing here is the action functional of torus-Chern-Simons theory.

Differential T-duality pairs form the homotopy fiber of the morphism (B n/Λ) conn×(B( n) */Λ^) connB 3U(1) conn(\mathbf{B}\mathbb{R}^n/\Lambda)_{conn}\times (\mathbf{B}(\mathbb{R}^n)^*/\hat\Lambda)_{conn}\to \mathbf{B}^3\mathbf{U}(1)_{conn} It relates differential cohomological structures on n/Λ\mathbb{R}^n / \Lambda-principal bundles with that on certain dual ( n) */Λ^(\mathbb{R}^n)^* /\hat \Lambda-principal bundles.


A differential T-duality pair is

  • a smooth manifold XX;

  • a n/Λ\mathbb{R}^n/\Lambda-principal bundle PXP \to X with connection θ\theta and a n/Λ^\mathbb{R}^n / \hat \Lambda-principal bundle P^X\hat P \to X with connection θ^\hat \theta;

    such that the underlying topological class of the cup product (P,θ)(P^,θ^)(P, \theta) \cup (\hat P, \hat \theta ) is trivial;

  • a choice of trivialization

    σ:(0,C)(P,θ)(P^,θ^). \sigma : (0,C) \stackrel{\simeq}{\to} (P, \theta) \cup (\hat P , \hat \theta) \,.

This is (KahleValentino, def. 2.1). It is the evident differential generalization of the description in topological T-duality that appears for instance around (7.11) of (BunkeSchick).


We may refine this naturally to a 2-groupoid of twisted T-duality pairs TDualityPairs(X) connTDualityPairs(X)_{conn}, the homotopy pullback

TDualityPairs(X) conn tw H diff 4(X) σ H(X,B n/Λ×B n/Λ^) H(X,B 3U(1)) conn. \array{ TDualityPairs(X)_{conn} &\stackrel{tw}{\to}& H^4_{diff}(X) \\ \downarrow &\swArrow_{\sigma}& \downarrow \\ \mathbf{H}(X, \mathbf{B}\mathbb{R}^n/\Lambda \times \mathbf{B} \mathbb{R}^n/\hat \Lambda) &\stackrel{}{\to}& \mathbf{H}(X, \mathbf{B}^3 U(1))_{conn} } \,.

This is itself an example of twisted cohomology (as discussed there). (We use here the notation at differential cohomology in a cohesive topos .)

The differential T-duality pairs of def. 1 are those elements (P,P^,σ)TDualityPairs(X) conn(P,\hat P, \sigma) \in TDualityPairs(X)_{conn} for which the twist tw(P,P^,σ)H diff 4(X)tw(P,\hat P, \sigma) \in H^4_{diff}(X) in ordinary differential cohomology has an underlying trivial circle 3-bundle. We could restrict the homotopy pullback to these, but it seems natural to include the full collection of twists. (Notice that these “twist” here are not the twists in “twisted K-theory”, rather we are observing that already the notion of T-duality pairs itself is an example of cocycles in twisted cohomology in the general sense.)

Notice that the above analogous to the notion of differential string structures in StringBund(X) tw,connStringBund(X)_{tw,conn} over XX: as discussed in detail there, this is the homotopy pullback

StringBund(X) tw,conn H diff 4(X) σ H(X,BSpin×BSU) 12p^ 1146c^ H(X,B 3U(1)) conn. \array{ StringBund(X)_{tw,conn} &\to& H^4_{diff}(X) \\ \downarrow &\swArrow_{\sigma}& \downarrow \\ \mathbf{H}(X, \mathbf{B}Spin \times \mathbf{B} SU) &\stackrel{\frac{1}{2}\hat \mathbf{p}_1 - \frac{1}{46}\hat \mathbf{c}}{\to}& \mathbf{H}(X, \mathbf{B}^3 U(1))_{conn} } \,.

In particular the looping TStringTString of the homotopy fiber

BTString * B n/Λ×B n/Λ^) B 3U(1)) conn \array{ \mathbf{B}TString &\stackrel{}{\to}& \ast \\ \downarrow &\swArrow& \downarrow \\ \mathbf{B}\mathbb{R}^n/\Lambda \times \mathbf{B} \mathbb{R}^n/\hat \Lambda) &\stackrel{\langle - \cup -\rangle}{\to}& \mathbf{B}^3 U(1))_{conn} }

has the right to be called the T-duality 2-group or similar. The principal 2-bundles for this are T-folds (see there).

Definition (roughly)

Write H diff,2(X,B 3U(1))\mathbf{H}_{diff,2}(X, \mathbf{B}^3 U(1)) for a 3-groupoid whose objects are cocycles in ordinary differential cohomology in degree 4, but whose morphisms need not preserve connections and are instead such that the automorphism 2-groupoid of the 0-object is that of circle 2-bundles with connection H diff(X,B 2U(1))\mathbf{H}_{diff}(X, \mathbf{B}^2 U(1)).

A 1-groupoid truncation of this idea is the object denoted p(X)\mathcal{H}^p(X) in KahleValentino, A.2.


In terms of the notion of differential function complex we should simply set

H diff,2(X,B 3U(1)):=filt 1(H 4) X. \mathbf{H}_{diff,2}(X, \mathbf{B}^3 U(1)) := filt_1 ( H \mathbb{Z}_4)^X \,.

(Notice the filt 1filt_1 instead of filt 0filt_0. ) By this proposition this has the right properties.


The choice σ\sigma of the trivialization of the cup product of the two torus bundles induces canonically elments in degree 3 ordinary differential cohomology (two circle 2-bundles with connection) on PP and on P^\hat P, respectively, whose pullbacks to the fiber product P× XP^P \times_X \hat P are equivalent there.

This is (KahleValentino, 2.2, 2.3), where an explicit construction of the classes and their equivalence is given.


This is a special case of the general statement about extensions of higher bundles discussed here:

Let AB n/Λ×B n/Λ^A \to \mathbf{B} \mathbb{R}^n / \Lambda \times \mathbf{B} \mathbb{R}^n / \hat \Lambda be the homotopy fiber of the pairing class B n/Λ×B n/Λ^B 3U(1)\mathbf{B} \mathbb{R}^n / \Lambda \times \mathbf{B} \mathbb{R}^n / \hat \Lambda \to \mathbf{B}^3 U(1). This leads to the long fiber sequence (as discussed there)

B 2U(1)AB n/Λ×B n/Λ^B 3U(1) \cdots \to \mathbf{B}^2 U(1) \to A \to \mathbf{B} \mathbb{R}^n / \Lambda \times \mathbf{B} \mathbb{R}^n / \hat \Lambda \to \mathbf{B}^3 U(1)

The characteristic map XB n/Λ×B n/Λ^X \to \mathbf{B} \mathbb{R}^n / \Lambda \times \mathbf{B} \mathbb{R}^n / \hat \Lambda of a pair of torus bundles P,P^XP , \hat P \to X factors through AA precisely if these form a T-duality pair. Such a factorization induces a BU(1)\mathbf{B} U(1)-principal 2-bundle on the fiber product P× XP^P \times_X \hat P. This follows from the following pasting diagram of homotopy pullbacks

P× XP^ τ˜ B 2U(1) * X A B n/Λ×B n/Λ^. \array{ P \times_X \hat P &\stackrel{\tilde \tau}{\to}& \mathbf{B}^2 U(1) &\to& * \\ \downarrow && \downarrow && \downarrow \\ X &\to& A & \to & \mathbf{B} \mathbb{R}^n / \Lambda \times \mathbf{B} \mathbb{R}^n / \hat \Lambda } \,.

The τ˜\tilde \tau here is the class on the fiber product in question.

Notice that in the top left we indeed have P× XP^P \times_X \hat P: the bottom left homotopy pullback of the product coefficients is equivalently given by the following pasting composite of homotopy pullbacks

P× XP^ P P^ * X * B n/Λ B n/Λ^. \array{ && && P \times_X \hat P \\ && & \swarrow && \searrow \\ && P &&&& \hat P \\ & \swarrow && \searrow && \swarrow && \searrow \\ * && && X && && * \\ & \searrow && \swarrow && \searrow && \swarrow \\ && \mathbf{B}\mathbb{R}^n / \Lambda && && \mathbf{B}\mathbb{R}^n / \hat \Lambda } \,.

Notice also that this is again directly analogous to the situation for string structures: as discussed there, a string structure on XX induces a BU(1)\mathbf{B}U(1)-2-bundle on the total space of a SpinSpin-principal bundle over XX.

Statement of differential T-duality


category: reference

Revised on February 26, 2014 11:00:55 by Urs Schreiber (