nLab
T-Duality and Differential K-Theory

under construction

This entry is about the article

• Alexander Kahle, Alessandro Valentino, T-duality and Differential K-Theory (arXiv:0912.2516)

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

• Ulrich Bunke, Thomas Schick, Differential K-theory. A survey (arXiv:1011.6663).

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

• a formalization of the setup

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

• statement of differential T-duality,

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

Write

$$\hat{\Lambda }:={\mathrm{Hom}}_{\mathrm{Grp}}(\Lambda ,\mathbb{Z})$$\hat \Lambda := Hom_{Grp}(\Lambda, \mathbb{Z})

for the dual lattice and

$$(-,-):\Lambda \times \hat{\Lambda }\to \mathbb{Z}$$(-,-) : \Lambda \times \hat \Lambda \to \mathbb{Z}

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

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

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

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

$$\Lambda [2]\otimes \hat{\Lambda }[2]\cong (\Lambda \otimes \hat{\Lambda })[4]$$\Lambda[2]\otimes\hat{\Lambda}[2]\cong (\Lambda\otimes\hat{\Lambda})[4]

together with the map of complexes

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

induces the cup product

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

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

$$\Lambda [2{]}_D^{\mathrm{\infty }}:=(\Lambda \hookrightarrow C^{\mathrm{\infty }}(-,{ℝ}^n)\stackrel{d_{\Lambda }}{\to }{\Omega }^1(-,{ℝ}^n)),$$\Lambda[2]^\infty_D:=(\Lambda\hookrightarrow C^\infty(-,\mathbb{R}^n)\xrightarrow{d_\Lambda} \Omega^1(-,\mathbb{R}^n)),

where the differential $d_{\Lambda }$ is defined as follows: if $e_1,\dots e_n$ is a $\mathbb{Z}$-basis of $\Lambda$ and $e^1,\dots ,e^n$ are the corresponding projections $e^i:{ℝ}^n\to ℝ$, then

$$d_{\Lambda }=(d\circ e^i)\otimes e_i$$d_\Lambda=(d\circ e^i)\otimes e_i

(this is independent of the chosen basis). The definition of $d_{\Lambda }$ is clearly chosen so to have an isomorphism of complexes $(Z[2{]}_D^{\mathrm{\infty }}{)}^{\otimes n}\cong \Lambda [2{]}_D^{\mathrm{\infty }}$ induced by the choice of a $\mathbb{Z}$-basis of $\Lambda$.

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

$$(B{ℝ}^n/\Lambda {)}_{\mathrm{conn}}\times (B({ℝ}^n{)}^{*}/\hat{\Lambda }{)}_{\mathrm{conn}}\to B^{3}U(1{)}_{\mathrm{conn}}$$(\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

$$\Lambda [2{]}_D^{\mathrm{\infty }}\otimes \hat{\Lambda }[2{]}_D^{\mathrm{\infty }}\stackrel{\cup }{\to }(\Lambda \otimes \hat{\Lambda })[4{]}_D^{\mathrm{\infty }}\stackrel{}{\to }\mathbb{Z}[4{]}_D^{\mathrm{\infty }},$$\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 $(B{ℝ}^n/\Lambda {)}_{\mathrm{conn}}\times (B({ℝ}^n{)}^{*}/\hat{\Lambda }{)}_{\mathrm{conn}}\to B^{3}U(1{)}_{\mathrm{conn}}$ It relates differential cohomological structures on ${ℝ}^n/\Lambda$-principal bundles with that on certain dual $({ℝ}^n{)}^{*}/\hat{\Lambda }$-principal bundles.

Definition

A differential T-duality pair is

• a smooth manifold $X$;

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

  such that the underlying topological class of the cup product $(P,\theta )\cup (\hat{P},\hat{\theta })$ is trivial;

• a choice of trivialization

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

Remark

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

$$\begin{array}{ccc}\mathrm{TDualityPairs}(X{)}_{\mathrm{conn}}& \stackrel{\mathrm{tw}}{\to }& H_{\mathrm{diff}}^4(X)\\ \downarrow & {⇙}_{\sigma }& \downarrow \\ H(X,B{ℝ}^n/\Lambda \times B{ℝ}^n/\hat{\Lambda })& \stackrel{}{\to }& H(X,B^{3}U(1){)}_{\mathrm{conn}}\end{array}.$$\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 $(P,\hat{P},\sigma )\in \mathrm{TDualityPairs}(X{)}_{\mathrm{conn}}$ for which the twist $\mathrm{tw}(P,\hat{P},\sigma )\in H_{\mathrm{diff}}^4(X)$ 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 $\mathrm{StringBund}(X{)}_{\mathrm{tw},\mathrm{conn}}$ over $X$: as discussed in detail there, this is the homotopy pullback

$$\begin{array}{ccc}\mathrm{StringBund}(X{)}_{\mathrm{tw},\mathrm{conn}}& \to & H_{\mathrm{diff}}^4(X)\\ \downarrow & {⇙}_{\sigma }& \downarrow \\ H(X,B\mathrm{Spin}\times B\mathrm{SU})& \stackrel{\frac{1}{2}{\hat{p}}_1-\frac{1}{46}\hat{c}}{\to }& H(X,B^{3}U(1){)}_{\mathrm{conn}}\end{array}.$$\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} } \,.

Definition (roughly)

Write $H_{\mathrm{diff},2}(X,B^{3}U(1))$ 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 $H_{\mathrm{diff}}(X,B^{2}U(1))$.

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

Remark

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

$$H_{\mathrm{diff},2}(X,B^{3}U(1)):={\mathrm{filt}}_1(H{\mathbb{Z}}_4{)}^X.$$\mathbf{H}_{diff,2}(X, \mathbf{B}^3 U(1)) := filt_1 ( H \mathbb{Z}_4)^X \,.

(Notice the ${\mathrm{filt}}_1$ instead of ${\mathrm{filt}}_0$. ) By this proposition this has the right properties.

Lemma

The choice $\sigma$ 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 $P$ and on $\hat{P}$, respectively, whose pullbacks to the fiber product $P{\times }_X\hat{P}$ are equivalent there.

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

Remark

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

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

$$\cdots \to B^{2}U(1)\to A\to B{ℝ}^n/\Lambda \times B{ℝ}^n/\hat{\Lambda }\to B^{3}U(1)$$\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 $X\to B{ℝ}^n/\Lambda \times B{ℝ}^n/\hat{\Lambda }$ of a pair of torus bundles $P,\hat{P}\to X$ factors through $A$ precisely if these form a T-duality pair. Such a factorization induces a $BU(1)$-principal 2-bundle on the fiber product $P{\times }_X\hat{P}$. This follows from the following pasting diagram of homotopy pullbacks

$$\begin{array}{ccccc}P{\times }_X\hat{P}& \stackrel{\tilde{\tau }}{\to }& B^{2}U(1)& \to & *\\ \downarrow & & \downarrow & & \downarrow \\ X& \to & A& \to & B{ℝ}^n/\Lambda \times B{ℝ}^n/\hat{\Lambda }\end{array}.$$\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 $\tilde{\tau }$ here is the class on the fiber product in question.

Notice that in the top left we indeed have $P{\times }_X\hat{P}$: the bottom left homotopy pullback of the product coefficients is equivalently given by the following pasting composite of homotopy pullbacks

$$\begin{array}{ccccc}& & & & P{\times }_X\hat{P}\\ & & & \swarrow & & \searrow \\ & & P& & & & \hat{P}\\ & \swarrow & & \searrow & & \swarrow & & \searrow \\ *& & & & X& & & & *\\ & \searrow & & \swarrow & & \searrow & & \swarrow \\ & & B{ℝ}^n/\Lambda & & & & B{ℝ}^n/\hat{\Lambda }\end{array}.$$\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 $X$ induces a $BU(1)$-2-bundle on the total space of a $\mathrm{Spin}$-principal bundle over $X$.

Statement of differential T-duality

(…)

category: reference