# nLab T-Duality and Differential K-Theory

under construction

### Context

#### Differential cohomology

differential cohomology

## Application to gauge theory

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

Write

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

for the dual lattice and

$\left(-,-\right):\Lambda ×\stackrel{^}{\Lambda }\to ℤ$(-,-) : \Lambda \times \hat \Lambda \to \mathbb{Z}

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

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

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

$\Lambda \left[2\right]\otimes \stackrel{^}{\Lambda }\left[2\right]\cong \left(\Lambda \otimes \stackrel{^}{\Lambda }\right)\left[4\right]$\Lambda[2]\otimes\hat{\Lambda}[2]\cong (\Lambda\otimes\hat{\Lambda})[4]

together with the map of complexes

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

induces the cup product

${H}^{k}\left(X,\Lambda \right)×{H}^{l}\left(X,\stackrel{^}{\Lambda }\right)\to {H}^{k+l}\left(X,\Lambda \otimes \stackrel{^}{\Lambda }\right)\stackrel{\left(-,-\right)}{\to }{H}^{k+l}\left(X,ℤ\right)\phantom{\rule{thinmathspace}{0ex}}.$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

${\overline{H}}^{k}\left(X,\Lambda \right)×{\overline{H}}^{l}\left(X,\stackrel{^}{\Lambda }\right)\to {\overline{H}}^{k+l}\left(X,\Lambda \otimes \stackrel{^}{\Lambda }\right)\stackrel{\left(-,-\right)}{\to }{\overline{H}}^{k+l}\left(X,ℤ\right)\phantom{\rule{thinmathspace}{0ex}}.$\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

$\Lambda \left[2{\right]}_{D}^{\infty }:=\left(\Lambda ↪{C}^{\infty }\left(-,{ℝ}^{n}\right)\stackrel{{d}_{\Lambda }}{\to }{\Omega }^{1}\left(-,{ℝ}^{n}\right)\right),$\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 $ℤ$-basis of $\Lambda$ and ${e}^{1},\dots ,{e}^{n}$ are the corresponding projections ${e}^{i}:{ℝ}^{n}\to ℝ$, then

${d}_{\Lambda }=\left(d\circ {e}^{i}\right)\otimes {e}_{i}$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 $\left(Z\left[2{\right]}_{D}^{\infty }{\right)}^{\otimes n}\cong \Lambda \left[2{\right]}_{D}^{\infty }$ induced by the choice of a $ℤ$-basis of $\Lambda$.

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

$\left(B{ℝ}^{n}/\Lambda {\right)}_{\mathrm{conn}}×\left(B\left({ℝ}^{n}{\right)}^{*}/\stackrel{^}{\Lambda }{\right)}_{\mathrm{conn}}\to {B}^{3}U\left(1{\right)}_{\mathrm{conn}}$(\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

$\Lambda \left[2{\right]}_{D}^{\infty }\otimes \stackrel{^}{\Lambda }\left[2{\right]}_{D}^{\infty }\stackrel{\cup }{\to }\left(\Lambda \otimes \stackrel{^}{\Lambda }\right)\left[4{\right]}_{D}^{\infty }\stackrel{}{\to }ℤ\left[4{\right]}_{D}^{\infty },$\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 $\left(B{ℝ}^{n}/\Lambda {\right)}_{\mathrm{conn}}×\left(B\left({ℝ}^{n}{\right)}^{*}/\stackrel{^}{\Lambda }{\right)}_{\mathrm{conn}}\to {B}^{3}U\left(1{\right)}_{\mathrm{conn}}$ It relates differential cohomological structures on ${ℝ}^{n}/\Lambda$-principal bundles with that on certain dual $\left({ℝ}^{n}{\right)}^{*}/\stackrel{^}{\Lambda }$-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 ${ℝ}^{n}/\stackrel{^}{\Lambda }$-principal bundle $\stackrel{^}{P}\to X$ with connection $\stackrel{^}{\theta }$;

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

• a choice of trivialization

$\sigma :\left(0,C\right)\stackrel{\simeq }{\to }\left(P,\theta \right)\cup \left(\stackrel{^}{P},\stackrel{^}{\theta }\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 $\mathrm{TDualityPairs}\left(X{\right)}_{\mathrm{conn}}$, the homotopy pullback

$\begin{array}{ccc}\mathrm{TDualityPairs}\left(X{\right)}_{\mathrm{conn}}& \stackrel{\mathrm{tw}}{\to }& {H}_{\mathrm{diff}}^{4}\left(X\right)\\ ↓& {⇙}_{\sigma }& ↓\\ H\left(X,B{ℝ}^{n}/\Lambda ×B{ℝ}^{n}/\stackrel{^}{\Lambda }\right)& \stackrel{}{\to }& H\left(X,{B}^{3}U\left(1\right){\right)}_{\mathrm{conn}}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\left(P,\stackrel{^}{P},\sigma \right)\in \mathrm{TDualityPairs}\left(X{\right)}_{\mathrm{conn}}$ for which the twist $\mathrm{tw}\left(P,\stackrel{^}{P},\sigma \right)\in {H}_{\mathrm{diff}}^{4}\left(X\right)$ 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 $\mathrm{StringBund}\left(X{\right)}_{\mathrm{tw},\mathrm{conn}}$ over $X$: as discussed in detail there, this is the homotopy pullback

$\begin{array}{ccc}\mathrm{StringBund}\left(X{\right)}_{\mathrm{tw},\mathrm{conn}}& \to & {H}_{\mathrm{diff}}^{4}\left(X\right)\\ ↓& {⇙}_{\sigma }& ↓\\ H\left(X,B\mathrm{Spin}×B\mathrm{SU}\right)& \stackrel{\frac{1}{2}{\stackrel{^}{p}}_{1}-\frac{1}{46}\stackrel{^}{c}}{\to }& H\left(X,{B}^{3}U\left(1\right){\right)}_{\mathrm{conn}}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 ${H}_{\mathrm{diff},2}\left(X,{B}^{3}U\left(1\right)\right)$ 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 ${H}_{\mathrm{diff}}\left(X,{B}^{2}U\left(1\right)\right)$.

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

###### Remark

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

${H}_{\mathrm{diff},2}\left(X,{B}^{3}U\left(1\right)\right):={\mathrm{filt}}_{1}\left(H{ℤ}_{4}{\right)}^{X}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\stackrel{^}{P}$, respectively, whose pullbacks to the fiber product $P{×}_{X}\stackrel{^}{P}$ 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 $A\to B{ℝ}^{n}/\Lambda ×B{ℝ}^{n}/\stackrel{^}{\Lambda }$ be the homotopy fiber of the pairing class $B{ℝ}^{n}/\Lambda ×B{ℝ}^{n}/\stackrel{^}{\Lambda }\to {B}^{3}U\left(1\right)$. This leads to the long fiber sequence (as discussed there)

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

$\begin{array}{ccccc}P{×}_{X}\stackrel{^}{P}& \stackrel{\stackrel{˜}{\tau }}{\to }& {B}^{2}U\left(1\right)& \to & *\\ ↓& & ↓& & ↓\\ X& \to & A& \to & B{ℝ}^{n}/\Lambda ×B{ℝ}^{n}/\stackrel{^}{\Lambda }\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\stackrel{˜}{\tau }$ here is the class on the fiber product in question.

Notice that in the top left we indeed have $P{×}_{X}\stackrel{^}{P}$: the bottom left homotopy pullback of the product coefficients is equivalently given by the following pasting composite of homotopy pullbacks

$\begin{array}{ccccc}& & & & P{×}_{X}\stackrel{^}{P}\\ & & & ↙& & ↘\\ & & P& & & & \stackrel{^}{P}\\ & ↙& & ↘& & ↙& & ↘\\ *& & & & X& & & & *\\ & ↘& & ↙& & ↘& & ↙\\ & & B{ℝ}^{n}/\Lambda & & & & B{ℝ}^{n}/\stackrel{^}{\Lambda }\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $BU\left(1\right)$-2-bundle on the total space of a $\mathrm{Spin}$-principal bundle over $X$.

## Statement of differential T-duality

(…)

category: reference

Revised on November 22, 2013 05:40:34 by Urs Schreiber (82.169.114.243)