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, Communications in Contemporary Mathematics, Volume 16, Issue 02, April 2014 (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 $\Lambda \subset \mathbb{R}^n$ be a lattice (a discrete subgroup of the abelian group of real numbers to the $n$th cartesian power).

Write

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

for the dual lattice and

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

for the canonical pairing (the evaluation map). Notice that $\hat\Lambda$ is canonically identified with the lattice of $(\mathbb{R}^n)^*$ consisting of those linear functionals $\varphi:\mathbb{R}^n\to\mathbb{R}$ such that $\varphi(\Lambda)\subseteq \mathbb{Z}$. 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 $\mathbb{R}^n\otimes(\mathbb{R}^n)^*\to \mathbb{R}$.

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

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

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

together with the map of complexes

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

induces the cup product

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

This can be refined to a pairing in differential cohomology

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

by considering the Deligne complex of sheaves

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

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 $e^i:\mathbb{R}^n\to \mathbb{R}$, then

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

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

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

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

induced by the composition of morphisms of complexes

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

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 $(\mathbf{B}\mathbb{R}^n/\Lambda)_{conn}\times (\mathbf{B}(\mathbb{R}^n)^*/\hat\Lambda)_{conn}\to \mathbf{B}^3\mathbf{U}(1)_{conn}$ It relates differential cohomological structures on $\mathbb{R}^n / \Lambda$-principal bundles with that on certain dual $(\mathbb{R}^n)^* /\hat \Lambda$-principal bundles.

Definition

A differential T-duality pair is

• a smooth manifold $X$;

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

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

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 $TDualityPairs(X)_{conn}$, the homotopy pullback

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

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. are those elements $(P,\hat P, \sigma) \in TDualityPairs(X)_{conn}$ for which the twist $tw(P,\hat P, \sigma) \in H^4_{diff}(X)$ 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 $StringBund(X)_{tw,conn}$ over $X$: as discussed in detail there, this is the homotopy pullback

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

In particular the looping $TString$ of the homotopy fiber

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

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 $\mathbf{H}_{diff,2}(X, \mathbf{B}^3 U(1))$ 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 $\mathbf{H}_{diff}(X, \mathbf{B}^2 U(1))$.

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

Remark

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

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

(Notice the $filt_1$ instead of $filt_0$. ) 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 $A \to \mathbf{B} \mathbb{R}^n / \Lambda \times \mathbf{B} \mathbb{R}^n / \hat \Lambda$ be the homotopy fiber of the pairing class $\mathbf{B} \mathbb{R}^n / \Lambda \times \mathbf{B} \mathbb{R}^n / \hat \Lambda \to \mathbf{B}^3 U(1)$. This leads to the long fiber sequence (as discussed there)

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

The characteristic map $X \to \mathbf{B} \mathbb{R}^n / \Lambda \times \mathbf{B} \mathbb{R}^n / \hat \Lambda$ 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 $\mathbf{B} U(1)$-principal 2-bundle on the fiber product $P \times_X \hat P$. This follows from the following pasting diagram of homotopy pullbacks

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

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

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

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

Statement of differential T-duality

(…)

category: reference