nLab twisted differential c-structure

Idea

For $c$ any characteristic class, its homotopy fibers on cocycle ∞-groupoids represent $c$-twisted cohomology (for instance twisted bundles, twisted spin structures, etc.).

If $c$ is refined to a characteristic class $\mathbf{c}$ in Smooth∞Grpd there may exist further refinements $\hat {\mathbf{c}}$ to ordinary differential cohomology. The twisted cohomology of these differential characteristic classes may be called twisted differential structures . For instance differential string structures . See below for more examples.

These structures have a natural interpretation and play a natural roles as physical fields (see there for a comprehensive discussion).

Definition

Let $\mathbf{H}$ be a cohesive (∞,1)-topos, usually $\mathbf{H} =$ Smooth∞Grpd or SynthDiff∞Grpd or the like.

Let $K, G$ be ∞-group objects in $\mathbf{H}$ and let

$$\mathbf{c} : \mathbf{B}G \to \mathbf{B}K$$

be a morphism of their delooping objects / moduli stacks.

Definition

For $X \in \mathbf{H}$ any object and $P \to X$ an $K$-principal ∞-bundle over $X$, the ∞-groupoid

$$\mathbf{c}Struc_{[P]}(X) := \mathbf{H}(X, \mathbf{B}G) \times_{\mathbf{H}(X, \mathbf{B}K)} \{P\} \,,$$

hence the (∞,1)-pullback

$$\array{ \mathbf{c}Struc_{[P]}(X) &\to& * \\ \downarrow^{\mathrlap{P}} && \downarrow \\ \mathbf{H}(X, \mathbf{B}G) &\stackrel{\mathbf{H}(X, \mathbf{c})}{\to}& \mathbf{H}(X, \mathbf{B}K) }$$

we may call equivalently

• the $\infty$-groupoid of $K$-structures on $P$ (with respect to the given $\mathbf{c}$);

• the $\infty$-groupoid of $[P]$-twisted $\mathbf{c}$-structures.

Remark

As discussed at twisted cohomology, we may think of an object in $\mathbf{c}Struc_{[P]}(X)$ as a section (up to homotopy) $\sigma$

$$\array{ && \mathbf{B}G \\ & {}^{\sigma}\nearrow& \downarrow^{\mathbf{c}} \\ X &\stackrel{g}{\to}& \mathbf{B}K }$$

where we think of $\mathbf{c}$ as being the universal twisting $\infty$-bundle and where $g : X \to \mathbf{B}K$ is a morphism presenting $P$.

The following definition looks at a differential refinement of this situation.

Definition

For $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ a characteristic map in $\mathbf{H}$ and $\hat {\mathbf{c}} : \mathbf{B}G_{\mathrm{conn}} \to \mathbf{B}^n U(1)_{\mathrm{conn}}$ its differential refinement, sending connections on ∞-bundles to circle n-bundles with connection (see ∞-Chern-Weil homomorphism, we may think of this also as an extended Lagrangian for a higher gauge theory).

We write $\hat {\mathbf{c}}\mathrm{Struc}_{\mathrm{tw}}(X)$ for the corresponding twisted cohomology,

$$\array{ \hat {\mathbf{c}}Struc_{tw}(X) &\stackrel{tw}{\to}& H^{n+1}_{diff}(X) \\ {}^{\mathllap{\chi}}\downarrow && \downarrow \\ \mathbf{H}(X, \mathbf{B}G_{conn}) & \stackrel{\hat \mathbf{c}}{\to} & \mathbf{H}(X, \mathbf{B}^n U(1)_{conn}) } \,.$$

Examples

Twisted differential $\mathbf{c}$-structures appear in various guises in the background gauge fields of string theory application.

• reduction of structure groups

  • orthogonal structure / Riemannian metric; see the discussion at vielbein .

  • generalized complex geometry

  • exceptional generalized geometry

• lift of structure groups

• Higher differential spin structures

  • twisted spin structure, differential spin structure

  • twisted spin^c-structure?

  • twisted differential string structure

  • supergravity C-field

  • twisted differential fivebrane structure

  • twisted Wu structure

• p1-structure

  2-framing

• w4-structure

  w4-orientation of EO(2)-theory

• differential T-duality

Related concepts

• reduction and lift of structure groups

• field (physics)

References

The notion was introduced in

• Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted Differential String and Fivebrane Structures

and expanded on in

• Domenico Fiorenza, Urs Schreiber, Jim Stasheff, Cech cocycles for differential characteristic classes

An exposition is in

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, A higher stacky perspective on Chern-Simons theory

Lecture notes include

• Urs Schreiber, twisted smooth cohomology in string theory, Lectures at the ESI-institute (June 2012)

A general account is in section 5.2 of

• Urs Schreiber, differential cohomology in a cohesive topos

In

• Daniel Freed, Constantin Teleman, Relative quantum field theory (arXiv:1212.1692)

it is proposed to call such twisted structures “relative fields”.