# nLab twisted differential c-structure

### Context

#### Differential cohomology

differential cohomology

## Application to gauge theory

#### $\infty$-Chern-Weil theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

# Contents

## 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 $c$ in Smooth∞Grpd there may exist further refinements $\stackrel{^}{c}$ 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 $H$ be a cohesive (∞,1)-topos, usually $H=$ Smooth∞Grpd or SynthDiff∞Grpd or the like.

Let $K,G$ be ∞-group objects in $H$ and let

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

be a morphism of their delooping objects / moduli stacks.

###### Definition

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

$c{\mathrm{Struc}}_{\left[P\right]}\left(X\right):=H\left(X,BG\right){×}_{H\left(X,BK\right)}\left\{P\right\}\phantom{\rule{thinmathspace}{0ex}},$\mathbf{c}Struc_{[P]}(X) := \mathbf{H}(X, \mathbf{B}G) \times_{\mathbf{H}(X, \mathbf{B}K)} \{P\} \,,

hence the (∞,1)-pullback

$\begin{array}{ccc}c{\mathrm{Struc}}_{\left[P\right]}\left(X\right)& \to & *\\ {↓}^{P}& & ↓\\ H\left(X,BG\right)& \stackrel{H\left(X,c\right)}{\to }& H\left(X,BK\right)\end{array}$\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 $c$);

• the $\infty$-groupoid of $\left[P\right]$-twisted $c$-structures.

###### Remark

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

$\begin{array}{ccc}& & BG\\ & {}^{\sigma }↗& {↓}^{c}\\ X& \stackrel{g}{\to }& BK\end{array}$\array{ && \mathbf{B}G \\ & {}^{\sigma}\nearrow& \downarrow^{\mathbf{c}} \\ X &\stackrel{g}{\to}& \mathbf{B}K }

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

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

###### Definition

For $c:BG\to {B}^{n}U\left(1\right)$ a characteristic map in $H$ and $\stackrel{^}{c}:B{G}_{\mathrm{conn}}\to {B}^{n}U\left(1{\right)}_{\mathrm{conn}}$ 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 $\stackrel{^}{c}{\mathrm{Struc}}_{\mathrm{tw}}\left(X\right)$ for the corresponding twisted cohomology,

$\begin{array}{ccc}\stackrel{^}{c}{\mathrm{Struc}}_{\mathrm{tw}}\left(X\right)& \stackrel{\mathrm{tw}}{\to }& {H}_{\mathrm{diff}}^{n+1}\left(X\right)\\ {}^{\chi }↓& & ↓\\ H\left(X,B{G}_{\mathrm{conn}}\right)& \stackrel{\stackrel{^}{c}}{\to }& H\left(X,{B}^{n}U\left(1{\right)}_{\mathrm{conn}}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $c$-structures appear in various guises in the background gauge fields of string theory application.

## References

The notion was introduced in

and expanded on in

An exposition is in

Lecture notes include

A general account is in section 4.2 of

In

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

Revised on January 8, 2013 06:25:36 by Urs Schreiber (82.113.99.249)