nLab
twisted differential c-structure

Context

Differential cohomology

\infty-Chern-Weil theory

Contents

Idea

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

If cc is refined to a characteristic class c\mathbf{c} in Smooth∞Grpd there may exist further refinements c^\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 H\mathbf{H} be a cohesive (∞,1)-topos, usually H=\mathbf{H} = Smooth∞Grpd or SynthDiff∞Grpd or the like.

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

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

be a morphism of their delooping objects / moduli stacks.

Definition

For XHX \in \mathbf{H} any object and PXP \to X an KK-principal ∞-bundle over XX, the ∞-groupoid

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

hence the (∞,1)-pullback

cStruc [P](X) * P H(X,BG) H(X,c) H(X,BK) \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 KK-structures on PP (with respect to the given c\mathbf{c});

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

Remark

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

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

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

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

Definition

For c:BGB nU(1)\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1) a characteristic map in H\mathbf{H} and c^:BG connB nU(1) conn\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 c^Struc tw(X)\hat {\mathbf{c}}\mathrm{Struc}_{\mathrm{tw}}(X) for the corresponding twisted cohomology,

c^Struc tw(X) tw H diff n+1(X) χ H(X,BG conn) c^ H(X,B nU(1) conn). \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\mathbf{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)