twisted differential c-structure


Differential cohomology

\infty-Chern-Weil theory



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).


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.


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.


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.


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}) } \,.


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


The notion was introduced in

and expanded on in

An exposition is in

Lecture notes include

A general account is in section 5.2 of


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

Revised on October 28, 2014 09:08:45 by Urs Schreiber (