# nLab gerbe (in differential geometry)

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

#### Differential cohomology

differential cohomology

## Application to gauge theory

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

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

## Theorems

This is a sub-entry for gerbe.

For related entries see

# Contents

## Idea

Gerbes give a nice way to group together bundle data on a smooth manifold, but gerbes also naturally define degree two cohomology. Thus the idea of using gerbes in differential geometry is to have a nice language that relates geometric concepts such as connections and curvature to cohomological classifications.

In addition, one can use the analogies above that are made precise with gerbes to define other new concepts such as 3-curvature and “local” fiberwise connections.

## Definitions

• Let $X$ be a smooth manifold. A Dixmier-Douady sheaf of groupoids over $X$ is a $\underline{\mathbb{C}}_X^*$- gerbe on $X$ where $\underline{\mathbb{C}}_X^*$ is the sheaf of smooth $\mathbb{C}^*$-valued functions (not to be confused with the constant sheaf $\mathbb{C}_X^*$).

• We define $\mathbb{Z}(1)$ to be the term in the exponential sequence on $X$: $0\to \mathbb{Z}(1)\to \underline{\mathbb{C}}_X \to \underline{\mathbb{C}}_X^*\to 0$.

## Preliminaries

Taking the associated sequence in cohomology to the exponential sequence gives us an isomorphism $H^2(X, \underline{\mathbb{C}}_X^*)\overset\sim\to H^3(X, \mathbb{Z}(1))$.

We have a canonical isomorphism between the group of equivalence classes of Dixmier-Douady sheaves of groupoids over $X$ (basically by the definition of $\mathcal{A}$-gerbe) and $H^2(X, \underline{\mathbb{C}}_X^*)\simeq H^3(X, \mathbb{Z}(1))$.

## Geometric Interpretation of $H^3(X, \mathbb{Z}(1))$

Idea: Classes in $H^3(X, \mathbb{Z}(1))$ correspond to principal $G$- bundles over $X$ where $G$ is the projective linear group of a separable Hilbert space, namely $C^\infty (\mathbb{T})$.

Matt: Actually, a slight issue has arisen. Most of the things I thought would go here actually already appear in other places even though they aren’t grouped as coming from the same idea.

For instance, bundle gerbe contains the geometric interpretation of $H^3(X, \mathbb{Z}(1))$. Also, 3-curvature and fiber-wise connections occur at connection on a bundle gerbe. Although I think there is still a lot to say, I’m not convinced that “gerbe (in differential geometry)” is necessary anymore…

## References

Further references are given in the other entries on gerbes.