The derivative of global surface-holonomy for a non-abelian gerbe

doi:10.1016/j.difgeo.2021.101737

Differential Geometry and its Applications

Volume 75, April 2021, 101737

https://doi.org/10.1016/j.difgeo.2021.101737 Get rights and content

Abstract

Starting with a non-abelian gerbe represented by a non-abelian differential cocycle, with values in a given crossed-module, this paper explicitly calculates a formula for the derivative of the associated surface holonomy of squares mapped into the base manifold; with spheres later considered as a special case. While the definitions in this paper used for gerbes, their connections, and the induced holonomy will initially be simplicial, translations into a cubical setting will be provided to aide in explicit coordinate-based calculations. While there are many previously published results on the properties of these non-abelian gerbes, including some calculations of the derivative over a single open set, this paper endeavors to take these local calculations and glue them together across multiple open sets in order to obtain a single expression for the change in surface holonomy with respect to a one-parameter family of squares.

Introduction

In [18], an equivariantly closed differential form is associated to an abelian gerbe with connection by considering the derivative of the induced 2-holonomy. Originating as a first step in generalizing their work, this paper focuses on differentiating (Theorem 4.1) the global 2-holonomy (Definition 3.1) for a non-abelian $G$ -gerbe (Definition 2.24).

Following Schreiber and Waldorf ([13], [14], [15], and [16]), the local cocycle description for a non-abelian gerbe with connection on a smooth manifold is reviewed and adopted. This paper uses their local transport data for bigons implicitly in Section 2.4.2, but translates that data into local transport data for squares as the process of differentiation became more manageable and organized in a cubical setting.

The method for glueing this local data together to provide a global definition of Hol (Definition 3.1) in a cubical setting was borrowed from Martins and Picken (specifically, Fig. 3 in [11]). Their papers proved many properties for a global holonomy on the group-level which can be found in Section 3.2, where these properties are reviewed in addition to some other relevant observations. This paper adds to those references by providing for a global formula, comprised of local data, for the derivative of global surface holonomy.

In Section 4, the main theorem of this paper, Theorem 4.1, is stated and proven, which essentially reads $d (H o l) = H o l \cdot \int_{S q} H (modulo terms on the boundary of S q),$ where $\int_{S q} H$ represents fiber-integration of the 3-curvature terms for the given non-abelian gerbe through the interior of the square. The proof of this theorem, given in Section 4.4, amounts to considering a 1-parameter family of squares, considering the associated arrangement of cubes from the local data, and organizing the terms in the derivative accordingly.

Finally, in Sections 5.1 and 5.2, some examples are offered where Theorem 4.1 has an even cleaner representation. In the case where the surfaces which are integrated over are spheres, $d (H o l)$ has only a boundary term at the base point. In the case where the gerbe is abelian, the well known situation where there are no terms for $d (H o l)$ on the boundary is reproduced.

The hope after this paper is to continue the work of finding a non-abelian analogue of the work done in [18], via a subsequent paper which will use the result of Theorem 4.1, and the appropriate cohomology theory, to find some equivariantly closed element representing a non-abelian gerbe with connection. In a separate project, the goal is to extend this paper's derivative for 2-dimensional holonomy (landing in crossed-modules) to the derivative of 3-dimensional holonomy (landing in the appropriate version of a 2-crossed module). Furthermore, some of the geometric comments regarding the “globalness” of this 2-holonomy are planned to be formalized in current joint work with Micah Miller, Thomas Tradler, and Mahmoud Zeinalian.

The author would like to thank Thomas Tradler for many helpful conversations about this paper.

Section snippets

Conventions, notation, and setup

In order to arrive at a definition for global 2-holonomy, Definition 3.1, it is necessary to introduce some preliminary definitions and conventions.

Glueing together local 2-holonomy

In this section, a definition is provided for 2-holonomy of a square mapped into M, $Σ \in N \subset M^{S q}$ (Definition 2.1), which lands in multiple open sets $U_{i} \subset M$ . Recall that in Section 2.1, following the notation in [18], an open cover of $M^{S q}$ is given by sets $N_{I} = {Σ \in M^{S q} | for each (p, q) \in I, Σ (S q_{(p, q)}) \subset U_{i_{(p, q)}}} .$ While the notation and inspiration for this particular paper is attributed to [18], the broader scope of the ideas in this section (in particular the non-abelian approach to glueing squares) can be

$d (H o l)$ for squares

The focus of this section, is the proof of Theorem 4.1, the main result of this paper, which says that the deRham differential applied to global 2-holonomy amounts to replacing one $B_{i}$ in any summand with one $H_{i}$ ; in addition to some terms associated to the boundary of Σ.

Special cases

The results of this paper are now briefly stated as three special cases:

1.
The surface holonomy of spheres, $M^{S^{2}} \to H$ .
2.
The surface holonomy, $M^{S q} \to H$ , which uses a crossed module $(H \overset{t}{\to} G, α)$ , whose α-action is given by inner-automorphisms.
3.
The surface holonomy for abelian gerbes.

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The derivative of global surface-holonomy for a non-abelian gerbe

Abstract

Introduction

Section snippets

Conventions, notation, and setup

Glueing together local 2-holonomy

d(Hol) for squares

Special cases

Adv. Math.

Convenient categories of smooth spaces

Trans. Am. Math. Soc.

Higher algebra - 2 groups

Theory Appl. Categ.

Higher gauge theory

Groupoids and crossed objects in algebraic topology

Homol. Homotopy Appl.

On the connection between the second relative homotopy groups of some related spaces

Proc. Lond. Math. Soc.

Double groupoids and crossed modules

Cah. Topol. Geom. Differ.

Iterated integrals of differential forms and loop space homology

Ann. Math. (2)

Higher gauge theory – differential versus integral formulation

J. Math. Phys.

$d (H o l)$ for squares