Infinite-Dimensional Representations of 2-Groups
John C. Baez, Aristide Baratin, Laurent Freidel, Derek K. Wise
TL;DR
This work advances the representation theory of 2-groups by embedding representations into the infinite-dimensional setting of measurable categories, yielding a geometric, measure-theoretic picture of skeletal 2-group representations. It builds the 2-category ${f{Meas}}$ as the target for representations, detailing measurable fields, direct integrals, and matrix functors, and then characterizes representations, intertwiners, and 2-intertwiners in terms of equivariant bundles and cocycle data. Key results include a complete classification of indecomposable and irretractable representations via orbit and stabilizer data, and a Schur-type result for 2-intertwiners. The framework connects higher gauge theory concepts to Mackey-style induced representations and lays groundwork toward separable 2-Hilbert spaces and 2-Hilbertian structure within a von Neumann algebra-inspired setting, with anticipated physical applications in spin foams and quantum gravity.
Abstract
A "2-group" is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2-groups have representations on "2-vector spaces", which are categories analogous to vector spaces. Unfortunately, Lie 2-groups typically have few representations on the finite-dimensional 2-vector spaces introduced by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain infinite-dimensional 2-vector spaces called "measurable categories" (since they are closely related to measurable fields of Hilbert spaces), and used these to study infinite-dimensional representations of certain Lie 2-groups. Here we continue this work. We begin with a detailed study of measurable categories. Then we give a geometrical description of the measurable representations, intertwiners and 2-intertwiners for any skeletal measurable 2-group. We study tensor products and direct sums for representations, and various concepts of subrepresentation. We describe direct sums of intertwiners, and sub-intertwiners - features not seen in ordinary group representation theory. We study irreducible and indecomposable representations and intertwiners. We also study "irretractable" representations - another feature not seen in ordinary group representation theory. Finally, we argue that measurable categories equipped with some extra structure deserve to be considered "separable 2-Hilbert spaces", and compare this idea to a tentative definition of 2-Hilbert spaces as representation categories of commutative von Neumann algebras.