One theme of some of my recent work is generalizing results from the intersection of differential geometry and algebraic topology, which traditionally had some abelian-ness to the setup, to a non-abelian context. This theme is found in (II) and (III) below. The second theme is to generate higher cohomological invariants (i.e. higher chern characters) by sheafifying classical invariants. This theme is found in (I) below.
(i) higher chern character data via sheafification
A simplicial presheaf is defined on a site (such as smooth or complex manifolds), whose (hyper-)sheafification has as its vertices something analogous to bundles with connection (such as holomorphic bundles with connection, coherent sheaves, gerbes, etc) and as its edges the appropriate equivalences between them. Then we take a classic map (such as the Chern character) from the simplicial presheaf above to the appropriate simplicial presheaf of differential forms (such as Dold-Kan of the Čech de Rham complex) and sheafify it. The result is that on vertices we recover the classical invariant but we obtain maps of higher simplices which yield higher invariants (such as Chern-Simons theory for a path of connections).
The papers [GMTZ0] , [GMTZ1], [GMTZ2], and [GMTZ3] below are part of this work.
(II) modeling bundle valued forms on mapping spaces
In joint work [GR] with C. Redden, building off of my PhD Thesis [M], we aim to model bundle valued forms on the loop space with some analog of the Hochschild complex. We begin by modeling bundle valued forms on the free path space with an analog of the two-sided bar construction on differential forms. There are two main generalizations to overcome: the first is that our values take place in a non-commutative ring and the second is that we allow for our bundle to have a non-flat connection and so we do not have a cochain complex of forms. In the paper [GR] below, we overcome both of these generalizations for forms on the path space with values in the endormorphism bundle with connection associated to a vector bundle with connection. We are now working on extending this initial success to the loop space, as well as to higher dimensional mapping spaces (i.e. squares, spheres, and tori).
(III) higher holonomy and its derivative
Given a non-abelian gerbe with connection (given by cocycle data), one can define a notion of 2-holonomy which depends on some choices (namely, which open sets in your cover a square is mapped into). Locally, the derivative of this 2-holonomy has boundary terms for each open set, intersections of these open sets, and triple intersections. However, in [G] we see that much of these terms cancel and we get a formula for the derivative which only has terms on the boundary of the square no matter how we choose to consider the square as landing in a particular “grid” of open sets. One future direction for this project is to consider higher-gerbes and holonomy and generalize the computation and statement. Another future direction is to construct an equivariant extension of 2-holonomy as an invariant analogous to Bismut’s equivariant extension of the Chern character for a vector bundle with connection.
[G] C. Glass, “The derivative of global surface-holonomy for a non-abelian gerbe”, Differential Geometry and its Applications. 75, (2021). DOI 10.1016/j.difgeo.2021.101737.
[GMTZ0] C. Glass, M. Miller, T. Tradler, and M. Zeinalian (2021). “The Hodge Chern character of holomorphic connections as a map of simplicial presheaves“. arXiv:1905.07674 [math.AT]. [accepted by Algebraic and Geometric Topology; forthcoming]
Papers in progress
[GR] C. Glass, C. Redden. “Modeling Bundle-Valued Forms on the Path Space with a Curved Iterated Integral”. [submitted to Journal of Homotopy and Related Structures]
[GMTZ1] C. Glass, M. Miller, T. Tradler, and M. Zeinalian. “Smooth Chern Simons Forms as a Map of Presheaves”.
[GMTZ2] C. Glass, M. Miller, T. Tradler, and M. Zeinalian. “A Chern Character for Infinity Vector Bundles”.
[GMTZ3] C. Glass, M. Miller, T. Tradler, and M. Zeinalian. “A Chern Character for Non-Abelian Gerbes“.
[M] C. Glass (Miller), “On the derivative of 2-holonomy for a non-abelian gerbe.”, Phd Thesis. CUNY Graduate Center.