Scott O. Wilson

Equivariant Holonomy for Bundles and Abelian Gerbes

This paper generalizes Bismut’s equivariant Chern character to the setting of abelian gerbes. In particular, associated to an abelian gerbe with connection, an equivariantly closed differential form is constructed on the space of maps of a torus into the manifold. These constructions are made explicit using a new local version of the higher Hochschild complex, resulting in differential forms given by iterated integrals. Connections to two dimensional topological field theories are indicated. Similarly, this local higher Hochschild complex is used to calculate the 2-holonomy of an abelian gerbe along any closed oriented surface, as well as the derivative of 2-holonomy, which in the case of a torus fits into a sequence of higher holonomies and their differentials.

Differential forms, fluids, and finite models

By rewriting the Navier-Stokes equation in terms of differential forms we give a formulation which is abstracted and reproduced in a finite dimensional setting. We give two examples of these finite models and, in the latter case, prove some approximation results. Some useful properties of these finite models are derived.

Addendum to “Conformal cochains”

The purpose of this note is to provide further details for a key step in the proof of the main result of Theorem 7.2 of [W], where the author states that the described sequence of combinatorial period matrices converges to the Riemann period matrix of the surface. The limit in equation (3), line 11, does not follow from Lemma 7.1 and analysis alone, since the convergence in Lemma 7.1 is with respect to the L-norm, and integration is not a bounded operator on smooth forms with respect to this norm. Nevertheless, the limit in (3) holds since we are considering smooth differential forms that are closed. We state this as a lemma below.