Elliptic Chern Characters and Elliptic Atiyah--Witten Formula

TL;DR

This paper develops elliptic Chern characters and an elliptic Atiyah--Witten formula for principal G-bundles with connections, linking loop group representations, gerbe modules, and moduli of G-bundles over elliptic curves.

Contribution

It introduces elliptic Chern characters and refines them to the elliptic Bismut--Chern character, extending the Atiyah--Witten formula to double loop spaces and connecting to conformal blocks.

Findings

Construction of elliptic Chern characters on loop spaces.

Establishment of an elliptic Atiyah--Witten formula.

Identification of elliptic holonomies with positive-energy representations.

Abstract

Let $G$ be a compact, connected, and simply connected Lie group. A principal $G$-bundle over a manifold $X$, equipped with a connection, together with a positive-energy representation of the loop group $LG$, gives rise to a circle-equivariant gerbe module on the free loop space $LX$. From this data we construct the elliptic Chern character on $LX$, and a refinement, the elliptic Bismut--Chern character, on the double loop space $L^2X$. Generalizing the classical Atiyah--Witten formula from the free loop space $LX$ to the double loop space $L^2X$, we establish an elliptic Atiyah--Witten formula. The elliptic holonomy on $L^2X$ is defined by $\tau$-deformed equivariant twisted parallel transport on $LX$. We show that the four Pfaffian sections, corresponding to the four spin structures on an elliptic curve, are identified with the four elliptic holonomies arising from the four virtual level-one positive-energy representations when $G=\mathrm{Spin}(2n)$. These constructions are intimately connected to the moduli of $G_{\mathbb{C}}$-bundles over elliptic curves and conformal blocks in the context of Chern--Simons gauge theory.