Elliptic Chern Characters and Elliptic Atiyah--Witten Formula

TL;DR

This work develops a coherent elliptic refinement of loop-space index theory by constructing elliptic Chern characters on the free loop space $LX$ and an elliptic Bismut--Chern character on the double loop space $L^2X$, built from lifting gerbes and positive-energy loop-group representations. It extends the classical Atiyah--Witten and Bismut--Chern frameworks to an elliptic setting, introducing the elliptic holonomy and an elliptic Atiyah--Witten formula at level one for $G= ext{Spin}(2n)$, and linking these to Chern--Simons theory via 2-transgression and moduli of holomorphic bundles on elliptic curves. The constructions reveal deep connections between loop-space geometry, modular forms (theta and eta functions), and genus-one conformal blocks, providing a unifying perspective on elliptic cohomology and conformal blocks. The paper also develops a broader cohomological framework (exotic twisted cohomologies) to accommodate the infinite-dimensional, twisted structures arising from positive-energy representations. Together, these results illuminate the elliptic analogues of holonomy, Pfaffians, and refined characteristic forms, with potential implications for string structures and the Stolz--Teichner program. The analytic and algebro-geometric facets underscore a profound link between double-loop geometry, Chern--Simons quantization, and elliptic genera in a coherent, higher-genus setting.

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 $τ$-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.

Theorems & Definitions (94)

• Remark 1.1
• Theorem 1.2: Elliptic Atiyah--Witten formula
• Remark 1.3
• Corollary 1.4
• Theorem 1.5: Preparation
• Theorem 1.6: Looijenga1976RootSA
• Remark 1.7
• Definition 2.1
• Lemma 2.2: pressley_loop_1988,Coquereaux1989StringSO
• Corollary 2.3