Elliptic Chern Characters and Elliptic Atiyah--Witten Formula
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912848930668544 |
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| author | Dai, Geyang Han, Fei |
| author_facet | Dai, Geyang Han, Fei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18126 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Elliptic Chern Characters and Elliptic Atiyah--Witten Formula Dai, Geyang Han, Fei Differential Geometry Mathematical Physics Algebraic Topology 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. |
| title | Elliptic Chern Characters and Elliptic Atiyah--Witten Formula |
| topic | Differential Geometry Mathematical Physics Algebraic Topology |
| url | https://arxiv.org/abs/2601.18126 |