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Mathematics > Algebraic Topology

arXiv:1805.04146 (math)
[Submitted on 10 May 2018 (v1), last revised 24 Aug 2020 (this version, v2)]

Title:A model for complex analytic equivariant elliptic cohomology from quantum field theory

Authors:Daniel Berwick-Evans, Arnav Tripathy
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Abstract:We construct a global geometric model for complex analytic equivariant elliptic cohomology for all compact Lie groups. Cocycles are specified by functions on the space of fields of the two-dimensional sigma model with background gauge fields and $\mathcal{N} = (0, 1)$ supersymmetry. We also consider a theory of free fermions valued in a representation whose partition function is a section of a determinant line bundle. We identify this section with a cocycle representative of the (twisted) equivariant elliptic Euler class of the representation. Finally, we show that the moduli stack of $U(1)$-gauge fields carries a multiplication compatible with the complex analytic group structure on the universal (dual) elliptic curve, with the Euler class providing a choice of coordinate. This provides a physical manifestation of the elliptic group law central to the homotopy-theoretic construction of elliptic cohomology.
Comments: Significant updates informed by the de Rham model in 1908.02868
Subjects: Algebraic Topology (math.AT); High Energy Physics - Theory (hep-th); Algebraic Geometry (math.AG); Differential Geometry (math.DG); Representation Theory (math.RT)
Cite as: arXiv:1805.04146 [math.AT]
  (or arXiv:1805.04146v2 [math.AT] for this version)
  https://doi.org/10.48550/arXiv.1805.04146

Submission history

From: Daniel Berwick-Evans [view email]
[v1] Thu, 10 May 2018 19:36:22 UTC (115 KB)
[v2] Mon, 24 Aug 2020 14:31:44 UTC (150 KB)
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