Equivariant elliptic cohomology via loop spaces
Abstract
We construct a cohomology theory HA(−) canonically attached to any 1-dimensional commutative group scheme A over a connective E∞-ring. The additive group in characteristic zero gives 2-periodic de Rham cohomology, the multiplicative group gives topological periodic cyclic homology, and an elliptic curve gives elliptic cohomology. The construction extends to the G-equivariant setting for an algebraic group G, yielding in the elliptic case a theory of equivariant elliptic cohomology on algebraic varieties with G-action that takes values in sheaves on the stack of G-bundles. Over the complex numbers, and for G connected reductive, this lifts Grojnowski's theory from the moduli space of G-bundles to the stack. The key player is the category of perfect complexes on A with 0-dimensional support, equipped with the convolution product induced by the group law of A. It plays the role of a categorified circle and is used to define a loop space LAG(X), whose structure sheaf gives rise to HAG(X) after periodization. Sheaves on LAG(X) equivariant under loop rotation provide a category of coefficients for HAG(−) in the same way that D-modules provide coefficients for de Rham cohomology. At the point X = *, the theory also gives rise to an E∞-ring spectrum (or sheaf thereof) KA, which recovers complex K-theory for A = Gm, and the sheaf 𝒪top of topological modular forms for the universal elliptic curve.
Related
- On equivariant affineness in elliptic cohomology
- The ℤ/2-equivariant Seiberg--Witten Floer cohomology of the $(2, 1)$-cable is an unknot detector
- Local coefficients for genuine equivariant cohomology I
- Formal Neighborhood Equivariant Cohomology and Kirwan Surjectivity at Singular Values
- Bounded cohomology property on Jacobian elliptic surfaces with simplicial Mori cones
BibTeX
@misc{bouaziz2026equivariant,
title = {{Equivariant elliptic cohomology via loop spaces}},
author = {Emile Bouaziz and Adeel A. Khan},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05707},
eprint = {2611.05707},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05707},
}
RIS
TY - UNPB
TI - Equivariant elliptic cohomology via loop spaces
AU - Emile Bouaziz
AU - Adeel A. Khan
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05707
UR - https://arxiv.org/abs/2611.05707
AB - We construct a cohomology theory H_A(−) canonically attached to any 1-dimensional commutative group scheme A over a connective E_∞-ring. The additive group in characteristic zero gives 2-periodic de Rham cohomology, the multiplicative group gives topological periodic cyclic homology, and an elliptic curve gives elliptic cohomology. The construction extends to the G-equivariant setting for an algebraic group G, yielding in the elliptic case a theory of equivariant elliptic cohomology on algebraic varieties with G-action that takes values in sheaves on the stack of G-bundles. Over the complex numbers, and for G connected reductive, this lifts Grojnowski's theory from the moduli space of G-bundles to the stack. The key player is the category of perfect complexes on A with 0-dimensional support, equipped with the convolution product induced by the group law of A. It plays the role of a categorified circle and is used to define a loop space L_A^G(X), whose structure sheaf gives rise to H_A^G(X) after periodization. Sheaves on L_A^G(X) equivariant under loop rotation provide a category of coefficients for H_A^G(−) in the same way that D-modules provide coefficients for de Rham cohomology. At the point X = *, the theory also gives rise to an E_∞-ring spectrum (or sheaf thereof) K_A, which recovers complex K-theory for A = G_m, and the sheaf 𝒪^top of topological modular forms for the universal elliptic curve.
ER -