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A construction of complex analytic elliptic cohomology from double free loop spaces

Part of: Lie groups Homology and cohomology theories

Published online by Cambridge University Press:  26 July 2021

Matthew Spong
Matthew Spong*
Affiliation:
School of Mathematics and Physics, The University of Queensland, St Lucia, QLD4067, Australiamatt.spong@gmail.com
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Abstract

We construct a complex analytic version of an equivariant cohomology theory which appeared in a paper of Rezk, and which is roughly modelled on the Borel-equivariant cohomology of the double free loop space. The construction is defined on finite, torus-equivariant CW complexes and takes values in coherent holomorphic sheaves over the moduli stack of complex elliptic curves. Our methods involve an inverse limit construction over all finite-dimensional subcomplexes of the double free loop space, following an analogous construction of Kitchloo for single free loop spaces. We show that, for any given complex elliptic curve $\mathcal {C}$, the fiber of our construction over $\mathcal {C}$ is isomorphic to Grojnowski's equivariant elliptic cohomology theory associated to $\mathcal {C}$.

Keywords

elliptic cohomologydouble loop spacesdouble loop groups

MSC classification

Primary: 55N34: Elliptic cohomology
Secondary: 22E67: Loop groups and related constructions, group-theoretic treatment
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 8 , August 2021 , pp. 1853 - 1897
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Copyright
© 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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References

Ando, M., Power operations in elliptic cohomology and representations of loop groups, Trans. Amer. Math. Soc. 352 (2000), 5619–5666.10.1090/S0002-9947-00-02412-0CrossRefGoogle Scholar
Atiyah, M. and Bott, R., The moment map and equivariant cohomology, Topology 23 (1984), 1–28.10.1016/0040-9383(84)90021-1CrossRefGoogle Scholar
Berwick-Evans, D. and Tripathy, A., A de Rham model for complex analytic equivariant elliptic cohomology, Adv. Math. 380 (2021), 107575.10.1016/j.aim.2021.107575CrossRefGoogle Scholar
Chen, H., Torus equivariant elliptic cohomology and sigma orientations, PhD thesis (2010), http://hdl.handle.net/2142/16822.Google Scholar
Etingof, P. I. and Frenkel, I. B., Central extensions of current groups in two dimensions, Commun. Math. Phys. 165 (1994), 429–444.10.1007/BF02099419CrossRefGoogle Scholar
Ganter, N., The elliptic Weyl character formula, Compos. Math. 150 (2014), 1196–1234.10.1112/S0010437X1300777XCrossRefGoogle Scholar
Ginzburg, V., Kapranov, M. and Vasserot, E., Elliptic algebras and equivariant elliptic cohomology, Preprint (1995), arXiv:q-alg/9505012.Google Scholar
Grojnowski, I., Delocalized equivariant elliptic cohomology, in Elliptic cohomology: geometry, applications, and higher chromatic analogues, London Mathemetical Lecture Note Series, vol. 342 (Cambridge University Press, Cambridge, 2007), 114–121.10.1017/CBO9780511721489.007CrossRefGoogle Scholar
Kitchloo, N., Quantization of the modular functor and equivariant elliptic cohomology, in Homotopy theory: tools and applications, Contemporary Mathematics, vol. 729 (American Mathematical Society, Providence, RI, 2019), 157–177.10.1090/conm/729/14694CrossRefGoogle Scholar
Lewis, L. G. Jr., May, J. P., Steinberger, M. and McClure, J. E., Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213 (Springer, Berlin, 1986).10.1007/BFb0075778CrossRefGoogle Scholar
Rezk, C., Looijenga line bundles in complex analytic elliptic cohomology, Tunis. J. Math. 2 (2020), 1–42.10.2140/tunis.2020.2.1CrossRefGoogle Scholar
Rosu, I., Equivariant elliptic cohomology and rigidity, Amer. J. Math. 123 (2001), 647–677.10.1353/ajm.2001.0027CrossRefGoogle Scholar
Rosu, I. and Knutson, A., Equivariant K-theory and equivariant cohomology, Math. Z. 243 (2003), 423–448.10.1007/s00209-002-0447-1CrossRefGoogle Scholar
Spong, M., Comparison theorems for torus-equivariant elliptic cohomology theories, PhD thesis (2019), http://hdl.handle.net/11343/225560.Google Scholar
The Stacks Project Authors, Stacks Project, https://stacks.math.columbia.edu.Google Scholar