Looijenga line bundles in complex analytic elliptic cohomology

作者:Rezk, Charles 刊名:Tunisian Journal of Mathematics 上传者:邓爱清

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Tunisian Journal of Mathematics an international publication organized by the Tunisian Mathematical Society msp Looijenga line bundles in complex analytic elliptic cohomology Charles Rezk 2020 vol. 2 no. 1 msp TUNISIAN JOURNAL OF MATHEMATICS Vol. 2, No. 1, 2020 dx.doi.org/10.2140/tunis.2020.2.1 Looijenga line bundles in complex analytic elliptic cohomology Charles Rezk We present a calculation that shows how the moduli of complex analytic elliptic curves arises naturally from the Borel cohomology of an extended moduli space of U(1)-bundles on a torus. Furthermore, we show how the analogous calcula-tion, applied to a moduli space of principal bundles for a K(Z,2) central exten-sion of U(1)d, gives rise to Looijenga line bundles. We then speculate on the relation of these calculations to the construction of complex analytic equivariant elliptic cohomology. 1. Introduction In this note, we describe some aspects of how complex analytic elliptic curves arise naturally from the cohomology of certain spaces which parametrize principal bundles on orientable genus-1 surfaces. This suggests how elliptic cohomology emerges from certain derived complex analytic spaces associated to dimensional reduction applied to 2-dimensional field theories. 1.1. Complex analytic elliptic cohomology. Complex analytic equivariant elliptic cohomology was first defined by Grojnowski [2007].1 In its most basic formulation, given • a compact connected abelian Lie group G (i.e., G ≈ U(1)d), with cocharacter lat

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