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Register a new userLooijenga line bundles in complex analytic elliptic cohomology
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Authors
Charles Rezk
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We present a calculation, which 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 calculation, applied to a moduli space of principal bundles for a $K(mathbb{Z},2)$ central extension of $U(1)^d$ give rise to Looijenga line bundles. We then speculate on the relation of these calculations to the construction of complex analytic equivariant elliptic cohomology.
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We construct a cocycle model for complex analytic equivariant elliptic cohomology that refines Grojnowskis theory when the group is connected and Devotos when the group is finite. We then construct Mathai--Quillen type cocycles for equivariant elliptic Euler and Thom classes, explaining how these are related to positive energy representations of loop groups. Finally, we show that these classes give a unique equivariant refinement of Hopkins theorem of the cube construction of the ${rm MString}$-orientation of elliptic cohomology.
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.
There is a standard method to calculate the cohomology of torus-invariant sheaves $L$ on a toric variety via the simplicial cohomology of associated subsets $V(L)$ of the space $N_{mathbb R}$ of 1-parameter subgroups of the torus. For a line bundle $L$ represented by a formal difference $Delta^+-Delta^-$ of polyhedra in the character space $M_{mathbb R}$, [ABKW18] contains a simpler formula for the cohomology of $L$, replacing $V(L)$ by the set-theoretic difference $Delta^- setminus Delta^+$. Here, we provide a short and direct proof of this formula.
In this paper, we study questions of Demailly and Matsumura on the asymptotic behavior of dimensions of cohomology groups for high tensor powers of (nef) pseudo-effective line bundles over non-necessarily projective algebraic manifolds. By generalizing Sius $partialoverline{partial}$-formula and Berndtssons eigenvalue estimate of $overline{partial}$-Laplacian and combining Bonaveros technique, we obtain the following result: given a holomorphic pseudo-effective line bundle $(L, h_L)$ on a compact Hermitian manifold $(X,omega)$, if $h_L$ is a singular metric with algebraic singularities, then $dim H^{q}(X,L^kotimes Eotimes mathcal{I}(h_L^{k}))leq Ck^{n-q}$ for $k$ large, with $E$ an arbitrary holomorphic vector bundle. As applications, we obtain partial solutions to the questions of Demailly and Matsumura.
We present a cocycle model for elliptic cohomology with complex coefficients in which methods from 2-dimensional quantum field theory can be used to rigorously construct cocycles. For example, quantizing a theory of vector bundle-valued fermions yields a cocycle representative of the elliptic Thom class. This constructs the complexified string orientation of elliptic cohomology, which determines a pushfoward for families of rational string manifolds. A second pushforward is constructed from quantizing a supersymmetric $sigma$-model. These two pushforwards agree, giving a precise physical interpretation for the elliptic index theorem with complex coefficients. This both refines and supplies further evidence for the long-conjectured relationship between elliptic cohomology and 2-dimensional quantum field theory. Analogous methods in supersymmetric mechanics recover path integral constructions of the Mathai--Quillen Thom form in complexified ${rm KO}$-theory and a cocycle representative of the $hat{A}$-class for a family of oriented manifolds.
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