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Elliptic loop spaces

Emile Bouaziz, Adeel A. Khan

TL;DR

This work constructs elliptic loop spaces $L_E(\mathcal{Y})$, an elliptic analogue of rational and trigonometric loop spaces, via a tensor-categorical framework built around the convolution category $\mathcal{H}_E$ of 0-dimensional sheaves on an elliptic curve $E$. It develops the machinery of stacks of tensor functors, Tannakian perspectives, and integral-domain tensor categories to realize $L_E(\mathcal{Y})$ as classifying tensor functors $\operatorname{QCoh}(\mathcal{Y})\to\mathcal{H}_E$, yielding an $E$-action and robust descent properties such as Mayer--Vietoris and K"unneth. The paper then studies tangent complexes, push-forwards, and the functor $\Theta$, establishing a differential-geometric backbone for elliptic loop spaces and giving explicit descriptions in the scheme case where $L_E(X)\simeq \mathbf{T}[-1]X$. It further defines equivariant elliptic Hodge cohomology $HH_E^G(X)$ from structure sheaves of elliptic loop spaces $L_E^G(X)$ and connects to moduli of semistable bundles on $E$ (e.g. $\operatorname Bun_r^{ss,0}(E)$) and to the elliptic Springer resolution, situating the construction within the landscape of elliptic cohomology and geometric representation theory. The framework aligns with and extends Grojnowski, Lurie, and GKV perspectives, and points toward a uniform, spectral elliptic cohomology theory for all groups via derived algebraic geometry.

Abstract

We introduce an elliptic avatar of loop spaces in derived algebraic geometry, completing the familiar trichotomoy of rational, trigonometric and elliptic objects. Heuristically, the elliptic loop space of $\mathcal{Y}$ is the stack of maps to $\mathcal{Y}$ from a certain exotic avatar $\mathcal{S}_{E}$ of the elliptic curve $E$, such that the category of quasi-coherent sheaves on $\mathcal{S}_{E}$ is the convolution category of zero-dimensionally supported coherent sheaves on $E$. For quotient stacks, the structure sheaf of the elliptic loop space gives rise to a theory of equivariant elliptic Hodge cohomology.

Elliptic loop spaces

TL;DR

This work constructs elliptic loop spaces , an elliptic analogue of rational and trigonometric loop spaces, via a tensor-categorical framework built around the convolution category of 0-dimensional sheaves on an elliptic curve . It develops the machinery of stacks of tensor functors, Tannakian perspectives, and integral-domain tensor categories to realize as classifying tensor functors , yielding an -action and robust descent properties such as Mayer--Vietoris and K"unneth. The paper then studies tangent complexes, push-forwards, and the functor , establishing a differential-geometric backbone for elliptic loop spaces and giving explicit descriptions in the scheme case where . It further defines equivariant elliptic Hodge cohomology from structure sheaves of elliptic loop spaces and connects to moduli of semistable bundles on (e.g. ) and to the elliptic Springer resolution, situating the construction within the landscape of elliptic cohomology and geometric representation theory. The framework aligns with and extends Grojnowski, Lurie, and GKV perspectives, and points toward a uniform, spectral elliptic cohomology theory for all groups via derived algebraic geometry.

Abstract

We introduce an elliptic avatar of loop spaces in derived algebraic geometry, completing the familiar trichotomoy of rational, trigonometric and elliptic objects. Heuristically, the elliptic loop space of is the stack of maps to from a certain exotic avatar of the elliptic curve , such that the category of quasi-coherent sheaves on is the convolution category of zero-dimensionally supported coherent sheaves on . For quotient stacks, the structure sheaf of the elliptic loop space gives rise to a theory of equivariant elliptic Hodge cohomology.

Paper Structure

This paper contains 54 sections, 29 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Rational and trigonometric loop spaces
  3. 1-categorical Cartier duality
  4. Elliptic loop spaces
  5. Formal syntax
  6. Construction
  7. Example: schemes
  8. Variation of the elliptic curve
  9. Example: classifying stacks
  10. Tensor categories
  11. Equivariant elliptic Hodge cohomology
  12. Other perspectives on elliptic cohomology
  13. Acknowledgments
  14. Stacks of tensor functors
  15. Recollections on derived geometry
  16. ...and 39 more sections

Key Result

Lemma 2.2

Given a diagram of tensor functors the induced square \begin{tikzcd} \Fun(C_{0}\otimes_{C_{01}}C_{1},D)\arrow{r} \arrow[swap]{d} & \Fun(C_{0},D) \arrow{d} \\ \Fun(C_{1},D)\arrow{r} & \Fun(C_{01},D) \end{tikzcd}is cartesian for every tensor category $D$.

Theorems & Definitions (96)

  • Example
  • Example
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • ...and 86 more