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Localization theorem for homological vector fields

Vera Serganova, Alexander Sherman

TL;DR

We develop a localization theorem for the cohomology $DS_Q\Bbbk[X]$ of a homological odd vector field $Q$ on a smooth affine supervariety $X$, under the assumption that $Q^2$ acts semisimply and that $Q$ vanishes along a smooth subvariety $Y$ with nondegenerate action on the conormal bundle. The main result identifies $DS_Q\Gamma(X,\mathcal{V})$ with $DS_Q\Gamma(Y,\mathcal{V}|_Y)$ for any $\mathfrak{q}$-equivariant vector bundle $\mathcal{V}$, via a local Koszul-type analysis near $Z(Q)$ and a suitable splitting at points of $Z(Q)$; this recovers simplified descriptions such as $DS_Q\Bbbk[X]=\Bbbk[Y]$ in key cases. The localization is then used to compute the odd-quotient groups $\widetilde{G_u}$ for classical and exceptional supergroups (e.g. $GL(m|n)$ and $Q(n)$), to study homogeneous and symmetric spaces, and to relate $DS_u$ to de Rham-type invariants in Appendix computations. Overall, the work provides a powerful, coordinate-free mechanism to reduce $DS_Q$-cohomology problems on supervarieties to tractable subschemes, with broad implications for representation theory of Lie superalgebras and related physical formalisms.

Abstract

We present a general theorem which computes the cohomology of a homological vector field on global sections of vector bundles over smooth affine supervarieties. The hypotheses and results have the clear flavor of a localization theorem.

Localization theorem for homological vector fields

TL;DR

We develop a localization theorem for the cohomology of a homological odd vector field on a smooth affine supervariety , under the assumption that acts semisimply and that vanishes along a smooth subvariety with nondegenerate action on the conormal bundle. The main result identifies with for any -equivariant vector bundle , via a local Koszul-type analysis near and a suitable splitting at points of ; this recovers simplified descriptions such as in key cases. The localization is then used to compute the odd-quotient groups for classical and exceptional supergroups (e.g. and ), to study homogeneous and symmetric spaces, and to relate to de Rham-type invariants in Appendix computations. Overall, the work provides a powerful, coordinate-free mechanism to reduce -cohomology problems on supervarieties to tractable subschemes, with broad implications for representation theory of Lie superalgebras and related physical formalisms.

Abstract

We present a general theorem which computes the cohomology of a homological vector field on global sections of vector bundles over smooth affine supervarieties. The hypotheses and results have the clear flavor of a localization theorem.
Paper Structure (26 sections, 27 theorems, 98 equations)

This paper contains 26 sections, 27 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Relation to physics
  3. Outlook for non-affine varieties
  4. Summary of sections
  5. Acknowledgments
  6. Supergeometric preliminaries
  7. Notation for supervarieties
  8. Coordinates
  9. Subvarieties
  10. q-supervarieties
  11. q-modules
  12. q-supervarieties
  13. Equivariant sheaves
  14. Main theorem
  15. Corollaries of Theorem \ref{['main_thm']}
  16. ...and 11 more sections

Key Result

Lemma 2.1

Suppose that $X=\operatorname{Spec} A$ is a smooth affine supervariety and $\xi_{1},\dots,\xi_{n}\in A_{\overline{1}}$ are global odd coordinates on $X$. Then the derivations $\partial_{\xi_{1}},\dots,\partial_{\xi_n}$ are well-defined, and if we set $A_0=\operatorname{ker}\partial_{\xi_1}\cap\cdots is an isomorphism.

Theorems & Definitions (65)

  • Remark 1.1
  • Lemma 2.1
  • proof
  • Definition 3.1: Duflo-Serganova functor
  • Lemma 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Remark 3.6
  • Lemma 3.7
  • ...and 55 more