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Supersymmetric Deformations of Maximally Supersymmetric Gauge Theories

M. Movshev, A. Schwarz

TL;DR

This work develops a comprehensive homological-algebraic framework for classifying and constructing supersymmetric deformations of maximally supersymmetric gauge theories, specifically ten-dimensional SYM and its dimensional reductions. By employing $L_{\infty}$/$A_{\infty}$ formalisms, Koszul duality, and Hochschild/Lie cohomology, the authors reduce deformation classifications to computable algebraic invariants, yielding explicit Spin$(10)$-invariant deformation classes, including the notable $\delta\mathcal{L}_{16}$ and related higher-order terms. The BV formalism provides a geometric perspective, relating deformations to $Q$-manifolds and $\mathfrak{g}$-equivariant cohomology, and enabling the extension of infinitesimal deformations to formal $\alpha'$-series deformations. Together, these results illuminate how stringy corrections to D-brane actions emerge from algebraic deformations of maximally supersymmetric gauge theories and establish a robust toolkit for analyzing such deformations across dimensions. The approach is deeply connected to pure spinor methods and offers a principled path to understanding higher-derivative SUSY completions in a manner compatible with gauge symmetry and Poincaré invariance.

Abstract

We study supersymmetric and super Poincaré invariant deformations of ten-dimensional super Yang-Mills theory and of its dimensional reductions. We describe all infinitesimal super Poincaré invariant deformations of equations of motion of ten-dimensional super Yang-Mills theory and its reduction to a point; we discuss the extension of them to formal deformations. Our methods are based on homological algebra, in particular, on the theory of L-infinity and A-infinity algebras. The exposition of this theory as well as of some basic facts about Lie algebra homology and Hochschild homology is given in appendices.

Supersymmetric Deformations of Maximally Supersymmetric Gauge Theories

TL;DR

This work develops a comprehensive homological-algebraic framework for classifying and constructing supersymmetric deformations of maximally supersymmetric gauge theories, specifically ten-dimensional SYM and its dimensional reductions. By employing / formalisms, Koszul duality, and Hochschild/Lie cohomology, the authors reduce deformation classifications to computable algebraic invariants, yielding explicit Spin-invariant deformation classes, including the notable and related higher-order terms. The BV formalism provides a geometric perspective, relating deformations to -manifolds and -equivariant cohomology, and enabling the extension of infinitesimal deformations to formal -series deformations. Together, these results illuminate how stringy corrections to D-brane actions emerge from algebraic deformations of maximally supersymmetric gauge theories and establish a robust toolkit for analyzing such deformations across dimensions. The approach is deeply connected to pure spinor methods and offers a principled path to understanding higher-derivative SUSY completions in a manner compatible with gauge symmetry and Poincaré invariance.

Abstract

We study supersymmetric and super Poincaré invariant deformations of ten-dimensional super Yang-Mills theory and of its dimensional reductions. We describe all infinitesimal super Poincaré invariant deformations of equations of motion of ten-dimensional super Yang-Mills theory and its reduction to a point; we discuss the extension of them to formal deformations. Our methods are based on homological algebra, in particular, on the theory of L-infinity and A-infinity algebras. The exposition of this theory as well as of some basic facts about Lie algebra homology and Hochschild homology is given in appendices.

Paper Structure

This paper contains 20 sections, 36 theorems, 270 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic algebras
  4. Calculation of the Lie algebra cohomology
  5. The group $\mathrm{Spin}(10, \mathbb{C})$ and the space of pure spinors
  6. Euler characteristics
  7. Infinitesimal SUSY Deformations of $\mathcal{L}_{SYM}$
  8. The Homological Approach to Infinitesimal Deformations
  9. Calculation of the Cohomology
  10. The BV formalism: a geometric approach
  11. Formal SUSY deformations
  12. L$_{\infty}$ and A$_{\infty}$ algebras
  13. Duality
  14. On the relation of the Lie algebra and the BV approaches to the deformation problem
  15. The L$_{\infty}$ action of the supersymmetry algebra in the BV formulation
  16. ...and 5 more sections

Key Result

Proposition 1

The algebra $YM$ is isomorphic to $\bigoplus_{k\geq 2}L^k$. The obvious map $\bigoplus_{k\geq 2}L^k\rightarrow (L +S,d)$ is a quasi-isomorphism. Similarly $U(YM)$ is quasi-isomorphic to $B^!_0$.

Theorems & Definitions (38)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Corollary 5
  • Corollary 6
  • Proposition 7
  • Corollary 8
  • Theorem 9
  • Theorem 10
  • ...and 28 more