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Chiral forms and their deformations

Xavier Bekaert, Marc Henneaux, Alexander Sevrin

TL;DR

The paper shows that the local BRST cohomology of a system of free chiral 2-forms in six dimensions is rigid: there are no non-abelian, local deformations that modify the gauge algebra. By computing the BRST cohomology $H^{0,n}(s|d)$ order by order, it demonstrates that allowed first-order deformations are confined to gauge-invariant curvature terms or Chern–Simons–type contributions, which do not alter the original abelian gauge transformations. The analysis leverages the antifield (BV) formalism and detailed cohomological tools, including the algebraic Poincaré lemma and the γ and δ cohomologies, both modulo $d$ and modulo the spatial differential $ ilde{d}$. Consequently, coincident M5-branes cannot be described by a local non-abelian extension of chiral 2-forms, reinforcing the uniqueness of the abelian chiral-form dynamics and informing the search for non-perturbative or non-local frameworks.

Abstract

We systematically study deformations of chiral forms with applications to string theory in mind. To first order in the coupling constant, this problem can be translated into the calculation of the local BRST cohomological group at ghost number zero. We completely solve this cohomology and present detailed proofs of results announced in a previous letter. In particular, we show that there is no room for non-abelian, local, deformations of a pure system of chiral p-forms.

Chiral forms and their deformations

TL;DR

The paper shows that the local BRST cohomology of a system of free chiral 2-forms in six dimensions is rigid: there are no non-abelian, local deformations that modify the gauge algebra. By computing the BRST cohomology order by order, it demonstrates that allowed first-order deformations are confined to gauge-invariant curvature terms or Chern–Simons–type contributions, which do not alter the original abelian gauge transformations. The analysis leverages the antifield (BV) formalism and detailed cohomological tools, including the algebraic Poincaré lemma and the γ and δ cohomologies, both modulo and modulo the spatial differential . Consequently, coincident M5-branes cannot be described by a local non-abelian extension of chiral 2-forms, reinforcing the uniqueness of the abelian chiral-form dynamics and informing the search for non-perturbative or non-local frameworks.

Abstract

We systematically study deformations of chiral forms with applications to string theory in mind. To first order in the coupling constant, this problem can be translated into the calculation of the local BRST cohomological group at ghost number zero. We completely solve this cohomology and present detailed proofs of results announced in a previous letter. In particular, we show that there is no room for non-abelian, local, deformations of a pure system of chiral p-forms.

Paper Structure

This paper contains 12 sections, 10 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Constructing consistent couplings as a deformation problem
  3. System of free chiral 2-forms in 6 dimensions
  4. Fields - Antifields - Solution of the master equation
  5. BRST operator
  6. Local forms - Algebraic Poincaré lemma
  7. Cohomology of $\gamma$
  8. Cohomology of $\gamma$ modulo $d$ at positive antighost number
  9. Cohomology of $\gamma$ modulo $d$ at zero antighost number
  10. Invariant cohomology of $\delta$ modulo $\tilde{d}$ in antighost number $2,4,6,\ldots$
  11. Decomposition of the Wess-Zumino equation
  12. Final comments and conclusions

Key Result

Theorem 6.1

The cohomology of $d$ in the algebra of local forms of degree $q<n$ is given by

Theorems & Definitions (10)

  • Theorem 6.1
  • Theorem 6.2
  • Theorem 7.1
  • Lemma 7.1
  • Theorem 8.1
  • Theorem 8.2
  • Theorem 9.1
  • Theorem 10.1
  • Lemma 10.1
  • Theorem 10.2