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Fermionic Spencer Cohomologies of D=11 Supergravity

C. A. Cremonini, P. A. Grassi, R. Noris, L. Ravera, A. Santi

TL;DR

This work computes the fermionic Spencer cohomology of the $D=11$ Poincaré superalgebra by fusing Cartan–Tanaka prolongations with the Molien–Weyl method and Hilbert–Poincaré series, revealing a nontrivial odd cohomology $H^{1,2}(\mathfrak{m},\mathfrak{p})\cong S$ and a vanishing $H^{3,2}(\mathfrak{m},\mathfrak{p})=0$, along with the standard even result $H^{2,2}(\mathfrak{m},\mathfrak{p})\cong \Lambda^{4}V$. The paper develops a general Molien–Weyl framework for graded Lie superalgebras, applies it to the $D=11$ case to obtain collapsed Hilbert–Poincaré series, and uses the Spencer complex to study filtered deformations of maximally supersymmetric subalgebras, culminating in a no-go theorem for maximally supersymmetric odd deformations along timelike directions. These results constrain the algebraic possibilities for Killing superalgebras of highly supersymmetric $D=11$ backgrounds and guide future explorations of FDA-like structures and nonholonomic superspace formulations. The findings connect cohomological data to supersymmetry variations and background geometry, informing the algebraic backbone of 11D supergravity.

Abstract

We combine the theory of Cartan-Tanaka prolongations with the Molien-Weyl integral formula and Hilbert-Poincaré series to compute the Spencer cohomology groups of the $D=11$ Poincaré superalgebra $\mathfrak p$, relevant for superspace formulations of $11$-dimensional supergravity in terms of nonholonomic superstructures. This includes novel fermionic Spencer groups, providing with new cohomology classes of $\mathbb Z$-grading $1$ and form number $2$. Using the Hilbert-Poincaré series and the Euler characteristic, we also explore Spencer cohomology contributions in higher form numbers. We then propose a new general definition of filtered deformations of graded Lie superalgebras along first-order fermionic directions and investigate such deformations of $\mathfrak p$ that are maximally supersymmetric. In particular, we establish a no-go type theorem for maximally supersymmetric filtered subdeformations of $\mathfrak p$ along timelike (i.e., generic) first-order fermionic directions.

Fermionic Spencer Cohomologies of D=11 Supergravity

TL;DR

This work computes the fermionic Spencer cohomology of the Poincaré superalgebra by fusing Cartan–Tanaka prolongations with the Molien–Weyl method and Hilbert–Poincaré series, revealing a nontrivial odd cohomology and a vanishing , along with the standard even result . The paper develops a general Molien–Weyl framework for graded Lie superalgebras, applies it to the case to obtain collapsed Hilbert–Poincaré series, and uses the Spencer complex to study filtered deformations of maximally supersymmetric subalgebras, culminating in a no-go theorem for maximally supersymmetric odd deformations along timelike directions. These results constrain the algebraic possibilities for Killing superalgebras of highly supersymmetric backgrounds and guide future explorations of FDA-like structures and nonholonomic superspace formulations. The findings connect cohomological data to supersymmetry variations and background geometry, informing the algebraic backbone of 11D supergravity.

Abstract

We combine the theory of Cartan-Tanaka prolongations with the Molien-Weyl integral formula and Hilbert-Poincaré series to compute the Spencer cohomology groups of the Poincaré superalgebra , relevant for superspace formulations of -dimensional supergravity in terms of nonholonomic superstructures. This includes novel fermionic Spencer groups, providing with new cohomology classes of -grading and form number . Using the Hilbert-Poincaré series and the Euler characteristic, we also explore Spencer cohomology contributions in higher form numbers. We then propose a new general definition of filtered deformations of graded Lie superalgebras along first-order fermionic directions and investigate such deformations of that are maximally supersymmetric. In particular, we establish a no-go type theorem for maximally supersymmetric filtered subdeformations of along timelike (i.e., generic) first-order fermionic directions.

Paper Structure

This paper contains 21 sections, 27 theorems, 119 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Molien-Weyl formula
  3. Invariant supersymmetric polynomials
  4. Spencer complex and Euler characteristic
  5. Hilbert-Poincaré series of the $D=11$ Poincaré superalgebra
  6. The $D=11$ ingredients
  7. A warm up: the trivial representation
  8. Analysis of the Hilbert-Poincaré series
  9. Fermionic Spencer cohomology of maximally supersymmetric subalgebras of the $D=11$ Poincaré superalgebra
  10. The deformation complex
  11. Proof of Theorem \ref{['thm:1']} (stated on page 3)
  12. Normalization conditions for $2$-cocycles in $\mathbb Z$-grading $1$
  13. The group $H^{1,2}(\mathfrak{m},\mathfrak{p})$
  14. The group $H^{3,2}(\mathfrak{m},\mathfrak{p})$
  15. Maximally supersymmetric even/odd filtered subdeformations of the $D=11$ Poincaré superalgebra
  16. ...and 6 more sections

Key Result

Theorem 1

Figueroa-OFarrill:2015rfh The Killing superalgebra of an 11-dimensional bosonic supergravity background is a filtered subdeformation of the $D=11$ Poincaré superalgebra.

Theorems & Definitions (56)

  • Theorem
  • Theorem
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.3
  • ...and 46 more