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Symplectic scalar curvature on supermanifolds

R Hernández-Amador, JA Vallejo, Yu Vorobiev

Abstract

We study the notion of symplectic scalar curvature on the supermanifold over an ordinary Fedosov manifold whose structural sheaf is that of differential forms. In this purely geometric context, we introduce two families of odd super-Fedosov structures, the first one is very general and uses a graded symmetric connection, leading to a vanishing odd symplectic scalar curvature, while the second one is based on a graded non-symmetric connection and has a non-trivial odd symplectic scalar curvature. As a simple example of the second case, we determine that curvature when the base Fedosov manifold is the torus.

Symplectic scalar curvature on supermanifolds

Abstract

We study the notion of symplectic scalar curvature on the supermanifold over an ordinary Fedosov manifold whose structural sheaf is that of differential forms. In this purely geometric context, we introduce two families of odd super-Fedosov structures, the first one is very general and uses a graded symmetric connection, leading to a vanishing odd symplectic scalar curvature, while the second one is based on a graded non-symmetric connection and has a non-trivial odd symplectic scalar curvature. As a simple example of the second case, we determine that curvature when the base Fedosov manifold is the torus.

Paper Structure

This paper contains 10 sections, 12 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Geometry of supermanifolds
  3. Graded symplectic forms
  4. Graded connections
  5. Graded curvature and Ricci tensors. Scalar supercurvature
  6. Fedosov structures on $(M,\Omega(M))$
  7. Vanishing odd symplectic scalar curvature
  8. The case of non-symmetric graded connections
  9. An example
  10. Fedosov structures on the torus

Key Result

Proposition 2.1

Let $\nabla \nabla$ be a graded connection on $(M,\Omega(M))$. Then, the graded cyclic sum of its graded curvature with respect its arguments vanishes:

Theorems & Definitions (12)

  • Proposition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 3.1
  • Theorem 3.2
  • Corollary 3.2.1
  • Lemma 4.1
  • ...and 2 more