The Graded Extension of Thomas-Whitehead Gravity

TL;DR

This paper extends Thomas-Whitehead gravity into a supersymmetric framework called Super TW Gravity, using supermanifolds to incorporate supersymmetry and exploring its connection to super-Virasoro algebra.

Contribution

It introduces the graded extension of TW Gravity within supermanifold formalism, laying groundwork for future supergravity theories with supersymmetry.

Findings

Constructed the Lagrangian for Super TW Gravity.

Derived classical field equations for the supersymmetric extension.

Discussed the graded projective connection and its implications.

Abstract

Thomas-Whitehead (TW) gravity was recently introduced as a projective gauge theory of gravity over a d-dimensional manifold that embeds reparameterization invariance into the action functional for gravitation through the use of the Thomas-Whitehead connection. The projective invariance in this d-dimensional theory enjoys an intimate relationship with the Virasoro coadjoint elements found in string theory as one of the components of the connection, $\mathcal{D}_{ab}$, is directly related to the coadjoint elements of the Virasoro algebra. TW Gravity exploits projective Gauss-Bonnet terms in the action functional which allows the theory to collapse to Einstein's theory of General Relativity in the limit that $\mathcal{D}_{ab }$ vanishes. In this note we develop the graded extension of TW Gravity, Super TW Gravity, in the framework of a DeWitt supermanifold. We construct the Lagrangian for Super TW Gravity, give a detailed derivation of the classical field equations and discuss the graded extension of the projective connection as a prelude to a future understanding of TW-Supergravity (which has manifest supersymmetry) and its relationship to the Super-Virasoro algebra.