A Geometric Action for the Courant Bracket

TL;DR

This paper constructs a geometric action based on the Courant bracket in generalized complex geometry, extending concepts from quantum gravity and affine algebras to explore new algebraic and geometric structures.

Contribution

It introduces a geometric action for the Courant bracket, generalizing known structures like the Virasoro algebra and Kac-Moody algebra, and explores their geometric and physical implications.

Findings

The action generalizes Polyakov's 2D quantum gravity.

The geometric action resembles the semi-direct product of Virasoro and U(1) Kac-Moody algebras.

Extensions to Dirac structures and almost complex structures are proposed.

Abstract

An important operation in generalized complex geometry is the Courant bracket which extends the Lie bracket that acts only on vectors to a pair given by a vector and a p-form. We explore the possibility of promoting the elements of the Courant bracket to physical fields by constructing a geometric action based on the Kirillov-Kostant symplectic form. For the $p=0$ forms, the action generalizes Polyakov's two-dimensional quantum gravity when viewed as the geometric action for the Virasoro algebra. We show that the geometric action arising from the centrally extended Courant bracket for the vector and zero form pair is similar to the geometric action obtained from the semi-direct product of the Virasoro algebra with a U(1) affine Kac-Moody algebra. For arbitrary $p$ restricted to a Dirac structure, we derived the geometric action and exhibit generalizations for almost complex structures built on the Kirillov-Kostant symplectic form. In the case of p+1 dimensional submanifolds, we also discuss a generalization of a Kahler structure on the orbits.