Towards a double field theory on para-Hermitian manifolds

TL;DR

This paper extends the geometric framework of double field theory from flat para-Kähler to general para-Hermitian manifolds, introducing new structures and reductions relevant to string theory and geometry.

Contribution

It generalizes the geometric interpretation of double field theory to all para-Hermitian manifolds, defining new brackets, connections, and reductions.

Findings

Bracket extends C-bracket on para-Hermitian manifolds

Defined a canonical connection and action for the field

Provided examples on well-known para-Hermitian manifolds

Abstract

In a previous paper, we have shown that the geometry of double field theory has a natural interpretation on flat para-K\"ahler manifolds. In this paper, we show that the same geometric constructions can be made on any para-Hermitian manifold. The field is interpreted as a compatible (pseudo-)Riemannian metric. The tangent bundle of the manifold has a natural, metric-compatible bracket that extends the C-bracket of double field theory. In the para-K\"ahler case this bracket is equal to the sum of the Courant brackets of the two Lagrangian foliations of the manifold. Then, we define a canonical connection and an action of the field that correspond to similar objects of double field theory. Another section is devoted to the Marsden-Weinstein reduction in double field theory on para-Hermitian manifolds. Finally, we give examples of fields on some well-known para-Hermitian manifolds.