# Geometric Low-Energy Effective Action in a Doubled Spacetime

## TL;DR

This paper develops a geometric low-energy effective action in doubled spacetime, integrating double field theory concepts with star products, and explores its consistency and extensions to sigma models and string theory.

## Contribution

It introduces a new geometric effective action using star products that implement the weak constraint in double field theory, ensuring gauge symmetry closure and extending to string models.

## Key findings

- The proposed action is consistent for d ≥ 1 due to orthogonality conditions.
- Orthogonality of momenta ensures gauge symmetry closure in the cubic terms.
- Extensions to double sigma models and non-commutative geometry are achieved.

## Abstract

The ten-dimensional supergravity theory is a geometric low-energy effective theory and the equations of motion for its fields can be obtained from string theory by computing $\beta$ functions. With $d$ compact dimensions, we can add to it an $O(d, d;\mathbb{Z})$ geometric structure and construct the supergravity theory inspired by double field theory through the use of a suitable commutative star product. The latter implements the weak constraint of the double field theory on its fields and gauge parameters in order to have a closed gauge symmetry algebra. The consistency of the action here proposed is based on the orthogonality of the momenta associated with fields in their triple star products in the cubic terms defined for $d\ge1$. This orthogonality holds also for an arbitrary number of star products of fields for $d=1$. Finally, we extend our analysis to the double sigma model, non-commutative geometry and open string theory.