$\beta$-deformations, potentials and KK modes

TL;DR

This paper investigates the geometric and physical effects of $eta$-deformations on compactification volumes and potentials, revealing non-trivial changes and marginal deformation properties in string theory backgrounds.

Contribution

It analyzes volume changes of cycles under $eta$-transformations and introduces a new potential arising from $SL(3,R)$ transformations, linking geometric deformations to physical potentials.

Findings

Volumes of 3-cycle and 5-cycle generally decrease under $eta$-transformation.

A non-trivial potential emerges due to $SL(3,R)$ mixing of fields.

The $eta$-transformation acts as a marginal deformation in gravity.

Abstract

We have studied volumes of the 3-cycle and the compact 5-volumes for the $\beta$ transformed geometry and it comes out to be decreasing except one choice for which the torus do not stay inside the 3-cycle and ``5-cycle.'' There are 3 possible ways to construct these cycles. one is as mentioned above and the other two are, when the torus stay inside the cycle and when both the torus and the cycle shares a common direction. Also, we have argued that under $\beta$ deformation there arises a non-trivial ``potential'' as the $SL(3,R)$ transformation mixes up the fields. If we start with a flat space after the $SL(3,R)$ transformation the Ricci-scalar of the transformed geometry do not vanishes but the transformed solution is reminiscent of NS5-brane. We have explicitly, checked that $\beta$-transformation indeed is a marginal deformation in the gravity side.