Connections, Field Redefinitions and Heterotic Supergravity

TL;DR

This paper explores the ambiguity in connection choices in heterotic supergravity at order alpha', analyzing how field redefinitions affect supersymmetric solutions and proposing a symmetry between gauge and tangent bundle connections that extends to higher orders.

Contribution

It introduces a detailed analysis of the moduli space of field redefinitions in heterotic supergravity and proposes a higher-order correction to the Hull connection based on symmetry considerations.

Findings

The moduli space of infinitesimal field redefinitions is given by H^{(0,1)}(X, End(TX)).

A symmetry between gauge and tangent bundle connections is identified and should be preserved at higher orders.

The Hull connection should be corrected at second and higher orders in alpha'.

Abstract

We study heterotic supergravity at $\mathcal{O}(\alpha')$, first described in detail in 1989 by Bergshoeff and de Roo. In particular, we discuss an ambiguity of a connection choice on the tangent bundle. It is well known that at $\mathcal{O}(\alpha')$ the Hull connection gives a consistent supergravity theory with supersymmetry transformations given in the usual way. We consider deformations of this connection corresponding to field redefinitions, and the necessary corrections to the supersymmetry transformations. We are interested in the moduli space of such field redefinitions which allow for supersymmetric solutions to the equations of motion. We show that for solutions on $M_4\times X$, where $M_4$ is Minkowski and $X$ is compact, the moduli space of infinitesimal field redefinitions is given by $H^{(0,1)}(X,\textrm{End}(TX))$. This space corresponds to infinitesimally close connections for which the equations of motion are satisfied. The setup suggests a symmetry between the gauge connection and the tangent bundle connection, as also employed by Bergshoeff and de Roo. We argue that this symmetry should be kept to higher orders in $\alpha'$, and propose a natural choice for the corresponding tangent bundle connection used in curvature computations. In particular, the Hull connection should be corrected at second and higher orders in $\alpha'$ from this point of view.