Duality symmetry in four-dimensional string actions

TL;DR

This paper explores duality symmetries in four-dimensional string actions derived from ten-dimensional supergravity, identifying specific symmetry groups and their effects on the equations of motion and actions.

Contribution

It derives the $SL(2,R)\times O(6,6+n)$ symmetry transformations for the reduced theory and clarifies their action on fields and the invariance of equations of motion.

Findings

Identifies $SL(2,R)\times O(6,6+n)$ as symmetry groups of the reduced theory.

Shows $SL(2,R)$ is a symmetry of the equations of motion, but only a subset leaves the action invariant for $n>0$.

Relates the four-dimensional theory to $N=4$ supergravity via duality transformation.

Abstract

We reduce the dual version of $D=10$, $N=1$ supergravity coupled to $n$ vector fields to four dimensions, and derive the $SL(2,R)\times O(6,6+n)$ transformations which leave the equations of motion invariant. For $n=0$ $SL(2,R)$ is also a symmetry of the action, but for $n>0$ only those $SL(2,R)$ transformations which act linearly on all fields leave the action invariant. The resulting four-dimensional theory is related to the bosonic part of the usual formulation of $N=4$ supergravity coupled to matter by a duality transformation.