Consistent Group and Coset Reductions of the Bosonic String

TL;DR

This paper explores conditions under which consistent dimensional reductions of bosonic string theory on cosets and group manifolds are possible, revealing new methods and extending known results with potential applications to supergravity solutions.

Contribution

It introduces a new understanding of how Kaluza and Pauli reductions can be combined for consistent string theory truncations, including cases with non-supersymmetric models.

Findings

Demonstrates a two-step reduction process for SU(2) group manifolds.

Shows that any theory from a U(1) reduction admits a G/U(1) coset reduction.

Identifies superpotential structures in non-supersymmetric models with specific group dimensions.

Abstract

Dimensional reductions of pure Einstein gravity on cosets other than tori are inconsistent. The inclusion of specific additional scalar and p-form matter can change the situation. For example, a D-dimensional Einstein-Maxwell-dilaton system, with a specific dilaton coupling, is known to admit a consistent reduction on S^2= SU(2)/U(1), of a sort first envisaged by Pauli. We provide a new understanding, by showing how an S^3=SU(2) group-manifold reduction of (D+1)-dimensional Einstein gravity, of a type first indicated by DeWitt, can be broken into in two steps; a Kaluza-type reduction on U(1) followed by a Pauli-type coset reduction on S^2. More generally, we show that any D-dimensional theory that itself arises as a Kaluza U(1) reduction from (D+1) dimensions admits a consistent Pauli reduction on any coset of the form G/U(1). Extensions to the case G/H are given. Pauli coset reductions of the bosonic string on G= (G\times G)/G are believed to be consistent, and a consistency proof exists for S^3=SO(4)/SO(3). We examine these reductions, and arguments for consistency, in detail. The structures of the theories obtained instead by DeWitt-type group-manifold reductions of the bosonic string are also studied, allowing us to make contact with previous such work in which only singlet scalars are retained. Consistent truncations with two singlet scalars are possible. Intriguingly, despite the fact that these are not supersymmetric models, if the group manifold has dimension 3 or 25 they admit a superpotential formulation, and hence first-order equations yielding domain-wall solutions.