Consistent Group and Coset Reductions of the Bosonic String

TL;DR

This work shows that many consistent group and coset reductions of the bosonic string can be understood as a two-step process: a DeWitt group-manifold reduction that can be decomposed into a Kaluza–U(1) reduction followed by a Pauli coset reduction on $G/U(1)$ (and extends to $G/H$). It proves that any $D$-dimensional theory arising from a circle reduction of $(D+1)$ dimensions admits a consistent Pauli reduction on cosets of the form $G/U(1)$, and provides explicit constructions and a consistency proof for Pauli reductions on $S^3$ and related group manifolds. The paper further develops the DeWitt reduction framework for the bosonic string, studies truncations that retain singlets, and derives explicit superpotentials for special group-dimension values ($q=3,25$) that yield domain-wall brane solutions, while also addressing branes without supersymmetry. Collectively, these results clarify the delicate mechanisms that enable (or preclude) consistent Pauli reductions in non-supersymmetric settings and offer a blueprint for exploring consistent reductions in broader string-theoretic contexts.

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.

Figures (2)

• Figure 1: DeWitt reduction on a group manifold $G$, composed as a Kaluza reduction on $U(1)$ followed by a Pauli reduction on $G/U(1)$.
• Figure 2: DeWitt reduction on $G$, composed as a DeWitt reduction on $H$ followed by a Pauli reduction on $G/H$.