Consistent S^2 Pauli Reduction of Six-dimensional Chiral Gauged Einstein-Maxwell Supergravity

TL;DR

This work demonstrates a fully consistent Pauli reduction on $S^2$ of six-dimensional $N=(1,0)$ gauged Einstein-Maxwell supergravity, producing a four-dimensional $N=1$ theory consisting of the supergravity multiplet, an $SU(2)$ Yang-Mills vector multiplet, and a scalar multiplet. Crucially, the reduction decouples all massive Kaluza-Klein modes, leaving only the breathing-mode scalar $\phi$ as the internal degree of freedom, and a dualised axion $\sigma$ enters the 4D theory alongside the Yang-Mills sector. The authors also show how four-dimensional black-hole solutions uplift to six dimensions, derive the explicit 4D-4D SUSY transformation rules from the higher-dimensional theory, and analyze the four-dimensional physics in terms of the compactification scale and cosmological constant problem, finding that very small bulk gauge couplings (e.g., $g_{YM}\sim 10^{-31}$ in Planck units for $M_K\sim 10^{-3}$ eV) are required unless brane-world modifications or orbifold limits are invoked. The results provide a concrete realization of Pauli’s idea to obtain Yang-Mills fields via reduction on a coset and open avenues connecting to brane-world scenarios and possible M-theory embeddings.

Abstract

Six-dimensional N=(1,0) Einstein-Maxwell gauged supergravity is known to admit a (Minkowski)_4\times S^2 vacuum solution with four-dimensional N=1 supersymmetry. The massless sector comprises a supergravity multiplet, an SU(2) Yang-Mills vector multiplet, and a scalar multiplet. In this paper it is shown that, remarkably, the six-dimensional theory admits a fully consistent dimensional reduction on the 2-sphere, implying that all solutions of the four-dimensional N=1 supergravity can be lifted back to solutions in six dimensions. This provides a striking realisation of the idea, first proposed by Pauli, of obtaining a theory that includes Yang-Mills fields by dimensional reduction on a coset space. We address the cosmological constant problem within this model, and find that if the Kaluza-Klein mass scale is taken to be 10^{-3} eV (as has recently been suggested) then four-dimensional gauge-coupling constants for bulk fields must be of the order of 10^{-31}. We also suggest a link between a modification of the model with 3-branes, and a five-dimensional model based on an S^1/Z_2 orbifold.