Gauging the Heisenberg algebra of special quaternionic manifolds

TL;DR

This work analyzes gauging the Heisenberg algebra present in the special quaternionic moduli space of N=2 supergravity arising from Calabi–Yau compactifications. By encoding RR scalars in a symplectic vector and introducing symplectic charge-vectors, the authors derive a precise cocycle condition $V_I\times V_J=0$ that governs abelian gaugings, and they provide explicit expressions for covariant derivatives and the RR-induced mass terms. They extend to non-abelian gaugings, showing the necessity of a central cocycle and offering a concrete construction when the vector space dimension matches the Heisenberg algebra, outlining how inconsistencies are avoided via cocycle and coboundary constraints. The analysis connects to half-flat flux compactifications, generalizing flux couplings to arbitrary index ranges and revealing a symplectic, mirror-symmetric structure between Type II theories, with implications for the resulting scalar potential and possible non-perturbative couplings.

Abstract

We show that in N=2 supergravity, with a special quaternionic manifold of (quaternionic) dimension h_1+1 and in the presence of h_2 vector multiplets, a h_2+1 dimensional abelian algebra, intersecting the 2h_1+3 dimensional Heisenberg algebra of quaternionic isometries, can be gauged provided the h_2+1 symplectic charge--vectors V_I, have vanishing symplectic invariant scalar product V_I X V_J=0. For compactifications on Calabi--Yau three--folds with Hodge numbers (h_1,h_2) such condition generalizes the half--flatness condition as used in the recent literature. We also discuss non--abelian extensions of the above gaugings and their consistency conditions.