Gauge theory for string algebroids
Mario Garcia-Fernandez, Roberto Rubio, Carl Tipler
TL;DR
This work places holomorphic string algebroids inside a moment-map framework, where inner automorphisms of Courant algebroids yield a Calabi system whose zeros unify the classical Calabi problem and Hull-Strominger. It constructs a (possibly degenerate) pseudo-Kähler metric on the moduli space of Calabi-system solutions, with the dilaton functional acting as a Kähler potential, and establishes an infinitesimal Donaldson-Uhlenbeck-Yau type correspondence under a natural gauge-fixing Condition A. The paper develops a robust gauge-theoretic foundation via complex gauge groups, Bott-Chern theory, and a Chern correspondence, then analyzes the moduli geometry, including fibre-wise structures and stability-type considerations, with explicit non-Kähler examples. These results provide a bridge between higher gauge theory, complex geometry, and string-theoretic moduli, offering a rigorous framework for studying deformations of BC and Aeppli classes in relation to stability and moduli metrics.
Abstract
We introduce a moment map picture for holomorphic string algebroids where the Hamiltonian gauge action is described by means of inner automorphisms of Courant algebroids. The zero locus of our moment map is given by the solutions of the Calabi system, a coupled system of equations which provides a unifying framework for the classical Calabi problem and the Hull-Strominger system. Our main results are concerned with the geometry of the moduli space of solutions, and assume a technical condition which is fulfilled in examples. We prove that the moduli space carries a pseudo-Kähler metric with Kähler potential given by the dilaton functional, a topological formula for the metric, and an infinitesimal Donaldson-Uhlenbeck-Yau type theorem.