Mario García-Fernández (ICMAT), Roberto Rubio (Autonomous University of Barcelona) and Carl Tipler (Université de Bretagne Occidentale, France) sign ‘Gauge theory for string algebroids’, published in the Journal of Differential Geometry.

Back to the work of Atiyah and Bott (1983), the interaction of mathematical gauge theories with symplectic geometry and, in particular, the idea of Hamiltonian reduction, has had an important impact in our understanding of moduli theory in algebraic geometry. The main result revolving this idea is the Donaldson-Uhlenbeck-Yau Theorem, initially conjectured by Hitchin and Kobayashi, which establishes a correspondence between the moduli space of solutions of the gauge theoretical Hermite-Yang-Mills equations and the moduli space of bundles on a compact Kähler manifold, satisfying a suitable numerical condition called “stability”. A key upshot of this important result is that certain moduli spaces in algebraic geometry, constructed via Mumford’s Geometric Invariant Theory, are endowed with natural Kähler metrics.

In a publication in the Journal of Differential Geometry, Mario Garcia-Fernandez (ICMAT), Roberto Rubio (UAB) and Carl Tipler (Université de Bretagne Occidentale) explore a new scenario where the Hamiltonian reduction picture arises. For this, the authors consider a class of holomorphic bundle-like objects which have appeared very recently in higher gauge theory. In this setup, the structure group of the bundle is replaced by a mild category, dubbed as complex Lie 2-group (2023). The authors go on to apply their construction to the moduli theory for the Hull-Strominger System. This system of partial differential equations has its origins in string theory (1986) and was first studied in mathematics by Shing-Tung Yau (Tsinghua) and Jun Li (Stanford) (2005), having an important impact in complex geometry ever since. The main results of the manuscript are concerned with the geometry of the moduli space of solutions for these equations. Under natural hypothesis, the authors prove that the moduli space carries a pseudo-Kähler metric with an explicit formula for the Kähler potential, a cohomological formula for the metric, and an infinitesimal version of the Donaldson-Uhlenbeck-Yau theorem.

Reference: Mario Garcia-Fernandez, Roberto Rubio, Carl Tipler. “Gauge theory for string algebroids.” Journal of Differential Geometry, 128(1) 77-152 September 2024. https://doi.org/10.4310/jdg/1721075260