A system of equations born in string theory has become one of the central problems of modern complex geometry. Recent work by Mario García-Fernández and Raúl González Molina provides new conceptual tools for understanding its solutions, its obstructions, and its role in the search for canonical geometries beyond the Kähler world.

Mathematician Shing-Tung Yau with the Hull-Strominger system written on the blackboard.

The Hull–Strominger system is a set of partial differential equations that originated in string theory. Its purpose is to characterize certain special geometries that arise in the description of the most realistic models of the universe within this theoretical framework.

In string theory, spacetime is originally formulated in ten dimensions. In order to recover the observable four-dimensional physics, it is postulated that the six extra dimensions are compactified, that is, curled up into very small geometric shapes that cannot be detected directly. The Hull–Strominger system provides the equations that these internal geometries must satisfy in order for the resulting physical model to be consistent. In this sense, it not only connects string theory with effective four-dimensional physics, but also describes a very broad class of geometric spaces with particularly favorable predictive properties.

These equations were first studied from a purely mathematical perspective in a 2005 article by Fields Medalist Shing-Tung Yau and Jun Li, from Stanford University. In that work, the authors proposed their study as a generalization of the Calabi problem, which had been solved by Yau in 1977. Unlike the classical setting of the Calabi problem, the geometric constraints imposed by the Hull–Strominger system go beyond the realm of algebraic geometry and even that of Kähler geometry.

By analogy with the Calabi problem, Yau proposed a conjecture—still open—based on the large classes of examples known to date. The main difficulties in addressing this conjecture stem from the impossibility of applying techniques from Kähler geometry and from the limited understanding of the geometry underlying the equations.

In recent years, the problem has attracted the attention of mathematicians from a wide range of areas, including non-Kähler Hermitian geometry, geometric analysis, algebraic topology, and Hitchin’s theory of generalized geometry. At present, the search for solutions to this system has become a guiding principle in the study of canonical geometries on these more flexible complex spaces. Recently, Mario García-Fernández (ICMAT-CSIC) and Raúl González Molina (SISSA and UPM) have made significant progress on this problem. Their results have been published in the Journal für die reine und angewandte Mathematik.

Building on previous work in generalized geometry, García-Fernández and González Molina reformulate the Hull–Strominger system in terms of natural equations for complex and geometric data associated with Courant algebroids in the holomorphic category, which are the basic building blocks of generalized geometry. In this way, they show that the system can be reinterpreted as a Hermite–Einstein equation for a generalized metric, providing a rigorous mathematical proof of a statement that had previously been put forward by physicists using perturbative methods.

As an application, the authors construct an invariant that detects the impossibility of solving the Hull–Strominger system. This invariant takes the form of a holomorphic character derived from a moment-map-type equation, in close analogy with the invariant introduced by Akito Futaki in 1983 to measure the obstruction for a complex manifold to admit a Kähler–Einstein metric.

These new results provide a powerful tool for addressing Yau’s conjecture in the context of the Hull–Strominger system. More broadly, this conjecture is related to another open problem in complex geometry: the geometrization of surgeries (transitions and flops) that arise when passing from algebraic varieties to non-Kähler Calabi–Yau varieties in complex dimension three, a phenomenon known as Reid’s fantasy. The appearance of a Futaki-type invariant sheds new light on this picture and suggests that the original expectations may need to be refined.

On the other hand, the interpretation of the Hull–Strominger system as a higher-order instanton represents an important step toward a Hitchin–Kobayashi-type correspondence. In this formulation, solutions of the system, within an extended version of gauge theory, should be equivalent to certain geometric objects—yet to be identified—that satisfy an appropriate notion of stability in the sense of geometric invariant theory in algebraic geometry.