Research

I focus on mathematical problems inspired by supergravity and string theory, primarily in the areas of differential geometry, topology, and geometric analysis. The central goal of my research is to understand and develop the mathematical structure of supergravity and investigate its potential applications in differential geometry and topology.

Supergravity theories are supersymmetric theories of gravity that describe the low-energy dynamics of string theory. Thanks to supersymmetry and their connection to string theory, they are crucial for a wide range of mathematical applications, ranging from mirror symmetry to the quaternionic c-map. Supergravity theories encompass a diverse range of state-of-the-art mathematical structures that interact in novel and unexpected ways, achieving an elegant equilibrium dictated by supersymmetry. This includes, but is not limited to, arithmetic groups, generalized metrics on Courant algebroids, Riemannian manifolds of special holonomy, spinorial Lipschitz structures, invariant Riemannian metrics on exceptional Lie groups, or connections in principal bundles and their categorizations, making the mathematical study of supergravity remarkably rich. Despite supergravity theories being extensively studied in the theoretical physics literature, especially as a window into advanced areas of string theory, the mathematical theory of supergravity is still in its early stages. As a result, most of the outstanding mathematical wonders that supergravity contains, as well as their mathematical applications, remain to be discovered. The mathematical study of supergravity draws from a broad spectrum of mathematical areas, tools, and techniques. As a mathematical discipline, it lies at the intersection of mathematical gauge theory and mathematical general relativity, in accordance with supergravity being a natural extension of Yang-Mills theory and general relativity.

My primary research goal is to develop the mathematical theory of supergravity, with a particular emphasis on the supersymmetric evolution flows and initial data moduli spaces that are determined by their class of globally hyperbolic supersymmetric solutions. One of my main objects of interest is the moduli space of supersymmetric initial data. Such moduli space provides a natural generalization of various celebrated moduli problems in differential geometry and mathematical gauge theory, such as the moduli of anti-self-dual instantons or Seiberg-Witten monopoles. This suggests that supersymmetric initial data moduli spaces may enjoy applications in the construction of smooth invariants in low-dimensional differential topology, a tantalizing possibility that constitutes the ideal final milestone of this line of research. Among the plethora of mathematical approaches and open problems surrounding supergravity, my research mainly focuses on the following four distinct but related topics.

The mathematical theory of four-dimensional supergravity

There are many types of supergravity theories, each with its own set of mathematical structures. To make progress, we have chosen to focus on ungauged supergravity in four Lorentzian dimensions. We started by studying its most fundamental building block: the bosonic sector, which describes the fundamental macroscopic gravitational solutions of the theory. This class of four-dimensional theories is closely related to the moduli space of Calabi-Yau structures on a six-dimensional compact manifold via string theory compactification, a procedure that involves deep mathematical structures in complex geometry and arithmetic that manifest in the mathematical structure of the corresponding four-dimensional supergravity theory. In particular, we have shown that the bosonic sector of four-dimensional ungauged supergravity admits a universal description in terms of a sophisticated gauge theory, containing a novel type of self-duality condition, coupled to a Lorentzian metric and to a section of a certain vertically Riemannian flat fibration whose fiber can be interpreted as the moduli of the theory. Applications in progress include the construction of the universal c-map or the supergravity attractor flow, among others.

Articles and preprints within this research area.

The geometry and DSZ quantization of four-dimensional supergravity, C. Lazaroiu, C. S. Shahbazi. Letters in Mathematical Physics, 2023.
The duality covariant geometry and DSZ quantization of abelian gauge theory, C. Lazaroiu, C. S. Shahbazi. Advances in Theoretical and Mathematical Physics, 2022.
Four-dimensional geometric supergravity and electromagnetic duality: a brief guide for mathematicians, C. I. Lazaroiu, C. S. Shahbazi. Rev. Roumaine Math. Pures Appl, 2021.
Geometric Supergravity and Chiral Triples on Riemann Surfaces, V. Cortés, C. I. Lazaroiu, C. S. Shahbazi. Communications in Mathematical Physics, 2020.
Section sigma models coupled to symplectic duality bundles on Lorentzian four-manifolds, C. I. Lazaroiu, C. S. Shahbazi. Journal of Geometry and Physics, 2018.
Generalized Einstein-Scalar-Maxwell theories and locally geometric U-folds, C. I. Lazaroiu, C. S. Shahbazi. Reviews in Mathematical Physics, 2018.
The global formulation of generalized Einstein-Scalar-Maxwell theories, C. I. Lazaroiu, C. S. Shahbazi. Quantum Theory And Symmetries, 2017.
Geometric U-folds in four dimensions. C. I. Lazaroiu, C. S. Shahbazi. Journal of Physics A: Mathematical and Theoretical, 2017.

Spin geometry and differential spinors

One of the most important concepts that supersymmetry and supergravity have introduced into mathematics is the notion of a supersymmetry solution, also known as a BPS state. Supersymmetric solutions in supergravity are described as solutions to certain spinorial systems of equations of mixed order, known as the Killing spinor equations or supersymmetry conditions, which always involve a parallelicity condition for a spinor. This immediately connects the study of supersymmetric solutions in supergravity with spin geometry and its associated natural differential operators. The type of spinorial equations occurring as supersymmetry conditions in supergravity is often beyond the cases considered in the mathematical literature, and it involves relatively exotic ingredients as a consequence of supersymmetry. Therefore, the mathematical study of supergravity requires developing an adequate geometric framework to describe the type of spinors occurring in supergravity and studying the differential equations that they satisfy. Our approach to this problem is based on the following two pillars:

The general theory of spinorial structures, initiated by Friedrich and Trautmann under the term Lipschitz structures. This theory characterizes topologically and classifies the most general spinorial structures associated with spinor bundles, the latter defined as bundles of Clifford modules over the bundle of Clifford algebras of the underlying pseudo-Riemannian manifold. Controlling such general spinorial structures is fundamental to defining natural differential operators, such as those occurring in supergravity Killing spinor equations, on general spinor bundles.
The theory of spinorial polyforms, by which we equivalently describe spinors in terms of their algebraic square, which we characterize as a differential polyform satisfying a prescribed system of algebraic relations. This, in turn, can be used to translate parallelicity conditions for a spinor into an equivalent differential system for its square, which is typically easier to handle and to interpret both geometrically and analytically, especially in relation to moduli problems that involve metrics, upon which spinor bundles formally depend, as variables.

Both the theory of Lipschitz structures and spinorial polyforms need to be developed for all types of spinor bundles relevant to supergravity, and then applied to the supersymmetry conditions in supergravity. Aside from the aforementioned applications to supergravity, this would lead to a comprehensive theory of spin geometry on manifolds.

Articles and preprints within this research area.

Parallel spinors for G2* and isotropic structures, A. Gil-García, C. S. Shahbazi. Annals of Global Analysis and Geometry, 2025.
Differential spinors and Kundt three-manifolds with skew-torsion, C. S. Shahbazi. arXiv preprint, 2024.
A functional for Spin(7) forms, C. I. Lazaroiu, C. S. Shahbazi. arXiv preprint, 2024.
Dirac operators on real spinor bundles of complex type, C. I. Lazaroiu, C. S. Shahbazi. Differential Geometry and its Applications, 2022.
Spinors of real type as polyforms and the generalized Killing equation, V. Cortés, C. Lazaroiu, C. S. Shahbazi. Mathematische Zeitschrift, 2021.
On the spin geometry of supergravity and string theory, C. I. Lazaroiu, C. S. Shahbazi. Geometric Methods in Physics, 2019.
Real pinor bundles and real Lipschitz structures, C. I. Lazaroiu, C. Shahbazi. Asian Journal of Mathematics, 2019.
Complex Lipschitz structures and bundles of complex Clifford modules,. C. I. Lazaroiu, C. S. Shahbazi. Differential Geometry and its Applications, 2018.

Supersymmetric evolution flows and initial data moduli spaces

Supergravity theories originated as supersymmetric theories of gravity and are therefore naturally Lorentzian, with all the consequences this implies for their mathematical study. In particular, supergravity theories are controlled by differential systems of hyperbolic type, in contrast to the differential systems of elliptic type that typically occur in mathematical gauge theory on Riemannian manifolds, and which can be recovered as particular cases of Lorentzian supergravity restricted to metrics of Euclidean signature. Since the class of all Lorentzian manifolds is vast (compare the classification of Riemann surfaces with the classification of Lorentzian surfaces), when addressing the mathematical study of supergravity in Lorentzian signature, it is convenient to consider the theory as defined on a particular class of Lorentzian manifolds akin to the complete or compact manifolds in Euclidean signature. Compact Lorentzian manifolds are typically avoided in the physics literature, as they appear to be necessarily non-realistic, in the sense that they contain closed time-like curves, which give rise to various physical paradoxes. On the other hand, imposing Lorentzian manifolds to be complete, for instance, geodesically complete, is in general too strict, since many remarkable gravitational space-times are known to develop finite-time singularities. There is, however, a natural class of Lorentzian manifolds that constitute the analog of complete (compact) Riemannian manifolds in Euclidean signature: these are the globally hyperbolic Lorentzian manifolds with a complete (compact) Cauchy hypersurface. From a practical point of view, globally hyperbolic Lorentzian manifolds constitute the class of Lorentzian manifolds on which the Cauchy problem for hyperbolic systems of equations can be naturally studied. In particular, a globally hyperbolic Lorentzian manifold admits a space-like submanifold on which we can define the initial data associated with a given hyperbolic system, and copies of this space-like submanifold foliate the entire space-time. The celebrated results of Choquet-Bruhat show that the Cauchy problem for vacuum General Relativity is well-posed, establishing the modern paradigm to understand the evolution and Cauchy problem of certain second-order hyperbolic systems that define gravitational theories. Within this paradigm, these gravitational theories can be understood, in their globally hyperbolic regime, in terms of a system of evolution equations in Euclidean signature for certain constrained initial data sets that satisfy a Riemannian system of equations.

Given that supergravity is a landmark of a gravitational theory, it is natural to study the supergravity initial value or Cauchy problem on globally hyperbolic Lorentzian configurations, namely configurations whose associated metric is globally hyperbolic. It should be noted that the Cauchy problem in supergravity is richer than in other, more standard, gravitational theories since a given supergravity involves two different systems of equations. A supergravity theory, by which we mean a bosonic supergravity theory, consists of its equations of motion, which follow from a local Lagrangian principle, together with its Killing spinor equations, which consist of a spinorial differential system, typically of first order. The equations of motion determine a Cauchy problem, defining the corresponding supergravity evolution flow and constraint equations, whereas the supersymmetric Killing spinor equations define in turn their own Cauchy problem, giving rise to their first-order supersymmetric evolution flow and constraint equations. Hence, we obtain two different systems of evolution and constraint equations that are intimately related as a consequence of the Killing spinor equations providing partial first-order integrability of the supergravity equations of motion. Many natural mathematical questions arise in this rich framework of interacting evolution flows, ranging from the study of the moduli of initial data admissible for both evolution flows, to the classification of the diffeomorphism types of supersymmetric compact Cauchy surfaces, or the development of the conformal method for supersymmetric constrained initial data.

Articles and preprints within this research area.

The local moduli space of the Einstein-Yang-Mills system, S. Bunk, V. Muñoz, C. S. Shahbazi. Asian Journal of Mathematics, 2025.
Parallel spinor flows on three-dimensional Cauchy hypersurfaces, Á. Murcia, C. S. Shahbazi. Journal of Physics A: Mathematical and Theoretical, 2023.
Supersymmetric Kundt four manifolds and their spinorial evolution flows, Á. Murcia, C. S. Shahbazi. Letters in Mathematical Physics, 2023.
Heterotic solitons on four-manifolds, A. Moroianu, Á. Murcia, C. S. Shahbazi. New York Journal of Mathematics, 2022.
Parallel spinors on globally hyperbolic Lorentzian four-manifolds, Á. Murcia, C. S. Shahbazi. Annals of Global Analysis and Geometry, 2021.
Contact metric three manifolds and Lorentzian geometry with torsion in six-dimensional supergravity, Á. Murcia, C. S. Shahbazi. Journal of Geometry and Physics, 2021.
Transversality for the moduli space of Spin (7)-instantons, V. Muñoz, C. S. Shahbazi. Reviews in Mathematical Physics, 2020.
Orientability of the moduli space of Spin (7)-instantons, V. Muñoz, C. S. Shahbazi. Pure and Applied Mathematics Quarterly, 2017.

Categorified geometry and higher-dimensional supergravity

Beyond four dimensions, my research also focuses on the mathematical theory of higher-dimensional supergravity, especially in ten and eleven Lorentzian dimensions. In these dimensions, supergravity describes the low-energy dynamics of the massless sector of string/M theory, and can be used to access various highly non-trivial non-perturbative aspects of these theories. Unsurprisingly, these supergravity theories involve remarkably sophisticated mathematics; in particular, they naturally involve higher gauge theories and categorified principal bundles.

The need for categorified geometry in supergravity has an arguably very innocent origin. The type of fields that can occur as part of a given supergravity theory is highly constrained by supersymmetry, sometimes to the point of being uniquely determined by supersymmetry at the local level, as it happens with the maximally supersymmetric supergravities. Physicists quickly realized that some supergravity theories required for their consistency gauge fields that did not correspond to local one-forms taking values on a Lie algebra, which were the type of gauge fields that physicists had studied up to that point in the context of Yang-Mills theories. Instead, imposing supersymmetry required the existence of fields that were locally represented by higher-order differential forms. These forms enjoyed a gauge principle and admitted a notion of curvature, similar to that of the traditional one-form gauge fields, albeit being of higher order. At the local level, it is possible to work with these higher gauge fields by keeping track of the hierarchy of gauge transformations that they involve and the higher curvatures they define, which are sometimes coupled to each other, giving rise to deep algebraic structures. A completely different story emerges when trying to elucidate the global mathematical structure behind these higher gauge fields. By this, I refer to constructing the type of global geometric objects on which these higher gauge fields can be defined as connections involving the correct notion of curvature, gauge transformations, and parallel transport. This turns out to be a surprisingly sophisticated mathematical problem that requires some of the latest developments in the categorification of principal bundles, that is, in higher geometry, a very active area of research at the intersection of geometry, topology, and higher category theory. Regarding supergravity, my main goal is to construct ten-dimensional supergravity globally on the appropriate notion of categorified principal bundle and study its moduli space of globally hyperbolic solutions.

Articles and preprints within this research area.

The Heterotic-Ricci flow and its three-dimensional solitons, A. Moroianu, Á. J. Murcia, C. S. Shahbazi. The Journal of Geometric Analysis, 2024.
Higher Geometric Structures on Manifolds and the Gauge Theory of Deligne Cohomology,. S. Bunk, C. S. Shahbazi. arXiv preprint, 2023.
Canonical metrics on holomorphic Courant algebroids, M. Garcia-Fernandez, R. Rubio, C. Shahbazi, C. Tipler. Proceedings of the London Mathematical Society, 2022.
Products of multisymplectic manifolds and homotopy moment maps, C. S. Shahbazi, M. Zambon. Journal of Lie Theory, 2016.

Miscellanea

In this section, I include the part of my research that does not fit into the previous four research topics, and which corresponds to my prior work on theoretical and mathematical physics in the areas of supergravity and string theory.