THESIS DEFENSE -- Supersymmetric Vertex Algebras and Killing Spinors

Speaker:  Andoni De Arriba De La Hera (ICMAT-CSIC) ()
Advisor/s:  Luis Álvarez Cónsul (ICMAT-CSIC) and Mario García Fernández (UAM-ICMAT)
Date:  Friday, 05 May 2023 - 09:15
Place:  Sala de Grados, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid


The goal of the present thesis is to construct embeddings of the N=2 superconformal vertex algebra, motivated by mirror symmetry, into the Chiral de Rham complex, provided that we have solutions to the Killing spinor equations. Our approach to the Chiral de Rham complex is based on the universal construction by Bressler and Heluani, which applies to any Courant algebroid over a smooth manifold. In fact, the main results are based on the approach to SUSY vertex algebras given by Heluani and Kac, and furthermore extend the techniques developed by Heluani and Zabzine to construct N = 2 superconformal structures on the Chiral de Rham complex. The Killing spinor equations that are considered come from the approach to special holonomy based on Courant algebroids in generalized geometry, and are inspired by the physics of heterotic supergravity and string theory.

The embeddings are constructed in two different set-ups. Firstly, for equivariant Courant algebroids over homogeneous manifolds, where the construction reduces to embeddings into the superaffinization of a quadratic Lie algebra, and the Killing spinor equations become purely algebraic conditions that can be checked on explicit examples. Secondly, for transitive Courant algebroids over complex manifolds, where these equations are equivalent to the Hull-Strominger system, with origins in the heterotic sigma-model studied by physicists. Several examples have been studied where the obtained results are applied.

As an application, the thesis presents the first examples of (0,2) mirror symmetry on compact non-Kähler complex manifolds via the Chiral de Rham complex. In fact, this thesis lays the ground to Borisov's vertex algebra approach to (0,2) mirror symmetry on non-Kähler manifolds.

More information: