I will explain how preservation of space-time supersymmetry, expressed as the existence of certain class of Killing spinors, determines embeddings of canonical supersymmetric vertex algebras in the space of sections of a sheaf of vertex algebras called the chiral de Rham complex. The Killing spinor equations considered here are part of an approach to special holonomy based on Courant algebroids in generalized geometry, and are inspired by heterotic supergravity. The link with the chiral de Rham complex is provided by a universal construction given by Bressler and Heluani that attaches a sheaf of vertex algebras to any Courant algebroid.
More specifically, I will consider three types of embeddings:
(1) Embeddings of the N=2 superconformal vertex algebra in two different set-ups, namely in the superaffinization of a quadratic Lie algebra, satisfying appropriate algebraic conditions, and in the space of sections of the chiral de Rahm complex of a Courant algebroid appearing in supergravity. The first set-up is applied to obtain the first examples of (0,2) mirror symmetry on compact non-Kähler complex manifolds. This part is based on my PhD thesis and joint work with Luis Álvarez-Cónsul and Mario Garcia-Fernandez (IRMN 2024 and arXiv:2305.06836).
(2) Embeddings of two commuting N=2 superconformal vertex algebras in the chiral de Rham complex of a generalized Kähler manifold, which is a joint work in progress with Luis Álvarez-Cónsul, Mario Garcia-Fernandez and Jethro van Ekeren.
(3) Examples of Embeddings of the G2 Shatashvili-Vafa vertex algebra in the chiral de Rham complex of a seven-dimensional Riemannian manifold, which is a joint work with Mateo Galdeano and Mario Garcia-Fernandez (arxiv.2502.02769).

Place: Room 206, RIMS