Higher-dimensional dg-Virasoro algebra from an ambitwistor space in terms of generators and relations

Critical phenomena are quite widespread in the world around us: from the Curie point of the magnets to the merger of liquid and vapour phases. These phenomena are known to be described by the conformal field theory (CFT). It is a field theory of a very special kind, which captures the scale invariance property of the system in question, arising naturally, as the system approaches the critical point. The study of CFTs achieved its greatest success in dimension $d = 2$, due to the fact that the algebra of conformal symmetries in this case is an infinite-dimensional Virasoro algebra. In $d > 2$, this is no longer the case; hence, the analytic results for higher-dimensional CFTs are mostly limited to supersymmetric systems, and people have to resort to numerical methods. In a recent paper, Mikhail Kapranov proposed an idea to reclaim a version of this infinite-dimensionality in $d > 2$ CFT by studying the derived analogue of the Virasoro algebra, which ties together both infinitesimal symmetries and deformations of the conformal structure. To describe it explicitly, he used the ambitwistor space construction: the space of all null geodesics, encoding the conformal geometry of the original manifold. Concretely, consider the sheaf of holomorphic contact Hamiltonian vector fields on the ambitwistor space of a $d$-dimensional complex plane. Kapranov’s higher-dimensional Virasoro algebra is a derived functor of the global sections of this sheaf. It is endowed with the structure of a dg-Lie algebra through Dold–Kan/Thom–Sullivan correspondence. I will write this algebra down explicitly in terms of generators and relations in the $d = 2$ and $d = 3$ cases, and explain how to find its central extension, which is crucial for physical applications.