Holomorphic field theory in one dimension (chiral conformal field theory, for example) has been of central importance in theoretical and mathematical physics for many years, and has had enormous influence in mathematics as well. These models are closely connected to the representation theory of certain infinite-dimensional (Virasoro and Kac-Moody) Lie algebras, which enhance finite-dimensional conformal and global symmetries in this setting. I will advocate for the viewpoint that higher-dimensional holomorphic theories ought to be studied in the same manner and with the same relish. In particular, I will discuss some infinite-dimensional Lie algebras, recently considered by Gwilliam-Williams and Faonte-Hennion-Kapranov, which generalize Kac-Moody and Virasoro geometrically to any complex dimension. In similar fashion, these algebras enhance conformal and global symmetries in holomorphic twists of supersymmetric theories. In the context of four-dimensional $\mathcal{N}=2$ superconformal theories, the higher Virasoro and Kac-Moody algebras localize to lines after a further twist and reproduce two-dimensional chiral algebras of recent interest, together with the correct central extensions; however, additional deformations of the higher Virasoro are available, which (for example) can localize the theory to any complex curve.

(This work is joint with Brian Williams)