Holomorphic field theories and higher-dimensional vertex algebras

Chiral conformal field theory in one complex dimension spurred Beilinson and Drinfeld's invention of
factorization algebras, a global counterpart to vertex algebras, which they then deployed in geometric
Langlands theory. I will outline a close cousin of their definitions that applies in arbitrary dimension,
but with the flavor of complex differential geometry. Along the way, I will describe several field theories
(including some twists of supersymmetric gauge theories) that provide examples in physics.