The most well known examples of holomorphic field theories come from chiral CFTs in complex dimension one. Of the symmetries of these theories, a special role is played by both the Kac-Moody and Virasoro vertex algebras. In this talk I will discuss generalizations of these symmetries to higher dimensions in the language of factorization algebras. For example, holomorphic gauge theories in complex dimension d (theories depending on the data of a holomorphic connection) generically have the symmetry of a certain dg Lie algebra studied recentlyby Faonte, Hennion, and Kapranov. My talk will focus on this higher Kac-Moody algebra and I will relate it to a shadow of a richer OPE algebra.I will give interpretations of generalizations of certain objects in representation theory, such as the KZ connection. Time permitting, I will discuss how to understand these higher algebras as living on the boundaryof twists of certain supersymmetric gauge theories, mirroring the relationship between Chern-Simons theory and WZW.