High Energy Theory Seminar

In a TQFT of homological type, such as arises from the topological twist of a supersymmetric theory, it is well known that the local operators form an algebra, associative in dimension d >= 1 and commutative in d >= 2. The product comes from collision of operators. I will explain how modern mathematical ideas in TQFT (in particular, the notion of $E_d$ algebras) lead to the existence of a *secondary* product in any dimension d, which acts as a generalized Lie bracket, and has a simple physical definition in terms of topological descent. In d=2, the secondary product is familiar as part of the L-infinity structure of local operators; in d >= 3 it has been relatively unexplored. I will use the secondary product to give a topological interpretation of the physical idea that "Omega background leads to quantization." Time permitting, I will explain some generalizations of the secondary product to line and surface operators, with interesting applications in d=3 and d=4. (Based on joint work with C. Beem, D. Ben-Zvi, M. Bullimore, A. Neitzke.)