Deformed Conformal Field Theory and the Symmetric Group

Abstract

By a proposal of Dijkgraaf, the generating function of simple Hurwitz numbers admits a striking reformulation: it is the partition function of a W1+∞-deformed conformal field theory. We make this claim explicit via a series of Fock space isomorphisms. In so doing, we discover a Clifford and Heisenberg action on the class functions of the symmetric group, easily computable by a Young-diagrammatic calculus. Finally, to present the advantages of the physical perspective, we elaborate on Zagier’s recent proof of the quasimodularity of the Bloch-Okounkov q-bracket to show that, in a CFT context, quasimodularity is a consequence of spin structures and spectral flow.