A New Topological Term in 2d Field Theory

This talk describes some work done jointly with L. Alvarez-Gaume, J.B. Bost, G. Moore, and C. Vafa [1][2], building on [3] and [4). We were concerned with establishing bosonization results on two-dimensional surfaces with complicated topology. Far from being a mere curiosity, bosoniza­tion is of great interest in string theory. For example, bosonization has been used in light-cone gauge to prove the equivalence of the Green-Schwarz and NSR superstring [5][6]. Bosonization also plays a key role in understanding the gauge­ and super-symmetry of the heterotic string [7] and in formulating the covariant fermion emission vertex [8][9]. The papers [1], [2] generalize existing results on Fermi-Bose equivalence for Fermi fields of any spin on the sphere [10]-[13]. In this talk I will only discuss a subproblem, that of bosonizing spin 1/2 on the torus. It turns out that this problem is only slightly more difficult than the sphere case. One needs a way to tell the bosonic theory which of the various spin structures it is to mimic; this is accomplished by adding to the bosonic action a new global term. The new term is already familiar to mathematicians as the parity of a spin structure; it has an immediate generalization to any genus surface