We reconsider the problem of deforming a conformal ﬁeld theory to a neighboring theory which is again critical. An invariant formulation of this problem is important for understanding the underlying symmetry of string theory. We give a simple derivation of A. Sen’s recent formula for the change in the stress tensor and show that, when correctly interpreted, it is coordinate-invariant. We give the corresponding superconformal perturbation for superﬁeld backgrounds and explain why it has no direct analog for spin-ﬁeld backgrounds. ]]>

such allowed coupling (up to boundary terms); this term will be present for any lipid bilayer composed of tilted chiral molecules. We calculate the renormalization-group behavior of this relevant coupling in a simpliﬁed model and show how thermal ﬂuctuations eﬀectivelyreduce it in the infrared. ]]>

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.

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