Geometry and analysis of spin equations

We introduce W ‐spin structures on a Riemann surface Σ and give a precise definition to the corresponding W ‐spin equations for any quasi‐homogeneous polynomial W . Then we construct examples of nonzero solutions of spin equations in the presence of Ramond marked points. The main result of the paper is a compactness theorem for the moduli space of the solutions of W ‐spin equations when W = W ( x 1 , …, x t ) is a nondegenerate, quasi‐homogeneous polynomial with fractional degrees (or weights) q i < ½ for all i . In particular, the compactness theorem holds for the superpotentials E 6 , E 7 , E 8 or A n − 1 , D n + 1 for n ≥ 3. © 2008 Wiley Periodicals, Inc.