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(x₁, …, 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 E₆, E₇, E₈ or A_{n ? 1}, D_{n + 1} for n ≥ 3. © 2008 Wiley Periodicals, Inc.