Cohomology of multipoint connections on complex curves

Abstract

Given complex functions on complex curves satisfying recursion relations with respect to the number of marked points at which they are evaluated, we construct a cohomology theory governing such recursions, expressed in terms of a generalization of holomorphic connections which we call multipoint connections. We identify precisely the sense in which the associated coboundary operators square to zero, as an integrability condition on the recursion kernels rather than as an identity forcing the correlation functions themselves to vanish, and prove, following the general theory of systems of functional equations, that this condition has non-trivial solution spaces. The recursion cohomology is computed explicitly, in Proposition \ref{duda}, in terms of the coefficient data of the defining recursion. We illustrate the theory on the rational, elliptic, Jacobi-form, genus-two, and Schottky-uniformized genus-$g$ recursions governing Zhu-reduction-type correlation functions, recovering the higher-genus counterparts of the Weierstrass functions as analytic continuations of solutions of the corresponding functional equations. We then develop applications of the theory: an identification of the recursion cohomology with the flat sections of the Knizhnik-Zamolodchikov connection, an interpretation of the coboundary data in terms of the cohomology of continual contragredient Lie algebras and the associated Toda hierarchies, a comparison with the Verlinde bundle of conformal blocks over the moduli space of curves, and a discussion of the Gauss-Manin-type flatness of the resulting bundle over Schottky space.