Tautological relations and the r-spin Witten conjecture

In a series of two preprints, Y.-P. Lee studied relations satisfied by all formal Gromov-Witten potentials, as defined by A. Givental. He called them "universal relations" and studied their connection with tautological relations in the cohomology ring of moduli spaces of stable curves. Building on Y.-P. Lee's work, we give a simple proof of the fact that every tautological relation gives rise to a universal relation (which was also proved by Y.-P. Lee modulo certain results announced by C. Teleman). In particular, this implies that in any semi-simple Gromov-Witten theory where arbitrary correlators can be expressed in genus 0 correlators using only tautological relations, the formal and the geometric Gromov-Witten potentials coincide. As the most important application, we show that our results suffice to deduce the statement of a 1991 Witten conjecture on r-spin structures from the results obtained by Givental for the corresponding formal Gromov-Witten potential. The conjecture in question states that certain intersection numbers on the moduli space of r-spin structures can be arranged into a power series that satisfies the r-KdV (or r-th higher Gelfand-Dikii) hierarchy of partial differential equations.Comment: 46 pages, 7 figures, A discussion of the analyticity of Gromov-Witten potentials and a more careful description of Givental's group action added in Section 5; minor changes elsewher

Conformal Invariance of Spin Correlations in the Planar Ising Model

We rigorously prove the existence and the conformal invariance of scaling limits of the magnetization and multi-point spin correlations in the critical Ising model on arbitrary simply connected planar domains. This solves a number of conjectures coming from the physical and the mathematical literature. The proof relies on convergence results for discrete holomorphic spinor observables and probabilistic techniques.Comment: Changes in this version: the explicit formula for n-point spin correlations is proved in full generality. The appendix is rewritten completely and contains this new proof, the introduction is changed accordingly, the presentation in Sections 2.5 and 2.7(2.8) is rearranged slightly. 42 pages, 2 figure

Computation of the one-loop Symanzik coefficients for the square action

We compute the one-loop coefficients for an alternative Symanzik improved pure gauge SU{N} lattice action (N=2 and N=3). For the standard Symanzik improved action we confirm previous results by L\"{u}scher and Weisz.Comment: 45 pages, LaTeX, includes library.ps for generating Feynman diagram