John Palmer - US grants

The two dimensional Ising model is a mathematical model for magnetic spins on a lattice which interact by nearest neighbor coupling. There is a vast literature in physics concerning the behavior of statistical systems near critical points (phase transitions), but there are few models in which it is possible to verify the conjectured behavior analytically. The importance of the Ising model is that it is a notable exception (in two dimensions). There, one has sufficient knowledge of the spin correlations to examine some of the more detailed conjectures about critical phenomena. This is of importance for statistical mechanics, quantum field theory, and the mathematical theories upon which the analysis rests. Professor Palmer is an expert on solvable models, such as the Ising model, in statistical mechanics. He is in the forefront in exploiting the relevant infinite dimensional group theory and algebraic geometry that has been so successful in these physical applications. In prior work he introduced a family of lattice models which are mathematical generalizations of the Ising model. In the current project he proposes to prove scaling hypotheses that rely upon prior work on infinite spin groups. He is also investigating the possibility of interpreting a certain invariance in the Ising model as the manifestation of a gauge group acting of the determinant bundle associated with a family of finite difference operators. This could clarify the appearance of conformal symmetry in solvable lattice models.

9401594 Palmer Holonomic fields are quantum field theories with an intimate connection to elliptic operators with isolated singularities. The correlation function or tau-function of such fields have appeared in a wide variety of contexts including, the scaling limit of the two dimensional Ising model, the Riemann-Hilbert problem on the complex sphere, level spacing distributions for random matrices, the KdV hierarchy and its relatives, the asymptotics of Toeplitz determinants, and recently intersection theory on the moduli space of Riemann surfaces. A reformulation of the original Sato, Miwa and Jimbo theory that has been worked out by Professor Palmer makes it possible to consider a variety of generalizations. The proposal here is to complete the analysis of the Dirac operators on the hyperbolic disk and to make the field theory connection by developing an analytic version of the theory of vertex operators. These vertex operators should also prove useful in simplifying (and generalizing) some of the algebraic analysis that goes into the theory of vertex operator algebras. Quantum field theories are mathematical models for quantum mechanics. In addition the use in mathematical physics in these theories often represent the beginning of new developments in mathematics. The proposal here involves explicit description of invariants of earlier quantum field theories. These descriptions lead to surprising connections with important recent mathematical developments in algebra and geometry. ***