A symmetry reduction technique for higher order Painlev\'e systems

The symmetry reduction of higher order Painlev\'e systems is formulated in terms of Dirac procedure. A set of canonical variables that admit Dirac reduction procedure is proposed for Hamiltonian structures governing the ${A^{(1)}_{2M}}$ and ${A^{(1)}_{2M-1}}$ Painlev\'e systems for $M=2,3,...$.Comment: to appear in Phys. Lett.

Macdonald's symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces

Quantum analogues of the homogeneous spaces \GL(n)/\SO(n) and \GL(2n)/\Sp(2n) are introduced. The zonal spherical functions on these quantum homogeneous spaces are represented by Macdonald's symmetric polynomials P_{\ld}=P_{\ld}(x_1,\cdots,x_n;q,t) with $t=q^{1 \over 2}$ or $t=q^2$

Higgs Mechanism in Nonlocal Field Theories

We study spontaneous gauge symmetry breaking and the Higgs mechanism in nonlocal field theories. Motivated by the level truncated action of string field theory, we consider a class of nonlocal field theories with an exponential factor of the d'Alembertian attached to the kinetic and mass terms. Modifications of this kind are known to make mild the UV behavior of loop diagrams and thus have been studied not only in the context of string theory but also as an alternative approach to quantum gravity. In this paper we argue that such a nonlocal theory potentially includes a ghost mode near the nonlocal scale in the particle spectrum of the symmetry broken phase. This is in sharp contrast to local field theories and would be an obstruction to making a simple nonlocal model a UV complete theory. We then discuss a possible way out by studying nonlocal theories with extra symmetries such as gauge symmetries in higher spacetime dimensions.Comment: 19 pages, 4 figures; v2: references added, version published in JHE

Birational Weyl group action arising from a nilpotent Poisson algebra

We propose a general method to realize an arbitrary Weyl group of Kac-Moody type as a group of birational canonical transformations, by means of a nilpotent Poisson algebra. We also give a Lie theoretic interpretation of this realization in terms of Kac-Moody Lie algebras and Kac-Moody groups.Comment: 31 pages, LaTe

Symmetries in the fourth Painleve equation and Okamoto polynomials

We propose a new representation of the fourth Painlev\'e equation in which the $A^{(1)}_2$-symmetries become clearly visible. By means of this representation, we clarify the internal relation between the fourth Painlev\'e equation and the modified KP hierarchy. We obtain in particular a complete description of the rational solutions of the fourth Painlev\'e equation in terms of Schur functions. This implies that the so-called Okamoto polynomials, which arise from the $\tau$-functions for rational solutions, are in fact expressible by the 3-reduced Schur functions.Comment: 25 pages, amslate