Yasuhiko Yamada

Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries

Publisher Summary This chapter focuses on the conformal field theory (CFT) on universal family of stable curves with gauge symmetries. CFT has not only useful application to string theory and two-dimensional critical phenomena but also has beautiful and rich mathematical structure, and it has interested many mathematicians. CFT is characterized by infinite-dimensional symmetry such as Virasoro algebra. Especially, its correlation functions are characterized by differential equations arising from representations of infinite-dimensional Lie algebras. Physically, correlation functions should have the properties such as locality, holomorphic factorization, and monodromy invariance (duality). To build conformal field theory having such properties, usual approach is to construct holomorphic (chiral) conformal blocks, which are the half of the theory and to study its monodromy.

Five-dimensional AGT conjecture and the deformed Virasoro algebra

We study an analog of the AGT (Alday-Gaiotto-Tachikawa) relation in five dimensions. We conjecture that the instanton partition function of 5D

Elliptic genera and N=2 superconformal field theory

\mathcal{N} = 1

Affine Weyl groups, discrete dynamical systems and Painleve equations

pure SU(2) gauge theory coincides with the inner product of the Gaiotto-like state in the deformed Virasoro algebra. In four-dimensional case, a relation between the Gaiotto construction and the theory of Braverman and Etingof is also discussed.

Five-Dimensional AGT Relation and the Deformed β-Ensemble

Abstract Recently Witten proposed to consider the elliptic genus in N = 2 superconformal field theory to understand the relation between N = 2 minimal models and Landau-Ginzburg theories. In this paper we first discuss the basic properties satisfied by elliptic genera in N = 2 theories. These properties are confirmed by some fundamental class of examples. Then we introduce a generic procedure to compute the elliptic genera of a particular class of orbifold theories, i.e. the ones orbifoldized by e2πiJ0 in the Neveu-Schwarz sector. This enables us to calculate the elliptic genera for Landau-Ginzburg orbifolds. When the Landau-Ginzburg orbifolds allow an interpretation as target manifolds with SU (N) holonomy we can compare the expressions with the ones obtained by orbifoldizing tensor products of N = 2 minimal models. We also give sigma model expressions of the elliptic genera for manifolds of SU (N) holomony.

Singular vectors of the Virasoro algebra in terms of Jack symmetric polynomials

Abstract:A new class of representations of affine Weyl groups on rational functions are constructed, in order to formulate discrete dynamical systems associated with affine root systems. As an application, some examples of difference and differential systems of Painlevé type are discussed.

A study on the fourth q-Painlevé equation

We discuss an analog of the AGT relation in five dimensions. We define a q-deformation of the beta-ensemble which satisfies q-W constraint. We also show a relation with the Nekrasov partition function of 5D SU(N) gauge theory with N_f=2N.

ENERGY FUNCTIONS IN BOX BALL SYSTEMS

We present an explicit formula of the Virasoro singular vectors in terms of Jack symmetric polynomials. The parametert in the Virasoro central chargec=13-6(t+1/t) is just identified with the deformation parameter α of Jack symmetric polynomialsJγ(α). As a by-product, we obtain an integral representation of Jack symmetric polynomials indexed by the rectangular Young diagrams.

Hypergeometric solutions to the q-Painlevé equations

A q-difference analogue of the fourth Painleve equation is proposed. Its symmetry structure and some particular solutions are investigated.

Integral formulas for the WZNW correlation functions

The box ball system is studied in the crystal theory formulation. New conserved quantities and the phase shift of the soliton scattering are obtained by considering the energy function (or H function) in the combinatorial R matrix.