Hidetoshi Awata

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

Instanton counting, Macdonald function and the moduli space of D-branes

mathcal{N} = 1

REFINED BPS STATE COUNTING FROM NEKRASOV'S FORMULA AND MACDONALD FUNCTIONS

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

We argue the connection of Nekrasovs partition function in the Ω background and the moduli space of D-branes, suggested by the idea of geometric engineering and Gopakumar-Vafa invariants. In the instanton expansion of = 2 SU(2) Yang-Mills theory the Nakrasovs partition function with equivariant parameters 1,2 of toric action on 2 factorizes correctly as the character of SU(2)L × SU(2)R spin representation. We show that up to two instantons the spin contents are consistent with the Lefschetz action on the moduli space of D2-branes on (local) F0. We also present an attempt at constructing a refined topological vertex in terms of the Macdonald function. The refined topological vertex with two parameters of T2 action allows us to obtain the generating functions of equivariant χy and elliptic genera of the Hilbert scheme of n points on 2 by the method of topological vertex.

The partition function of ABJ theory

It has been argued that Nekrasovs partition function gives the generating function of refined BPS state counting in the compactification of M theory on local Calabi–Yau spaces. We show that a refined version of the topological vertex we previously proposed (arXiv:hep-th/0502061) is a building block of Nekrasovs partition function with two equivariant parameters. Compared with another refined topological vertex by Iqbal, Kozcaz and Vafa (arXiv:hep-th/0701156), our refined vertex is expressed entirely in terms of the specialization of the Macdonald symmetric functions which is related to the equivariant character of the Hilbert scheme of points on ℂ2. We provide diagrammatic rules for computing the partition function from the web diagrams appearing in geometric engineering of Yang–Mills theory with eight supercharges. Our refined vertex has a simple transformation law under the flop operation of the diagram, which suggests that homological invariants of the Hopf link are related to the Macdonald functions.

Volume Conjecture: Refined and Categorified

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.

Quantum Algebraic Approach to Refined Topological Vertex

This article has been accepted for publication in . Prog. Theor. Exp. Phys. (2013) 053B04 ndoi: 10.1093/ptep/ptt014. Published by Oxford University Press.

Explicit examples of DIM constraints for network matrix models

The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the A-polynomial

FUSION RULES FOR THE FRACTIONAL LEVEL

A(x,y)

\widehat{{\rm sl}(2)}

. Another family version of the volume conjecture depends on a quantization parameter, usually denoted