Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

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.

Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

We study an analog of the AGT relation in five dimensions. We conjecture that the instanton partition function of 5D N=1 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.

Authorship：Lead author   Language：Japanese  

位相的頂点は３つのヤング図でパラメトライズされたある関数で、
位相的弦の分配関数（もしくは グロモフ-ウィッテン不変量など）
を計算するための強力な道具である。
しかし、ゲージ理論/弦理論対応などを通じ、
ゲージ理論のネクラソフの公式、絡み目不変量、
結晶融解や3次元分割など、色々な量と関係している。

Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

We consider a topological quiver matrix model which is
expected to give a dual description of the instanton dynamics
of topological U(N) gauge theory on D6 branes.
The model is a higher dimensional analogue of the ADHM matrix model
that leads to Nekrasov's partition function.
The fixed points of the toric action
on the moduli space are labeled by colored plane partitions.
Assuming the localization theorem, we compute
the partition function as an equivariant index.
It turns out that the partition function does not depend on
the vacuum expectation values of Higgs fields that break
U(N) symmetry to U(1)^N at low energy.
We conjecture a general formula of the partition function,
which reduces to a power of the MacMahon function,
if we impose the Calabi-Yau condition.
For non Calabi-Yau case we prove the conjecture
up to the third order in the instanton expansion.

Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

It has been argued that Nekrasov's 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 (hep-th/0502061) is a building block of
Nekrasov's partition function with two equivariant parameters.
Compared with another refined topological vertex
by Iqbal, Kozcaz and Vafa (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 C^2.
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.

Authorship：Lead author   Language：English   Publishing type：Research paper (scientific journal)  

We argue the connection of Nekrasov's partition function
in the $\Omega$ background and the moduli space of $D$-branes,
suggested by the idea of geometric engineering and Gopakumar-Vafa
invariants. In the instanton expansion of ${\cal N} =2$ $SU(2)$ Yang-Mills theory
the Nakrasov's partition function with equivariant parameters
$\epsilon_1, \epsilon_2$ of toric action on ${\mathbb C}^2$ factorizes
correctly as the character of $SU(2)_L \times SU(2)_R$ 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) ${\bf F}_0$.
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 $T^2$ action allows us
to obtain the generating functions of equivariant $\chi_y$
and elliptic genera of the Hilbert scheme of $n$ points on ${\mathbb C}^2$
by the method of topological vertex.