Updated on 2018/03/26

AWATA, Hidetoshi

Organization
Graduate School of Mathematics Computational Mathematics Associate professor
School of Science

### Degree 1

1. PhD （ 1993.3   Hokkaido University ）

### Research Interests 2

1. Integrable Model

2. Quantum Field Theory

### Research Areas 1

1. Others / Others  / Elementary Particle/Atomic Nucleus/Cosmic Ray/Space Physics

### Current Research Project and SDGs 1

1. Analysis of Quantum Field Theories and Mathematical Physics

### Education 1

1. Hokkaido University   Graduate School, Division of Natural Science   Physics

1990.4 - 1993.3

More details

Country： Japan

1. 日本数学会

### Papers 11

1. The partition function of ABJ theory Reviewed

Hidetoshi Awata, Shinji Hirano and Masaki Shigemori

Progress of Theoretical and Experimental Physics   Vol. 2013 ( 053B04 )   2013

More details

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

2. Quantum Algebraic Approach to Refined Topological Vertex Reviewed

H. Awata, B. Feigin and J. Shiraishi

JHEP   Vol. 1203   page： 041   2012.3

More details

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

3. Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String Reviewed

Hidetoshi Awata, Hiroyuki Fuji, Hiroaki Kanno, Masahide Manabe and Yasuhiko Yamada

Advances in Theoretical and Mathematical Physics   Vol. 16   page： 725--804   2012

More details

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

4. Notes on Ding-Iohara algebra and AGT conjecture

H. Awata, B. Feigin, A. Hoshino, M. Kanai, J. Shiraishi, S. Yanagida

RIMS Kôkyûroku, Proceeding of RIMS Conference 2010 "Diversity of the Theory of Integrable Systems" (ed. Masahiro Kanai)   Vol. 1765   page： 12-32   2011

More details

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

We study the representation theory of the Ding-Iohara algebra $\calU$ to find $q$-analogues of the Alday-Gaiotto-Tachikawa (AGT) relations. We introduce the endomorphism $T(u,v)$ of the Ding-Iohara algebra, having two parameters $u$ and $v$. We define the vertex operator $\Phi(w)$ by specifying the permutation relations with the Ding-Iohara generators $x^\pm(z)$ and $\psi^\pm(z)$ in terms of $T(u,v)$. For the level one representation, all the matrix elements of the vertex operators with respect to the Macdonald polynomials are factorized and written in terms of the Nekrasov factors for the $K$-theoretic partition functions as in the AGT relations. For higher levels $m=2,3,...$, we present some conjectures, which imply the existence of the $q$-analogues of the AGT relations.

5. *Macdonald operators and homological invariants of the colored Hopf link Reviewed

Hidetoshi Awata, Hiroaki Kanno

Journal of Physics   Vol. A44 ( 375201 )   2011

More details

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

Using a power sum (boson) realization for the Macdonald operators, we investigate the Gukov, Iqbal, Kozcaz and Vafa (GIKV) proposal for the homological invariants of the colored Hopf link, which include Khovanov-Rozansky homology as a special case. We prove the polynomiality of the invariants obtained by GIKV's proposal for arbitrary representations. We derive a closed formula of the invariants of the colored Hopf link for antisymmetric representations. We argue that a little amendment of GIKV's proposal is required to make all the coefficients of the polynomial non-negative integers.

6. *Five-dimensional AGT Conjecture and the Deformed beta-ensemble Reviewed

Progress of Theoretical Physics   Vol. 124   page： 227-269   2010.9

More details

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.

7. *Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra Reviewed

Journal of High Energy Physics   Vol. 1001 ( 125 )   2010.1

More details

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.

8. 位相的頂点とその周辺の話題 ---位相的場の理論や結び目理論など--- Invited

粟田英資

2009年度日本数学会秋期総合分科会　総合講演・企画特別講演アブストラクト     page： 44-53   2009.9

More details

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。

9. *Quiver Matrix Model and Topological Partition Function in Six Dimensions Reviewed

Hidetoshi Awata, Hiroaki Kanno

Journal of High Energy Physics   Vol. 0907 ( 076 )   2009.7

More details

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.

10. *Refined BPS state counting from Nekrasov's formula and Macdonald functions Reviewed

Hidetoshi Awata, Hiroaki Kanno

International Journal of Modern Physics A   Vol. 24 ( 12 ) page： 2253-2306   2009.5

More details

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.

11. *Instanton counting, Macdonald function and the moduli space of D-branes, Reviewed

H. Awata and H. Kanno

Journal of High Energy Physics   Vol. 0505 ( 039 )   2005

More details

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.

### Books 2

1. 数理物理への誘い６、最新の動向をめぐって、

小嶋泉、粟田英資、他、全11名（ Role： Joint author）

2006.8

More details

Language：Japanese

2. 数理物理への誘い3、最新の動向をめぐって、

江沢洋、粟田英資、他、全8名 （ Role： Joint author）

遊星社  2000.8

More details

Language：Japanese

### Presentations 2

1. 位相的頂点 ---インタートワイナーを中心に--- Invited

粟田英資

日本数学会　2017年秋の分科会　特別講演

More details

Event date： 2017.9

Language：Japanese   Presentation type：Oral presentation (invited, special)

Country：Japan

Ding-Iohara-Miki（丁-庵原-三木）代数\cite{DI:1997,Mi:2007}は、
微分作用素の代数\cite{rKR,Aw}の拡張であり、
$gl(1)$ タイプの量子トロイダル代数\cite{GKV}で、
2つの方向のアフィン性から2次元の中心を持つ。
$q$-ビラソロ代数\cite{SKAO:1996}や $q$-$W_N$ 代数\cite{rf:Feigin-Frenkel,rf:AKOS}
等の複雑な代数もその特殊な場合として内包し、
多くの対称性を持ち、又ホップ代数でもあり、非常に性質の良い代数である。
更に、ゲージ理論のネクラソフ関数\cite{rf:Nekrasov}
とも関係しており\cite{AFS}非常に興味深い。
ここではその幾つかを紹介する。

2. 位相的頂点とその周辺の話題 ---位相的場の理論や結び目理論など---

粟田英資

日本数学会　2009年秋の分科会　企画特別講演

More details

Event date： 2009.9

Language：Japanese   Presentation type：Oral presentation (invited, special)

Country：Japan

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。
ここではその幾つかを紹介する。

### Teaching Experience (On-campus) 2

1. Linear Algebra II

2011

2. Linear Algebra I

2011