これまでのセミナー (2012年度)
第1回（2012年4月23日 16:40-18:10）
講師： 細道和夫 氏 [京都大学基礎物理学研究所]
タイトル： M-theory and exact results in SUSY gauge theories
概要：
In superstring theory and M-theory, there are various spatially extended solitons called branes. Understanding their worldvolume dynamics has been a major problem in formal quantum field theory for the last 15 years.
In this talk I will first review some recent developments in supersymmetric gauge theories based on explicit path integration using SUSY localization principle. I will then discuss some application of the results which led to a new understanding of branes in M-theory.
第2回（2012年5月7日 17:00-18:30）
講師： 笹本智弘 氏 [千葉大学理学部]
タイトル： Exact solutions of the 1D KPZ equation
概要：
In 1986, Kardar, Parisi and Zhang proposed a nonlinear Langevin equation to describe surface growth phenomena. This KPZ equation has been studied extensively by various methods such as the dynamical renormalization group, mode-coupling etc. Two years ago the first exact solution for this equation was obtained for the one dimensional version with narrow wedge initial condition [1]. Almost at the same time, there appeared a very nice experiment by Takeuchi and Sano who observed that the height distribution of a surface in a liquid crystal turbulence is well described by the Tracy-Widom distribution from random matrix theory [2]. Since then there have been renewed interests in this equation and several generalizations have been obtained.
In this talk after reviewing these developments we present our recent results of the height distribution and the two point correlation function for the KPZ equation in the steady state [3]. We explain basic ideas for the derivation based on replica method and discuss its importance from the point of view of statistical mechanics. If time permits, we may also discuss connections to quantum Toda lattice and Macdonald symmetric functions.
[1] T. Sasamoto and H. Spohn, " One-Dimensional Kardar-Parisi-Zhang Equation: An Exact Solution and its Universality", Phys. Rev. Lett. 104, 230602 (2010).
[2] K.A. Takeuchi and M. Sano,"Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals", Phys. Rev. Lett. 104, 230601 (2010).
[3] T. Imamura, T. Sasamoto, "Exact solution for the stationary KPZ equation", arXiv:1111.4634, to appear in PRL.
臨時セミナー (2012年5月18日 15:00-16:30)
講演者: Ivan Corwin (Microsoft Research and MIT)
題目: Integrability in the Kardar-Parisi-Zhang universality class
概要:
We will study a variety of disordered systems (including directed polymers, stochastic growth models and interacting particle systems). The exact solvability of some of these systems is due to connections with objects from integrable systems, including Macdonald symmetric functions, tropical Robinson-Schensted-Knuth correspondence, and quantum many body systems. The purpose of this talk is to explain how these objects arise and highlight some of the results which come from this structure.
第3回（2012年5月21日 17:00-18:30）
講師： 立川裕二 氏 [東京大学理学部]
題目： The ABCDEFG of Instantons and W-algebras
概要：
It has been proved that the W-algebra of type SU(N) acts on the cohomology of the moduli space of instantons of SU(N).
In this talk, this correspondence will be formulated in a form applicable for any gauge group; we will see that the instantons of a non-simply-laced group give the twisted sector of a W-algebra for an associated corresponding simply-laced algebra.
This talk is based on the speaker's collaboration with Keller, Mekareeya and Song,
http://arxiv.org/abs/1111.5624 .
第4回（2012年6月4日 17:00-18:30）
講師： 木村太郎 氏 [理化学研究所]
題目： Gauge theory and matrix models: β-deformation and orbifolding
概要：
Matrix model is one of the quite useful tool to explore non-perturbative aspects of quantum field theory. Recently it has been pointed out the importance of the β-deformed matrix model in supersymmetric gauge theory. In this talk we would like to show that such a β-deformed matrix model is naturally arising from Nekrasov's instanton partition function, and also its generalization. In particular we deal with the matrix model corresponding to the instanton counting on orbifold, and then derive Seiberg-Witten curve as the spectral curve of the matrix model. We will also show that the root of unity limit of the q-deformed partition function plays a crucial role on the orbifolding procedure. This talk is based on the following papers,
T. Kimura, JHEP 1109 (2011) 015 [arXiv:1105.6091];
第5回（2012年6月8日 17:00-18:30）
講師： 藤 博之 氏 [名古屋大学理学部]
題目： 体積予想とその一般化に関して
概要：
体積予想は結び目に対する色付きJones多項式の漸近的振る舞いと，結び目補空間の幾何的不変量とを関係付け，Kashaev氏によって提案され，村上順氏と村上斉氏によって提唱された予想である．特に双曲結び目に対しては，幾何的不変量として双曲体積が知られており，色付きJones多項式の漸近値とこの双曲体積が一致するという現象が様々な結び目に対して検証されてきた．
本講演では，この体積予想の物理的側面を概観した後，1-パラメータ拡張された体積予想と結び目に対する特性多項式である「A-多項式」との関係を基に，その弦理論的記述やホモロジカルな結び目不変量への一般化などを論じたい．
尚、本講演は、Sergei Gukov氏とPiotr Sulkowski氏との以下の共同研究に基づいている：
Hiroyuki Fuji, Sergei Gukov, Piotr Sułkowski,
"Volume Conjecture: Refined and Categorified",
第6回（2012年7月2日 17:00-18:30）
講師： 村上 斉 氏 [東京工業大学]
題目： 色付きJones多項式と結び目の基本群の表現
概要：
リー環 sl(2;C) に付随した結び目の量子不変量である色付きJones多項式の漸近挙動が，その結び目補空間の双曲的体積を決定するだろう，というのがKashaev氏，村上順氏，講演者によって提唱された体積予想である．
本講演では，体積予想の一般化として，複素パラメータで動かしたときの色付きJones多項式の漸近挙動を考え，そのパラメータが結び目補空間の双曲構造を決めるであろうという予想を紹介する．特に，その双曲構造に対応した双曲体積，Chern-Simons不変量，Reidemeister torsionがどのように得られるかを解説する．
この講演は次の論文に基づいている．
"The colored Jones polynomial, the Chern--Simons invariant, and the Reidemeister torsion of the figure-eight knot", arXiv:1102.3530
また，体積予想とその一般化については，次の解説も参照してください．
"An introduction to the volume conjecture and its generalizations", arXiv:0802.0039
第7回（2012年7月9日 17:30-19:00）
講師： 今村 洋介 氏 [東京工業大学]
題目： Exact results and dualities
概要：
近年、超対称ゲージ理論の分配関数が様々な背景時空上で厳密に計算され、ゲージ理論の双対性の研究に役立っている。本講演では、これまでに計算された分配関数のうち幾つかについて、その導出と応用例について簡単に触れた後、私自身が最近研究している３次元球面上のオービフォールド上の分配関数に関する話題について解説する。
臨時セミナー (2012年9月26日 13:30-15:00)
講演者: 鈴木了 氏 [Utrecht University]
題目: Spectrum for Y=0 brane in planar AdS/CFT
概要:
The spectrum of states with boundaries in AdS/CFT correspondence is revisited from an integrability point of view. It is believed that open strings ending on Y=0 maximal giant graviton brane in AdS_5xS^5 are dual to determinant-like gauge-invariant operators in N=4 super Yang-Mills theory. We conjecture that the exact spectrum of these states is described by the same Y-system as in the periodic case.
第8回（2012年9月26日 17:00-18:30）
講師： 酒井一博 氏 [京都大学基礎物理学研究所]
題目： Counting BPS states in E-string theory
概要：
The E-string theory is one of the simplest interacting quantum field theories with minimal (1,0) supersymmetry in six dimensions. It is obtained as the world-volume theory of a small E_8 instanton in heterotic string theory. The theory shows extremely rich properties when toroidally compactified down to lower dimensions.
In this talk, I will discuss the counting of BPS states in the E-string theory compactified on T^2 and its relation to Seiberg-Witten theory and the counting of holomorphic curves in half K3 and del Pezzo surfaces. I will then introduce a Nekrasov-type explicit expression for the BPS partition function which we have found recently and discuss its properties. (arXiv:1111.3967, 1203.2921, 1207.5739)
第9回（2012年10月10日 17:00-18:30）
講師： Prof. Ivan Cherednik [University of North Carolina]
題目： New Theory of q, t-hypergeometric and q-Whittaker Functions
概要：
The lecture will be devoted to the new theory of global difference hypergeometric and Whittaker functions, one of the major applications of the double affine Hecke algebras and a breakthrough in the classical harmonic analysis. They integrate the Ruijsenaars-Macdonald QMBP and the eigenvalue problem for the Q-Toda operators (any root systems) and are analytic everywhere ("global") with superb asymptotic behavior.
The definition of the global functions was suggested about 13 years ago; it is conceptually different from the definition Heine gave in 1846, which remained unchanged and unchallenged since then. Algebraically, the new functions are closer to Bessel functions than to the classical hypergeometric and spherical functions. The analytic theory of these functions and their q-Whittaker limits was completed very recently (the speaker and Jasper Stokman).
The construction is based on DAHA, which are deformations of the classical Heisenberg-Weyl algebras. The global functions are defined as the reproducing kernels of Fourier-DAHA transforms. The nonsymmetric Macdonald polynomials (eigenfunctions of difference Dunkl operators) play a key role. The nonsymmetric theory is a new powerful tool in the representations theory and the theory of special functions, generally, beyond the Lie theory.
The case of sl(2) will be mainly considered; no special knowledge of representation theory is assumed. The purpose is to reach (if time allows) the global q-Whittaker functions, which connect the Givental-Lee theory (Gromov-Witten invariants of flag varieties) with the algebraic theory of affine flag varieties, an exact link between the physics A-model and B-model for flag varieties at level of generating functions.
第10回（2012年10月24日 17:00-18:30）
講師： 奥田拓也 氏 [東京大学駒場]
題目： Vortex loop operators and mirror symmetry in three dimensions
概要：
Mirror symmetry in three dimensions is a supersymmetric generalization of particle/vortex duality that exchanges elementary particles with vortex solitons. We consider "vortex loop operators", defined by vortex-like singularities along a curve, and explain how these operators and Wilson loop operators transform under mirror symmetry. We also compute the expectation values of the loop operators by localization, and confirm the predictions of mirror symmetry.
第11回（2012年11月07日 17:00-18:30）
講師： Dr. Ivan Chi-Ho Ip (葉智皓) [Kavli IPMU]
題目： Positive Representations of Split Real Quantum Groups
概要：
In this talk, I will introduce the family of positive principal series representations for split real quantum groups by positive self-adjoint operators. The construction of these representations gives the starting point of a new research program devoted to the representation theory of split real quantum groups initiated in the joint work with Igor Frenkel.
It is a generalization of the special class of representations considered by J. Teschner for Uq(sl(2,R)) in Liouville theory, where it exhibits a strong parallel to the finite-dimensional representation theory of quantum groups.
Recently from the construction of the positive representations, a direct analytic relation between modular duality and Langlands duality is also discovered, which should have deep consequences in the Langlands program.
第12回（2012年11月14日 17:00-18:30）
講師： 山口哲 氏 (大阪大学)
題目： 境界のある3次元N=2超対称理論
概要：
3次元N=2超対称性を持つ場の理論において、超対称性を半分保つ境界条件について考える。これには、大雑把に分けて2種類のタイプがあり、それぞれA型、B型と名付ける。具体的な模型として、まずN=2超対称Landau-Ginzburg理論の境界条件の分類を行った。また、N=2自由Maxwell理論においても考察を行い、双対性との関係を見た。さらにN=2超対称量子電磁力学の場合においてmirror 対称性によって境界条件がどのように移るかについても考察する。
臨時セミナー（2012年11月15日 17:00-18:30）
講師： Basil Grammaticos 氏 [Universite Paris Diderot-Paris 7]
題目： Linearisable mappings: their taxonomy and their integration
概要：
We present a series of results on linearisable second-order mappings. Three distinct families of such mappings do exist: projective, mappings of Gambier type and mappings which we have dubbed "of third kind".
Our starting point are the linearisable mappings belonging to the QRT family. We show how they can be linearised and how in some cases their explicit solution can be constructed. We discuss also the growth property of these mappings, a property intimately related to linearisability.
In the second part of the talk we address the question of the extension of these mappings to a non-autonomous form. We show that the QRT invariant can also be extended (to a quantity which depends explicitly on the independent variable). Using this non-autonomous form we show that it is possible to construct the explicit solution of all third-kind mappings. We discuss also the relation of mappings of the third kind to Gambier-type mappings. We show that a large subclass of third-kind mappings can be considered as the discrete derivative of Gambier-type ones.
第13回（2012年12月05日 17:00-18:30）
講師： 小森靖 氏 [立教大学]
題目： ルート系に付随する多重ゼータ関数について
概要：
Wittenは2次元量子ゲージ理論の研究において、分配関数がある種のディリクレ級数になることを発見した。これを多変数関数化したものをルート系に付随する多重ゼータ関数とよぶ。本講演ではこの特殊値をベルヌーイ多項式の拡張を用いて記述する方法を紹介する。また、この関数を用いてEuler-Zagier型の多重ゼータ値やそれらの関係式のいくつかが導出できることを示す。
第14回（2013年01月23日 17:00-18:30）
講師： 瀧 雅人 氏 [理研]
題目： AGT Corespondence & Irregular Whittaker Vectors
概要：
The original AGT correspondence for the 4d theories of class S is the equivalence between 4d instanton partition functions and 2d conformal blocks. We extend this correspondence to more extended class of theories such as isolated SCFT, and the 2d side is then a irregular conformal block which is defined by using irregular vectors. An irregular vector is a certain extension of the primary state of 2d CFT, and we demonstrate that this object plays a key role to establish the AGT correspondence for wider class of 4d theories. This talk is based on the following collaborations with Hiroaki Kanno, Kazunobu Maruyoshi and Shotaro Shiba:
(1) H.Kanno-M.T., arXiv:1203.1427
(2) H.Kanno-K.Maruyoshi-S.Shiba-M.T., arXiv:1301.0721