一般社団法人 日本数学会 Application Server

# アブストラクト事後公開 — 2018年度年会(於:東京大学)

## 無限可積分系特別セッション

 特別講演 箙ゲージ理論と箙W代数 木村太郎 (慶大自然科学研究教育センター) Quiver W-algebra is a gauge theory construction of (q-deformed) W-algebra associated with a quiver. In this formalism, the generating current of the W-algebra is obtained through double quantization of Seiberg–Witten geometry, describing the moduli space of supersymmetric vacua, and the gauge theory partition function, known as the Nekrasov function, is explicitly given by a correlator of the screening charge. The formalism of quiver W-algebra naturally reproduces the construction of W$_{q,t}(g)$ by Frenkel–Reshetikhin, and also gives rise to several generalized situations for W-algebra and gauge theory: (1) affine quiver W-algebra (2) elliptic deformation of W-algebra (3) non-simply-laced (fractional) quiver variety. msjmeeting-2018mar-11i001 PDF 特別講演 シューベルト・カルキュラスの視点からのHall–Littlewood函数の一般化・母函数表示と応用 成瀬　弘 (山梨大教育) We generalize Hall–Littlewood function in the framework of generalized cohomology theory. We get a generating function expression for the generalized Hall–Littlewood functions. For the case of connective K-theory we recover determinantal or Pfaffian formula for K-theoretic Schur or Schur Q-function. msjmeeting-2018mar-11i002 PDF 1. $q$ホイン方程式の多項式解について 小嶋健太郎 (中大理工)・​佐藤　司 (中大理工)・​竹村剛一 (中大理工) We study polynomial solutions of $q$-Heun equation. In particular we investigate the condition for the accessory parameter $E$ of $q$-Heun equation which admits a non-zero polynomial solution. PDF 2. Real-root property of the spectral polynomial of the Treibich–Verdier potential and related problems Zhijie Chen (Yau Math. Sci. Center)・​Ting-Jung Kuo (Nat. Taiwan Normal Univ.)・​Chang-Shou Lin (Nat. Taiwan Univ.)・​竹村剛一 (中大理工) We study the spectral polynomial of the Treibich–Verdier potential. Such spectral polynomial, which is a generalization of the classical Lame polynomial, plays fundamental roles in both the finite-gap theory and the ODE theory of Heun’s equation. In this talk, we prove that all the roots of such spectral polynomial are real and distinct under some assumptions. The proof uses the classical concept of Sturm sequence and isomonodromic theories. We also prove an analogous result for a polynomial associated with a generalized Lame equation. Differently, our new approach is based on the viewpoint of the monodromy data. PDF 3. 一列型$C$, $D$型Macdonald多項式の明示公式 星野　歩 (広島工大工)・​白石潤一 (東大数理) We present explicit formulas for the Macdonald polynomials of types $C_n$ and $D_n$ in the one-column case. PDF 4. $G_2$ 型Weyl群不変な $q$ 超幾何積分の行列式公式 伊藤雅彦 (琉球大理)・​宮永愛子 (神戸大理)・​野海正俊 (神戸大理) We present some determinant formulas for the $q$-hypergeometric integrals associated with the root system of type $G_2$, which generalize Macdonald’s constant term formula. We introduce a method of deriving the $q$-difference equation satisfied by the determinant and finding its special value. PDF 5. $q$サイクルのホモロジー 伊藤公毅 (豊橋技科大) We introduce homology to be dual of the $q$-de Rham cohomology. Conventional $q$-cycles correspond to $q$-analogues of (noncompact) locally finite chains. We need regularize Jackson integrals over such $q$-cycles. Essentially, a regularization of such a $q$-cycle has been introduced. Nevertheless, such a regularization has not been understood as a compact chain. Thus, we introduce $q$-cycles including compact ones in the case of dimension 1. PDF 6. $q$超幾何関数の一般化と, それを特殊解に持つモノドロミー保存変形 朴　佳南 (神戸大理) Tsuda obtained a monodromy preserving deformation which has a special solution represented by a generalization of Gauss hypergeometric function. Our purpose is to obtain its $q$-analog. We define a series $\mathcal{F}_{M,N}$ as an extension of a $q$-hypergeometric series. In this talk, we give such a monodromy preserving deformation when $N=1$. PDF 7. $q$-超幾何函数${}_r\phi_{r-1}(\mbox{\boldmath$0$}; \mbox{\boldmath$b$}; q, x)$の接続問題 大山陽介 (徳島大理工) We show a connection formula of a linear $q$-differential equation satisfied by ${}_r\phi_{r-1}(\mbox{\boldmath$0$}; \mbox{\boldmath$b$};q,x)$. We use a $q$-Laplace transformation to obtain an integral representation of solutions of the $q$-differential equation. PDF 8. $q$-超幾何函数${}_3\phi_{2}(a_1,a_2,a_3;b_1,0;q,x)$の満たす差分方程式の$q$-Stokes係数 大山陽介 (徳島大理工) We study a resummation of a divergent solution of a $q$-difference equation satisfied by ${}_3\phi_{2}(a_1,a_2,a_3;b_1,0;q,x)$. For the divergent series which is not hypergeometric type, we determine the $q$-Stokes coefficients. PDF 9. モノドロミー保存変形へのKZ理論的アプローチにおける解の多重対数関数による展開 神原 北斗 ・​竹田悠人・​上野喜三雄 (早大理工) We introduce a system of nonlinear differential equations which is the integrable condition of deformation of the KZ equation of two variables $(z,w)$. We denote this system by 1DE which is equations in the variable $w$. We consider solutions holomorphic at the origin $w=0$ of 1DE. In this talk, we will show that these solutions are expanded in terms of multiple polylogarithms. PDF 10. モノドロミー保存変形へのKZ理論的アプローチとSchlesinger方程式との関係 上野喜三雄 (早大理工) We consider the relation between 1DE and the Schlesinger equation of one variable, 1SE. Particularly, we show that from constant solutions to 1DE, one can construct solutions to 1SE. Moreover an example related to Appell $F_1(\alpha,\beta,\beta',\gamma;\,z,zw)$ are discussed. PDF 11. アフィン非例外型のパスと艤装配位の全単射 尾角正人 (阪市大理)・​A. Schilling (UC Davis)・​T. Scrimshaw (Univ. of Queensland) We establish a bijection between rigged configurations and highest weight elements of a tensor product of Kirillov–Reshetikhin crystals for all nonexceptional types. A key idea for the proof is to embed both objects into bigger sets for simply-laced types $A_n^{(1)}$ or $D_n^{(1)}$, whose bijections have already been established. As a consequence we settle the $X=M$ conjecture in full generality for nonexceptional types. PDF 13. 楕円Felderhof模型と楕円Schur関数 茂木康平 (東京海洋大海洋工) We apply the recently developed Izergin–Korepin analysis on the wavefunctions of integrable lattice models to the elliptic Felderhof model. We prove that the wavefunctions are expressed as the product of a deformed elliptic Vandermonde determinant and elliptic Schur functions. As an application of the correspondence between the wavefunctions and the elliptic Schur functions, we derive dual Cauchy formula for the elliptic Schur functions. PDF 14. Bruhat order of Weyl groupoids 山根宏之 (富山大理工)・​I. Angiono (Nat. Univ. of C\'ordoba) We introduce Bruhat order of Weyl groupoids. We use nil-Hecke algebras of Weyl groupoids. PDF 15. Screening operators and $\mathfrak{sl}_2$ action on the lattice vertex operator algebras of type $A_1$ 橋本義武 (東京都市大知識工)・​松本拓也 (名大多元数理)・​土屋昭博 (Kavli IPMU) In this talk, we shall consider the marginal deformations of the Belavin–Polyakov–Zamolodchikov (BPZ) minimal models, which are the fundamental models of the two-dimensional conformal field theory. These deformations preserve the Virasoro symmetries and parametrized by the formal deformation parameter $\epsilon$. In particular, by formulating the deformed theories over the pair $({\cal K},{\cal O})$ of the ring of formal power series ${\cal O}=\mathbb{C}[[\epsilon]]$ and the quotient field ${\cal K}=\mathbb{C}((\epsilon))$, we discuss the characteristic features of the BPZ minimal models and their extensions. PDF 16. Modular transformation properties and the Verlinde formula 佐藤　僚 (東大数理) The classification of simple modules over the $\mathcal{N}=2$ vertex operator superalgebra (VOSA) of central charge $3(1-\frac{2p'}{p})$ is obtained by D. Adamovi\’c via the Kazama–Suzuki coset construction. When $p'=1$, the simple modules coincide with the $\mathcal{N}=2$ unitary minimal series. On the other hand, when $p'>1$, there are uncountably many simple modules and they are non-unitary. In this talk we give the modular transformation law of the characters of the simple non-unitary modules. As an application, we propose a conjectural Verlinde formula for the non-unitary $\mathcal{N}=2$ VOSA. Note that this result is an analogue to the conjectural Verlinde formula for the admissible affine $\mathfrak{sl}(2)$ VOA proposed by T. Creutzig and D. Ridout. PDF