2018年度年会(於:東京大学)

SUMMARY: 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.

SUMMARY: 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.

SUMMARY: 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.

SUMMARY: 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.

SUMMARY: We present explicit formulas for the Macdonald polynomials of types \(C_n\) and \(D_n\) in the one-column case.

SUMMARY: 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.

SUMMARY: 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.

SUMMARY: 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 \(N=1\).

SUMMARY: We show a connection formula of a linear \(q\)-differential equation satisfied by \({}_r\phi _{r-1}(\bm {0}; \bm {b};q,x)\). We use a \(q\)-Laplace transformation to obtain an integral representation of solutions of the \(q\)-differential equation.

SUMMARY: 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.

SUMMARY: 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.

SUMMARY: 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.

SUMMARY: 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.

SUMMARY: 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.

SUMMARY: We introduce Bruhat order of Weyl groupoids. We use nil-Hecke algebras of Weyl groupoids.

SUMMARY: 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.

SUMMARY: 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. Adamović 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.