2018年度秋季総合分科会(於:岡山大学)

SUMMARY: Exact WKB analysis, initiated by Voros, is an effective method to study the global properties of Schrödinger-type linear ODEs with a small parameter \(\hbar \). A fundamental result in exact WKB analysis established by Aoki–Kawai–Takei–Sato claims that monodromy matrices of (Borel resumed) WKB solutions are described combinatorially by period integrals over a Riemann surface defined as the classical limit of the Schrödinger-type ODE. Such period integrals are called Voros coefficients, and they play important role in the theory of exact WKB analysis. On the other hand, topological recursion, introduced by Eynard–Orantin, is a remarkable algorithm which computes certain correlation functions and free energy (of matrix models) from a given spectral curve. Correlation functions and free energy are expected to encode information of various geometric or enumerative invariants, and \(\tau \)-functions of integrable hierarchies.

A surprising connection between WKB analysis and topological recursion was discovered recently by many people including Gukov–Sulkowski, Dumitrescu–Mulase and Bouchard–Eyanrd. They claim that a generating functions of the correlation functions gives the WKB solution of a Schrödinger-type equation whose classical limit coincides with the spectral curve for the topological recursion, and hence the resulting Schrödinger-type equation is called “quantum (spectral) curve”. In my talk, I’ll give brief introductions to exact WKB analysis, topological recursion and quantum curves. After that, I’ll explain our recent result on the realization of Voros coefficients in terms of the free energy of topological recursion, obtained in a joint work with T. Koike (Kobe) and Y. Takei (Kobe). If time allows, I’ll also show results on Painlevé equations.

SUMMARY: The \(K\)-homology ring of the affine Grassmannian of \(SL_n(\mathbb {C})\) was studied by Lam, Schilling, and Shimozono. It is realized as a certain concrete Hopf subring of the ring of symmetric functions. On the other hand, a presentation for the quantum \(K\)-theory of the flag variety \(Fl_n\) is given by Anderson–Chen–Tseng and Koroteev–Pusukar–Smirnov–Zeitlin. We construct an explicit birational morphism between the spectrums of these two rings. Our method relies on Ruijsenaars’s relativistic Toda lattice with unipotent initial condition. From this result, we obtain a \(K\)-theory analogue of the so-called Peterson isomorphism for (co)homology. We provide a conjecture on the detailed relationship between the Schubert bases, and, in particular, we determine the image of Lenart–Maeno’s quantum Grothendieck polynomial associated with a Grassmannian permutation.

SUMMARY: Eisenstein derived the addition formula for the Weierstrass zeta function from the addition formula for the cotangent function and the fact that the Weierstrass zeta function can be represented as an infinite series of the cotangent function.

In this talk, we apply this Eisenstein’s idea to the addition type formula for the double cotangent function, established by the speaker. We show that the elliptic digamma function, defined by the logarithmic derivative of the elliptic gamma function, satisfies an addition type formula. This formula includes the addition formula for the Weierstrass zeta function, evaluation formulas for the double Eisenstein series introduced by Tsumura and the double shuffle relations for the double Eisenstein series, proved by Gangl–Kaneko–Zagier.

SUMMARY: Buchstaber and Mikhailov constructed the polynomial dynamical systems on the basis of commuting vector fields on the symmetric square of hyperelliptic curves. The zero set of the sigma function in the Jacobian of a curve is called sigma divisor. We constructed solutions of the systems for genus 3 in terms of the meromorphic functions on the sigma divisor of the hyperelliptic curves of genus 3. In this talk, we derive a partial differential equation from the dynamical systems, which is integrable by the meromorphic functions on the sigma divisor of the hyperelliptic curves of genus 3. When the curves degenerate to certain singular curves, the solution of the partial differential equation tends to that of the KdV-equation.

SUMMARY: The Drinfeld–Sokolov hierarchy is an extension of the KP hierarchy for the affine Lie algebra. In the case of type \(A\), it can be characterized by partitions of natural numbers. In the previous work, we investigated the hierarchy corresponding to the partition \((n+1,n+1)\) and expressed its similarity reduction as the polynomial Hamiltonian system. In this talk, we will consider the Drinfeld–Sokolov hierarchy of type \(A^{(1)}_{3n+2}\) corresponding to the partition \((n+1,n+1,n+1)\).

SUMMARY: The \(q\)-Heun equation and its variants arise as degenerations of Ruijsenaars–van Diejen operators with one particle. We investigate local properties of these equations. In particular we characterize the variants of the \(q\)-Heun equation by using analysis of regular singularities. We also consider the quasi-exact solvability of the \(q\)-Heun equation and its variants. Namely we investigate finite-dimensional subspaces which are invariant under the action of the \(q\)-Heun operator or variants of the \(q\)-Heun operator.

SUMMARY: We give a connection formula of a difference equation satisfied by \(q\)-hypergeometric series \({}_3\phi _{2}(a_1,a_2,a_3;0,0;q,x)\). This equation has an irregular singular point at the origin, we need a resummation of a divergent power series. We also study other connection formulae of \(q\)-hypergeometric equations.

SUMMARY: We would like to talk about a way to extend Gustafson’s \(q\)-beta integral of type \(G_2\) to its elliptic form. We will also present the explicit expression for its infinite product in terms of Ruijsenaars’ elliptic gamma functions.

SUMMARY: We study the Jacobian matrices associated with sequences of Y-seed mutations in universal semifields. For the case where these mutations occur at least once for all indices, we present a formula for a special value of their characteristic polynomials using mutation networks, which are combinatorial objects that describe the data of mutation sequences.

SUMMARY: We define \(r\)-inferior regular partition which is a restriction of partition. Its generating function equals to that of the number of operations in Glaisher correspondence. Using this result, we prove Mizukawa–Yamada’s identity. And extend this identity to \(m\)-tuple version.

SUMMARY: We introduce a matrix inversion for Koornwinder polynomials with one-column diagram.

SUMMARY: It is a well-known fact that the KZ equation which relates the differential to the algebraic action can be obtained from the representation theory of the Kac–Moody algebra. We show that the same idea can be applied to the quantum toroidal algebra and propose the difference analogue, namely the \((q,t)\)-KZ equation. Furthermore, we can prove that the solution to the \((q,t)\)-KZ equation is the Nekrasov function for the instanton counting of the super symmetric gauge theory. This story realizes the celebrated AGT correspondence.

SUMMARY: We give a detailed proof of the existence of evaluation map for affine Yangians of type A to clarify that it needs an assumption on parameters. This map was first found by Guay but a proof of its well-definedness and the assumption have not been written down in the literature.

SUMMARY: A conjectural formula expressing the generating series of three-partition Hodge integrals in terms of topological vertex of topological string theory is proved. The proof is given by utilizing the recent result on quantum torus symmetry of random skew plane partition, which is a generalization of the previous study on random plane partition or melting crystal.

SUMMARY: We formulate a quantized reflection equation in which \(q\)-boson valued \(L\) and \(K\) matrices satisfy the reflection equation up to conjugation by a solution to the Isaev–Kulish 3D reflection equation. By forming its \(n\)-concatenation along the \(q\)-boson Fock space followed by suitable reductions, we construct families of solutions to the reflection equation in a matrix product form connected to the 3D integrability. They involve the quantum \(R\) matrices of the antisymmetric tensor representations of \(U_p(A^{(1)}_{n-1})\) and the spin representations of \(U_p(B^{(1)}_{n})\), \(U_p(D^{(1)}_{n})\) and \(U_p(D^{(2)}_{n+1})\). Similar results on the \(G_2\) reflection equation will also be presented.

SUMMARY: The generalized quantum group of type \(A\) is an affine analogue of quantum group associated to a general linear Lie superalgebra, which appears in the study of solutions to the tetrahedron equation. We construct Kirillov–Reshetikhin (KR) modules of this algebra, that is, a family of irreducible modules which have crystal bases. We also give a combinatorial description of the crystal structure, the combinatorial \(R\) matrix, and energy function on their tensor products.