arXiv subjects

Hiroaki Kanno

Publications and source records attributed to Hiroaki Kanno.

At least 19 recordsLinked to original sources

arXiv2026

Non-stationary difference equation and affine Laumon space III : Generalization to $\widehat{\mathfrak{gl}}_N$

In a series of papers we have considered a non-stationary difference equation which was originally discovered for the deformed Virasoro conformal block. The equation involves mass parameters and, when they are tuned appropriately, the equation is regarded as a quantum KZ equation for $U_q(A_{1}^{(1)})$. We introduce a $\widehat{\mathfrak{gl}}_N$ generalization of the non-stationary difference equation. The Hamiltonian is expressed in terms of $q$-commuting variables and allows both factorized forms and a normal ordered form. By specializing the mass parameters appropriately, the Hamiltonian can be identified with the $R$-matrix of the symmetric tensor representation of $U_q(A_{N-1}^{(1)})$, which in turn comes from the 3D (tetrahedron) $R$-matrix. We conjecture that the affine Laumon partition function of type $A_{N-1}^{(1)}$ gives a solution to our $\widehat{\mathfrak{gl}}_N$ non-stationary difference equation. As a check of our conjecture, we work out the four dimensional limit and find that the non-stationary difference equation reduces to the Fuji-Suzuki-Tsuda system.

math.QA↗

arXiv2026

Shifted quantum toroidal algebra of type $\mathfrak{gl}_{1|1}$ and the Pieri rule of the super Macdonald polynomials

The super Macdonald polynomials indexed by the super partitions form a basis of the level zero super Fock module (combinatorial representation) of the quantum toroidal algebra $\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_{1|1})$. The action of the super charges of $\mathcal{U}_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_{1|1})$ implies the Pieri rule of the super Macdonald polynomials. We can express the Pieri rule in terms of differential operators in the power sums $p_k$ and the fermionic power sums $π_k$, which leads to the operators on the Fock space of a free boson and a free fermion. From the Pieri rule we compute the supersymmetric Hamiltonians given by the anti-commutator of the super charges and recover the results previously obtained in the literature. It is remarkable that we have to deal with a shifted quantum toroidal algebra.

math.QA↗

arXiv2025

Super Macdonald polynomials and BPS state counting on the blow-up

We explore the relation of the super Macdonald polynomials and the BPS state counting on the blow-up of $\mathbb{P}^2$, which is mathematically described by framed stable perverse coherent sheaves. Fixed points of the torus action on the moduli space of BPS states are labeled by super partitions. From the equivariant character of the tangent space at the fixed points we can define the Nekrasov factor for a pair of super partitions, which is used for the localization computation of the partition function. The Nekrasov factor also allows us to compute matrix elements of the action of the quantum toroidal algebra of type $\mathfrak{gl}_{1|1}$ on the $K$ group of the moduli space. We confirm that these matrix elements are consistent with the Pieri rule of the super Macdonald polynomials.

hep-th↗

arXiv2024

Non-Stationary Difference Equation and Affine Laumon Space II: Quantum Knizhnik-Zamolodchikov Equation

We show that Shakirov's non-stationary difference equation, when it is truncated, implies the quantum Knizhnik-Zamolodchikov ($q$-KZ) equation for $U_{\mathsf v}\bigl(A_1^{(1)}\bigr)$ with generic spins. Namely, we can tune mass parameters so that the Hamiltonian acts on the space of finite Laurent polynomials. Then the representation matrix of the Hamiltonian agrees with the $R$-matrix, or the quantum $6j$ symbols. On the other hand, we prove that the $K$ theoretic Nekrasov partition function from the affine Laumon space is identified with the well-studied Jackson integral solution to the $q$-KZ equation. Combining these results, we establish that the affine Laumon partition function gives a solution to Shakirov's equation, which was a conjecture in our previous paper. We also work out the base-fiber duality and four-dimensional limit in relation with the $q$-KZ equation.

math.QA↗

arXiv2023

Non-Stationary Difference Equation and Affine Laumon Space: Quantization of Discrete Painlevé Equation

We show the relation of the non-stationary difference equation proposed by one of the authors and the quantized discrete Painlevé VI equation. The five-dimensional Seiberg-Witten curve associated with the difference equation has a consistent four-dimensional limit. We also show that the original equation can be factorized as a coupled system for a pair of functions $\bigl(\mathcal{F}^{(1)},\mathcal{F}^{(2)}\bigr)$, which is a consequence of the identification of the Hamiltonian as a translation element in the extended affine Weyl group. We conjecture that the instanton partition function coming from the affine Laumon space provides a solution to the coupled system.

nlin.SI↗

arXiv2023

Solitons in Open N=2 String Theory

The open N=2 string theory is defined on the four-dimensional space-time with the split signature (+,+,-,-). The string field theory action of the open N=2 string theory is described by the four-dimensional Wess-Zumino-Witten (WZW_4) model. Equation of motion of the WZW_4 model is the Yang equation which is equivalent to the anti-self-dual Yang-Mills equation. In this paper, we study soliton-type classical solutions of the WZW_4 model in the split signature by calculating the action density of the WZW_4 model. We find that the action density of the one-soliton solutions is localized on a three-dimensional hyperplane. This shows that there would be codimension-one-solitonic objects, or equivalently, some kind of three-branes in the open N=2 string theory. We also prove that in the asymptotic region of the space-time, the action density of the n-soliton solutions is a ``nonlinear superposition'' of n one-solitons. This suggests the existence of intersecting n three-branes in the N=2 strings. Finally we make a reduction to a (1+2)-dimensional real space-time to calculate energy densities of the soliton solutions. We can successfully evaluate the energy distribution for the two-soliton solutions and find that there is no singularity in the interacting region. This implies the existence of smooth intersecting codimension-one branes in the whole region. Soliton solutions in the Euclidean signature are also discussed.

hep-th↗

arXiv2021

Quasi-Hopf twist and elliptic Nekrasov factor

We investigate the quasi-Hopf twist of the quantum toroidal algebra of $\mathfrak{gl}_1$ as an elliptic deformation. Under the quasi-Hopf twist the underlying algebra remains the same, but the coproduct is deformed, where the twist parameter $p$ is identified as the elliptic modulus. Computing the quasi-Hopf twist of the $R$ matrix, we uncover the relation to the elliptic lift of the Nekrasov factor for instanton counting of the quiver gauge theories on $\mathbb{R}^4 \times T^2$. The same $R$ matrix also appears in the commutation relation of the intertwiners, which implies an elliptic quantum KZ equation for the trace of intertwiners. We also show that it allows a solution which is factorized into the elliptic Nekrasov factors and the triple elliptic gamma function.

hep-th↗

arXiv2021

MacMahon KZ equation for Ding-Iohara-Miki algebra

We derive a generalized Knizhnik-Zamolodchikov equation for the correlation function of the intertwiners of the vector and the MacMahon representations of Ding-Iohara-Miki algebra. These intertwiners are cousins of the refined topological vertex which is regarded as the intertwining operator of the Fock representation. The shift of the spectral parameter of the intertwiners is generated by the operator which is constructed from the universal $R$ matrix. The solutions to the generalized KZ equation are factorized into the ratio of two point functions which are identified with generalizations of the Nekrasov factor for supersymmetric quiver gauge theories

hep-th↗

arXiv2020

Quiver matrix model of ADHM type and BPS state counting in diverse dimensions

We review the problem of BPS state counting described by the generalized quiver matrix model of ADHM type. In four dimensions the generating function of the counting gives the Nekrasov partition function and we obtain generalization in higher dimensions. By the localization theorem, the partition function is given by the sum of contributions from the fixed points of the torus action, which are labeled by partitions, plane partitions and solid partitions. The measure or the Boltzmann weight of the path integral can take the form of the plethystic exponential. Remarkably after integration the partition function or the vacuum expectation value is again expressed in plethystic form. We regard it as a characteristic property of the BPS state counting problem, which is closely related to the integrability.

hep-th↗

arXiv2018

Equivariant U(N) Verlinde algebra from Bethe/Gauge correspondence

We compute the topological partition function (twisted index) of $\mathcal{N}=2$ $U(N)$ Chern-Simons theory with an adjoint chiral multiplet on $Σ_g \times S^1$. The localization technique shows that the underlying Frobenius algebra is the equivariant Verlinde algebra which is obtained from the canonical quantization of the complex Chern-Simons theory regularized by $U(1)$ equivariant parameter $t$. Our computation relies on a Bethe/Gauge correspondence which allows us to represent the equivariant Verlinde algebra in terms of the Hall-Littlewood polynomials $P_λ(x_B, t)$ with a specialization by Bethe roots $x_B$ of the $q$-boson model. We confirm a proposed duality to the Coulomb branch limit of the lens space superconformal index of four dimensional $\mathcal{N}=2$ theories for $SU(2)$ and $SU(3)$ with lower levels. In $SU(2)$ case we also present more direct computation based on Jeffrey-Kirwan residue operation.

hep-th↗

arXiv2017

(q,t)-KZ equation for Ding-Iohara-Miki algebra

We derive the generalization of the Knizhnik-Zamolodchikov equation (KZE) associated with the Ding-Iohara-Miki (DIM) algebra U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1). We demonstrate that certain refined topological string amplitudes satisfy these equations and find that the braiding transformations are performed by the R-matrix of U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1). The resulting syste is the uplifting of the \widehat{\mathfrak{u}}_1 Wess-Zumino-Witten model. The solutions to the (q,t)-KZE are identified with the (spectral dual of) building blocks of the Nekrasov partition function for 5d linear quiver gauge theories. We also construct an elliptic version of the KZE and discuss its modular and monodromy properties, the latter being related to a dual version of KZE.

hep-th↗

arXiv2016

Explicit examples of DIM constraints for network matrix models

Dotsenko-Fateev and Chern-Simons matrix models, which describe Nekrasov functions for SYM theories in different dimensions, are all incorporated into network matrix models with the hidden Ding-Iohara-Miki (DIM) symmetry. This lifting is especially simple for what we call balanced networks. Then, the Ward identities (known under the names of Virasoro/W-constraints or loop equations or regularity condition for qq-characters) are also promoted to the DIM level, where they all become corollaries of a single identity.

hep-th↗

arXiv2016

Toric Calabi-Yau threefolds as quantum integrable systems. R-matrix and RTT relations

R-matrix is explicitly constructed for simplest representations of the Ding-Iohara-Miki algebra. The calculation is straightforward and significantly simpler than the one through the universal R-matrix used for a similar calculation in the Yangian case by A.~Smirnov but less general. We investigate the interplay between the R-matrix structure and the structure of DIM algebra intertwiners, i.e.\ of refined topological vertices and show that the R-matrix is diagonalized by the action of the spectral duality belonging to the SL(2,Z) group of DIM algebra automorphisms. We also construct the T-operators satisfying the RTT relations with the R-matrix from refined amplitudes on resolved conifold. We thus show that topological string theories on the toric Calabi-Yau threefolds can be naturally interpreted as lattice integrable models. Integrals of motion for these systems are related to q-deformation of the reflection matrices of the Liouville/Toda theories.

hep-th↗

arXiv2014

Hyperkahler Metrics from Monopole Walls

We present ALH hyperkahler metrics induced from well-separated SU(2) monopole walls which are equivalent to monopoles on T^2 x R. The metrics are explicitly obtained due to Manton's observation by using explicit monopole solutions. These are doubly-periodic and have the modular invariance with respect to the complex structure of the complex torus T^2. We also derive metrics from monopole walls with Dirac-type singularities.

hep-th↗

arXiv2013

W_3 irregular states and isolated N=2 superconformal field theories

We explore the proposal that the six-dimensional (2,0) theory on the Riemann surface with irregular punctures leads to a four-dimensional gauge theory coupled to the isolated N=2 superconformal theories of Argyres-Douglas type, and to two-dimensional conformal field theory with irregular states. Following the approach of Gaiotto-Teschner for the Virasoro case, we construct W_3 irregular states by colliding a single SU(3) puncture with several regular punctures of simple type. If n simple punctures are colliding with the SU(3) puncture, the resulting irregular state is a simultaneous eigenvector of the positive modes L_n, ..., L_{2n} and W_{2n}, ..., W_{3n} of the W_3 algebra. We find the corresponding isolated SCFT with an SU(3) flavor symmetry as a nontrivial IR fixed point on the Coulomb branch of the SU(3) linear quiver gauge theories, by confirming that its Seiberg-Witten curve correctly predicts the conditions for the W_3 irregular states. We also compare these SCFT's with the ones obtained from the BPS quiver method.

hep-th↗

arXiv2012

Changing the preferred direction of the refined topological vertex

We consider the issue of the slice invariance of refined topological string amplitudes, which means that they are independent of the choice of the preferred direction of the refined topological vertex. We work out two examples. The first example is a geometric engineering of five-dimensional U(1) gauge theory with a matter. The slice invariance follows from a highly non-trivial combinatorial identity which equates two known ways of computing the chi_y genus of the Hilbert scheme of points on C^2. The second example is concerned with the proposal that the superpolynomials of the colored Hopf link are obtained from a refinement of topological open string amplitudes. We provide a closed formula for the superpolynomial, which confirms the slice invariance when the Hopf link is colored with totally anti-symmetric representations. However, we observe a breakdown of the slice invariance for other representations.

hep-th↗

arXiv2012

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

Following a recent paper by Alday and Tachikawa, we compute the instanton partition function in the presence of the surface operator by the localization formula on the moduli space. For SU(2) theories we find an exact agreement with CFT correlation functions with a degenerate operator insertion, which enables us to work out the decoupling limit of the superconformal theory with four flavors to asymptotically free theories at the level of differential equations for CFT correlation functions (irregular conformal blocks). We also argue that the K theory (or five dimensional) lift of these computations gives open topological string amplitudes on local Hirzebruch surface and its blow ups, which is regarded as a geometric engineering of the surface operator. By computing the amplitudes in both A and B models we collect convincing evidences of the agreement of the instanton partition function with surface operator and the partition function of open topological string.

hep-th↗

arXiv2012

Generalized Whittaker states for instanton counting with fundamental hypermultiplets

M-theoretic construction of N=2 gauge theories implies that the instanton partition function is expressed as the scalar product of coherent states (Whittaker states) in the Verma module of an appropriate two dimensional conformal field theory. We present the characterizing conditions for such states that give the partition function with fundamental hypermultiplets for SU(3) theory and SU(2) theory with a surface operator. We find the states are no longer the coherent states in the strict sense but we can characterize them in terms of a few annihilation operators of lower levels combined with the zero mode (Cartan part) of the Virasoro algebra L_0 or the sl(2) current algebra J_0^0.

hep-th↗