arXiv subjects

Hajime Nagoya

Publications and source records attributed to Hajime Nagoya.

16 recordsLinked to original sources

arXiv2026

Neveu--Schwarz Irregular Vertex Operators, Decomposition Theorems, and Bilinear Operators

We construct rank-zero irregular vertex operators for the Neveu--Schwarz algebra as linear maps between irregular Verma modules of the same rank satisfying the usual superconformal commutation relations and an irregular asymptotic condition. Under the nondegeneracy assumption $Λ_{2p}\neq0$, we prove their existence and uniqueness. We then extend the decomposition theorem for the Neveu--Schwarz algebra to the irregular setting: the tensor product of the free-fermion Fock module with a rank-$p$ Neveu--Schwarz irregular Verma module decomposes into an infinite direct sum of tensor products of two Virasoro irregular Verma modules. We also prove a compatible decomposition of the irregular vertex operators into tensor products of Virasoro irregular vertex operators. Using suitable pairings and mode insertions, we derive bilinear differential equations for weighted sums of products of Virasoro irregular conformal blocks of types $(0,0,1)$ and $(0,2)$. After explicit parameter identifications and a gauge transformation in the Painlevé V case, the resulting bilinear differential operators agree with those appearing in the quantum Painlevé V and IV tau-function equations.

math-ph↗

arXiv2026

Existence and Uniqueness of Irregular Vectors of Integer and Half-Integer Ranks for the Virasoro Algebra

Although irregular vectors for the Virasoro algebra are widely used in modern mathematical physics, a rigorous existence and uniqueness theorem in arbitrary rank has not been available in the literature. In this paper, we develop an algebraic framework, based on Virasoro differential operators on the parameter space, which gives such a theorem for arbitrary integer and half-integer ranks. A key ingredient is the construction of a canonical operator $L_*$ from the coefficient matrix of the vector-field part of a truncated Virasoro realization. This operator closes the recursive system by isolating the derivative with respect to the highest irregular parameter. Using this mechanism, we prove the existence and uniqueness of formal irregular vectors of arbitrary integer rank. We then construct the truncated Virasoro vector fields required in the half-integer-rank setting and prove the existence and uniqueness of the corresponding half-integer-rank formal irregular vectors. We also prove that, after a scalar gauge normalization, the canonical solutions satisfy the full lower Virasoro deformation equations. These results provide an algebraic foundation for the rigorous construction of irregular conformal blocks built from higher-rank irregular vectors. After passing to eigenvalue coordinates, the vector-field part of the half-integer construction is identified with the differential realizations appearing in the literature, while the zeroth-order terms are explained by scalar gauge freedom.

math-ph↗

arXiv2026

Degeneration limits of Virasoro vertex operators and Painlevé tau functions

We construct degeneration limits of vertex operators for the Virasoro algebra. Our method relies on the rearranged expansion of compositions of vertex operators together with their integral representations. Using this framework, we obtain a vertex operator between Verma modules of rank $r+1$ as a degeneration of a composition of two vertex operators between Verma modules of rank $r$ ($r\in\mathbb{Z}_{\geq 0}$). Furthermore, we apply these degeneration limits to prove the conjectural expansions of the $τ$ functions of the fifth and fourth Painlevé equations in terms of irregular conformal blocks [H. Nagoya, J. Math. Phys. 56, 123505 (2015)].

math-ph↗

arXiv2026

Accessory Parameter of Confluent Heun Equations, Voros Periods and classical irregular conformal blocks

For the Heun differential equation and all of its confluent equations, we derive formal series expansions of the accessory parameters using the Voros periods. We then compare these expansions with the classical conformal blocks recently obtained by Bonelli--Shchechkin--Tanzini, and examine the Zamolodchikov-type conjecture expected to hold between them, allowing for irregular singularities. In particular, as an extension of the previous works of Mironov--Morozov, Piatek--Pietrykowski and Lisovyy--Naidiuk, we provide a detailed prescription for choosing cycles on the spectral curve that yield the Voros period which corresponds to the classical (regular or irregular) conformal blocks through the accessory parameter.

math-ph↗

arXiv2021

On $q$-Isomonodromic Deformations and $q$-Nekrasov Functions

We construct a fundamental system of a $q$-difference Lax pair of rank $N$ in terms of 5d Nekrasov functions with $q=t$. Our fundamental system degenerates by the limit $q\to 1$ to a fundamental system of a differential Lax pair, which yields the Fuji-Suzuki-Tsuda system. We introduce tau functions of our system as Fourier transforms of 5d Nekrasov functions. Using asymptotic expansions of the fundamental system at $0$ and $\infty$, we obtain several determinantal identities of the tau functions.

math.CA↗

arXiv2019

Combinatorial expressions for the tau functions of $q$-Painlevé V and III equations

We derive series representations for the tau functions of the $q$-Painlevé V, $\mathrm{III_1}$, $\mathrm{III_2}$, and $\mathrm{III_3}$ equations, as degenerations of the tau functions of the $q$-Painlevé VI equation in [Jimbo M., Nagoya H., Sakai H., J. Integrable Syst. 2 (2017), xyx009, 27 pages]. Our tau functions are expressed in terms of $q$-Nekrasov functions. Thus, our series representations for the tau functions have explicit combinatorial structures. We show that general solutions to the $q$-Painlevé V, $\mathrm{III_1}$, $\mathrm{III_2}$, and $\mathrm{III_3}$ equations are written by our tau functions. We also prove that our tau functions for the $q$-Painlevé $\mathrm{III_1}$, $\mathrm{III_2}$, and $\mathrm{III_3}$ equations satisfy the three-term bilinear equations for them.

math-ph↗

arXiv2018

Remarks on irregular conformal blocks and Painlevé III and II tau functions

We prove a conjecture on uniqueness and existence of the irregular vertex operators of rank $r$ introduced in our previous paper. We also introduce ramified irregular vertex operators of the Virasoro algebra. As applications, we give conjectural formulas for series expansions of Painlevé III and II tau functions in terms of our ramified irregular conformal blocks.

math-ph↗

arXiv2016

Conformal blocks and Painlevé functions

This paper is based on my presentation at RIMS workshop on "Theory of Integrable Systems and Its Applications in Various Fields" held in Kyoto on 19--21, August 2015. The aim of the present paper is to give a short account of recent studies on relations between conformal blocks in the two-dimensional conformal field theory and Painlevé functions. In addition, we present a conjecture on a combinatorial expansion formula of the three-point irregular conformal block at an irregular singular point, with two regular singular points and one irregular singular point. Our conjectural expansion formula is written in terms of pairs of skew Young diagrams, while the four-point regular conformal block, by AGT correspondence, is written in terms of pairs of Young diagrams.

math-ph↗

arXiv2015

Irregular conformal blocks, with an application to the fifth and fourth Painlevé equations

We develop the theory of irregular conformal blocks of the Virasoro algebra. In previous studies, expansions of irregular conformal blocks at regular singular points were obtained as degeneration limits of regular conformal blocks; however, such expansions at irregular singular points were not clearly understood. This is because precise definitions of irregular vertex operators had not been provided previously. In this paper, we present precise definitions of irregular vertex operators of two types and we prove that one of our vertex operators exists uniquely. Then, we define irregular conformal blocks with at most two irregular singular points as expectation values of given irregular vertex operators. Our definitions provide an understanding of expansions of irregular conformal blocks and enable us to obtain expansions at irregular singular points. As an application, we propose conjectural formulas of series expansions of the tau functions of the fifth and fourth Painlevé equations, using expansions of irregular conformal blocks at an irregular singular point.

math-ph↗

arXiv2013

Symmetries of quantum Lax equations for the Painlevé equations

The Painlevé equations can be written as Hamiltonian systems with affine Weyl group symmetries. A canonical quantization of the Painlevé equations preserving the affine Weyl group symmetries has been studied. While, the Painlevé equations are isomonodromic equations for certain second-order linear differential equations. In this paper, we introduce a canonical quantization of Lax equations for the Painlevé equations and construct symmetries of the quantum Lax equations. We also show that our quantum Lax equations are derived from Virasoro conformal field theory.

math-ph↗

arXiv2012

Integral formulas for quantum isomonodromic systems

We conisder time-dependent Schrödinger systems, which are quantizations of the Hamiltonian systems obtained from a similarity reduction of the Drinfeld-Sokolov hierarchy by K. Fuji and T. Suzuki, and a similarity reduction of the UC hierarchy by T.Tsuda, independently. These Hamiltonian systems describe isomonodromic deformations for certain Fuchsian systems. Thus, our Schrödinger systems can be regarded as quantum isomonodromic systems. Y. Yamada conjectured that our quantum isomonodromic systems determine instanton partition functions in N=2 SU(L) gauge theory. The main purpose of this paper is to present integral formulas as particular solutions to our quantum isomonodromic systems. These integral formulas are generalizations of the generalized hypergeometric function.

math.QA↗

arXiv2011

Hypergeometric solutions to Schrödinger equations for the quantum Painlevé equations

We consider Schrödinger equations for the quantum Painlevé equations. We present hypergeometric solutions of the Schrödinger equations for the quantum Painlevé equations, as particular solutions. We also give a representation theoretic correspondence between Hamiltonians of the Schrödinger equations for the quantum Painlevé equations and those of the KZ equation or the confluent KZ equations.

math-ph↗

arXiv2010

Confluent KZ equations for sl_N with Poincare rank 2 at infinity

We construct confluent KZ equations with Poincare rank 2 at infinity for the case of sl_N and the integral representation for the solutions. Hamiltonians of these confluent KZ equations are derived from suitable quantization of dlog tau constructed in the theory of monodromy preserving deformation by Jimbo, Miwa and Ueno. Our confluent KZ equations may be viewed as a quantization of monodromy preserving deformation with Poincare rank 2 at infinity.

math-ph↗

arXiv2010

Confluent primary fields in the conformal field theory

For any complex simple Lie algebra, we generalize primary fileds in the Wess-Zumino-Novikov-Witten conformal field theory with respect to the case of irregular singularities and we construct integral representations of hypergeometric functions of confluent type, as expectation values of products of generalized primary fields. In the case of sl(2), these integral representations coincide with solutions to confluent KZ equations. Computing the operator product expansion of the energy-momentum tensor and the generalized primary field, new differential operators appear in the result. In the case of sl(2), these differential operators are the same as those of the confluent KZ equations.

math-ph↗

arXiv2008

Quantum Painlevé Equations: from Continuous to Discrete

We examine quantum extensions of the continuous Painlevé equations, expressed as systems of first-order differential equations for non-commuting objects. We focus on the Painlevé equations II, IV and V. From their auto-Bäcklund transformations we derive the contiguity relations which we interpret as the quantum analogues of the discrete Painlevé equations.

math.QA↗

arXiv2004

Quantum Painlevé systems of type $A^{(1)}_l$

We propose quantum Painlevé systems of type $A_l^{(1)}$. These systems, for $l=1$ and $l\ge 2$, should be regarded as quantizations of the second Painlevé equation and the differential systems with the affine Weyl group symmetries of type $A_l^{(1)}$ studied by M. Noumi and Y. Yamada \cite{NYhigherorder}, respectively. These quantizations enjoy the affine Weyl group symmetries of type $A_l^{(1)}$ as well as the Lax representations. The quantized systems of type $A_1^{(1)}$ and type $A_l^{(1)}$ ($l= 2n$) can be obtained as the continuous limits of the discrete systems constructed from the affine Weyl group symmetries of type $A_2^{(1)}$ and $A_{l+1}^{(1)}$, respectively.

math.QA↗