arXiv subjects

Hiroshi Itoyama

Publications and source records attributed to Hiroshi Itoyama.

At least 19 recordsLinked to original sources

arXiv2026

Notes on phase structure and non-vanishing $β$ functions of one-unitary matrix model

Unitary matrix models, among other things, play an important role in representing the irregular conformal block and hence LEEA of some asymptotically free susy gauge theories. Here, we develop a method of how to determine the qualitative structure of phase diagram by the deformation of classical potential, and illustrate this by the examples that contain term up to $\cos nα$, $n \leq 3$. We point out that a set of $β$ functions on critical ``lines" is nowhere a vanishing vector. This generalizes the $n =1$ GWW case and tells us the third order (rather than second order) phase transition in conformity with the range of values for the susceptibility exponent.

hep-th↗

arXiv2024

Phases and triple(multiple) point: critical phenomena around the AD singularity

Continuing with our previous series of work, we present a case study of the critical phenomena around Argyres-Douglas singularity of ${\cal N} =2$ susy made at $(A_1, A_{4k-1} ), k =1, 2$ realized by one-unitary matrix model. We determine the phase diagram, which is recast into LEEA of $\mathcal{N}=2$, 4d gauge theory by the 0d-4d connection. There are three distinct phases, each corresponding to an eigenvalue distribution with 0, 1, and 2 gaps. These form an entire phase diagram with a triple point. Examining the behavior of the planar free energy, we show, among other things,that the transition line between 1- and 2-gap phases ending at the triple point is the $k=2$ multicritical one.

hep-th↗

arXiv2024

Large order behavior near the AD point: the case of $\mathcal{N} =2$, $su(2)$, $N_f =2$

A non-perturbative effect in $κ$ (renormalized string coupling) obtained from the large order behavior in the vicinity of the prototypical Argyres-Douglas critical point of $su(2)$, $N_f =2$, $\mathcal{N} =2$ susy gauge theory can be studied in the GWW unitary matrix model with the log term: the one as the work done against the barrier of the effective potential by a single eigenvalue lifted from the sea and the other as a non-perturbative function contained in the solutions of the nonlinear differential equation PII that goes beyond the asymptotic series. The leading behaviors are of the form $\exp (-\frac{4}{3}\frac{1}κ \, (1, \left(\frac{s}{K}\right)^{\frac{3}{2}} ))$ respectively. We make comments on their agreement.

hep-th↗

arXiv2023

Construction of irregular conformal/W block and flavor mass relations of $\mathcal{N}=2$ SUSY gauge theory from the $A_{n-1}$ quiver matrix model

A sequence of massive scaling limits of the $β$-deformed $A_{n-1}$ quiver matrix model that keeps the size of the matrices finite and that corresponds to the $N_{f} =2n \rightarrow 2n-1, 2n-2$ limits on the number of flavors at 4d $su(n)$ ${\cal N} = 2$ SUSY gauge theory side is carried out to provide us with the integral representation of $su(n)$ irregular conformal/W block. The original paths are naturally deformed into those in the complex plane, permitting us to convert into an $su(n)$ extension of the unitary matrix model of GWW type with a set of log potentials for all species of eigenvalues. Looking at the region in the parameter space that enjoys the maximal symmetry of the model, we derive a set of relations among the mass parameters which may serve as evidence for the existence of the Argyres-Douglas critical hypersurface.

hep-th↗

arXiv2022

A-D hypersurface of $su(n)$ $\mathcal{N}=2$ supersymmetric gauge theory with $N_f = 2n-2$ flavors

In the previous letter, arXiv:2210.16738[hep-th], we found a set of flavor mass relations as constraints that the $β$-deformed $A_{n-1}$ quiver matrix model restores the maximal symmetry in the massive scaling limit and reported the existence of Argyres-Douglas critical hypersurface. In this letter, we derive the concrete conditions on moduli parameters which maximally degenerates the Seiberg-Witten curve while maintaining the flavor mass relations. These conditions define the A-D hypersurface.

hep-th↗

arXiv2021

Theory space of one unitary matrix model and its critical behavior associated with Argyres-Douglas theory

The lowest critical point of one unitary matrix model with cosine plus logarithmic potential is known to correspond with the $(A_1, A_3)$ Argyres-Douglas (AD) theory and its double scaling limit derives the Painlevé II equation with parameter. Here, we consider the critical points associated with all cosine potentials and determine the scaling operators, their vevs and their scaling dimensions from perturbed string equations at planar level. These dimensions agree with those of $(A_1,A_{4k-1})$ AD theory.

hep-th↗

arXiv2019

Discrete Painleve system for the partition function of $N_f =2$ $SU(2)$ supersymmetric gauge theory and its double scaling limit

We continue to study the matrix model of the $N_f =2$ $SU(2)$ case that represents the irregular conformal block. What provides us with the Painlevé system is not the instanton partition function per se but rather a finite analog of its Fourier transform that can serve as a generating function. The system reduces to the extension of the Gross-Witten-Wadia unitary one-matrix model by the logarithmic potential while keeping the planar critical behavior intact. The double scaling limit to this critical point is a constructive way to study Argyres-Douglas type theory from IR. We elaborate upon the method of orthogonal polynomial and its relevance to these problems, developing it further for the case of a generic unitary matrix model and that of a special case with the logarithmic potential.

hep-th↗

arXiv2019

Generalized cut operation associated with higher order variation in tensor models

The cut and join operations play important roles in tensor models in general. We introduce a generalization of the cut operation associated with the higher order variations and demonstrate how they generate operators in the Aristotelian tensor model. We point out that, by successive choices of appropriate variations, the cut operation generalized this way can generate those operators which do not appear in the ring of the join operation, providing a tool to enumerate the operators by a level by level analysis recursively. We present a set of rules that control the emergence of such operators.

hep-th↗

arXiv2019

Discrete Painleve system and the double scaling limit of the matrix model for irregular conformal block and gauge theory

We study the partition function of the matrix model of finite size that realizes the irregular conformal block for the case of the ${\cal N}=2$ supersymmetric $SU(2)$ gauge theory with $N_f =2$. This model has been obtained in [arXiv:1008.1861 [hep-th]] as the massive scaling limit of the $β$ deformed matrix model representing the conformal block. We point out that the model for the case of $β=1$ can be recast into a unitary matrix model with log potential and show that it is exhibited as a discrete Painlevé system by the method of orthogonal polynomials. We derive the Painlevé II equation, taking the double scaling limit in the vicinity of the critical point which is the Argyres-Douglas type point of the corresponding spectral curve. By the $0$d-$4$d dictionary, we obtain the time variable and the parameter of the double scaled theory respectively from the sum and the difference of the two mass parameters scaled to their critical values.

hep-th↗

arXiv2017

Elliptic algebra, Frenkel-Kac construction and root of unity limit

We argue that the level-$1$ elliptic algebra $U_{q,p}(\widehat{\mathfrak{g}})$ is a dynamical symmetry realized as a part of 2d/5d correspondence where the Drinfeld currents are the screening currents to the $q$-Virasoro/W block in the 2d side. For the case of $U_{q,p}(\widehat{\mathfrak{sl}}(2))$, the level-$1$ module has a realization by an elliptic version of the Frenkel-Kac construction. The module admits the action of the deformed Virasoro algebra. In a $r$-th root of unity limit of $p$ with $q^2 \rightarrow 1$, the $\mathbb{Z}_r$-parafermions and a free boson appear and the value of the central charge that we obtain agrees with that of the 2d coset CFT with para-Virasoro symmetry, which corresponds to the 4d $\mathcal{N}=2$ $SU(2)$ gauge theory on $\mathbb{R}^4/\mathbb{Z}_r$.

hep-th↗

arXiv2016

Cubic constraints for the resolvents of the ABJM matrix model and its cousins

A set of Schwinger-Dyson equations forming constraints for at most three resolvent functions are considered for a class of Chern-Simons matter matrix models with two nodes labelled by a non-vanishing number $n$. The two cases $n=2$ and $n= -2$ label respectively the ABJM matrix model, which is the hyperbolic lift of the affine $A_1^{(1)}$ quiver matrix model, and the lens space matrix model. In the planar limit, we derive two cubic loop equations for the two planar resolvents. One of these reduces to the quadratic one when $n = \pm 2$.

hep-th↗

arXiv2016

q-Vertex Operator from 5D Nekrasov Function

The five dimensional AGT correspondence implies the connection between the q-deformed Virasoro block and the 5d Nekrasov partition function. In this paper, we determine a q-deformation of the four-point block in the Coulomb gas representation from the 5d Nekrasov function, and obtain an expression of the q-deformed vertex operator. If we use only one kind of the q-vertex operators, one of the insertion points of them must be modified in order to hold the 2d/5d correspondence.

hep-th↗

arXiv2015

Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models

This is a semi-pedagogical review of a medium size on the exact determination of and the role played by the low energy effective prepotential ${\cal F}$ in QFT with (broken) extended supersymmetry, which began with the work of Seiberg and Witten in 1994. While paying an attention to an overall view of this subject lasting long over the two decades, we probe several corners marked in the three major stages of the developments, emphasizing uses of the deformation theory on the attendant Riemann surface as well as its close relation to matrix models. Examples picked here in different contexts tell us that the effective prepotential is to be identified as the suitably defined free energy $F$ of a matrix model: ${\cal F} = F$. To be submitted to PTEP as an invited review article and based in part on the talk delivered by one of the authors (H.I.) in the workshop held at Shizuoka University, Shizuoka, Japan, on December 5, 2014.

hep-th↗

arXiv2014

$q$-Virasoro/W Algebra at Root of Unity and Parafermions

We demonstrate that the parafermions appear in the $r$-th root of unity limit of $q$-Virasoro/$W_n$ algebra. The proper value of the central charge of the coset model $ \frac{\widehat{\mathfrak{sl}}(n)_r \oplus \widehat{\mathfrak{sl}}(n)_{m-n}}{\widehat{\mathfrak{sl}}(n)_{m-n+r}}$ is given from the parafermion construction of the block in the limit.

hep-th↗

arXiv2013

2d-4d Connection between q-Virasoro/W Block at Root of Unity Limit and Instanton Partition Function on ALE Space

We propose and demonstrate a limiting procedure in which, starting from the q-lifted version (or K-theoretic five dimensional version) of the (W)AGT conjecture to be assumed in this paper, the Virasoro/W block is generated in the r-th root of unity limit in q in the 2d side, while the same limit automatically generates the projection of the five dimensional instanton partition function onto that on the ALE space R^4/Z_r. This circumvents case-by-case conjectures to be made in a wealth of examples found so far. In the 2d side, we successfully generate the super-Virasoro algebra and the proper screening charge in the q -> -1, t -> -1 limit, from the defining relation of the q-Virasoro algebra and the q-deformed Heisenberg algebra. The central charge obtained coincides with that of the minimal series carrying odd integers of the N=1 superconformal algebra. In the r-th root of unity limit in q in the 2d side, we give some evidence of the appearance of the parafermion-like currents. Exploiting the q-analysis literatures, q-deformed su(n) block is readily generated both at generic q, t and the r-th root of unity limit. In the 4d side, we derive the proper normalization function for general (n, r) that accomplishes the automatic projection through the limit.

hep-th↗

arXiv2012

D-term Dynamical SUSY Breaking

We consider an N=1 supersymmetric U(N) gauge theory with an adjoint chiral multiplet. By developing a self-consistent Hartree-Fock approximation to the leading order which is reminiscent of that of the BCS/NJL in the superconductivity/chiral symmetry, we show that the N=1 supersymmetry is spontaneously broken, giving a mixed Majorana-Dirac mass term for gaugino due to the nonvanishing D-term VEV and F-term one induced by D-term.

hep-ph↗

arXiv2011

An(1) Affine Quiver Matrix Model

We introduce An(1) (n=1,2,...) affine quiver matrix model by simply adopting the extended Cartan matrices as incidence matrices and study its finite N Schwinger-Dyson equations as well as their planar limit. In the case of n=1, we extend our analysis to derive the cubic planar loop equation for one-parameter family of models labelled by alpha: alpha=1 and alpha=2 correspond to the non-affine A2 case and the affine A1(1) case respectively. In the case of n=2, we derive three sets of constraint equations for the resolvents which are quadratic, cubic and quartic respectively.

hep-th↗

arXiv2011

ε-Corrected Seiberg-Witten Prepotential Obtained From Half Genus Expansion in beta-Deformed Matrix Model

We consider the half-genus expansion of the resolvent function in the $β$-deformed matrix model with three-Penner potential under the AGT conjecture and the $0d-4d$ dictionary. The partition function of the model, after the specification of the paths, becomes the DF conformal block for fixed $c$ and provides the Nekrasov partition function expanded both in $g_s = \sqrt{-ε_1 ε_2}$ and in $ε= ε_1+ε_2$. Exploiting the explicit expressions for the lower terms of the free energy extracted from the above expansion, we derive the first few $ε$ corrections to the Seiberg-Witten prepotential in terms of the parameters of SU(2), $N_{f} =4$, ${\cal N}= 2$ supersymmetric gauge theory.

hep-th↗