arXiv subjects

Taro Kimura

Publications and source records attributed to Taro Kimura.

At least 19 recordsLinked to original sources

arXiv2026

Gauge Origami of the Conifold $\times$ $\mathbb{C}$

We construct gauge-origami systems on the toric Calabi--Yau fourfold $\mathcal{C}\times\mathbb{C}$ from its brane brick model. Brick matchings and phase boundaries determine elementary D6- and D4-brane framings, respectively, realizing the magnificent-four, tetrahedron-instanton, and spiked-instanton sectors in a unified framed quiver quantum mechanics. We define their partition functions by Jeffrey--Kirwan residues and describe the fixed points in terms of colored BPS crystals. General gauge-origami configurations are obtained by combining these elementary sectors through their common fractional D0-brane quiver. This construction provides a first step toward gauge origami on general toric Calabi--Yau fourfolds.

hep-th↗

arXiv2026

M2-brane matrix models and Fermi gas on affine ABCD quivers

We study a new class of M2-brane matrix models associated with affine BC and twisted affine AD quivers. These models exhibit the characteristic $N^{3/2}$ scaling of the free energy in the large $N$ limit. We establish a Fermi gas formalism and demonstrate that their grand canonical sum is given by a Fredholm Pfaffian, which is in parallel with the affine D quiver matrix models. The resulting Fermi gas density matrix shows a rational dependence on the canonical position/momentum operators in addition to the standard hyperbolic dependence. This leads to unconventional sub-leading corrections, which obstruct the Airy function structure in the $1/N$ expansion of the free energy observed in the previously studied M2-brane matrix models.

hep-th↗

arXiv2026

Random matrices

We provide a self-contained introduction to random matrices. While some applications are mentioned, our main emphasis is on three different approaches to random matrix models: the Coulomb gas method and its interpretation in terms of algebraic geometry, loop equations and their solution using topological recursion, orthogonal polynomials and their relation with integrable systems. Each approach provides its own definition of the spectral curve, a geometric object which encodes all the properties of a model. We also introduce the two peripheral subjects of counting polygonal surfaces, and computing angular integrals.

math-ph↗

arXiv2026

Higher-Rank Orthogonal Twists, APS Boundary Conditions, and $O(2)$-Equivariant Spectral Flow on a Warped Cylinder

We study $O(2)$-equivariant spectral flow for Dirac operators on a finite warped cylinder equipped with fixed admissible regularized APS boundary conditions. The twisting bundle is a real higher-rank orthogonal bundle, and reflection symmetry is implemented by a fiber involution. After complexifying the twisting bundle and diagonalizing the orthogonal twist, the Dirac equation decomposes into a scalar Fourier-mode radial equation, with moving rotating blocks and stationary neutral blocks. After regrouping conjugate and reflection-paired blocks, the crossing contributions define real $RO(O(2))$-classes. Consequently, we obtain an explicit blockwise formula for the $RO(O(2))$-valued spectral flow of the resulting regularized APS family. Under the standard self-adjoint Fredholm, endpoint-invertibility, and regular-crossing hypotheses, together with a fixed neutral-sector convention, this formula is obtained by assembling the local crossing contributions of the separated blocks. It refines ordinary integer-valued spectral flow and shows explicitly how the dimension map $RO(O(2))\to\mathbb Z$ loses representation-theoretic information. We also discuss the rank-three case, including the role of the fixed neutral sector, and the corresponding endpoint $η$/APS index interpretation.

math-ph↗

arXiv2026

Multicritical Scaling Limit of Shifted Schur Measure

We investigate the multicritical scaling limit of the shifted Schur measures. Under an appropriate scaling limit and specific conditions on the continuous parameters, we explicitly determine the limit shape of strict partitions distributed according to the shifted Schur measure. We then show that, under a multicritical condition, the edge scaling limit of the correlation function converges to a determinant of the higher-order Airy kernel. This rigorously demonstrates a transition from a Pfaffian point process to a determinantal distribution in the scaling limit.

math.CO↗

arXiv2026

Reflection Symmetry, APS Boundary Conditions, and Equivariant Spectral Flow on a Warped Cylinder

We study reflection symmetry and Atiyah-Patodi-Singer (APS) boundary conditions for twisted Dirac operators on a finite warped cylinder. For a complex line twist with holonomy parameter $A$, we show that the reflection lifts to a unitary symmetry of the twisted Dirac setting if and only if $2A\in\mathbb Z$. In the resulting reflection-compatible fixed-holonomy case, reflection pairs opposite shifted angular modes, and the paired APS blocks are unitarily equivalent. The reflection trace on the APS harmonic space localizes to the unique self-paired zero-mode sector. We then turn to parameter-dependent versions of the model. For fixed gauge-trivial holonomy, the family remains pointwise $O(2)$-equivariant, and its spectral flow admits an $RO(O(2))$-valued decomposition. For genuinely varying holonomy, pointwise $O(2)$-equivariance is lost along the path. The representation-ring-valued invariant is then replaced by a residual sign-level invariant: the mod-two parity of the APS crossing events.

math-ph↗

arXiv2026

Wall-crossing of Instantons on the Blow-up

We study the instanton counting in four dimensional $\mathcal{N}=2$ supersymmetric gauge theories on the blow-up of $\mathbb{C}^2$: we start by formulating the instanton moduli space as a quiver variety, which we regularise by introducing two stability parameters, thus endowing it with a structure of infinitely many chambers separated by walls. Within a given chamber, we formulate the instanton partition function as a contour integral, which can be evaluated using the Jeffrey-Kirwan residue prescription. We characterise the physically relevant contributions in terms of bipartite oriented graphs and show that they can more efficiently be classified in terms of combinatorial objects called super-partitions. Within a given chamber, only certain types of super-partitions contribute and we show that the corresponding selection criteria are equivalent to stability conditions that have previously been proposed in the literature. We use this formalism to compare how the instanton counting changes when moving across walls between neighbouring chambers and provide explicit expressions for the corresponding partition functions. In a limiting chamber and using our approach, we show how to reproduce the Nakajima-Yoshioka blow-up formula.

hep-th↗

arXiv2026

Dirac Operators, APS Boundary Conditions, and Spectral Flow on a Finite Warped Cylinder

We study the Dirac operator on a finite warped cylinder coupled to a background $U(1)$ gauge field. We identify the intrinsic endpoint operators defining the Atiyah-Patodi-Singer (APS) boundary condition and derive a determinant characterization of the modewise APS spectrum. In the constant-gauge, invertible setting, the endpoint reduced $η$ contributions cancel, so the APS index vanishes. For smooth gauge families, the APS projector becomes discontinuous when a boundary mode crosses zero. We therefore introduce a regularized APS-type family of self-adjoint endpoint conditions that remains continuous across such crossings. This regularized family admits a real-symplectic boundary formulation within the standard spectral-flow/Maslov framework: for nondegenerate regularization, the zero-mode set coincides with the boundary-zero set, and transverse boundary zeros give isolated regular crossings.

math-ph↗

arXiv2026

Determinantal formulas for Sklyanin-Whittaker integrals

We study multi-variable integrals, that we name Sklyanin-Whittaker integrals, and prove their determinantal formulas. We also discuss a $q$-deformation, a determinantal point process, and associated Mellin--Barnes integrals.

math-ph↗

arXiv2026

The 4-fold Pandharipande--Thomas vertex and Jeffrey--Kirwan residue

We present a contour integral formalism for computing the K-theoretic equivariant Pandharipande--Thomas (PT) 4-vertex. Within the Jeffrey--Kirwan (JK) residue framework, we show that the PT 4-vertex can be obtained from the same integrand as the Donaldson--Thomas (DT) 4-vertex by choosing a different reference vector. We illustrate the formalism through examples involving curves and surfaces on the 4-fold. Furthermore, we investigate the DT/PT correspondence for the 4-fold setting together with its higher rank and supergroup-like generalizations.

hep-th↗

arXiv2026

Gauge origami and quiver W-algebras IV: Pandharipande--Thomas $qq$-characters

We develop a contour integral formalism for computing the K-theoretic equivariant 3-vertex. Within the Jeffrey--Kirwan (JK) residue framework, we show that, by an appropriate choice of the reference vector, both the equivariant Donaldson--Thomas (DT) and Pandharipande--Thomas (PT) 3-vertices can be extracted from the same integrand. We analyze three distinct limits of the PT 3-vertex, recovering the unrefined topological vertex, the refined topological vertex, and the Macdonald refined topological vertex. Higher-rank extensions of PT counting and the DT/PT correspondence are also explored. From a quantum algebraic perspective, we construct an operator version of the equivariant PT 3-vertex and term it the Pandharipande--Thomas $qq$-character. We then discuss its connection with the quantum toroidal $\mathfrak{gl}_{1}$.

hep-th↗

arXiv2025

$q$-deformation of random partitions, determinantal structure, and Riemann-Hilbert problem

We study $q$-deformation of probability measures on partitions, i.e., $q$-deformed random partitions. We in particular consider the $q$-Plancherel measure and show a determinantal formula for the correlation function using a $q$-deformation of the discrete Bessel kernel. We also investigate Riemann-Hilbert problems associated with the corresponding orthogonal polynomials and obtain $q$-Painlevé equations from the $q$-difference Lax formalism.

math.CO↗

arXiv2025

Higgsing $qq$-character and irreducibility

We study the $qq$-character of quantum affine and toroidal algebra modules, with a focus on the role of spectral parameters. In particular, we revisit how their specialization affects the irreducibility of these modules.

math.QA↗

arXiv2025

Surface Defects in $A$-type Little String Theories

$A$-type Little String Theories (LSTs) are engineered from parallel M5-branes on a circle $\mathbb{S}_\perp^1$, probing a transverse $\mathbb{R}^4/\mathbb{Z}_M$ background. Below the scale of the radius of $\mathbb{S}_\perp^1$, these theories resemble a circular quiver gauge theory with $M$ nodes of gauge group $U(N)$ and matter in the bifundamental representation (or adjoint in the case of $M=1$). In this paper, we study these LSTs in the presence of a surface defect, which is introduced through the action of a $\mathbb{Z}_N$ orbifold that breaks the gauge groups into $[U(1)]^N$. We provide a combinatoric expression for the non-perturbative BPS partition function for this system. This form allows us to argue that a number of non-perturbative symmetries, that have previously been established for the LSTs, are preserved in the presence of the defect. Furthermore, we discuss the Nekrasov-Shatashvili (NS) limit of the defect partition function: focusing in detail on the case $(M,N)=(1,2)$, we analyse two distinct proposals made in the literature. We unravel an algebraic structure that is responsible for the cancellation of singular terms in the NS limit, which we generalise to generic $(M,N)$. In view of the dualities of higher dimensional gauge theories to quantum many-body systems, we provide indications that our combinatoric expression for the defect partition are useful in constructing and analysing quantum integrable systems in the future.

hep-th↗

arXiv2025

Gauge origami and quiver W-algebras II: Vertex function and beyond quantum $q$-Langlands correspondence

We continue the study of generalized gauge theory called gauge origami, based on the quantum algebraic approach initiated in [arXiv:2310.08545]. In this article, we in particular explore the D2 brane system realized by the screened vertex operators of the corresponding W-algebra. The partition function of this system given by the corresponding conformal block is identified with the vertex function associated with quasimaps to Nakajima quiver varieties and generalizations, that plays a central role in the quantum $q$-Langlands correspondence. Based on the quantum algebraic perspective, we address three new aspects of the correspondence: (i) Direct equivalence between the electric and magnetic blocks by constructing stable envelopes from the chamber structure of the vertex operators, (ii) Double affine generalization of quantum $q$-Langlands correspondence, and (iii) Conformal block realization of the origami vertex function associated with intersection of quasimaps, that realizes the higher-rank multi-leg Pandharipande-Thomas vertices of 3-fold and 4-fold.

hep-th↗

arXiv2025

Gauge origami and quiver W-algebras III: Donaldson--Thomas $qq$-characters

We further develop the BPS/CFT correspondence between quiver W-algebras/$qq$-characters and partition functions of gauge origami. We introduce $qq$-characters associated with multi-dimensional partitions with nontrivial boundary conditions which we call Donaldson--Thomas (DT) $qq$-characters. They are operator versions of the equivariant DT vertices of toric Calabi--Yau three and four-folds. Moreover, we revisit the construction of the D8 $qq$-characters with no boundary conditions and give a quantum algebraic derivation of the sign rules of the magnificent four partition function. We also show that under the proper sign rules, the D6 and D8 $qq$-characters with no boundary conditions all commute with each other and discuss its physical interpretation.

hep-th↗

arXiv2024

Fredholm Determinants from Schrödinger Type Equations, and Deformation of Tracy-Widom Distribution

We undertake an analysis of Fredholm determinants arising from kernels whose defining functions satisfy a Schrödinger type equation. When this defining function is the Airy one, the evaluation of the corresponding Fredholm determinant yields the notorious Tracy-Widom distribution [hep-th/9211141], which has found many applications in numerous domains. In this paper, we unveil a generalization of the Tracy-Widom distribution for a generic class of defining functions. Furthermore, we bring forth a direct application of our upshot and survey the relation between the framework which we employ and isomonodromic systems.

math-ph↗

arXiv2024

Seiberg-Witten curves of $\widehat{D}$-type Little Strings

Little Strings are a type of non-gravitational quantum theories that contain extended degrees of freedom, but behave like ordinary Quantum Field Theories at low energies. A particular class of such theories in six dimensions is engineered as the world-volume theory of an M5-brane on a circle that probes a transverse orbifold geometry. Its low energy limit is a supersymmetric gauge theory that is described by a quiver in the shape of the Dynkin diagram of the affine extension of an ADE-group. While the so-called $\widehat{A}$-type Little String Theories (LSTs) are very well studied, much less is known about the $\widehat{D}$-type, where for example the Seiberg-Witten curve (SWC) is only known in the case of the $\widehat{D}_4$ theory. In this work, we provide a general construction of this curve for arbitrary $\widehat{D}_{M}$ that respects all symmetries and dualities of the LST and is compatible with lower-dimensional results in the literature. For $M=4$ our construction reproduces the same curve as previously obtained by other methods. The form in which we cast the SWC for generic $\widehat{D}_M$ allows to study the behaviour of the LST under modular transformations and provides insights into a dual formulation as a circular quiver gauge theory with nodes of $Sp(M-4)$ and $SO(2M)$.

hep-th↗