Taro Kimura

University of Burgundy | UB · Institut de Mathématiques de Bourgogne (IMB)

We show that the double quantization of Seiberg-Witten spectral curve for $\Gamma$-quiver gauge theory defines the generating current of W$(\Gamma)$-algebra in the free field realization. We also show that the partition function is given as a correlator of the corresponding W$(\Gamma)$-algebra, which is equivalent to the AGT relation under the gaug...

We apply conformal field theory analysis to the $k$-channel SU($N$) Kondo system, and find a peculiar behavior in the cases $N > k > 1$, which we call Fermi/non-Fermi mixing: The low temperature scaling is described as the Fermi liquid, while the zero temperature IR fixed point exhibits the non-Fermi liquid signature. We also show that the Wilson r...

We find that generic boundary conditions of Weyl semimetal is dictated by only a single real parameter, in the continuum limit. We determine how the energy dispersions (the Fermi arcs) and the wave functions of edge states depend on this parameter. Lattice models are found to be consistent with our generic observation. Furthermore, the enhanced par...

We define elliptic generalization of W-algebras associated with arbitrary quiver using the formalism of arXiv:1512.08533 applied to six-dimensional quiver gauge theory compactified on elliptic curve.

For a quiver with weighted arrows we define gauge-theory K-theoretic W-algebra generalizing the definition of Shiraishi et al., and Frenkel and Reshetikhin. In particular, we show that the qq-character construction of gauge theory presented by Nekrasov is isomorphic to the definition of the W-algebra in the operator formalism as a commutant of scre...

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...

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 vertice...

We undertake an analysis of Fredholm determinants arising from kernels whose defining functions satisfy a Schr\"odinger 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 dom...

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 energ...

A bstract We explore the quantum algebraic formalism of the gauge origami system in ℂ ⁴ , where D2/D4/D6/D8-branes are present. We demonstrate that the contour integral formulas have free field interpretations, leading to the operator formalism of qq -characters associated with each D-brane. The qq -characters of D2 and D4-branes correspond to scre...

A bstract We consider Little String Theories (LSTs) that are engineered by N parallel M5-branes probing a transverse ℤ M geometry. By exploiting a dual description in terms of F-theory compactified on a toric Calabi-Yau threefold X N , M , we establish numerous symmetries that leave the BPS partition function 𝒵 N , M invariant. They furthemore act...

We study the universal scaling limit of random partitions obeying the Schur measure. Extending our previous analysis (Kimura and Zahabi in Lett. Math. Phys. 111:48, 2021. https://doi.org/10.1007/s11005-021-01389-y), we obtain the higher-order Pearcey kernel describing the multi-critical behavior in the cusp scaling limit. We explore the gap probabi...

We extend the study of superinstantons presented in (Kimura and Pestun in superinstanton counting and localization, 2019) to include orthosymplectic supergroup gauge theories, \(B_{n_0|n_1}\), \(C_n\), and \(D_{n_0|n_1}\). We utilize equivariant localization to obtain the LMNS contour integral formula for the instanton partition function, and we in...

Scale evolution of interactions between a Weyl fermion and a heavy magnetic impurity is calculated nonperturbatively using the functional renormalization group technique. Using an expansion around the vanishing pairing gap, we derive the flow equations for all possible quartic couplings in the system. We find that, contrary to conventional perturba...

We investigate a deformation of $w_{1+\infty}$ algebra recently introduced in arxiv:2111.11356 in the context of Celestial CFT that we denote by $\widetilde{W}_{1+\infty}$ algebra. We obtain the operator product expansions of the generating currents of this algebra and explore its supersymmetric and topological generalizations.

Scale evolution of interactions between a Weyl fermion and a heavy magnetic impurity is calculated non-perturbatively using the functional renormalization group technique. Using an expansion around the vanishing pairing gap, we derive the flow equations for all possible quartic couplings in the system. We find that contrary to conventional perturba...

We extend the study of superinstantons presented in 1905.01513 to include orthosymplectic supergroup gauge theories, $B_{n_0|n_1}$, $C_n$, and $D_{n_0|n_1}$. We utilize equivariant localization to obtain the LMNS contour integral formula for the instanton partition function, and we investigate the Seiberg--Witten geometries associated with these th...

We provide a formalism using the q-Cartan matrix to compute the instanton partition function of quiver gauge theory on various manifolds. Applying this formalism to eight dimensional setups, we introduce the notion of double quiver gauge theory characterized by a pair of quivers. We also explore the BPS/CFT correspondence in eight dimensions based...

We study the surface defect in N=2⁎ U(N) gauge theory in four dimensions and its relation to quantum Hall states in two dimensions. We first prove that the defect partition function becomes the Jack polynomial of the variables describing the brane positions by imposing the Higgsing condition and taking the bulk decoupling limit. Further tuning the...

In this paper, provide a survey of recent studies of supergroup gauge theory. We first discuss the supermatrix model as a zero-dimensional toy model of supergroup gauge theory and its geometric and algebraic characterization. We then focus on four-dimensional Yang–Mills theory with supergroup gauge symmetry and explore its nonperturbative propertie...

We provide a survey of recent studies of supergroup gauge theory. We first discuss supermatrix model as a zero-dimensional toy model of supergroup gauge theory and its geometric and algebraic characterization. We then focus on four-dimensional Yang--Mills theory with supergroup gauge symmetry and explore its non-perturbative properties, including i...

We provide a formalism using the $q$-Cartan matrix to compute the instanton partition function of quiver gauge theory on various manifolds. Applying this formalism to eight dimensional setups, we introduce the notion of double quiver gauge theory characterized by a pair of quivers. We also explore the BPS/CFT correspondence in eight dimensions base...

We analytically study boundary conditions of the Dirac fermion models on a lattice, which describe the first and second order topological insulators. We obtain the dispersion relations of the edge and hinge states by solving these boundary conditions, and clarify that the Hamiltonian symmetry may provide a constraint on the boundary condition. We al...

We study the surface defect in $\mathcal{N}=2^*$ $U(N)$ gauge theory in four dimensions and its relation to quantum Hall states in two dimensions. We first prove that the defect partition function becomes the Jack polynomial of the variables describing the brane positions by imposing the Higgsing condition and taking the bulk decoupling limit. Furt...

A bstract We study generalizations of the Gross-Witten-Wadia unitary matrix model for the special orthogonal and symplectic groups. We show using a standard Coulomb gas treatment — employing a path integral formalism for the ungapped phase and resolvent techniques for the gapped phase with one coupling constant — that in the large N limit, the free...

We study the universal scaling limit of random partitions obeying the Schur measure. Extending our previous analysis [arXiv:2012.06424], we obtain the higher-order Pearcey kernel describing the multi-critical behavior in the cusp scaling limit. We explore the gap probability associated with the higher Pearcey kernel, and derive the coupled nonlinea...

We show that the $qq$-character of the irreducible highest weight module for finite-type and affine quivers is obtained by Higgsing, specialization of the equivariant parameters of the associated framing space in the quiver variety.

We analytically study boundary conditions of the Dirac fermion models on a lattice, which describe the first and second order topological insulators. We obtain the dispersion relations of the edge and hinge states by solving these boundary conditions, and clarify that the Hamiltonian symmetry may provide a constraint on the boundary condition. We a...

We study matrix models involving Pfaffian interactions as generalizations of the standard $\beta = 1$ and $\beta = 4$ matrix models. We present the Pfaffian formulas for the partition function and the characteristic polynomial averages. We also explore the matrix chain with the Pfaffian interaction, which realizes the BCD-type quiver matrix models.

We study generalizations of the Gross--Witten--Wadia unitary matrix model for the special orthogonal and symplectic groups. We show using a standard Coulomb gas treatment -- employing a path integral formalism for the ungapped phase and resolvent techniques for the gapped phase with one coupling constant -- that in the large $N$ limit, the free ene...

A bstract We study the root of unity limit of ( q , t )-deformed Virasoro matrix models, for which we call the resulting model Uglov matrix model. We derive the associated Virasoro constraints on the partition function, and find agreement of the central charge with the expression obtained from the level-rank duality associated with the parafermion...

We study correlation functions of the characteristic polynomials in coupled matrix models based on the Schur polynomial expansion, which manifests their determinantal structure.

We study the root of unity limit of $(\textbf{q}, \textbf{t})$-deformed Virasoro matrix models, for which we call the resulting model Uglov matrix model. We derive the associated Virasoro constraints on the partition function and find agreement of the central charge with the expression obtained from the level-rank duality associated with the parafe...

We develop a new framework to compute the exact correlators of characteristic polynomials, and their inverses, in random matrix theory. Our results hold for general potentials and incorporate the effects of an external source. In matrix model realizations of string theory, these correspond to correlation functions of exponentiated "(anti-)branes" i...

We consider 5d supersymmetric gauge theories with unitary groups in the Ω-background and study codim-2/4 BPS defects supported on orthogonal planes intersecting at the origin along a circle. The intersecting defects arise upon implementing the most generic Higgsing (geometric transition) to the parent higher dimensional theory, and they are describ...

Seiberg–Witten theory is a geometric framework to describe the low energy effective theory of N=2 supersymmetric gauge theory in four dimensions [58, 59]. This geometric point of view gives rise to various interesting insights on gauge theory, including dualities in gauge theory, the brane dynamics in string/M-theory, connections with integrable sy...

Together with a set of edges (arrows) Γ1, we define a quiver Γ=(Γ0,Γ1). Quiver For each edge e:i→j, in 4d N=1 gauge theory (4 SUSY) convention, we assign the chiral multiplet in the bifundamental representation of (Gi,Gj).

Supergroup is a natural generalization of the concept of group, which describes symmetry of systems involving both bosonic and fermionic degrees of freedom. Although it is typically considered as a global symmetry of QFT, there are several situations where we should consider a local supergroup symmetry, e.g., supergravity is interpreted as gauge th...

We have discussed classical and quantum geometric aspects of N=2 gauge theory in four dimensions in the relation to various fields of physics and mathematics. One of the key ingredients in such aspects is the non-perturbative symmetry of gauge theory incorporated by instantons, i.e., covariance of path integral partition function under the adding/r...

The aim of this Chapter is to introduce the Yang–Mills (YM) theory, and explain how the specific solution, called the instanton, plays an important role in four-dimensional gauge theory. We will explain a systematic method to describe the instanton solution, a.k.a. ADHM construction [2], and discuss how the moduli space of the instanton plays a rol...

The low energy effective behavior of N=2 gauge theory has the geometric characterization due to the Seiberg–Witten theory. In particular, the algebraic curve, called the Seiberg–Witten curve, geometrically encodes the information about the prepotential of N=2 theory. In this Chapter, we show how to obtain such an algebraic object from the microscop...

We have shown that the instanton partition function has a realization as a chiral correlation function of the vertex operators, whose algebraic structure depends on the quiver structure of gauge theory.

The quantum algebraic structure emerging from the moduli space of gauge theory is not unique to 5d N=1 gauge theory. As shown in Sect. 4.6.2, 6d N=(1,0) gauge theory compactified on a torus has a similar geometric description, and it is natural to explore the underlying quantum algebraic structure of it.

A bstract The generating functions for the gauge theory observables are often represented in terms of the unitary matrix integrals. In this work, the perturbative and non-perturbative aspects of the generic multi-critical unitary matrix models are studied by adopting the integrable operator formalism, and the multi-critical generalization of the Tr...

We consider 5d supersymmetric gauge theories with unitary groups in the $\Omega$-background and study codim-2/4 BPS defects supported on orthogonal planes intersecting at the origin along a circle. The intersecting defects arise upon implementing the most generic Higgsing (geometric transition) to the parent higher dimensional theory, and they are...

The generating functions for the supersymmetric indices of the gauge theory such as superconformal index are often represented in terms of the unitary matrix integrals with double trace potential. In the limit of weak interactions between the eigenvalues, they can be approximated by the matrix models with the single-trace potential, i.e. the genera...

We establish the universal edge scaling limit of random partitions with the infinite-parameter distribution called the Schur measure. We explore the asymptotic behavior of the wave function, which is a building block of the corresponding kernel, based on the Schrödinger-type differential equation. We show that the wave function is in general asympt...

A bstract In this article, we extend the work of [1] to a Bethe/Gauge correspondence between 2d (or resp. 3d) SO/Sp gauge theories and open XXX (resp. XXZ) spin chains with diagonal boundary conditions. The case of linear quiver gauge theories is also considered.

We propose a refined version of the Sato-Tate conjecture about the spacing distribution of the angle determined for each prime number. We also discuss its implications on $L$-function associated with elliptic curves in the relation to random matrix theory.

This book pedagogically describes recent developments in gauge theory, in particular four-dimensional N = 2 supersymmetric gauge theory, in relation to various fields in mathematics, including algebraic geometry, geometric representation theory, vertex operator algebras. The key concept is the instanton, which is a solution to the anti-self-dual Ya...

In this article, we extend the work of arXiv:0901.4744 to a Bethe/Gauge correspondence between 2d (or resp. 3d) SO/Sp gauge theories and open XXX (resp. XXZ) spin chains with diagonal boundary conditions. The case of linear quiver gauge theories is also considered.

The aim of this memoir for "Habilitation \`a Diriger des Recherches" is to present quantum geometric and algebraic aspects of supersymmetric gauge theory, which emerge from non-perturbative nature of the vacuum structure induced by instantons. We start with a brief summary of the equivariant localization of the instanton moduli space, and show how...

We establish the universal edge scaling limit of random partitions with the infinite-parameter distribution called the Schur measure. We explore the asymptotic behavior of the wave function, which is a building block of the corresponding kernel, based on the Schr{\"o}dinger-type differential equation. We show that the wave function is in general as...

We compute the partition functions of 𝒩 = 1 gauge theories on S2 × ℝ 𝜀2 using supersymmetric localization. The path integral reduces to a sum over vortices at the poles of S2 and at the origin of ℝ𝜀2. The exact partition functions allow us to test Seiberg duality beyond the supersymmetric index. We propose the 𝒩 = 1 partition functions on the Ω-bac...

We explore the Kondo effect incorporating the localized impurity transforming under generic symmetry group $G$, that we call the $G$-Kondo effect. We derive the one-dimensional effective model coupled with the impurity, and studied the thermodynamic properties based on the boundary conformal field theory approach. We in particular study the impurit...

We consider a large class of branes in toric strip geometries, both non-periodic and periodic ones. For a fixed background geometry we show that partition functions for such branes can be reinterpreted, on one hand, as quiver generating series, and on the other hand as wave-functions in various polarizations. We determine operations on quivers, as...

We consider a large class of branes in toric strip geometries, both non-periodic and periodic ones. For a fixed background geometry we show that partition functions for such branes can be reinterpreted, on one hand, as quiver generating series, and on the other hand as wave-functions in various polarizations. We determine operations on quivers, as...

A bstract In this note, we establish several interesting connections between the super- group gauge theories and the super integrable systems, i.e. gauge theories with supergroups as their gauge groups and integrable systems defined on superalgebras. In particular, we construct the super-characteristic polynomials of super-Toda lattice and elliptic...

We show the vertex operator formalism for the quiver gauge theory partition function and the qq-character of the highest weight module on quiver, both associated with the integral over the quiver variety.

A bstract We propose a new vertex formalism, called anti-refined topological vertex (anti-vertex for short), to compute the generalized topological string amplitude, which gives rise to the supergroup gauge theory partition function. We show the one-to-many correspondence between the gauge theory and the Calabi-Yau geometry, which is peculiar to th...

In this note, we establish several interesting connections between the supergroup gauge theoriesand the super integrable systems, i.e. gauge theories with supergroups as their gauge groups and integrablesystems defined on superalgebras. In particular, we construct the super-characteristic polynomials of super-Toda lattice and elliptic double Caloge...

A bstract We systematically study the interesting relations between the quantum elliptic Calogero-Moser system (eCM) and its generalization, and their corresponding supersymmetric gauge theories. In particular, we construct the suitable characteristic polynomial for the eCM system by considering certain orbifolded instanton partition function of th...

We propose a new vertex formalism, called anti-refined topological vertex (anti-vertex for short), to compute the generalized topological string amplitude, which gives rise to the supergroup gauge theory partition function. We show the one-to-many correspondence between the gauge theory and the Calabi--Yau geometry, which is peculiar to the supergr...

We show the vertex operator formalism for the quiver gauge theory partition function and the $qq$-character of highest-weight module on quiver, both associated with the integral over the quiver variety.

A bstract The topological vertex formalism for 5d $$ \mathcal{N} $$ N = 1 gauge theories is not only a convenient tool to compute the instanton partition function of these theories, but it is also accompanied by a nice algebraic structure that reveals various kinds of nice properties such as dualities and integrability of the underlying theories. T...

We systematically study the interesting relations between the quantum elliptic Calogero-Moser system (eCM) and its generalization, and their corresponding supersymmetric gauge theories. In particular, we construct the suitable characteristic polynomial for the eCM system by considering certain orbifolded instanton partition function of the correspo...

The topological vertex formalism for 5d $\mathcal{N}=1$ gauge theories is not only a convenient tool to compute the instanton partition function of these theories, but it is also accompanied by a nice algebraic structure that reveals various kinds of nice properties such as dualities and integrability of the underlying theories. The usual refined t...

We consider the $k$-twisted Nekrasov-Shatashvili limit (NS$_k$ limit) of 5d (K-theoretic) and 6d (elliptic) quiver gauge theory, where one of the multiplicative equivariant parameters is taken to be the $k$-th root of unity. We obtain the extended center of the associated $q$-deformed quiver W-algebras constructed by our formalism [arXiv:1512.08533...

We study the super instanton solution in the gauge theory with U$(n_{+}| n_{-})$ gauge group. Based on the ADHM construction generalized to the supergroup theory, we derive the instanton partition function from the super instanton moduli space through the equivariant localization. We derive the Seiberg-Witten geometry and its quantization for the s...

A bstract The mirror curves enable us to study B-model topological strings on noncompact toric Calabi-Yau threefolds. One of the method to obtain the mirror curves is to calculate the partition function of the topological string with a single brane. In this paper, we discuss two types of geometries: one is the chain of N ℙ ¹ ’s which we call “ N -c...

We study nonperturbative aspects of the QCD Kondo effect, which has been recently proposed for the finite density and strong magnetic field systems, using conformal field theory describing the low-energy physics near the IR fixed point. We clarify the symmetry class of the QCD Kondo effect for both the finite density and magnetic field systems and...

We compute the partition functions of $\mathcal{N} = 1$ gauge theories on $S^2 \times \mathbb{R}^2_\varepsilon$ using supersymmetric localization. The path integral reduces to a sum over vortices at the poles of $S^2$ and at the origin of $\mathbb{R}^2_\varepsilon$. The exact partition functions allow us to test Seiberg duality beyond the supersymm...

We introduce quiver gauge theory associated with the non-simply-laced type fractional quiver, and define fractional quiver W-algebras by using construction of arXiv:1512.08533 and arXiv:1608.04651 with representation of fractional quivers.

The mirror curves enable us to study B-model topological strings on non-compact toric Calabi--Yau threefolds. One of the method to obtain the mirror curves is to calculate the partition function of the topological string with single brane. In this paper, we discuss two types of geometries; one is the chain of $N$ $\mathbb{P}^1$'s which we call `$N$...

We study non-perturbative aspects of QCD Kondo effect, which has been recently proposed for the finite density and strong magnetic field systems, using conformal field theory describing the low energy physics near the IR fixed point. We clarify the symmetry class of QCD Kondo effect both for the finite density and magnetic field systems, and show h...

We study non-perturbative aspects of QCD Kondo effect, which has been recently proposed for the finite density and strong magnetic field systems, using conformal field theory describing the low energy physics near the IR fixed point. We clarify the symmetry class of QCD Kondo effect both for the finite density and magnetic field systems, and show h...

A bstract We consider the compactifcation of 5d non-simply laced fractional quiver gauge theory constructed in [1]. In contrast to the simply laced quivers, here two Ω-background parameters play different roles, so that we can take two possible Nekrasov-Shatashvili limits. We demonstrate how different quantum integrable systems can emerge from thes...

We consider the compactifcation of 5d non-simply laced fractional quiver gauge theory constructed in arXiv:1705.04410. In contrast to the simply laced quivers, here two $\Omega$-background parameters play different roles, so that we can take two possible Nekrasov-Shatashvili limits. We demonstrate how different quantum integrable systems can emerge...

We demonstrate how one can see quantization of geometry, and quantum algebraic structure in supersymmetric gauge theory.

We introduce quiver gauge theory associated with the non-simply-laced type fractional quiver, and define fractional quiver W-algebras by using construction of arXiv:1512.08533 and arXiv:1608.04651 with representation of fractional quivers.

We show the refinement of the prescription for the geometric transition in the refined topological string theory and, as its application, discuss a possibility to describe $qq$-characters from the string theory point of view. Though the suggested way to operate the refined geometric transition has passed through several checks, it is additionally f...

We show the refinement of the prescription for the geometric transition in the refined topological string theory and, as its application, discuss a possibility to describe $qq$-characters from the string theory point of view. Though the suggested way to operate the refined geometric transition has passed through several checks, it is additionally f...

We study an exotic state which is localized only at an intersection of edges of a topological material. This “edge-of-edge” state is shown to exist generically. We construct explicitly generic edge-of-edge states in five-dimensional Weyl semimetals and their dimensional reductions, such as four-dimensional topological insulators of class A and thre...

We study an exotic state which is localized only at an intersection of edges of a topological material. This "edge-of-edge" state is shown to exist generically. We construct explicitly generic edge-of-edge states in 5-dimensional Weyl semimetals and their dimensional reductions, such as 4-dimensional topological insulators of class A and 3-dimensio...

We study \(\mathrm {U}(N|M)\) character expectation value with the supermatrix Chern–Simons theory, known as the ABJM matrix model, with emphasis on its connection to the knot invariant. This average just gives the half-BPS circular Wilson loop expectation value in ABJM theory, which shall correspond to the unknot invariant. We derive the determina...

We apply conformal field theory analysis to the $k$-channel SU($N$) Kondo system, and find a peculiar behavior in the cases $N > k > 1$, which we call Fermi/non-Fermi mixing: The low temperature scaling is described as the Fermi liquid, while the zero temperature IR fixed point exhibits the non-Fermi liquid signature. We also show that the Wilson r...

We study the electron-electron interaction effects on topological phase transitions by the ab-initio quantum Monte Carlo simulation. We analyze two-dimensional class A topological insulators and three-dimensional Weyl semimetals with the long-range Coulomb interaction. The direct computation of the Chern number shows the electron-electron interacti...

We study the electron-electron interaction effects on topological phase transitions by the ab-initio quantum Monte Carlo simulation. We analyze two-dimensional class A topological insulators and three-dimensional Weyl semimetals with the long-range Coulomb interaction. The direct computation of the Chern number shows the electron-electron interacti...

We find that generic boundary conditions of Weyl semimetal is dictated by only a single real parameter, in the continuum limit. We determine how the energy dispersions (the Fermi arcs) and the wave functions of edge states depend on this parameter. Lattice models are found to be consistent with our generic observation. Furthermore, the enhanced par...

We define elliptic generalization of W-algebras associated with arbitrary quiver using the formalism of arXiv:1512.08533 applied to six-dimensional quiver gauge theory compactified on elliptic curve.