arXiv subjects

Hiroyuki Fuji

Publications and source records attributed to Hiroyuki Fuji.

At least 19 recordsLinked to original sources

arXiv2025

A Hamiltonian Formalism for Topological Recursion

We propose a string field Hamiltonian formalism that associates a class of spectral curves and provides their quantization through the Chekhov-Eynard-Orantin topological recursion. As illustrative examples, we present Hamiltonians for the $(2,2m-1)$ minimal discrete and continuum dynamical triangulation (DT) models, the supersymmetric analogue of minimal continuum DT models, the Penner model, and 4D $\mathcal{N}=2$ $SU(2)$ gauge theories in the self-dual $\Omega$-background.

math-ph↗

arXiv2025

Multicritical Dynamical Triangulations and Topological Recursion

We explore a continuum theory of multicritical dynamical triangulations and causal dynamical triangulations in two-dimensional quantum gravity from the perspective of the Chekhov-Eynard-Orantin topological recursion. The former model lacks a causal time direction and is governed by the two-reduced $W^{(3)}$ algebra, whereas the latter model possesses a causal time direction and is governed by the full $W^{(3)}$ algebra. We show that the topological recursion solves the Schwinger-Dyson equations for both models, and we explicitly compute several amplitudes.

hep-th↗

arXiv2025

Dynamical Triangulations for 2D Pure Gravity and Topological Recursion

We show that, in two-dimensional Euclidean quantum gravity without matter fields, the Schwinger-Dyson equations derived within the Hamiltonian framework of non-critical string field theory can be reformulated in terms of the Chekhov-Eynard-Orantin topological recursion, and we explicitly compute the associated low-order amplitudes. In particular, we establish this reformulation for two discrete models -- the basic type and the strip type -- as well as for the continuum limit of dynamical triangulations.

hep-th↗

arXiv2023

Some Generalizations of Mirzakhani's Recursion and Masur-Veech Volumes via Topological Recursions

Via Andersen-Borot-Orantin's geometric recursion, a twist of the topological recursion was proposed, and a recursion for the Masur-Veech polynomials was uncovered. The purpose of this article is to explore generalizations of Mirzakhani's recursion based on physical two-dimensional gravity models related to the Jackiw-Teitelboim gravity and to provide an introduction to various realizations of topological recursion. For generalized Mirzakhani's recursions involving a Masur-Veech type twist, we derive Virasoro constraints and cut-and-join equations, and also show some computations of generalized volumes for the physical two-dimensional gravity models.

math-ph↗

arXiv2021

Topology of protein metastructure and $β$-sheet topology

We introduce a new, simplified model of proteins, which we call protein metastructure. The metastructure of a protein carries information about its secondary structure and $β$-strand conformations. Furthermore, protein metastructure allows us to associate an object called a fatgraph to a protein, and a fatgraph in turn gives rise to a topological surface. It becomes thus possible to study the topological invariants associated to a protein. We discuss the correspondence between protein metastructures and fatgraphs, and how one can compute topological invariants, such as genus and the number of boundary components, from fatgraphs. We then describe an algorithm for generating likely candidate metastructures using the information obtained from topology of protein fatgraphs. This algorithm is further developed to predict $β$-sheet topology of proteins, with a possibility to combine it with an existing algorithm. We demonstrate the algorithm on the data from PDB, and improve the performance of and existing algorithm by combining with it.

q-bio.BM↗

arXiv2020

Witten-Reshetikhin-Turaev function for a knot in Seifert manifolds

In this paper, for a Seifert loop (i.e., a knot in a Seifert three-manifold), first we give a family of an explicit function $Φ(q; N)$ whose special values at roots of unity are identified with the Witten-Reshetikhin-Turaev invariants of the Seifert loop for the integral homology sphere. Second, we show that the function $Φ(q; N)$ satisfies a $q$-difference equation whose classical limit coincides with a component of the character varieties of the Seifert loop. Third, we give an interpretation of the function $Φ(q; N)$ from the view point of the resurgent analysis.

math.GT↗

arXiv2019

Reconstructing GKZ via topological recursion

In this article, a novel description of the hypergeometric differential equation found from Gel'fand-Kapranov-Zelevinsky's system (referred to GKZ equation) for Givental's $J$-function in the Gromov-Witten theory will be proposed. The GKZ equation involves a parameter $\hbar$, and we will reconstruct it as the WKB expansion from the classical limit $\hbar\to 0$ via the topological recursion. In this analysis, the spectral curve (referred to GKZ curve) plays a central role, and it can be defined as the critical point set of the mirror Landau-Ginzburg potential. Our novel description is derived via the duality relations of the string theories, and various physical interpretations suggest that the GKZ equation is identified with the quantum curve for the brane partition function in the cohomological limit. As an application of our novel picture for the GKZ equation, we will discuss the Stokes matrix for the equivariant $\mathbb{C}\textbf{P}^{1}$ model and the wall-crossing formula for the total Stokes matrix will be examined. And as a byproduct of this analysis we will study Dubrovin's conjecture for this equivariant model.

math-ph↗

arXiv2019

Janossy densities for chiral random matrix ensembles and their applications to two-color QCD

We compute individual distributions of low-lying eigenvalues of massive chiral random matrix ensembles by the Nyström-type quadrature method for evaluating the Fredholm determinant and Pfaffian that represent the analytic continuation of the Janossy densities (conditional gap probabilities). A compact formula for individual eigenvalue distributions suited for precise numerical evaluation by the Nyström-type method is obtained in an explicit form, and the $k^{\rm{\small th}}$ smallest eigenvalue distributions are numerically evaluated for chiral unitary and symplectic ensembles in the microscopic limit. As an application of our result, the low-lying Dirac spectra of the SU(2) lattice gauge theory with $N_F=8$ staggered flavors are fitted to the numerical prediction from the chiral symplectic ensemble, leading to a precise determination of the chiral condensateof a two-color QCD-like system in the future.

hep-lat↗

arXiv2017

Enumeration of chord diagrams via topological recursion and quantum curve techniques

In this paper we consider the enumeration of orientable and non-orientable chord diagrams. We show that this enumeration is encoded in appropriate expectation values of the $β$-deformed Gaussian and RNA matrix models. We evaluate these expectation values by means of the $β$-deformed topological recursion, and - independently - using properties of quantum curves. We show that both these methods provide efficient and systematic algorithms for counting of chord diagrams with a given genus, number of backbones and number of chords.

math-ph↗

arXiv2017

Partial chord diagrams and matrix models

In this article, the enumeration of partial chord diagrams is discussed via matrix model techniques. In addition to the basic data such as the number of backbones and chords, we also consider the Euler characteristic, the backbone spectrum, the boundary point spectrum, and the boundary length spectrum. Furthermore, we consider the boundary length and point spectrum that unifies the last two types of spectra. We introduce matrix models that encode generating functions of partial chord diagrams filtered by each of these spectra. Using these matrix models, we derive partial differential equations - obtained independently by cut-and-join arguments in an earlier work - for the corresponding generating functions.

math-ph↗

arXiv2017

The boundary length and point spectrum enumeration of partial chord diagrams using cut and join recursion

We introduce the boundary length and point spectrum, as a joint generalization of the boundary length spectrum and boundary point spectrum in arXiv:1307.0967. We establish by cut-and-join methods that the number of partial chord diagrams filtered by the boundary length and point spectrum satisfies a recursion relation, which combined with an initial condition determines these numbers uniquely. This recursion relation is equivalent to a second order, non-linear, algebraic partial differential equation for the generating function of the numbers of partial chord diagrams filtered by the boundary length and point spectrum.

math-ph↗

arXiv2013

Super-A-polynomial

We review a construction of a new class of algebraic curves, called super-A-polynomials, and their quantum generalizations. The super-A-polynomial is a two-parameter deformation of the A-polynomial known from knot theory or Chern-Simons theory with SL(2,C) gauge group. The two parameters of the super-A-polynomial encode, respectively, the t-deformation which leads to the "refined A-polynomial", and the Q-deformation which leads to the augmentation polynomial of knot contact homology. For a given knot, the super-A-polynomial encodes the asymptotics of the corresponding S^r-colored HOMFLY homology for large r, while the quantum super-A-polynomial provides recursion relations for such homology theories for each r. The super-A-polynomial also admits a simple physical interpretation as the defining equation for the space of SUSY vacua in a circle compactification of the effective 3d N=2 theory associated to a given knot (complement). We discuss properties of super-A-polynomials and illustrate them in many examples.

math.AG↗

arXiv2012

3d analogs of Argyres-Douglas theories and knot homologies

We study singularities of algebraic curves associated with 3d N=2 theories that have at least one global flavor symmetry. Of particular interest is a class of theories T_K labeled by knots, whose partition functions package Poincare polynomials of the S^r-colored HOMFLY homologies. We derive the defining equation, called the super-A-polynomial, for algebraic curves associated with many new examples of 3d N=2 theories T_K and study its singularity structure. In particular, we catalog general types of singularities that presumably exist for all knots and propose their physical interpretation. A computation of super-A-polynomials is based on a derivation of corresponding superpolynomials, which is interesting in its own right and relies solely on a structure of differentials in S^r-colored HOMFLY homologies.

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

Super-A-polynomial for knots and BPS states

We introduce and compute a 2-parameter family deformation of the A-polynomial that encodes the color dependence of the superpolynomial and that, in suitable limits, reduces to various deformations of the A-polynomial studied in the literature. These special limits include the t-deformation which leads to the "refined A-polynomial" introduced in the previous work of the authors and the Q-deformation which leads, by the conjecture of Aganagic and Vafa, to the augmentation polynomial of knot contact homology. We also introduce and compute the quantum version of the super-A-polynomial, an operator that encodes recursion relations for S^r-colored HOMFLY homology. Much like its predecessor, the super-A-polynomial admits a simple physical interpretation as the defining equation for the space of SUSY vacua (= critical points of the twisted superpotential) in a circle compactification of the effective 3d N=2 theory associated to a knot or, more generally, to a 3-manifold M. Equivalently, the algebraic curve defined by the zero locus of the super-A-polynomial can be thought of as the space of open string moduli in a brane system associated with M. As an inherent outcome of this work, we provide new interesting formulas for colored superpolynomials for the trefoil and the figure-eight knot.

hep-th↗

arXiv2012

Volume Conjecture: Refined and Categorified

The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the A-polynomial $A(x,y)$. Another "family version" of the volume conjecture depends on a quantization parameter, usually denoted $q$ or $\hbar$; this quantum volume conjecture (also known as the AJ-conjecture) can be stated in a form of a q-difference equation that annihilates the colored Jones polynomials and $SL(2,\C)$ Chern-Simons partition functions. We propose refinements / categorifications of both conjectures that include an extra deformation parameter $t$ and describe similar properties of homological knot invariants and refined BPS invariants. Much like their unrefined / decategorified predecessors, that correspond to $t=-1$, the new volume conjectures involve objects naturally defined on an algebraic curve $A^{ref} (x,y; t)$ obtained by a particular deformation of the A-polynomial, and its quantization $\hat A^{ref} (\hat x, \hat y; q, t)$. We compute both classical and quantum t-deformed curves in a number of examples coming from colored knot homologies and refined BPS invariants.

hep-th↗

arXiv2011

Causal Dynamical Triangulation with Extended Interactions in 1+1 Dimensions

We study the Causal Dynamical Triangulation (CDT) with extended interactions in 1+1 dimensions applying the method in the non-critical string field theory (SFT) constructed by Ishibashi and Kawai. For this model, we solve Schwinger-Dyson's equation (SDE) for disk amplitude perturbatively, and find a matrix model in the continuum limit reproducing the SDE in the non-critical SFT approach as the loop equation.

hep-th↗

arXiv2011

Summing Up All Genus Free Energy of ABJM Matrix Model

The localization technique allows us to compute the free energy of the U(N)_k x U(N)_{-k} Chern-Simons-matter theory dual to type IIA strings on AdS_4 x CP^3 from weak to strong 't Hooft coupling λ= N / k at finite N, as demonstrated by Drukker, Marino, and Putrov. In this note we study further the free energy at large 't Hooft coupling with the aim of testing AdS/CFT at the quantum gravity level and, in particular, sum up all the 1/N corrections, apart from the worldsheet instanton contributions. The all genus partition function takes a remarkably simple form -- the Airy function, Ai (k^{4/3} λ_r), with the renormalized 't Hooft coupling λ_r.

hep-th↗