arXiv subjects

Shinobu Hikami

Publications and source records attributed to Shinobu Hikami.

17 recordsLinked to original sources

arXiv2026

A relation between the HOMFLY-PT and Kauffman polynomials via characters

The HOMFLY-PT and Kauffman polynomials are related to each other for special classes of knots constructed by full twists and Jucys-Murphy twists. The conditions for this relation are articulated in terms of characters of the Birman-Murakami-Wenzl algebra. The latter are the coefficients in the expansion of the Kauffman polynomial involving the quantum dimensions of SO(N+1). This expansion allows to prove the conjectural 1-1 correspondence between the HOMFLY-PT/Kauffman relation and the Harer-Zagier (HZ) factorisability for a large family of 3-strand knots. The conjecture remains open for knots with 4 and higher strands.

hep-th↗

arXiv2025

Braid twists and HZ factorisation via character expansion

The HOMFLY-PT polynomial of a closed braid admits a character expansion, expressed as a sum of SU(N) characters over Young diagrams. The Harer-Zagier (HZ) transform, which converts the HOMFLY-PT polynomial into a rational function, is applied directly to the characters, yielding the HZ character expansion. We use this expansion to illumine the hidden structure of the HZ function that enables its decomposition into a sum of factorised terms. We further articulate explicit conditions for full HZ factorisation, which include that non-vanishing contributions come solely from hook-shaped Young diagrams. We show that these conditions remain invariant under three mutually commuting braid twists: full twists, partial full twists and Jucys-Murphy twists. We, hence, employ such twists to construct infinite, HZ-factorisable families of knots and links, which can often be thought of as a hyperbolic extension of torus knots. Remarkably, these families encompass the Coxeter links corresponding to E-type Dynkin diagrams.

math-ph↗

arXiv2024

The HOMFLY-PT polynomial and HZ factorisation

The Harer-Zagier (HZ) transform maps the HOMFLY-PT polynomial into a rational function. For some special knots and links, the latter admits a simple factorised form, which is referred to as HZ factorisation. This property is preserved under full twists and the Jucys-Murphy twists, which are hence used to generate infinite HZ-factorisable families of hyperbolic knots. For such families, the HOMFLY-PT polynomial can be fully encoded in two sets of integers, corresponding to the numerator and denominator exponents, which turn out to be related to the double-grading in Khovanov homology. Moreover, a relation between the HOMFLY-PT and Kauffman polynomials, which was only known to hold for torus knots, is now proven for several of these hyperbolic families. Such a relation has a peculiar implication in topological string theory, namely, it is equivalent to the vanishing of the two-crosscap BPS invariants. It is conjectured that the HOMFLY-PT/Kauffman relation provides a criterion for HZ factorisability.

math-ph↗

arXiv2023

Harer-Zagier formulas for families of twisted hyperbolic knots

In an attempt to generalise knot matrix models for non-torus knots, which currently remains an open problem, we derived formulas for the Harer-Zagier transform of the HOMFLY-PT polynomial for some infinite families of twisted hyperbolic knots. Among them, we found a family of Pretzel knots for which the transform has a fully factorised form, while for the remaining families considered it consists of sums of factorised terms. Their zeros have a remarkable structure as the modulus of their product always equals unity.

math-ph↗

arXiv2023

Knots from the random matrix theory with a replica

A classical knot is described by a one-stroke trajectory with entanglements of a string. The replica method appears as a powerful tool in statistical mechanics for a polymer or self-avoiding walk. We consider this replica N to 0 limit in Gaussian means of the products of trace of N x N Hermitian matrices, which provides one-stroke graphs of a knot. The Seifert surfaces of knots and links are derived by a random matrix model. The zeros of Alexander polynomials on a unit circle are discussed for the case of n-vertices in the analogy of Yang-Lee edge singularity. The extension of one matrix model is considered for higher dimensional knots and for half integral level k in Chern-Simons gauge theory.

math-ph↗

arXiv2022

Punctures and p-spin curves from matrix models III. Dl type and logarithmic potential

The intersection numbers for p spin curves of the moduli space M(g,n) are considered for D type by a matrix model. The asymptotic behavior of the large genus g limit and large p limit are derived. The remarkable features of the cases of p= 1/2, - 1/2, -2, -3 are examined in the Laurent expansion for multiple correlation functions. The strong coupling expansions for the negative p cases are considered.

hep-th↗

arXiv2020

Spectral Form Factor for Time-dependent Matrix model

The quantum chaos is related to a Gaussian random matrix model, which shows a dip-ramp-plateau behavior in the spectral form factor for the large size $N$. The spectral form factor of time dependent Gaussian random matrix model shows also dip-ramp-plateau behavior with a rounding behavior instead of a kink near Heisenberg time. This model is converted to two matrix model, made of $M_1$ and $M_2$. The numerical evaluation for finite $N$ and analytic expression in the large $N$ are compared for the spectral form factor.

hep-th↗

arXiv2019

Conformal Bootstrap Analysis for Localization: Symplectic Case

The localization phenomena due to the random potential scattering is widely discussed in the electron and photon systems, where the theoretical approach is the nonlinear $σ$ model with the replica method or with the supersymmetry. In this article, we discuss the application of the conformal bootstrap method to the localization by the small determinants. The possible correspondence to the symplectic Anderson localization is discussed.

cond-mat.dis-nn↗

arXiv2019

Dimensional Reduction by Conformal Bootstrap

The dimensional reductions in the branched polymer and the random field Ising model (RFIM) are discussed by a conformal bootstrap method. The small size minors are applied for the evaluations of the scale dimensions of these two models and the results are compared to D'=D-2 dimensional Yang-Lee edge singularity and to pure D'=D-2 dimensional Ising model, respectively. For the former case, the dimensional reduction is shown to be valid for $3 \le D \le 8$, and for the latter case, the deviation from the dimensional reduction can be seen below five dimensions.

cond-mat.dis-nn↗

arXiv2016

Fractal dimensions of self-avoiding walks and Ising high-temperature graphs in 3D conformal bootstrap

The fractal dimensions of polymer chains and high-temperature graphs in the Ising model both in three dimension are determined using the conformal bootstrap applied for the continuation of the $O(N)$ models from $N=1$ (Ising model) to $N=0$ (polymer). The unitarity bound below $N=1$ of the scaling dimension for the the $O(N)$-symmetric-tensor develops a kink as a function of the fundamental field as in the case of the energy operator dimension in the Ising model. Although this kink structure becomes less pronounced as $N$ tends to zero, an emerging asymmetric minimum in the current central charge $C_J$ can be used to locate the CFT. It is pointed out that certain level degeneracies at the $O(N)$ CFT should induce these singular shapes of the unitarity bounds. As an application to the quantum and classical spin systems, we also predict critical exponents associated with the $\mathcal{N}=1$ supersymmetry, which could be relevant for locating the correspoinding fixed point in the phase diagram.

cond-mat.stat-mech↗

arXiv2008

Intersection numbers from the antisymmetric Gaussian matrix model

The matrix model of topological field theory for the moduli space of p-th spin curves is extended to the case of the Lie algebra of the orthogonal group. We derive a new duality relation for the expectation values of characteristic polynomials in the antisymmetric Gaussian matrix model with an external matrix source. The intersection numbers for non-orientable surfaces of spin curves with k marked points are obtained from the Fourier transform of the k-point correlation functions at the critical point where the gap is closing.

math-ph↗

arXiv2007

Instanton and Superconductivity in Supersymmetric CP(N-1) Model

The two dimensional supersymmetric CP(N-1) model has a striking similarity to the N=2 supersymmetric gauge theory in four dimensions. The BPS mass formula and the curve of the marginal stability (CMS), which exist in the four dimensional gauge theory, appears in this two dimensional CP(N-1) model. These two quntities are derived by a one-dimensional n-vector spin model in the large n limit for the N=2 case. This mapping is further investigated at the critical point. An application of the study of the BPS mass formula is proposed to the phenomena of the spin and charge separations in the Higgs phase.

cond-mat.stat-mech↗

arXiv1994

Renormalized expansion for matrix models

Matrix models of 2d quantum gravity coupled to matter field are investigated by the renormalized perturbational method, in which the matrix model Hamiltonian is represented by the equivalent vector model. By the saddle point method, the renormalization group beta-function is obtained in the successive approximation.

cond-mat↗

arXiv1994

Renormalized expansion for matrix models

hep-th↗

arXiv1993

Finite N analysis of matrix models for n-Ising spin on a random surface

The saddle point equation described by the eigenvalues of N by N Hermitian matrices is analized for a finite N case and the scaling relation for the large N is considered. The critical point and the critical exponents of matrix model are obtained by the finite N scaling. One matrix model and two matrix model are studied in detail. Small N behavior for n-Ising model on a random surface is investigated.

hep-th↗

arXiv1993

Exponent of n-Ising matter fields coupled to 2d gravity

n-Ising spins on a random surface represented by a matrix model is studied as a model of the 2D gravity coupled to matter field with the central charge c > 1. The magnetic field is introduced to discuss the scaling exponent $Δ$, and the value of this magnetic field exponent is estimated by the series expansion.

hep-th↗

arXiv1992

Perturbative analysis of an n-Ising model on a random surface

Two dimensional quantum gravity coupled to a conformally invariant matter field of central charge c=n/2, is represented, in a discretized version, by n independent Ising spins per cell of the triangulations of a random surface. The matrix integral representation of this model leads to a diagrammatic expansion at large orders, when the Ising coupling constant is tuned to criticality, one extracts the values of the string susceptibility exponent. We extend our previous calculation to order eight for genus zero and investigate now also the genus one case in order to check the possibility of having a well-defined double scaling limit even c>1.

hep-th↗