arXiv subjects

Akifumi Sako

Publications and source records attributed to Akifumi Sako.

At least 19 recordsLinked to original sources

arXiv2026

Quantization of Algebraic Varieties Defined by Casimir Polynomials via Matrix Regularization: Fuzzy $S^7$ and Beyond

We study the quantization of algebraic varieties defined by equations involving Casimir polynomials of compact semisimple Lie algebras. The Casimir polynomials belong to the Poisson center of the corresponding Lie-Poisson algebra. For this purpose, we employ a recently developed matrix regularization of Lie-Poisson algebras. In particular, using its formulation based on reducible representations, we construct quantizations of these algebraic varieties through their decomposition into coadjoint orbits, including singular orbits. As a concrete example, we present the construction of fuzzy $S^7$ in detail.

hep-th↗

arXiv2026

Quantization of Lie-Poisson algebra and Lie algebra solutions of mass-deformed type IIB matrix model

A quantization of Lie-Poisson algebras is studied. Classical solutions of the mass-deformed Ishibashi-Kawai-Kitazawa-Tsuchiya (IKKT) matrix model can be constructed from semisimple Lie algebras whose dimension matches the number of matrices in the model. We consider the geometry described by the classical solutions of the Lie algebras in the limit where the mass vanishes and the matrix size tends to infinity. Lie-Poisson varieties are regarded as such geometric objects. We provide a quantization called ``weak matrix regularization'' of Lie-Poisson algebras (linear Poisson algebras) on the algebraic varieties defined by their Casimir polynomials. This quantization is a generalization of matrix regularization, and neither faithfulness of the map nor the correspondence between integration and trace in the commutative limit is required. Casimir polynomials correspond with Casimir operators of the Lie algebra by the quantization. This quantization is a generalization of the method for constructing the fuzzy sphere. In order to define the weak matrix regularization of the quotient space by the ideal generated by the Casimir polynomials, we take a fixed reduced Gröbner basis of the ideal. The Gröbner basis determines remainders of polynomials. The operation of replacing these remainders with representation matrices of a Lie algebra roughly corresponds to a weak matrix regularization. As concrete examples, we construct weak matrix regularization for $\mathfrak{su}(2)$ and $\mathfrak{su}(3)$. In the case of $\mathfrak{su}(3)$, we not only construct weak matrix regularization for the quadratic Casimir polynomial, but also construct weak matrix regularization for the cubic Casimir polynomial.

hep-th↗

arXiv2025

Deformation Quantization with Separation of Variables of $G_{2,4}(\mathbb{C})$

We construct a deformation quantization with separation of variables of the Grassmannian $G_{2,4}(\mathbb{C})$. A star product on $G_{2,4}(\mathbb{C})$ can be explicitly determined as the solution of the recurrence relations for $G_{2,4}(\mathbb{C})$ given by Hara and one of the authors (A. Sako). To provide the solution to the recurrence relations, it is necessary to solve a system of linear equations in each order. However, to give a concrete expression of the general term is not simple because the variables increase with the order of the differentiation of the star product. For this reason, there has been no formula to express the general term of the recurrence relations. In this paper, we overcome this problem by transforming the recurrence relations into simpler ones. We solve the recurrence relations using creation and annihilation operators on a Fock space. From this solution, we obtain an explicit formula of a star product with separation of variables on $G_{2,4}(\mathbb{C})$.

math-ph↗

arXiv2025

Relationship between a $Φ^4$ matrix model and harmonic oscillator systems

A Hermitian $Φ^4$ matrix model with a Kontsevich-type kinetic term is studied. It was recently discovered that the partition function of this matrix model satisfies the Schrödinger equation of the $N$-body harmonic oscillator, and that eigenstates of the Virasoro operators can be derived from this partition function. We extend these results and obtain an explicit formula for such eigenstates in terms of the free energy. Furthermore, the Schrödinger equation for the $N$-body harmonic oscillator can also be reformulated in terms of connected correlation functions. The $U(1)^N$-symmetry allows us to derive loop equations.

hep-th↗

arXiv2024

Explicit formula of deformation quantization with separation of variables for complex two-dimensional locally symmetric Kähler manifold

We give a complex two-dimensional noncommutative locally symmetric Kähler manifold via a deformation quantization with separation of variables. We present an explicit formula of its star product by solving the system of recurrence relations given by Hara-Sako. In the two-dimensional case, this system of recurrence relations gives two types of equations corresponding to the two coordinates. From the two types of recurrence relations, symmetrized and antisymmetrized recurrence relations are obtained. The symmetrized one gives the solution of the recurrence relation. From the antisymmetrized one, the identities satisfied by the solution are obtained. The star products for $\mathbb{C}^{2}$ and $\mathbb{C}P^{2}$ are constructed by the method obtained in this study, and we verify that these star products satisfy the identities.

math.DG↗

arXiv2024

Ollivier Ricci curvature of Cayley graphs for dihedral groups, generalized quaternion groups, and cyclic groups

Lin, Lu, and Yau formulated the Ricci curvature of edges in simple undirected graphs[2]. Using their formulations, we calculate the Ricci curvatures of Cayley graphs for the dihedral groups, the general quaternion groups, and cyclic groups with some generating sets that are chosen so that their cardinal numbers are less than or equal to four. For the dihedral group and the general quaternion group, we obtained the Ricci curvatures of all edges of the Cayley graph with generator sets consisting of the four elements that are the two generators defining each group and their inverses elements.For the cyclic group (Z/nZ, +), we have the Ricci curvatures of edges of the Cayley graph generating by S_{1, k} = {+1, -1, +k, -k}.

math.CO↗

arXiv2023

Real symmetric $Φ^4$-matrix model as Calogero-Moser model

We study a real symmetric $Φ^4$-matrix model whose kinetic term is given by $\mathrm{Tr}( E Φ^2)$, where $E$ is a positive diagonal matrix without degenerate eigenvalues. We show that the partition function of this matrix model corresponds to a zero-energy solution of a Schödinger type equation with Calogero-Moser Hamiltonian. A family of differential equations satisfied by the partition function is also obtained from the Virasoro algebra.

hep-th↗

arXiv2023

Integrability of $Φ^4$ Matrix Model as $N$-body Harmonic Oscillator System

We study a Hermitian matrix model with a kinetic term given by $ Tr (H Φ^2 )$, where $H$ is a positive definite Hermitian matrix, similar as in the Kontsevich Matrix model, but with its potential $Φ^3$ replaced by $Φ^4$. We show that its partition function solves an integrable Schrödinger-type equation for a non-interacting $N$-body Harmonic oscillator system.

math-ph↗

arXiv2023

Exact Solutions v.s. Perturbative Calculations of Finite $Φ^{3}$-$Φ^{4}$ Hybrid-Matrix-Model

There is a matrix model corresponding to a scalar field theory called Grosse-Wulkenhaar model, which is renormalizable by adding a harmonic oscillator potential to scalar $Φ^{4}$ theory on Moyal spaces. There are more unknowns in $Φ^{4}$ matrix model than in $Φ^{3}$ matrix model, for example, in terms of integrability. We then construct a one-matrix model ($Φ^{3}$-$Φ^{4}$ Hybrid-Matrix-Model) with multiple potentials, which is a combination of a $3$-point interaction and a $4$-point interaction, where the $3$-point interaction of $Φ^{3}$ is multiplied by some positive definite diagonal matrix $M$. This model is solvable due to the effect of this $M$. In particular, the connected $\displaystyle\sum_{i=1}^{B}N_{i}$-point function $G_{|a_{N_{1}}^{1}\cdots a_{N_{1}}^{1}|\cdots|a_{1}^{B}\cdots a_{N_{B}}^{B}|}$ of $Φ^{3}$-$Φ^{4}$ Hybrid-Matrix-Model is studied in detail. This $\displaystyle\sum_{i=1}^{B}N_{i}$-point function can be interpreted geometrically and corresponds to the sum over all Feynman diagrams (ribbon graphs) drawn on Riemann surfaces with $B$ boundaries (punctures). Each $|a_{1}^{i}\cdots a_{N_{i}}^{i}|$ represents $N_{i}$ external lines coming from the $i$-th boundary (puncture) in each Feynman diagram. First, we construct Feynman rules for $Φ^{3}$-$Φ^{4}$ Hybrid-Matrix-Model and calculate perturbative expansions of some multipoint functions in ordinary methods. Second, we calculate the path integral of the partition function $\mathcal{Z}[J]$ and use the result to compute exact solutions for $1$-point function $G_{|a|}$ with $1$-boundary, $2$-point function $G_{|ab|}$ with $1$-boundary, $2$-point function $G_{|a|b|}$ with $2$-boundaries, and $n$-point function $G_{|a^{1}|a^{2}|\cdots|a^{n}|}$ with $n$-boundaries. They include contributions from Feynman diagrams corresponding to nonplanar Feynman diagrams or higher genus surfaces.

hep-th↗

arXiv2023

Category of Quantizations and Inverse Problem

We introduce a category composed of all quantizations of all Poisson algebras. By the category, we can treat in a unified way the various quantizations for all Poisson algebras and develop a new classical limit formulation. This formulation proposes a new method for the inverse problem, that is, the problem of finding the classical limit from a quantized space. Equivalence of quantizations is defined by using this category, and the conditions under which the two quantizations are equivalent are investigated. Two types of classical limits are defined as the limits in the context of category theory, and they are determined by giving a sequence of objects. Using these classical limits, we discuss the inverse problem of determining the classical limit from some noncommutative Lie algebra. From a Lie algebra, we construct a sequence of quantized spaces, from which we determine a Poisson algebra. We also present a method to obtain this sequence of quantizations from the principle of least action by using matrix regularization. Apart from the above category of quantizations of all Poisson algebras, a category of quantizations of a fixed single Poisson algebra is also introduced. In this category, the other classical limit is defined, and it is automatically determined for the category.

math-ph↗

arXiv2022

Exact solution of the $Φ_{2}^{3}$ finite matrix model

We find the exact solutions of the $Φ_{2}^{3}$ finite matrix model (Grosse-Wulkenhaar model). In the $Φ_{2}^{3}$ finite matrix model, multipoint correlation functions are expressed as $G_{|a_{1}^{1}\ldots a_{N_{1}}^{1}|\ldots|a_{1}^{B}\ldots a_{N_{B}}^{B}|}$. The $\displaystyle \sum_{i=1}^{B}N_{i}$-point function denoted by $G_{|a_{1}^{1}\ldots a_{N_{1}}^{1}|\ldots|a_{1}^{B}\ldots a_{N_{B}}^{B}|}$ is given by the sum over all Feynman diagrams (ribbon graphs) on Riemann surfaces with $B$-boundaries, and each $|a^{i}_{1}\cdots a^{i}_{N_{i}}|$ corresponds to the Feynman diagrams having $N_{i}$-external lines from the $i$-th boundary. It is known that any $G_{|a_{1}^{1}\ldots a_{N_{1}}^{1}|\ldots|a_{1}^{B}\ldots a_{N_{B}}^{B}|}$ can be expressed using $G_{|a^{1}|\ldots|a^{n}|}$ type $n$-point functions. Thus we focus on rigorous calculations of $G_{|a^{1}|\ldots|a^{n}|}$. The formula for $G_{|a^{1}|\ldots|a^{n}|}$ is obtained, and it is achieved by using the partition function $\mathcal{Z}[J]$ calculated by the Harish-Chandra-Itzykson-Zuber integral. We give $G_{|a|}$, $G_{|ab|}$, $G_{|a|b|}$, and $G_{|a|b|c|}$ as the specific simple examples. All of them are described by using the Airy functions.

hep-th↗

arXiv2021

Homology Groups and Categorical Diagonalization

We discuss the relationship between (co)homology groups and categorical diagonalization. We consider the category of chain complexes in the category of finitely generated free modules on a commutative ring. For a fixed chain complex with zero maps as an object, a chain map from the object to another chain complex is defined, and the chain map introduce a mapping cone. We found that the fixed object is isomorphic to the (co)homology groups of the codomain of the chain map if and only if the chain map is injective to the cokernel of differentials of the codomain chain complex and the mapping cone is homotopy equivalent to zero. On the other hand, the fixed object is regarded as a categorified eigenvalue of the chain complex in the context of the categorical diagonalization introduced by B.Elias and M. Hogancamp arXiv:1801.00191v1. It is found that (co)homology groups are constructed as the eigenvalue of a chain complex.

math.CT↗

arXiv2020

Noncommutative Deformations of Locally Symmetric Kähler manifolds

We derive algebraic recurrence relations to obtain a deformation quantization with separation of variables for a locally symmetric Kähler manifold. This quantization method is one of the ways to perform a deformation quantization of Kähler manifolds, which is introduced by Karabegov. From the recurrence relations, concrete expressions of star products for one-dimensional local symmetric Kähler manifolds and ${\mathbb C}P^N$ are constructed. The recurrence relations for a Grassmann manifold $G_{2,2}$ are closely studied too.

math-ph↗

arXiv2019

Categorical Perspective on Quantization of Poisson Algebra

We propose a generalization of quantization as a categorical way. For a fixed Poisson algebra quantization categories are defined as subcategories of R-module category with the structure of classical limits. We construct the generalized quantization categories including matrix regularization, strict deformation quantization, prequantization, and Poisson enveloping algebra, respectively. It is shown that the categories of strict deformation quantization, prequantization, and matrix regularization with some conditions are categorical equivalence. On the other hand, the categories of Poisson enveloping algebra is not equivalent to the other categories.

math-ph↗

arXiv2018

Hermitian-Einstein metrics from noncommutative $U\left(1 \right)$ instantons

We show that Hermitian-Einstein metrics can be locally constructed by a map from (anti-)self-dual two-forms on Euclidean ${\mathbb R}^4$ to symmetric two-tensors introduced in "Gravitational instantons from gauge theory," H. S. Yang and M. Salizzoni, Phys. Rev. Lett. (2006) 201602, [hep-th/0512215]. This correspondence is valid not only for a commutative space but also for a noncommutative space. We choose $U(1)$ instantons on a noncommutative ${\mathbb C}^2$ as the self-dual two-form, from which we derive a family of Hermitian-Einstein metrics. We also discuss the condition when the metric becomes Kähler.

hep-th↗

arXiv2017

The $Φ^3_4$ and $Φ^3_6$ matricial QFT models have reflection positive two-point function

We extend our previous work (on $D=2$) to give an exact solution of the $Φ^3_D$ large-$\mathcal{N}$ matrix model (or renormalised Kontsevich model) in $D=4$ and $D=6$ dimensions. Induction proofs and the difficult combinatorics are unchanged compared with $D=2$, but the renormalisation - performed according to Zimmermann - is much more involved. As main result we prove that the Schwinger 2-point function resulting from the $Φ^3_D$-QFT model on Moyal space satisfies, for real coupling constant, reflection positivity in $D=4$ and $D=6$ dimensions. The Källén-Lehmann mass spectrum of the associated Wightman 2-point function describes a scattering part $|p|^2 \geq 2μ^2$ and an isolated fuzzy mass shell around $|p|^2=μ^2$.

math-ph↗

arXiv2017

Exact solution of matricial $Φ^3_2$ quantum field theory

We apply a recently developed method to exactly solve the $Φ^3$ matrix model with covariance of a two-dimensional theory, also known as regularised Kontsevich model. Its correlation functions collectively describe graphs on a multi-punctured 2-sphere. We show how Ward-Takahashi identities and Schwinger-Dyson equations lead in a special large-$\mathcal{N}$ limit to integral equations that we solve exactly for all correlation functions. Remarkably, these functions are analytic in the $Φ^3$ coupling constant, although bounds on individual graphs justify only Borel summability. The solved model arises from noncommutative field theory in a special limit of strong deformation parameter. The limit defines ordinary 2D Schwinger functions which, however, do not satisfy reflection positivity.

math-ph↗

arXiv2016

Twisted Fock Representations of Noncommutative Kähler Manifolds

We introduce twisted Fock representations of noncommutative Kähler manifolds and give their explicit expressions. The twisted Fock representation is a representation of the Heisenberg like algebra whose states are constructed by acting creation operators on a vacuum state. "Twisted" means that creation operators are not hermitian conjugate of annihilation operators in this representation. In deformation quantization of Kähler manifolds with separation of variables formulated by Karabegov, local complex coordinates and partial derivatives of the Kähler potential with respect to coordinates satisfy the commutation relations between the creation and annihilation operators. Based on these relations, we construct the twisted Fock representation of noncommutative Kähler manifolds and give a dictionary to translate between the twisted Fock representations and functions on noncommutative Kähler manifolds concretely.

math-ph↗