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Kiyokazu Nagatomo

Publications and source records attributed to Kiyokazu Nagatomo.

17 recordsLinked to original sources
arXiv2023

Minimal models, modular linear differential equations and modular forms of fractional weights

We show that modular forms of fractional weights on principal congruence subgroups of odd levels, which are found by T. Ibukiyama, naturally appear as characters being multiplied $η^{c_{\text{eff}}}$ of the so-called minimal models of type $(2, p)$, where $c_{\text{eff}}$ is the effective central charge of a minimal model. Using this fact and modular invariance property of the space of characters, we give a different proof of the result showned by Ibukiyama that is the explicit formula representing $\mathrm{SL}_{2}(\mathbb{Z})$ on the space of the ibukiyama modular forms. We also find several pairs of spaces of the Ibukiyama modular forms on $Γ(m)$ and $Γ(n)$ with $m|n$ having the property that the former are included in the latter. Finally, we construct vector-valued modular forms of weight $k\in\frac{1}{5}\mathbb{Z}_{>0}$ starting from the Ibukiyama modular forms of weight $k\in\frac{1}{5}\mathbb{Z}_{>0}$ with some multiplier system by a symmetric tensor product of this representation and show that the components functions coincides with the solution space of some monic modular linear differential equation of weight $k$ and the order $1+5k$.

math.NT↗
arXiv2022

Modular Linear Differential Operators and Generalized Rankin-Cohen Brackets

The aim in this paper is to give expressions for modular linear differential operators of any order. In particular, we show that they can all be described in terms of Rankin-Cohen brackets and a modified Rankin-Cohen bracket found by Kaneko and Koike. We also give more uniform descriptions of MLDOs in terms of canonically defined higher Serre derivatives and an extension of Rankin-Cohen brackets, as well as in terms of quasimodular forms and almost holomorphic modular forms. The last of these descriptions involves the holomorphic projection map. The paper also includes some general results on the theory of quasimodular forms on both cocompact and non-cocompact subgroups of $SL_2(\mathbb{R})$, as well as a slight sharpening of a theorem of Martin and Royer on Rankin-Cohen brackets of quasimodular forms

math.NT↗
arXiv2018

Vertex Operator Algebras with central charge 8 and 16

We will partially classify spaces of characters of vertex operator algebras $V$ with central charges 8 and 16, such that the spaces of characters is 3-dimensional and the characters forms a basis of the solution space of a third order monic modular linear differential equation with rational indicial roots. Assuming a mild arithmetic condition, we show that the space of characters of $V$ coincides with the space of characters of lattice vertex operators associated with integral lattices $\sqrt{2}E_8$ or the affine vertex operator algebra of type $D_{20}^{(1)}$ for $c=8$, and the Barnes--Wall lattice $Λ_{16}$, the affine vertex operator algebras of type $D_{16}^{(1)}$ with level 1 and type $D_{28}^{(1)}$ with level 1 for $c=16$. (The central charge of the affine vertex operator algebra of type $D_{28}^{(1)}$ with level 1 is 28, but the space of characters satisfies the differential equations for $c=16$.) Supposing a mild condition on characters of $V$, then it uniquely determines (up to isomorphism) the spaces of characters of the lattice $\sqrt{2}E_8$ and the Barnes--Wall lattice $Λ_{16}$, respectively. The reason why vertex operator algebras with central charges 8 and 16 are intensively studied is that there are solutions which do not depend on extra parameters (which represent conformal weights). This fact is well understood using the hypergeometric function $3F2$. Hence we cannot apply our standard method to classify vertex operator algebras in which we are interested. In appendix we classify $c=4$ vertex operator algebras with the same conditions mentioned above.

math.QA↗
arXiv2018

Vertex Operator Algebras with Two Simple Modules - the Mathur-Mukhi-Sen Theorem Revisited

Let $V$ be a strongly regular vertex operator algebra and let $\frak{ch}_V$ be the space spanned by the characters of the irreducible $V$-modules.\ It is known that $\frak{ch}_V$ is the space of solutions of a so-called \emph{modular linear differential equation (MLDE)}.\ In this paper we obtain a near-classification of those $V$ for which the corresponding MLDE is irreducible and monic of order $2$.\ As a consequence we derive the complete classification when $V$ has exactly two simple modules.\ It turns out that $V$ is either one of four affine Kac-Moody algebras of level $1$, or the Yang-Lee Virasoro model of central charge ${-}22/5$.\ Our proof establishes new connections between the characters of $V$ and Gauss hypergeometric series, and puts the finishing touches to work of Mathur, Mukhi and Sen who first considered this problem forty years ago.

math.QA↗
arXiv2011

Some remarks on pseudo-trace functions for orbifold models associated with symplectic fermions

We give a method to construct pseudo-trace functions for vertex operator algebras satisfying Zhu's finiteness condition not through higher Zhu's algebras and apply our method to the Z_2-orbifold model associated with d-pairs of symplectic fermions. For d=1, we determine the dimension of the space of one-point functions. For d>1, we construct 2^{2d-1}+3 linearly independent one-point functions and study their values at the vacuum vector.

math.QA↗
arXiv2005

Quasi-finite algebras graded by Hamiltonian and vertex operator algebras

A general notion of a quasi-finite algebra is introduced as an algebra graded by the set of all integers equipped with topologies on the homogeneous subspaces satisfying certain properties. An analogue of the regular bimodule is introduced and various module categories over quasi-finite algebras are described. When applied to the current algebras (universal enveloping algebras) of vertex operator algebras satisfying Zhu's $C_2$-finiteness condition, our general consideration derives important consequences on representation theory of such vertex operator algebras. In particular, the category of modules over such a vertex operator algebra is shown to be equivalent to the category of modules over a finite-dimensional associative algebra.

math.QA↗
arXiv2002

Conformal field theories associated to regular chiral vertex operator algebras I: theories over the projective line

Based on any chiral vertex operator algebra satisfying a suitable finiteness condition, the semisimplicity of the zero-mode algebra as well as a regularity for induced modules, we construct conformal field theory over the projective line with the chiral vertex operator algebra as symmetries of the theory. We appropriately generalize the argument in [TUY] so that we are able to define sheaves of conformal blocks for chiral vertex operator algebras and study them in detail. We prove the factorization theorem under the fairly general conditions for chiral vertex operator algebras and the zero-mode algebras.

math.QA↗
arXiv2002

Finiteness of conformal blocks over compact Riemann surfaces

We study conformal blocks (the space of correlation functions) over compact Riemann surfaces associated to vertex operator algebras which are the sum of highest weight modules for the underlying Virasoro algebra. Under the fairly general condition, for instance, $C_2$-finiteness, we prove that conformal blocks are of finite dimensional. This, in particular, shows the finiteness of conformal blocks for many well-known conformal field theories including WZNW model and the minimal model.

math.QA↗
arXiv2000

Quasi-modular forms and trace functions associated to free boson and lattice vertex operator algebras

We study graded traces of vectors in free bosonic vertex operator algebras and lattice vertex operator algebras. We show in particular that trace functions in these two theories always have the shape f(q)/η(q)^d where f(q) is quasi-modular in the case of d free bosons, and modular (i.e., a sum of holomorphic modular forms of various weights) in the case of theories based on a lattice L of rank d. We also show how spherical harmonic polynomials with respect to L are related to primary fields in lattice theories.

math.QA↗
arXiv1997

On axioms for a vertex algebra and the locality of quantum fields

The identities satisfied by two-dimensional chiral quantum fields are studied from the point of view of vertex algebras. The Cauchy-Jacobi identity (or the Borcherds identity) for three mutually local fields is proved and consequently a direct proof of Li's theorem on a local system of vertex operators is provided. Several characterizations of vertex algebras are also discussed.

hep-th↗