arXiv subjects

Yuto Moriwaki

Publications and source records attributed to Yuto Moriwaki.

16 recordsLinked to original sources

arXiv2026

Higher-dimensional full vertex algebras and systems of conformally covariant correlation functions

We formulate a class of systems of Euclidean correlation functions for compact $d$-dimensional conformal field theories which are not necessarily unitary. We also introduce conformal $d$-vertex algebras, $d$-dimensional analogues of full vertex algebras, and prove that such an algebra is obtained from every system of conformally covariant correlation functions. Finally, for each $d\geq 2$, we construct a family of examples satisfying these axioms, corresponding to generalized free scalar fields. For $d>2$, this family contains both unitary and non-unitary examples.

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arXiv2026

Conformally flat factorization homology in Ind-Hilbert spaces and Conformal field theory

We introduce a metric-dependent geometric variant of factorization homology in conformally flat Riemannian geometry for $d \geq 2$. Its coefficients are symmetric monoidal functors from a disk category in conformal Riemannian geometry to the ind-category of Hilbert spaces, which we call conformally flat $d$-disk algebras. We prove that their left Kan extensions define symmetric monoidal invariants of conformally flat manifolds. Under suitable positivity and continuity assumptions, the value on the standard sphere recovers the sphere partition function of the associated conformal field theory. For $d\geq 3$, we construct explicit examples using unitary representations of $\mathrm{SO}^+(d,1)$ and harmonic analysis, and show that their operadic structures do not extend to bounded operations on the natural Hilbert space completions.

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arXiv2026

Conformal blocks, parenthesized braid operad, and $c=1/2$ Virasoro vertex operator algebra

We review the construction of a pseudo-braided category structure on the $C_1$-cofinite module category of a vertex operator algebra using conformal blocks and analytic continuation along paths in configuration spaces. In the rational $C_2$-cofinite case, the pseudo-braided category is represented by tensor products and becomes a balanced braided tensor category. We then compute all four-point conformal blocks of the Virasoro vertex operator algebra of central charge $1/2$ in terms of hypergeometric functions. We explain how analytic continuation of these blocks determines the braiding and associator, and identify the resulting module category with the Tambara--Yamagami category over $\mathbb{Z}_2$ as a balanced braided tensor category.

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arXiv2026

Vertex operator algebra and parenthesized braid operad

We study conformal blocks of vertex operator algebras on configuration spaces from the viewpoint of the parenthesized braid operad, a combinatorial model of the fundamental groupoid of the little 2-disk operad. For each binary tree we introduce coordinates and a simply connected domain in the configuration space, and show that conformal blocks admit convergent expansions on these domains. Inserting one binary tree into a leaf of another gives gluing maps for the corresponding conformal blocks, while analytic continuation along paths in configuration spaces gives isomorphisms between conformal blocks associated with different trees. We prove that these operations are compatible with the operadic composition in the parenthesized braid operad. As a consequence, the category of $C_1$-cofinite modules whose contragredient modules are finitely generated carries a canonical unital pseudo-braided category structure, without assuming rationality or $C_2$-cofiniteness of the vertex operator algebra. In the rational $C_2$-cofinite case, this structure is represented by tensor products and recovers the balanced braided tensor category structure with twist $\exp(2πiL(0))$.

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arXiv2026

Subtraction Nim with Continuous Parameters

When $S$ is a finite set of positive integers, we can consider classical Subtraction Nim with $S$ as the set of removable numbers. Even when $S$ consists of three elements, many questions remain unanswered. For example, we do not have a period formula of the Nim value. In this paper, we generalize $S$ to be a finite set of positive real numbers. We found that in some regions, we can give concrete formulae for the period and the Nim value function. In particular when $S$ consists of three elements, we found sufficient conditions for the Nim value function to be purely periodic with the period which is equal to the sum of two of elements of $S$. To be more precise, let $S = {a,b,c}$ with $0 < a < b < c$, then for example when $a \leq b \leq 2a$ with $a+b \geq c$, the Nim value function is purely periodic with a period $a+c$. There are much more regions with precise period formulae. We have also some generalizations for the cases $|S| \geq 4$. Even when $S$ consists of integers, these results seem to be new.

math.CO↗

arXiv2026

Convergence and operadic compatibility of bulk and boundary OPEs in two-dimensional conformal field theory

We prove convergence and compatibility of iterated bulk and boundary operator product expansions (OPEs) in two-dimensional conformal field theory with locally $C_1$-cofinite chiral symmetry. For each tree, we give an explicit domain of convergence for the corresponding iterated OPE. These local expansions glue to single-valued real analytic functions on the configuration spaces, which are the correlation functions of the theory. The proof uses an action of the parenthesized permutation-braid operad on $C_1$-cofinite module categories of a vertex operator algebra. This operad models the fundamental groupoid of the two-dimensional Swiss-cheese operad, and under this action the operadic generators correspond to the genus-zero bootstrap equations of boundary CFT.

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arXiv2026

Cohomology ring of unitary $N=(2,2)$ full vertex algebra and mirror symmetry

We formulate two-dimensional $N=(2,2)$ supersymmetric conformal field theories in terms of unitary full vertex operator superalgebras and develop their cohomology theory. Cohomology rings, Hodge numbers, and the Witten index of a unitary $N=(2,2)$ full VOA are introduced. Using generalized full vertex operator superalgebras, spectral flow is constructed algebraically. Its periodicities are proved to be equivalent to the existence of top-degree cohomology classes, namely volume forms and holomorphic volume forms, and these characterizations yield Poincaré duality, T-duality, and Frobenius algebra structures on the cohomology rings, and thus two-dimensional topological field theories. A mirror construction for full VOAs and its relation to Hodge-theoretic mirror symmetry are also discussed. Finally, examples arising from abelian varieties, a special K3 surface, and a Landau-Ginzburg model are examined.

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arXiv2026

Prefactorization algebras for the conformal Laplacian: Central charge and Hilbert Fock space

Let $d \geq 2$. We consider the symmetric monoidal category of oriented Riemannian $d$-manifolds with conformal open embeddings. The prefactorization algebra associated with the conformal Laplacian defines a symmetric monoidal functor from this category to real vector spaces. For Euclidean domains $U\subset\mathbb{R}^d$, the value of this functor is identified, via the Green function, with the symmetric algebra on the topological dual of the space of harmonic functions. For $d \geq 3$ this identification is natural under all conformal transformations, while in dimension two, its failure of naturality is governed by a harmonic cocycle, which plays the role of a central charge. For the unit disk, the resulting vector space carries an algebra structure over the operad of conformal disk embeddings and admits a canonical dense embedding into the Hilbert Fock space. In dimension two, this statement holds after restricting to a codimension-one subspace, as suggested by logarithmic CFT.

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arXiv2026

Bergman space, Conformally flat 2-disk operads and affine Heisenberg vertex algebra

In this paper we consider the operad of holomorphic disk embeddings of the unit disk $\mathbb D \subset \mathbb C$. We introduce a suboperad $\mathbb{CE}_2^{HS}$ defined by square-integrability conditions and show that the symmetric algebra $\mathrm{Sym} A^{2}(\mathbb D)$ of the Bergman space carries a natural $\mathbb{CE}_2^{HS}$-algebra structure. Conformally flat factorization homology with coefficients in $\mathrm{Sym} A^{2}(\mathbb D)$ then yields metric-dependent invariants of two-dimensional Riemannian manifolds. Moreover, $\mathrm{Sym} A^{2}(\mathbb D)$ is identified with the ind-Hilbert space completion of the affine Heisenberg vertex operator algebra.

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arXiv2024

Osterwalder-Schrader axioms for unitary full vertex operator algebras

Full Vertex Operator Algebras (full VOA) are extensions of two commuting Vertex Operator Algebras, introduced to formulate compact two-dimensional conformal field theory. We define unitarity, polynomial energy bounds and polynomial spectral density for full VOA. Under these conditions and local $C_1$-cofiniteness of the simple full VOA, we show that the correlation functions of quasi-primary fields define tempered distributions and satisfy a conformal version of the Osterwalder-Schrader axioms, including the linear growth condition. As an example, we show that a family of full extensions of the Heisenberg VOA satisfies all these assumptions.

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arXiv2024

Two-dimensional conformal field theory, full vertex algebra and current-current deformation

The main purpose of this paper is a mathematical construction of a non-perturbative deformation of a two-dimensional conformal field theory. We introduce a notion of a full vertex algebra which formulates a compact two-dimensional conformal field theory. Then, we construct a deformation family of a full vertex algebra which serves as a current-current deformation of conformal field theory in physics. The parameter space of the deformation is expressed as a double coset of an orthogonal group, a quotient of an orthogonal Grassmannian. As an application, we consider a deformation of chiral conformal field theories, vertex operator algebras. A current-current deformation of a "vertex operator algebra" may produce new vertex operator algebras. We give a formula for counting the number of the isomorphic classes of vertex operator algebras obtained in this way. We demonstrate it for some holomorphic vertex operator algebra of central charge $24$.

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arXiv2021

Quantum coordinate ring in WZW model and affine vertex algebra extensions

In this paper, we construct various simple vertex superalgebras which are extensions of affine vertex algebras, by using abelian cocycle twists of representation categories of quantum groups. This solves the Creutzig and Gaiotto conjectures in the case of type ABC. If the twist is trivial, the resulting algebras correspond to chiral differential operators in the chiral case, and to WZW models in the non-chiral case.

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arXiv2021

Code conformal field theory and framed algebra

It is known that there are 48 Virasoro algebras acting on the monster conformal field theory. We call conformal field theories with such a property, which are not necessarily chiral, code conformal field theories. In this paper, we introduce a notion of a framed algebra, which is a finite-dimensional non-associative algebra, and showed that the category of framed algebras and the category of code conformal field theories are equivalent. We have also constructed a new family of integrable conformal field theories using this equivalence. These conformal field theories are expected to be useful for the study of moduli spaces of conformal field theories.

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arXiv2021

Genus of vertex algebras and mass formula

We introduce the notion of a genus and its mass for vertex algebras. For lattice vertex algebras, their genera are the same as those of lattices, which plays an important role in the classification of lattices. We derive a formula relating the mass for vertex algebras to that for lattices, and then give a new characterization of some holomorphic vertex operator algebras.

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arXiv2020

Full vertex algebra and bootstrap -- consistency of four point functions in 2d CFT

In physics, it is believed that the consistency of two dimensional conformal field theory follows from the bootstrap equation. In this paper, we introduce the notion of a full vertex algebra by analyzing the bootstrap equation, which is a "real analytic" generalization of a $\mathbb{Z}$-graded vertex algebra. We also give a mathematical formulation of the consistency of four point correlation functions in two dimensional conformal field theory and prove it for a full vertex algebra with additional assumptions on the conformal symmetry. In particular, we show that the bootstrap equation together with the conformal symmetry implies the consistency of four point correlation functions. As an application, a deformable family of full vertex algebras parametrized by the Grassmanian is constructed, which appears in the toroidal compactification of string theory. This give us examples satisfying the above assumptions.

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arXiv2018

On classification of conformal vectors in vertex operator algebra and the vertex algebra automorphism group

Herein we study conformal vectors of a Z-graded vertex algebra of (strong) CFT type. We prove that the full vertex algebra automorphism group transitively acts on the set of the conformal vectors of strong CFT type if the vertex algebra is simple. The statement is equivalent to the uniqueness of self-dual vertex operator algebra structures of a simple vertex algebra. As an application, we show that the full vertex algebra automorphism group of a simple vertex operator algebra of strong CFT type uniquely decomposes into the product of certain two subgroups and the vertex operator algebra automorphism group. Furthermore, we prove that the full vertex algebra automorphism group of the moonshine module over the field of real numbers is the Monster.

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