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Shintarou Yanagida

Publications and source records attributed to Shintarou Yanagida.

At least 19 recordsLinked to original sources
arXiv2026

BV construction of SUSY vertex algebras from SUSY factorization algebras

We construct $N=1$ supersymmetric (SUSY) vertex algebras from supersymmetric enhancements of Costello--Gwilliam factorization algebras on super Riemann surfaces. Introducing SUSY factorization algebras defined on embedded SUSY disks together with natural symmetry conditions, we prove a SUSY analogue of the Costello--Gwilliam extraction theorem. As an application, we study the holomorphic sigma model in the BV formalism. For a linear target, we obtain the free $bc$-$\beta\gamma$ system and recover its structure as a SUSY vertex algebra. For general complex targets, we describe the descent of the theory under coordinate changes and identify the resulting SUSY vertex algebra with the chiral de Rham complex. We further show that Ricci-flat K\"ahler and hyperk\"ahler targets give rise to $N=2$ and $N=4$ supersymmetric enhancements introduced by Ben-Zvi--Heluani--Szczesny.

math.QA↗
arXiv2025

Zhu algebras of superconformal vertex algebras

The purpose of this note is to demonstrate the advantages of Y.-Z.~Huang's definition of the Zhu algebra (Comm.\ Contemp.\ Math., 7 (2005), no.~5, 649--706) for an arbitrary vertex algebra, not necessarily equipped with a Hamiltonian operator or a Virasoro element, by achieving the following two goals: (1) determining the Zhu algebras of $N=1, 2, 3, 4$ and big $N=4$ superconformal vertex algebras, and (2) introducing the Zhu algebras of $N_K=N$ supersymmetric vertex algebras.

math.QA↗
arXiv2025

On the Hopf superalgebra of symmetric functions in superspace

We introduce a superspace analogue of combinatorial Hopf algebras (Aguiar-Bergeron-Sottile, 2006), and show that the Hopf superalgebra of quasi-symmetric (resp. symmetric) functions in superspace (Fishel-Lapointe-Pinto, 2019) is a terminal object in the category of all (resp. cocommutative) combinatorial Hopf superalgebras. We also introduce a superspace analogue of chromatic symmetric functions of graphs (Stanley, 1995) using the chromatic Hopf superalgebra of two-colored graphs.

math.CO↗
arXiv2024

$q$-Racah probability distribution

We introduce a certain discrete probability distribution $P_{n,m,k,l;q}$ having non-negative integer parameters $n,m,k,l$ and quantum parameter $q$ which arises from a zonal spherical function of the Grassmannian over the finite field $\mathbb{F}_q$ with a distinguished spherical vector. Using representation theoretic arguments and hypergeometric summation technique, we derive the presentation of the probability mass function by a single $q$-Racah polynomial, and also the presentation of the cumulative distribution function in terms of a terminating ${}_4 ϕ_3$-hypergeometric series.

math.RT↗
arXiv2023

Stochastic behavior of outcome of Schur-Weyl duality measurement

We focus on the measurement defined by the decomposition based on Schur-Weyl duality on $n$ qubits. As the first setting, we discuss the asymptotic behavior of the measurement outcome when the state is given as the permutation mixture $ρ_{mix,n,l}$ of the state $| 1^{l} \, 0^{n-l} \rangle := | 1 \rangle^{\otimes l} \otimes |0\rangle^{\otimes (n-l)}$. In contrast, when the state is given as the Dicke state $|Ξ_{n,l}\rangle$, the measurement outcome takes one deterministic value. These two cases have completely different behaviors. As the second setting, we study the case when the state is given as the tensor product of the permutation mixture $ρ_{mix,k,l}$ and the Dicke state $| Ξ_{n-k,m-l} \rangle$. We derive various types of asymptotic distribution including a kind of central limit theorem when $n$ goes to infinity.

math-ph↗
arXiv2023

Algebraic operad of SUSY vertex algebra

We introduce algebraic operads $\mathcal{P}^{\text{ch}N_W=N}$ and $\mathcal{P}^{\text{ch}N_K=N}$ encoding the structures of $N_W=N$ and $N_K=N$ SUSY vertex algebras, and study the corresponding cohomology theory. Our operad is a SUSY analogue of the operad $P^{\text{ch}}$ introduced by Bakalov, De Sole, Heluani and Kac (2019) as a purely algebraic translation of the chiral operad of Beilinson and Drinfeld.

math.QA↗
arXiv2023

Algebraic operad of SUSY Poisson vertex algebra

As a continuation of our study (Y.N., S.Y., arXiv:2209.14617) on the algebraic operad of SUSY vertex algebras, we introduce the SUSY coisson operad, which encodes the structures of SUSY Poisson vertex algebras. Our operad is a natural SUSY analogue of the operad encoding the structures of Poisson vertex algebras introduced by Bakalov, De Sole, Heluani and Kac (2019). We also give an embedding of the associated graded of the SUSY chiral operad into the SUSY coisson operad in the filtered case.

math.QA↗
arXiv2023

A review of rank one bispectral correspondence of quantum affine KZ equations and Macdonald-type eigenvalue problems

This note consists of two parts. The first part (§1 and §2) is a partial review of the works by van Meer and Stokman (2010), van Meer (2011) and Stokman (2014) which established a bispectral analogue of the Cherednik correspondence between quantum affine Knizhnik-Zamolodchikov equations and the eigenvalue problems of Macdonald type. In this review we focus on the rank one cases, i.e., on the reduced type $A_1$ and the non-reduced type $(C_1^\vee,C_1)$, to which the associated Macdonald-Koornwinder polynomials are the Rogers polynomials and the Askey-Wilson polynomials, respectively. We give detailed computations and formulas that may be difficult to find in the literature. The second part (§3) is a complement of the first part, and is also a continuation of our previous study (Y.-Y., 2022) on the parameter specialization of Macdonald-Koornwinder polynomials, where we found four types of specialization of the type $(C_1^\vee,C_1)$ parameters (which could be called the Askey-Wilson parameters) to recover the type $A_1$. In this note, we show that among the four specializations there is only one which is compatible with the bispectral correspondence discussed in the first part.

math.QA↗
arXiv2022

A dynamical analogue of Ding-Iohara quantum algebras

We introduce a family of dynamical Hopf algebroids $U_{q,p}(g,X_l)$ depending on a complex parameter $q$, a formal parameter $p$, a set $g$ of structure functions satisfying the so-called Ding-Iohara condition, and a finite root system of type $X_l$. If $g$ is set to be certain theta functions, then our family recovers the elliptic algebras $U_{q,p}(\widehat{\mathfrak{g}})$ for untwisted affine Lie algebras $\widehat{\mathfrak{g}}$ studied by Konno (1998, 2009), Jimbo-Konno-Odake-Shiraishi (1999) and Farghly-Konno-Oshima (2014). Also, taking the limit $p \to 0$ in the case $X_l=A_l$, we recover the Hopf algebras $U_q(\overline{g},A_l)$ of type $A_l$ with structure functions $\overline{g} := \lim_{p \to 0} g$, introduced by Ding-Iohara (1998) as a generalization of Drinfeld quantum affine algebras. Thus, our Hopf algebroid $U_{q,p}(g,X_l)$ can be regarded as a dynamical analogue of the Ding-Iohara quantum algebras. As a byproduct, we obtain an extension of the Ding-Iohara quantum algebras to those of non-simply-laced type.

math.QA↗
arXiv2022

Intermediate symplectic $Q$-functions

We introduce an intermediate family of Laurent polynomials between Schur's $Q$-functions and S. Okada's symplectic $Q$-functions. It can also be regarded as a $Q$-function analogue of Proctor's intermediate symplectic characters, and is named the family of intermediate symplectic $Q$-functions. We also derive a tableau-sum formula and a Józefiak-Pragacz-type Pfaffian formula of the Laurent polynomials.

math.CO↗
arXiv2021

Li filtrations of SUSY vertex algebras

Any vertex algebra has a canonical decreasing filtration, called Li filtration, whose associated graded space has a natural structure of a vertex Poisson algebra. In this note, we introduce an analogous filtration for any SUSY vertex algebra, which was introduced by Heluani and Kac as a superfield formalism of a supersymmetric vertex algebra. We prove that the associated graded superspace of our filtration has a structure of SUSY vertex Poisson algebras. We also introduce and discuss related notions, such as Zhu's $C_2$-Poisson superalgebras, associated superschemes and singular supports, for SUSY vertex algebras.

math.QA↗
arXiv2021

Specializing Koornwinder polynomials to Macdonald polynomials of type $B,C,D$ and $B C$

We study the specializations of parameters in Koornwinder polynomials to obtain Macdonald polynomials associated to the subsystems of the affine root system of type $(C_n^\vee,C_n)$ in the sense of Macdonald (2003), and summarize them in what we call the specialization table. As a verification of our argument, we check the specializations to type $B,C$ and $D$ via Ram-Yip type formulas of non-symmetric Koornwinder and Macdonald polynomials.

math.QA↗
arXiv2020

Derived gluing construction of chiral algebras

We discuss the gluing construction of class $\mathcal{S}$ chiral algebras in derived setting. The gluing construction in non-derived setting was introduced by Arakawa to construct a family of vertex algebras of which the associated varieties give genus zero Moore-Tachikawa symplectic varieties. Motivated by the higher genus case, we introduce a dg vertex algebra version $\mathsf{MT}_{\mathrm{ch}}$ of the category of Moore-Tachikawa symplectic varieties, where a morphism is given by a dg vertex algebra equipped with action of the universal affine vertex algebra, and composition of morphisms is given by the BRST reduction. We also show that the procedure taking the associated scheme of gives a functor from $\mathsf{MT}_{\mathrm{ch}}$ to the category $\mathsf{MT}$ of derived Moore-Tachikawa varieties, which would imply compatibility of gluing constructions in both categories.

math.QA↗
arXiv2019

Geometric derived Hall algebra

We give a geometric formulation of Toën's derived Hall algebra by constructing Grothendieck's six operations for the derived category of lisse-étale constructible sheaves on the derived stacks of complexes. Our formulation is based on an variant of Laszlo and Olsson's theory of derived categories and six operations for algebraic stacks. We also give an $\infty$-theoretic explanation of the theory of derived stacks, which was originally constructed by Toën and Vezzosi in terms of model theoretical language.

math.AG↗
arXiv2018

A study of symmetric functions via derived Hall algebra

We use derived Hall algebra of the category of nilpotent representations of Jordan quiver to reconstruct the theory of symmetric functions, focusing on Hall-Littlewood symmetric functions and various operators acting on them.

math.QA↗
arXiv2018

Operadic semi-infinite homology

We propose the notion of semi-infinite homology for algebras over operads using the relative homology theory for operadic algebras.

math.QA↗