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Yukiko Konishi

Publications and source records attributed to Yukiko Konishi.

18 recordsLinked to original sources
arXiv2025

Satake's good basic invariants for finite complex reflection groups

In arXiv:2004.01871 Satake introduced the notions of admissible triplets and good basic invariants for finite complex reflection groups. For irreducible finite Coxeter groups, he showed the existence and the uniqueness of good basic invariants. Moreover he showed that good basic invariants are flat in the sense of K.Saito's flat structure. He also obtained a formula for the multiplication of the Frobenius structure. In this article, we generalize his results to finite complex reflection groups. We first study the existence and the uniqueness of good basic invariants. Then for duality groups, we show that good basic invariants are flat in the sense of the natural Saito structure constructed in arXiv:1612.03643. We also give a formula for the potential vector fields of the multiplication in terms of the good basic invariants. Moreover, in the case of irreducible finite Coxeter groups, we derive a formula for the potential functions of the associated Frobenius manifolds.

math.AG↗
arXiv2025

A reduction theorem for good basic invariants of finite complex reflection groups

This is a sequel to our previous article arXiv:2307.07897. We describe a certain reduction process of Satake's good basic invariants. We show that if the largest degree $d_1$ of a finite complex reflection group $G$ is regular and if $δ$ is a divisor of $d_1$, a set of good basic invariants of $G$ induces that of the reflection subquotient $G_δ$. We also show that the potential vector field of a duality group $G$, which gives the multiplication constants of the natural Saito structure on the orbit space, induces that of $G_δ$. Several examples of this reduction process are also presented.

math.AG↗
arXiv2019

Almost duality for Saito structure and complex reflection groups II: the case of Coxeter and Shephard groups

It is known that the orbit spaces of the finite Coxeter groups and the Shephard groups admit two types of Saito structures without metric. One is the underlying structures of the Frobenius structures constructed by Saito and Dubrovin. The other is the natural Saito constructed by Kato-Mano-Sekiguchi and by Arsie-Lorenzoni. We study the relationship between these two Saito structures from the viewpoint of almost duality.

math.AG↗
arXiv2018

Mixed Frobenius Structure and Local A-model

We define the notion of mixed Frobenius structure which is a generalization of the structure of a Frobenius manifold. We construct a mixed Frobenius structure on the cohomology of weak Fano toric surfaces and that of the three dimensional projective space using local Gromov-Witten invariants. This is an analogue of the Frobenius manifold associated to the quantum cohomology in the local Calabi-Yau setting.

math.AG↗
arXiv2018

Almost duality for Saito structure and complex reflection groups

We reformulate Dubrovin's almost duality of Frobenius structures to Saito structures without metric. Then we formulate and study the existence and uniqueness problem of the natural Saito structure on the orbit spaces of finite complex reflection groups from the viewpoint of the almost duality. We give a complete answer to the problem for the irreducible groups.

math.AG↗
arXiv2015

Mixed Frobenius Structure and Local Quantum Cohomology

This paper is a sequel to arXiv:1209.5550 where the notion of mixed Frobenius structure (MFS) was introduced as a generalization of the structure of a Frobenius manifold. Roughly speaking, the MFS is defined by replacing a metric of the Frobenius manifold with a filtration on the tangent bundle equipped with metrics on its graded quotients. The purpose of the current paper is to construct a MFS on the cohomology of a smooth projective variety whose multiplication is the non-equivariant limit of the quantum product twisted by a concave vector bundle. We show that such a MFS is naturally obtained as the non-equivariant limit of the Frobenius structure in the equivariant setting.

math.AG↗
arXiv2009

Local B-model and Mixed Hodge Structure

We study the mixed Hodge theoretic aspects of the B-model side of local mirror symmetry. Our main objectives are to define an analogue of the Yukawa coupling in terms of the variations of the mixed Hodge structures and to study its properties. We also describe a local version of Bershadsky-Cecotti-Ooguri-Vafa's holomorphic anomaly equation.

math.AG↗
arXiv2006

Jacobian variety and Integrable system -- after Mumford, Beauville and Vanhaecke

Beauville introduced an integrable Hamiltonian system whose general level set is isomorphic to the complement of the theta divisor in the Jacobian of the spectral curve. This can be regarded as a generalization of the Mumford system. In this article, we construct a variant of Beauville's system whose general level set is isomorphic to the complement of the `intersection' of the translations of the theta divisor in the Jacobian. A suitable subsystem of our system can be regarded as a generalization of the even Mumford system introduced by Vanhaecke.

math-ph↗
arXiv2006

Flop invariance of the topological vertex

We prove transformation formulae for generating functions of Gromov-Witten invariants on general toric Calabi-Yau threefolds under flops. Our proof is based on a combinatorial identity on the topological vertex and analysis of fans of toric Calabi-Yau threefolds.

math.AG↗
arXiv2005

Integrality of Gopakumar-Vafa Invariants of Toric Calabi-Yau Threefolds

The Gopakumar-Vafa invariants are numbers defined as certain linear combinations of the Gromov-Witten invariants. We prove that the GV invariants of a toric Calabi-Yau threefold are integers and that the invariants for high genera vanish. The proof of the integrality is based on elementary number theory and that of the vanishing uses the operator formalism and the exponential formula.

math.AG↗
arXiv2005

Pole Structure of Topological String Free Energy

We show that the free energy of the topological string admits a certain pole structure by using the operator formalism. Combined with the results of Peng that proved the integrality, this gives a combinatoric proof of the Gopakumar-Vafa conjecture.

math.AG↗
arXiv2004

Topological Strings, Instantons and Asymptotic Forms of Gopakumar-Vafa Invariants

We calculate the topological string amplitudes of Calabi-Yau toric threefolds corresponding to 4D, N=2, SU(2) gauge theory with N_f=0,1,2,3,4 fundamental hypermultiplets by using the method of the geometric transition and show that they reproduce Nekrasov's formulas for instanton counting. We also determine the asymptotic forms of the Gopakumar-Vafa invariants of the Calabi-Yau threefolds including those at higher genera from instanton amplitudes of the gauge theory.

hep-th↗
arXiv2003

Asymptotic Form of Gopakumar-Vafa Invariants from Instanton Counting

We study the asymptotic form of the Gopakumar-Vafa invariants at all genera for Calabi-Yau toric threefolds which have the structure of fibration of the A_n singularity over P^1. We claim that the asymptotic form is the inverse Laplace transform of the corresponding instanton amplitude in the prepotential of N=2 SU(n+1) gauge theory coupled to external graviphoton fields, which is given by the logarithm of the Nekrasov's partition function.

hep-th↗
arXiv2003

Geometric Engineering of Seiberg-Witten Theories with Massive Hypermultiplets

We analyze the geometric engineering of the N=2 SU(2) gauge theories with $1\leq N_f\leq 3$ massive hypermultiplets in the vector representation. The set of partial differential equations satisfied by the periods of the Seiberg-Witten differential is obtained from the Picard-Fuchs equations of the local B-model. The differential equations and its solutions are consistent with the massless case. We show that the Yukawa coupling of the local A-model gives rise to the correct instanton expansion in the gauge theory, and propose the pattern of the distribution of the world-sheet instanton number from it. As a side result, we obtain the asymptotic form of the instanton number in the gauge theories with massless hypermultiplets.

hep-th↗