Quantum cohomology of the Springer resolution

Braverman, Alexander; Maulik, Davesh; Okounkov, Andreĭ

doi:10.1016/j.aim.2011.01.021

Cited by 82 publications

(111 citation statements)

References 43 publications

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“…This introduces surface operators in the gauge theory [62] and provides a set-up to compute the quantum cohomology of their moduli spaces, such as for example Laumon spaces [65] and partial flag varieties [66]. Our approach should be compared with the results of [67]. Furthermore, we remark that the above constitutes a useful set-up to study the AGT correspondence [49,63,64].…”

Section: Jhep01(2014)038mentioning

confidence: 99%

The stringy instanton partition function

Bonelli

Sciarappa

Tanzini

et al. 2014

J. High Energ. Phys.

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Abstract:We perform an exact computation of the gauged linear sigma model associated to a D1-D5 brane system on a resolved A 1 singularity. This is accomplished via supersymmetric localization on the blown-up two-sphere. We show that in the blow-down limit C 2 /Z 2 the partition function reduces to the Nekrasov partition function evaluating the equivariant volume of the instanton moduli space. For finite radius we obtain a tower of world-sheet instanton corrections, that we identify with the equivariant Gromov-Witten invariants of the ADHM moduli space. We show that these corrections can be encoded in a deformation of the Seiberg-Witten prepotential. From the mathematical viewpoint, the D1-D5 system under study displays a twofold nature: the D1-branes viewpoint captures the equivariant quantum cohomology of the ADHM instanton moduli space in the Givental formalism, and the D5-branes viewpoint is related to higher rank equivariant Donaldson-Thomas invariants of P 1 × C 2 .

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Section: Jhep01(2014)038mentioning

confidence: 99%

The stringy instanton partition function

Bonelli

Sciarappa

Tanzini

et al. 2014

J. High Energ. Phys.

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“…Наше отождествление алгебры Бете с алгеброй операторов умножения в эк-вивариантных когомологиях H * GL n (F λ , C) можно рассматривать как вырож-дение недавно полученного в работе [12] описания эквивариантных квантовых когомологий многообразия неполных флагов как алгебры Бете некоторой ян-гианной модели, связанной с V ⊗n , ср. [2]. В § 2 мы вводим алгебру Бете.…”

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Когомологии Многообразия Флагов Как Алгебра Бете

Varchenko¹,

Римани²,

Rimányi³

et al. 2011

Функц. анализ и его прил.

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“…Thus (1.1) and (1.3) are related. Recently it was realized that in some cases the KZ equations appear as quantum differential equations, see [2] and [11], and therefore the KZ equations are related to the Frobenius structures. On Frobenius structures see, for example, [4,5,14].…”

Section: Introductionmentioning

confidence: 99%

Arrangements and Frobenius like structures

Varchenko¹

2015

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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We consider a family of generic weighted arrangements of n hyperplanes in C k and show that the Gauss-Manin connection for the associated hypergeometric integrals, the contravariant form on the space of singular vectors, and the algebra of functions on the critical set of the master function define a Frobenius like structure on the base of the family.As a result of this construction we show that the matrix elements of the linear operators of the Gauss-Manin connection are given by the 2k + 1-st derivatives of a single function on the base of the family, the function called the potential of second kind, see formula (6.46).On considère une famille d'arrangements pondérés génériques de n hyperplans dans C k et montre que la connexion de Gauss -Manin pour les intégrales hypergéométriques associées, la forme contravariante sur l'espace des vecteurs singuliers et l'algébre de fonctions sur l'ensemble des points critiques définissent une structure du type Frobenius sur la base de la famille.Comme un résultat de cette construction nous montrons que les éléments matriciels des opérateurs linéaires de la connexion de Gauss -Manin sont donnés par les (2k + 1)-mes dérivées d'une seule fonction sur la base de la famille, cf. la formule (6.46).3.4. Bad fibers 3.5. Operators K j (z) : V → V , j ∈ J 3.6. Gauss-Manin connection on (C n − ∆) × (Sing V ) → C n − ∆ 3.7. Bundle of algebras 3.8. Quantum integrable model of the arrangement (C(z), a) 3.9. A remark. Asymptotically flat sections 3.10. Conformal blocks, period map, potential functions 3.11. Tangent bundle and a Frobenius like structure 4. Points on line 4.1. An arrangement in C n × C 4.2. Good fibers 4.3. Operators K j (z) : V → V 4.4. Conformal blocks 4.5. Canonical isomorphism and period map 4.6. Contravariant map as the inverse to the canonical map 4.7. Multiplication on Sing V and (Sing V ) * 4.8. Tangent morphism 4.9. Multiplication and potential function of second kind 4.10. Connections on the bundle ⊔ z∈C n −∆ B * (z) → C n − ∆ defined in (4.33) 4.11. Functoriality 4.12. Frobenius like structure 5. Generic lines on plane 5.1. An arrangement in C n × C 2 5.2. Good fibers 5.3. Operators K i (z) : V → V 5.4. Conformal blocks 5.5. Algebra A Φ (z) 5.6. Proof of Theorem 5.15 5.7. Contravariant map as the inverse to the canonical map 5.8. Corollaries of Theorem 5.15 5.9. Frobenius like structure 6. Generic arrangements in C k 6.1. An arrangement in C n × C k 6.2. Good fibers 6.3. Operators K i (z) : V → V 6.4. Algebra A Φ (z) 6.5. Canonical isomoprhism References

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Quantum cohomology of the Springer resolution

Cited by 82 publications

References 43 publications

The stringy instanton partition function

The stringy instanton partition function

Когомологии Многообразия Флагов Как Алгебра Бете

Arrangements and Frobenius like structures

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