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Ding, Jíntai; Iohara, Kenji

doi:10.1023/a:1007341410987

Cited by 174 publications

(162 citation statements)

References 7 publications

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“…This proves (1). The other two are proved in a similar way: (3) is proven using diagram (22), and replacing pX * j !X by pX ! j * X and a homological shifting; (2) is straightforward.…”

Section: Hyperbolic Localizationmentioning

confidence: 62%

Cohomological Hall Algebras, Vertex Algebras and Instantons

Rapčák

Soibelman

Yang

et al. 2019

Commun. Math. Phys.

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We define an action of the (double of) Cohomological Hall algebra of Kontsevich and Soibelman on the cohomology of the moduli space of spiked instantons of Nekrasov. We identify this action with the one of the affine Yangian of gl(1). Based on that we derive the vertex algebra at the corner Wr 1 ,r 2 ,r 3 of Gaiotto and Rapčák. We conjecture that our approach works for a big class of Calabi-Yau categories, including those associated with toric Calabi-Yau 3-folds.

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“…This proves (1). The other two are proved in a similar way: (3) is proven using diagram (22), and replacing pX * j !X by pX ! j * X and a homological shifting; (2) is straightforward.…”

Section: Hyperbolic Localizationmentioning

confidence: 62%

Cohomological Hall Algebras, Vertex Algebras and Instantons

Rapčák

Soibelman

Yang

et al. 2019

Commun. Math. Phys.

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“…, (A. 15) in the qq-character, where we used the blue color in the above equation to show that the residues for poles in Y (z 2 ) have not been included in the analysis so far. The S-function appears in the computation when we move x − to the right of the Y-operator.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Web construction of ABCDEFG and affine quiver gauge theories

Kimura

Zhu

2019

J. High Energ. Phys.

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The topological vertex formalism for 5d N = 1 gauge theories is not only a convenient tool to compute the instanton partition function of these theories, but it is also accompanied by a nice algebraic structure that reveals various kinds of nice properties such as dualities and integrability of the underlying theories. The usual refined topological vertex formalism is derived for gauge theories with A-type quiver structure (and A-type gauge groups). In this article, we propose a construction with a web of vertex operators for all ABCDEF G-type and affine quivers by introducing several new vertices into the formalism, based on the reproducing of known instanton partition functions and qq-characters for these theories. IntroductionThe exact computation of partition functions on various curved spaces via localization for supersymmetric Yang-Mills theories starting from [1] (see [2] for a nice review) provides extremely powerful ways to examine different types of dualities derived from the superstring theory. Among them, the Alday-Gaiotto-Tachikawa (AGT) relation [3,4], which states that the Nekrasov partition function of a 4d N = 2 gauge theory with gauge group SU(N) can be encoded in terms of some conformal block in 2d CFT with W N symmetry, is in particular intriguing for us. It can be even uplift to gauge theories with eight supercharges in 5d [5] and 6d [6,7,8], where the corresponding 2d CFT is respectively q-deformed and elliptically deformed. Thanks to the string duality [9] between topological string theory and 5d N = 1 gauge theory, we can reformulate the 5d version of the AGT relation in the language of topological vertices [10,11,12] on the toric Calabi-Yau geometry when we restrict our attention to 5d gauge theories constructed from (p, q) 5-brane webs introduced in [13]. In this context, one finds [14] a beautiful algebraic structure of the (refined) topological vertex, which is often called the Ding-Iohara-Miki (DIM) algebra [15,16] or the quantum toroidal algebra of gl 1 , and the dual q-deformed W-algebra can be obtained as a truncation of this algebra due to the finite number of D5 branes in the system. One can also perform a fiber-base duality [17] to the brane web to find a dual q-deformed W M -algebra associated to the gauge theory, where M is the number of NS5 branes in the construction. This W M -algebra is nothing but the quiver W-algebra discovered in [18].The ADHM construction [19], which lies behind the localization calculation [20,21,22] on Ω-background, can be extended to SO and Sp type gauge groups [23,24]. We expect the AGT relation holds for any gauge group, and indeed it has been examined for ABCDEF G-type (i.e. all Lie algebraic) gauge groups at one instanton level in [25] in 4d that this is true. The examination in 5d is even more difficult to perform and more non-trivial. One thus may want to rely on the topological vertex formalism and the associated algebraic structure to check and understand the underlying principle of this duality. The topological vertex formalism for SO and Sp...

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“…Finally, it is worth noting that the type of 3d/1d defects that we have considered in this paper appear in the Higgs branch localization approach to SQCD on S 5 [43], whose partition functions are identified with correlators in the q-Virasoro modular triple [80]. Therefore, another interesting route of investigation would be the study of type IIB SL(2, Z) symmetry from the viewpoint of the BPS/CFT and 5d AGT correspondences [81][82][83][84][85][86][87][88][89][90][91][92][93][94][95] and the DIM algebra [96,97], whose representation theory is known to govern the topological amplitudes associated to toric CY 3-folds or (p, q)-webs [98][99][100][101]. From this perspective, the SL(2, Z) symmetry group is identified with the automorphism group of the DIM algebra, and it would be interesting to systematically study how different q-deformed correlators are related to each other.…”

Section: Discussion and Outlookmentioning

confidence: 99%

3D mirror symmetry fromS-duality

2018

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We consider type IIB SL(2, Z) symmetry to relate the partition functions of different 5d supersymmetric Abelian linear quiver Yang-Mills theories in the Ω-background and squashed S 5 background. By Higgsing S-dual theories, we extract new and old 3d mirror pairs. Generically, the Higgsing procedure yields 3d defects on intersecting spaces, and we derive new hyperbolic integral identities expressing the equivalence of the squashed S 3 partition functions with additional degrees of freedom on the S 1 intersection.

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Cited by 174 publications

References 7 publications

Cohomological Hall Algebras, Vertex Algebras and Instantons

Cohomological Hall Algebras, Vertex Algebras and Instantons

Web construction of ABCDEFG and affine quiver gauge theories

3D mirror symmetry fromS-duality

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