(p, q)-webs of DIM representations, 5d $$ \mathcal{N}=1 $$ instanton partition functions and qq-characters

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Quiver 5D N = 1 gauge theories describe the low-energy dynamics on webs of (p, q)-branes in type IIB string theory. S-duality exchanges NS5 and D5 branes, mapping (p, q)-branes to branes of charge (−q, p), and, in this way, induces several dualities between 5D gauge theories. On the other hand, these theories can also be obtained from the compactification of topological strings on a Calabi-Yau manifold, for which the S-duality is realized as a fiber-base duality. Recently, a third point of view has emerged in which 5D gauge theories are engineered using algebraic objects from the Ding-Iohara-Miki (DIM) algebra. Specifically, the instanton partition function is obtained as the vacuum expectation value of an operator T constructed by gluing the algebra's intertwiners (the equivalent of topological vertices) following the rules of the toric diagram/brane web. Intertwiners and Toperators are deeply connected to the co-algebraic structure of the DIM algebra. We show here that S-duality can be realized as the twist of this structure by Miki's automorphism.

Section: 1 T -Operatorsmentioning

confidence: 99%

Fiber-base duality from the algebraic perspective

Bourgine

2019

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“…They span the quiver W-algebra [18], where the qq-character corresponding to the fundamental representation reduces to the generating current of the algebra, and in the Nekrasov-Shatashvili limit gives rise to the T-operator of the corresponding quantum integrable system. We review how to derive the qq-character, which is an important quantity to check, in the web diagram (or simplified diagram) by using the Ward identity in DIM algebra following [36,40] in Appendix A. We will argue later that this prescription gives the correct expressions of the qq-characters in the web diagram proposed in this article for ABCDEFG-type quiver.…”

Section: )mentioning

confidence: 99%

“…It plays a similar role to the reflection state, while the proposal in [30] works only for U(1) gauge group in the node built with the reflection state or if one wants to left up the rank of the gauge group, one will have to use the generalized vertex introduced in [40]. On the other hand, we can raise the rank of the gauge group in the ordinary way by simply adding more D5 branes by usinḡ Φ * (n) [u, v] instead.…”

Section: D-type Quiver Construction With Orientifoldmentioning

confidence: 99%

“…Even in the unrefined limit, where Φ * is simplified to Φ −1 , the web diagram for an SU(2) gauge group in the problem node looks like (7.3) and it is more complex and almost not tractable in the refined case. We can bypass this problem by introducing generalized trivalent vertices as how generalized topological vertices are defined in [40]. Essentially it was also the generalized trivalent vertex associated with the SU(2) gauge group that helped in [49] build the D, E-type quiver gauge theories with SU(2) gauge groups, which can be S-dualized to (A 1 quiver) gauge theories with D, E-type gauge groups.…”

Section: De-type Quiver Construction With Trivalent Vertexmentioning

confidence: 99%

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Web construction of ABCDEFG and affine quiver gauge theories

Kimura

Zhu

2019

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...

“…One of the research directions here is the interpretation of the corresponding Nekrasov functions in terms of the representation theory of DIM algebras [20,21] and network models [18,22], which generalize the Dotsenko-Fateev (conformal matrix model [23][24][25][26][27][28]) realization of conformal blocks, manifest an explicit spectral duality [16,17,[29][30][31][32][33][34] and satisfy the Virasoro/W-constraints in the form of the qq-character equations [18,21,[35][36][37]. Another direction is study of the underlying integrable systems, where the main unknown ingredient is the double-elliptic (DELL) generalization [38][39][40][41][42][43] of the Calogero-Ruijsenaars model [44][45][46][47][48][49][50][51].…”

Section: Introductionmentioning

confidence: 99%

Modular properties of 6d (DELL) systems

Aminov¹,

Миронов

Морозов

2017

If super-Yang-Mills theory possesses the exact conformal invariance, there is an additional modular invariance under the change of the complex bare charge τ = θ 2π + 4πı g 2 −→ − 1 τ . The low-energy Seiberg-Witten prepotential F (a), however, is not explicitly invariant, because the flat moduli also change a −→ a D = ∂F /∂a. In result, the prepotential is not a modular form and depends also on the anomalous Eisenstein series E 2 . This dependence is usually described by the universal MNW modular anomaly equation. We demonstrate that, in the 6d SU(N ) theory with two independent modular parameters τ andτ , the modular anomaly equation changes, because the modular transform of τ is accompanied by an (N -dependent!) shift ofτ and vice versa. This is a new peculiarity of double-elliptic systems, which deserves further investigation.