Web construction of ABCDEFG and affine quiver gauge theories

Abstract

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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“…The other is to establish the refinement of this generalized topological vertex formalism. By achieving that, we can start to discuss the relation with the formalism constructed in the S-dual setup [20,26], and to derive the qq-characters [60-69] 6 associated to BCD-type gauge theories. The fundamental qq-characters of BCD-type gauge theories was discussed in a recent work [70], and unlike the beautiful results obtained for A-type gauge theories, they contain infinite number of terms and there is no known closed form for these qq-characters.…”

“…Furthermore, web diagrams each of which are made by gluing three or four (dual) toric diagrams have been constructed in [19] and the method developed there computes the partition functions of SO(2N ) gauge theories and also the pure E 6 , E 7 , E 8 gauge theories. Recently the topolgical vertex formalims has been also extended in other directions [20][21][22][23][24][25][26][27][28][29].…”

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“…Beside, the elliptic genera of A-type chain of (−2) curves can be computed by refined topological vertex [10] or from the viewpoint of 2d sigma model [10,83]. The recently proposed elliptic topological vertex can also compute the partition function of these theories [84,85].…”

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“…For example, A k quiver gauge theory, U(m|n) ⊗k , is realized with k + 1 chains (NS5 branes): where we denote the gauge node by and the flavor node by , and all the nodes are associated with U(m|n) group. In addition to the A-type quiver, one can also consider more generic quivers [26].…”