Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver

Bourgine, Jean-Emile; Fukuda, Masayuki; Matsuo, Yutaka; Zhu, Rui-Dong

doi:10.1007/jhep12(2017)015

Cited by 45 publications

(63 citation statements)

References 57 publications

(141 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…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: 86%

“…Now we go to the next simplest example, D-type quiver gauge theories. It was reported in [30] that the topological vertex formalism can also be applied to the brane webs proposed in [44] constructed with an ON − orientifold plane for D-type quiver theories [45,46]. The web diagram of the simplest ON − Figure 5: The web diagram for the D 2 quiver theory.…”

Section: D-type Quiver Construction With Orientifoldmentioning

confidence: 99%

See 1 more Smart Citation

Web construction of ABCDEFG and affine quiver gauge theories

Kimura

Zhu

2019

J. High Energ. Phys.

View full text Add to dashboard Cite

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...

show abstract

Section: D-type Quiver Construction With Orientifoldmentioning

confidence: 86%

Section: D-type Quiver Construction With Orientifoldmentioning

confidence: 99%

Web construction of ABCDEFG and affine quiver gauge theories

Kimura

Zhu

2019

J. High Energ. Phys.

View full text Add to dashboard Cite

show abstract

“…In the algebraic engineering, vertical representations are associated to (multiple) D5-branes, and horizontal representations to NS5-branes (possibly dressed by extra D5-branes). These representations have already been presented in several papers, we reproduce them here for consistency and follow the conventions employed in [21]. Then, we investigate the S-dual representations and derive the expression of several matrices M ( ,¯ ) S .…”

Section: Representationsmentioning

confidence: 98%

“…This paper explores the realization of S-duality, also called fiber-base duality (see below), among 5D N = 1 quiver gauge theories using an algebraic formalism developed recently in the series of papers [15][16][17][18][19][20][21], and based on the Ding-Iohara-Miki (DIM) algebra [22,23]. This formalism, referred here as algebraic engineering, highlights the integrable properties of the gauge theories' BPS sector, and, at the same time, expresses in a very elegant manner the covariance properties under the (q-deformed) W-algebra at the root of AGT correspondence.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Fiber-base duality from the algebraic perspective

Bourgine

2019

J. High Energ. Phys.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Webs of Quantum Algebra Representations in 5d $${\mathcal {N}}=1$$ Super Yang–Mills

Bourgine

2018

Springer Proceedings in Mathematics &Amp; Statistics

View full text Add to dashboard Cite

Reflection states in Ding-Iohara-Miki algebra and brane-web for D-type quiver

Cited by 45 publications

References 57 publications

Web construction of ABCDEFG and affine quiver gauge theories

Web construction of ABCDEFG and affine quiver gauge theories

Fiber-base duality from the algebraic perspective

Webs of Quantum Algebra Representations in 5d $${\mathcal {N}}=1$$ Super Yang–Mills

Contact Info

Product

Resources

About