Fiber-base duality from the algebraic perspective

Abstract

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.

New quantum toroidal algebras from 5D $$ \mathcal{N} $$ = 1 instantons on orbifolds

“…This algebraic construction turns out to be useful in probing various properties of the partition function, e.g. in addressing the (q-deformed) AGT correspondence [37,38], or in studying strings' S-duality [39,40].…”

A note on the algebraic engineering of 4D N=2 super Yang–Mills theories

“…The weights of representations will be identified with the branes positions. In addition, automorphisms of the DIM algebra encode the invariance of the brane-web under geometric transformations [28]. These automorphisms can also be defined on the algebra 2.1, except for Miki's automorphism S [15] corresponding to a 90 • rotation of the brane-web that maps D5 on NS5 and vice versa, thus realizing the S-duality of type IIB string theory.…”

“…As in [28], the opposite (or permuted) coproduct ∆ ′ can also be obtained as a twist of ∆ by the automorphism S 2 .…”

“…In the algebraic engineering of 5D N = 1 gauge theories, the role of the topological vertex is devoted to the intertwiners of the DIM algebra, first introduced by Awata, Feigin and Shiraishi (AFS) in [19], and further generalized in [18,28]. In the 4D case, the intertwiners, denoted Φ (m) [u, a] and Φ * (m) [u, a], are obtained as solution to the following equations,…”

“…Furthermore, the automorphism S discovered by Miki [15] maps vertical to horizontal representations (and vice-versa), effectively implementing a rotation of the (p, q)-brane web, and thereby exchanging the role of the two different q-W-algebras. We refer to the upcoming paper [28] for a deeper discussion of this automorphism in the formalism of algebraic engineering.…”