On the quantization of Seiberg-Witten geometry

Abstract

We propose a double quantization of four-dimensional $$ \mathcal{N} $$ N = 2 Seiberg-Witten geometry, for all classical gauge groups and a wide variety of matter content. This can be understood as a set of certain non-perturbative Schwinger-Dyson identities, following the program initiated by Nekrasov [1]. The construction relies on the computation of the instanton partition function of the gauge theory on the so-called Ω-background on ℝ4, in the presence of half-BPS codimension 4 defects. The two quantization parameters are identified as the two parameters of this background. The Seiberg-Witten curve of each theory is recovered in the flat space limit. Whenever possible, we motivate our construction from type IIA string theory.

More on topological vertex formalism for 5-brane webs with O5-plane

“…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. Since the topological vertex formalism computes the partition function in a different basis (labeled by Young diagrams), it might simplify the expression of the qq-characters beyond the A-type gauge group.…”

“…The unrefined limit of the qq-characters is certainly also interesting, as one can expect things to be simplified in this limit from the computation of[70].…”

5d/6d Wilson loops from blowups

“…Wilson loop VEVs receive non-perturbative contribution on the instanton background. The instanton contributions to Wilson loop VEVs in various classical gauge groups have been computed using supersymmetric localization based on ADHM (or brane) constructions of instanton moduli space [54-56, 59, 60] or using auxiliary loop observables called qqcharacters [53,[71][72][73][74]. For instance, the 1-instanton correction to the fundamental Wilson loop in the pure SU(2) gauge theory at θ = 0 is [55,56,59]…”

“…There is yet another type of codimension-4 defects that can be defined systematically by coupling 1d (or 2d) degrees of freedom to 5d (or 6d) supersymmetric field theories. These operators are often referred to as qq-characters as discussed in [53,54,74]. The spectrum of such operators can also be obtained by employing the blowup approach, which we will talk about briefly from now on.…”

Instanton counting and O-vertex

“…In particular, partition functions with codimension-4 defects appear as the Y -operator in the qq-characters [8,40]. At the refined level, the qq-characters for gauge groups of type BCD have infinitely many terms [41]. However, they truncate to finite terms at the unrefined level.…”