5d AGT correspondence of supergroup gauge theories from quantum toroidal $$ \mathfrak{gl} $$1

Abstract

We discuss the 5d AGT correspondence of supergroup gauge theories with A-type supergroups. We introduce two intertwiners called positive and negative intertwiners to compute the instanton partition function. The positive intertwiner is the ordinary Awata-Feigin-Shiraishi intertwiner while the negative intertwiner is an intertwiner obtained by using central charges with negative levels. We show that composition of them gives the basic Nekrasov factors appearing in supergroup partition functions. We explicitly derive the instanton partition functions of supergroup gauge theories with A and D-type quiver structures. Using the intertwiners, we briefly study the Gaiotto state, qq-characters and the relation with quiver W-algebra. Furthermore, we show that the negative intertwiner corresponds to the anti-refined topological vertex recently defined by Kimura and Sugimoto. We also discuss how superquiver theories should appear in our formalism if they exist. The existence of the AGT correspondence of the theories we study in this paper implies that there is a broader 2d/4d (5d/q-algebra) correspondence, or more generally the BPS/CFT correspondence, where new non-unitary theories play important roles.

Gauge origami and quiver W-algebras

“…We can do the same analysis for the negative weights where supergroup analogs appear. The corresponding representations in the quantum toroidal gl 1 were constructed in [99,159,160], where vertical Fock representations with negative levels were introduced. One can show that the representations in [99] correspond with the D4 qq-character generated by X A (z) −1 .…”

“…The partition functions were first derived by [93] (see also [97] for a review). The quantum algebraic perspective was studied in [99] for A, D-quiver gauge theories. The discussion here is complementary to the discussion in [99].…”

“…The quantum algebraic perspective was studied in [99] for A, D-quiver gauge theories. The discussion here is complementary to the discussion in [99].…”

Aspects of Supergroup Gauge Theory

“…It has been known that the refined topological vertex is understood as the intertwiner of quantum troidal algebra of gl 1 [AFS12], known as Ding-Iohara-Miki algebra [DI97,Mik07]. In this context, the anti-vertex is realized as the intertwiner involving the negative level [Bou19,Nos22]. See also [Rap20] for a related realization of superalgebra from the CY geometry.…”

“…We have a similar realization of supergroup gauge theory partition function. In this case, we may use both the screening currents to obtain the Z-state [KN21,Nos22].…”