Gauge origami and quiver W-algebras II: Vertex function and beyond quantum $q$-Langlands correspondence

TL;DR

This paper advances the understanding of gauge origami and quiver W-algebras by exploring their connection to quantum q-Langlands correspondence through vertex functions, stable envelopes, and higher-rank geometric realizations.

Contribution

It introduces a quantum algebraic framework for gauge origami, establishing new links between electric and magnetic blocks, and extends the quantum q-Langlands correspondence to double affine and higher-dimensional cases.

Findings

Constructed stable envelopes from chamber structures of vertex operators.

Established direct equivalence between electric and magnetic blocks.

Realized higher-rank multi-leg Pandharipande-Thomas vertices via conformal blocks.

Abstract

We continue the study of generalized gauge theory called gauge origami, based on the quantum algebraic approach initiated in [arXiv:2310.08545]. In this article, we in particular explore the D2 brane system realized by the screened vertex operators of the corresponding W-algebra. The partition function of this system given by the corresponding conformal block is identified with the vertex function associated with quasimaps to Nakajima quiver varieties and generalizations, that plays a central role in the quantum $q$-Langlands correspondence. Based on the quantum algebraic perspective, we address three new aspects of the correspondence: (i) Direct equivalence between the electric and magnetic blocks by constructing stable envelopes from the chamber structure of the vertex operators, (ii) Double affine generalization of quantum $q$-Langlands correspondence, and (iii) Conformal block realization of the origami vertex function associated with intersection of quasimaps, that realizes the higher-rank multi-leg Pandharipande-Thomas vertices of 3-fold and 4-fold.