[Paper Review] Gauge origami and quiver W-algebras

Taro Kimura, Go Noshita|arXiv (Cornell University)|Oct 12, 2023

Black Holes and Theoretical PhysicsPhysics and Astronomy215 references3 citations

TL;DR

This paper establishes a quantum algebraic framework for gauge origami systems in C⁴ with D2/D4/D6/D8-branes, showing that contour integral formulas for instanton partition functions admit free field realizations, leading to operator formalisms of qq-characters. D2 and D4-brane qq-characters correspond to screening charges and affine quiver W-algebra generators, respectively, while D6 and D8-brane qq-characters are novel, built from plane and solid partitions. The fusion of lower-dimensional qq-characters yields higher-dimensional ones, and the construction realizes the BPS/CFT and Bethe/Gauge correspondences via quantum toroidal gl₁ representations, culminating in a conjecture for BPS qq-characters on general toric Calabi–Yau four-folds.

ABSTRACT

We explore the quantum algebraic formalism of the gauge origami system in $\mathbb{C}^{4}$, where D2/D4/D6/D8-branes are present. We demonstrate that the contour integral formulas have free field interpretations, leading to the operator formalism of $qq$-characters associated with each D-brane. The $qq$-characters of D2 and D4-branes correspond to screening charges and generators of the affine quiver W-algebra, respectively. On the other hand, the $qq$-characters of D6 and D8-branes represent novel types of $qq$-characters, where monomial terms are characterized by plane partitions and solid partitions. The composition of these $qq$-characters yields the instanton partition functions of the gauge origami system, eventually establishing the BPS/CFT correspondence. Additionally, we demonstrate that the fusion of $qq$-characters of D-branes in lower dimensions results in higher-dimensional D-brane $qq$-characters. We also investigate quadratic relations among these $qq$-characters. Furthermore, we explore the relationship with the representations, $q$-characters, and the Bethe ansatz equations of the quantum toroidal $\mathfrak{gl}_{1}$. This connection provides insights into the Bethe/Gauge correspondence of the gauge origami system from both gauge-theoretic and quantum-algebraic perspectives. We finally present conjectures regarding generalizations to general toric Calabi-Yau four-folds. These generalizations imply the existence of an extensive class of $qq$-characters, which we call BPS $qq$-characters. These BPS $qq$-characters offer a new systematic approach to derive a broader range of BPS/CFT correspondence and Bethe/Gauge correspondence.

Motivation & Objective

• To develop a quantum algebraic formalism for gauge origami systems in C⁴ with D2/D4/D6/D8-branes.
• To interpret contour integral formulas for instanton partition functions in terms of free field realizations and operator formalisms.
• To identify qq-characters of D-branes with algebraic structures: screening charges (D2), affine quiver W-algebra generators (D4), and novel monomial types based on plane/solid partitions (D6/D8).
• To establish the BPS/CFT correspondence and Bethe/Gauge correspondence through connections to quantum toroidal gl₁ algebras.
• To conjecture a generalization to BPS qq-characters on arbitrary toric Calabi–Yau four-folds, extending the scope of the correspondence.

Proposed method

• Derive instanton partition functions using equivariant index formalism and Nekrasov factorizations in the context of D-brane configurations in C⁴.
• Construct qq-characters from contour integrals, identifying them as generating functions for plane and solid partitions for D6 and D8-branes.
• Realize contour integral formulas via free field representations, mapping them to vertex operator algebras and screening currents.
• Establish commutation relations between qq-characters and screening currents, proving their consistency with quiver W-algebra structures.
• Use i-Weyl reflections and recursion relations of Nekrasov factors to derive fusion rules and consistency conditions for qq-characters.
• Connect the qq-character formalism to representations of quantum toroidal gl₁, including vector, Fock, and MacMahon representations, via q-characters and Bethe ansatz equations.

Experimental results

Research questions

• RQ1 How do contour integral formulas for instanton partition functions in gauge origami systems admit free field realizations?
• RQ2 What is the algebraic interpretation of D2-brane qq-characters in terms of screening charges?
• RQ3 How do D4-brane qq-characters realize generators of affine quiver W-algebras?
• RQ4 What is the structure of D6 and D8-brane qq-characters, and how do they generalize to plane and solid partitions?
• RQ5 How does the fusion of lower-dimensional D-brane qq-characters produce higher-dimensional ones, and what is the underlying algebraic mechanism?

Key findings

• D2-brane qq-characters are identified as screening charges in the quantum algebraic structure, with their operator formalism derived from free field realizations of contour integrals.
• D4-brane qq-characters generate the affine quiver W-algebra, and their consistency is proven via commutation with screening currents using i-Weyl reflections and residue recursion.
• D6-brane qq-characters are constructed from plane partitions and correspond to the MacMahon representation of quantum toroidal gl₁, with explicit one-instanton contributions like S₃₄(q¹).
• D8-brane qq-characters emerge via fusion of D6-brane qq-characters, suggesting a hierarchy of algebraic structures based on solid partitions.
• The fusion of D4-brane qq-characters yields D6-brane qq-characters, demonstrating a recursive construction of higher-dimensional algebraic objects from lower-dimensional ones.
• The full system realizes the BPS/CFT correspondence and Bethe/Gauge correspondence through isomorphisms with representations of quantum toroidal gl₁, including vector, Fock, and MacMahon modules.