Quantum toroidal algebras and solvable structures in gauge/string theory

Abstract

This is a review article on the quantum toroidal algebras, focusing on their roles in various solvable structures of 2d conformal field theory, supersymmetric gauge theory, and string theory. Using $\mathcal{W}$-algebras as our starting point, we elucidate the interconnection of affine Yangians, quantum toroidal algebras, and double affine Hecke algebras. Our exploration delves into the representation theory of the quantum toroidal algebra of $\mathfrak{gl}_1$ in full detail, highlighting its connections to partitions, $\mathcal{W}$-algebras, Macdonald functions, and the notion of intertwiners. Further, we also discuss integrable models constructed on Fock spaces and associated $\mathcal{R}$-matrices, both for the affine Yangian and the quantum toroidal algebra of $\mathfrak{gl}_1$. The article then demonstrates how quantum toroidal algebras serve as a unifying algebraic framework that bridges different areas in physics. Notably, we cover topological string theory and supersymmetric gauge theories with eight supercharges, incorporating the AGT duality. Drawing upon the representation theory of the quantum toroidal algebra of $\mathfrak{gl}_1$, we provide a rather detailed review of its role in the algebraic formulations of topological vertex and $qq$-characters. Additionally, we briefly touch upon the corner vertex operator algebras and quiver quantum toroidal algebras.

Figures (25)

• Figure 1: Flow chart of the entire note. The sections colored in red, blue, and purple are topics related to algebras, representations, and applications in physics, respectively.
• Figure 2: Flow chart of defining quantum toroidal algebras associated with toric quivers. The example drawn is the algebra associated with $\mathbb{C}^{3}$, which is simply the quantum toroidal $\mathfrak{gl}_{1}$.
• Figure 3: Generators of the elliptic braid group, which is the orbifold fundamental group of $(E^N)^{\textrm{reg}}/S_N$. The $i$-th blue dot represents a point in the $i$-th torus $E^{(i)}$. The braid group generator $T_i$ can be understood as the exchange of positions between the $i$-th and $(i+1)$-th blue dots. Additionally, the generators $X_i$ and $Y_{i}$ represent the meridian (horizontal) and longitude (vertical) cycles of the $i$-th torus $E^{(i)}$.
• Figure 4: The modes for the $W^{(d)}(z)$ current are expressed by generators $\mathsf{D}_{m,n}$ with $n<d$.
• Figure 5: Flow chart of §\ref{['sec:QTrep']}. The subsections colored in red, blue, and purple are topics related to vertical, horizontal, and both representations, respectively.

Theorems & Definitions (6)

• Definition 4.1
• Definition 4.2
• Theorem 4.1
• Theorem 5.1
• Theorem 5.2
• proof