[Paper Review] Deformed Double Current Algebras, Matrix Extended $\mathcal W_{\infty}$ Algebras, Coproducts, and Intertwiners from the M2-M5 Intersection

Davide Gaiotto, Miroslav Rapčák+1 more|arXiv (Cornell University)|Sep 29, 2023

Black Holes and Theoretical Physics4 citations

TL;DR

This paper constructs and analyzes deformed double current algebras (DDCAs) and matrix extended $\mathcal{W}_{\infty}$ algebras arising from the M2-M5 brane intersection in twisted M-theory. It provides a new presentation of the $\mathfrak{gl}_K$-DDCA, rigorously defines the $\mathfrak{gl}_K$-extended $\mathcal{W}_{\infty}$ vertex algebra, and establishes coproducts and intertwiners—specifically, matrix Miura operators—via bimodule structures, revealing a deep algebraic duality in 5d $\mathcal{N}=1$ SCFTs.

ABSTRACT

We study the algebraic structures which govern the deformation of supersymmetric intersections of M2 and M5 branes. The universal algebras on M2 and M5 branes are deformed double current algebra of $\mathfrak{gl}_K$ and $\mathfrak{gl}_K$-extended $\mathcal{W}_{\infty}$-algebra respectively. We give a new presentation of the deformed double current algebra of $\mathfrak{gl}_K$, and we give a rigorous mathematical construction of the $\mathfrak{gl}_K$-extended $\mathcal{W}_{\infty}$-algebra. A new presentation of the affine Yangian of $\mathfrak{gl}_K$ is also obtained. We construct various coproducts of these algebras, which are expected to encode the fusions of defects in twisted M-theory. The matrix extended Miura operators are identified as intertwiners in certain bimodules of these algebras.

Motivation & Objective

• To construct a rigorous mathematical framework for the $\mathfrak{gl}_K$-extended $\mathcal{W}_{\infty}$ vertex algebra arising from the M2-M5 brane intersection in twisted M-theory.
• To provide a new presentation of the deformed double current algebra $\mathsf{A}^{(K)}$ for $\mathfrak{gl}_K$, generalizing the $K=1$ case.
• To define and analyze coproduct structures on $\mathsf{A}^{(K)}$, $\mathsf{Y}^{(K)}$, and $\mathsf{L}^{(K)}$, encoding defect fusion in twisted M-theory.
• To identify matrix Miura operators as intertwiners in bimodules over the affine Yangian $\mathsf{Y}^{(K)}$ and the $\mathcal{W}_{\infty}^{(K)}$ algebra.
• To establish isomorphisms between the mode algebras of rectangular $\mathcal{W}_{\infty}$ algebras and the $\mathfrak{gl}_K$-DDCA, revealing duality structures.

Proposed method

• Introduces a new presentation of the deformed double current algebra $\mathsf{A}^{(K)}$ using unsymmetrized generators and a filtration to prove PBW theorems.
• Constructs the $\mathfrak{gl}_K$-extended $\mathcal{W}_{\infty}$ vertex algebra $\mathcal{W}^{(K)}_{\infty}$ via matrix-valued pseudodifferential symbols and proves polynomiality of structure constants.
• Defines a meromorphic coproduct on $\mathsf{A}^{(K)}$ using a point-splitting construction on configuration spaces, leading to a vertex coalgebra structure.
• Establishes a map from the affine Yangian $\mathsf{Y}^{(K)}$ to the mode algebra of $\mathcal{W}^{(K)}_{\infty}$, proving a PBW theorem and coproduct compatibility.
• Uses a gluing construction to define $\mathsf{L}^{(K)}$ as the intersection of positive and negative halves of $\mathsf{A}^{(K)}$, and constructs its coproducts.
• Identifies matrix Miura operators as intertwiners via a bimodule structure on $\operatorname{Hom}(\mathcal{V}, D(\mathbb{C}^{\times I}_{\mathrm{disj}}) \widetilde{\otimes} \mathcal{V})$, with actions from $\mathsf{Y}^{(K)}$ on both sides.

Experimental results

Research questions

• RQ1 How can the deformed double current algebra $\mathsf{A}^{(K)}$ for $\mathfrak{gl}_K$ be re-expressed in a new, more symmetric presentation with a clear filtration and PBW basis?
• RQ2 What is the precise mathematical construction of the $\mathfrak{gl}_K$-extended $\mathcal{W}_{\infty}$ vertex algebra $\mathcal{W}^{(K)}_{\infty}$, and how do its structure constants behave under polynomiality?
• RQ3 How do coproducts on $\mathsf{A}^{(K)}$, $\mathsf{Y}^{(K)}$, and $\mathsf{L}^{(K)}$ encode the fusion of defects in twisted M-theory?
• RQ4 In what sense are matrix Miura operators intertwiners between representations of the affine Yangian $\mathsf{Y}^{(K)}$ and the $\mathcal{W}^{(K)}_{\infty}$ algebra?
• RQ5 How do duality automorphisms and anti-involutions on the mode algebras of $\mathcal{W}^{(K)}_{\infty}$ and $\mathsf{Y}^{(K)}$ relate to the underlying physical symmetries in 5d $\mathcal{N}=1$ theories?

Key findings

• A new presentation of the deformed double current algebra $\mathsf{A}^{(K)}$ is given using unsymmetrized generators, with a PBW theorem established via a vertical and horizontal filtration.
• The $\mathfrak{gl}_K$-extended $\mathcal{W}_{\infty}$ vertex algebra $\mathcal{W}^{(K)}_{\infty}$ is rigorously constructed as a vertex algebra with polynomial structure constants.
• The meromorphic coproduct on $\mathsf{A}^{(K)}$ is defined via a point-splitting map on configuration spaces, yielding a vertex coalgebra structure compatible with the algebraic operations.
• Matrix Miura operators are identified as intertwiners in a $\mathsf{Y}^{(K)}$-bimodule, with the action defined through a composition of maps in $\operatorname{Hom}(\mathcal{V}, D(\mathbb{C}^{\times I}_{\mathrm{disj}}) \widetilde{\otimes} \mathcal{V})$.
• A map from the affine Yangian $\mathsf{Y}^{(K)}$ to the mode algebra of $\mathcal{W}^{(K)}_{\infty}$ is constructed, and its PBW theorem is proven, confirming the algebraic independence of generators.
• Duality isomorphisms and anti-involutions are established for both the rectangular $\mathcal{W}_{\infty}$ algebra and its mode algebra, revealing hidden symmetries in the algebraic structure.