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

Davide Gaiotto, Miroslav Rapčák, Yehao Zhou

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.

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

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 and -extended -algebra respectively. We give a new presentation of the deformed double current algebra of , and we give a rigorous mathematical construction of the -extended -algebra. A new presentation of the affine Yangian of 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.
Paper Structure (67 sections, 129 theorems, 624 equations, 1 figure)

This paper contains 67 sections, 129 theorems, 624 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results of the paper
  3. The Algebra Ak
  4. A filtration on Ak
  5. PBW theorems for Ak, Bk, and Dk, TEXT, and TEXT
  6. Relation to Costello's DDCA
  7. Duality automorphism
  8. Relations between various DDCAs
  9. Vertical and horizontal filtrations on Ak
  10. Unsymmetrized generators
  11. B-algebra and the Yangian algebra of gl(K)
  12. A simpler definition of A1
  13. A simpler definition of Ak, K>1
  14. A map from Ak to the Mode Algebras of Rectangular W-Algebras
  15. A map from Ak to the mode algebra of affine Kac-Moody vertex algebra
  16. ...and 52 more sections

Key Result

Lemma 2.0.2

$D_{\epsilon_2}(\mathbb C)\otimes \mathfrak{gl}_K^{\sim}$ is a Lie subalgebra of $D_{\epsilon_2}(\mathbb C)\otimes \mathfrak{gl}_K[\epsilon_2^{-1}]$.

Figures (1)

  • Figure 1: Relations between the algebras $\mathsf A^{(K)}$, $\mathsf B^{(K)}$, $\mathsf D^{(K)}$, $\widetilde{\mathsf D}^{(K)}$, and $\mathbb D^{(K)}$. Here all hook arrows are algebra embeddings and they become isomorphism after localizing the parameter $\epsilon_i$.

Theorems & Definitions (290)

  • Definition 2.0.1
  • Lemma 2.0.2
  • proof
  • Definition 2.0.3
  • Remark 2.0.4
  • Definition 2.0.5
  • Definition 2.0.6
  • Lemma 2.0.7
  • proof
  • Lemma 2.0.8
  • ...and 280 more