Gluing two affine Yangians of \U0001d524\U0001d5291

Wei Li,Pietro Longhi

doi:10.1007/jhep10(2019)131

Abstract

We construct a four-parameter family of affine Yangian algebras by gluing two copies of the affine Yangian of \U0001d524\U0001d5291. Our construction allows for gluing operators with arbitrary (integer or half integer) conformal dimension and arbitrary (bosonic or fermionic) statistics, which is related to the relative framing. The resulting family of algebras is a two-parameter generalization of the mathcal{N} = 2 affine Yangian, which is isomorphic to the universal enveloping algebra of \U0001d532 (1)⊕ \U0001d4b2 {}_{infty}^{mathcal{N}=2}left[lambda right] . All algebras that we construct have natural representations in terms of “twin plane partitions”, a pair of plane partitions appropriately joined along one common leg. We observe that the geometry of twin plane partitions, which determines the algebra, bears striking similarities to the geometry of certain toric Calabi-Yau threefolds.

Highlights

There is an interesting and useful triangle of relations among the W1+∞[λ] algebra, the affine Yangian of gl1, and the set of plane partitions.affine Y7 angiani of gl1 “iso”w W1+∞[λ] o irreps irreps plane partitions (1.1)The W1+∞[λ] algebra is a family of VOAs with higher spin currents parameterized by the central charge c and the ’t Hooft coupling λ
We observe that the geometry of twin plane partitions, which determines the algebra, bears striking similarities to the geometry of certain toric Calabi-Yau threefolds
With different formulations, by [8] and [9] in the process of proving the AGT conjecture [10]. It was proposed by [11] and later proven by [12] that there is an isomorphism between the affine Yangian of gl1 and the universal enveloping algebra of the W1+∞[λ] algebra

Summary

Introduction

The W1+∞[λ] algebra is a family of VOAs with higher spin currents (i.e. one current per spin from s ≥ 1) parameterized by the central charge c and the ’t Hooft coupling λ. The affine Yangian of gl1 — denoted by Y(gl1) — appeared later on the scene It was constructed independently, and with different formulations, by [8] and [9] in the process of proving the AGT conjecture [10]. With different formulations, by [8] and [9] in the process of proving the AGT conjecture [10] It was proposed by [11] and later proven by [12] that there is an isomorphism between the affine Yangian of gl and the universal enveloping algebra of the W1+∞[λ] algebra.

Results

Discussion

Conclusion