Twin-plane-partitions and $$ \mathcal{N} $$ = 2 affine Yangian

Abstract

The universal enveloping algebra of W 1+∞ is isomorphic to the affine Yangian of gl 1. We study the N = 2 supersymmetric version of this correspondence, and identify the full set of defining relations of the supersymmetric affine Yangian. These relations can be deduced by demanding that the algebra has a representation on twin-plane-partitions, which we construct by gluing pairs of plane partitions. We define the action of the algebra on these twin-plane-partitions explicitly.

Gluing two affine Yangians of 𝔤𝔩1

“…In the attempt to supersymmetrize the triangle (1.1) for the N = 2 supersymmetric W ∞ [λ] algebra, [22,23] developed a construction of new affine Yangians by gluing two distinct plane partitions. The construction is based on the fact that u(1) ⊕ W N =2 ∞ [λ] has two commuting bosonic W 1+∞ subalgebras, and that all fermionic generators transform in irreducible representations derived from tensor products of ( , ) and ( , ) w.r.t.…”

“…In this paper, we will generalize the gluing construction of [22,23] to produce a fourparameter family of VOAs with a natural action on (suitably generalized) twin plane partitions. In addition to c, λ, we introduce a "shifting" parameter ρ ∈ 1 Z ≥0 and a "framing" p that can take values 0 or ±1.…”

“…Once these are fixed, the final algebra can be completely determined, by demanding that the action of both corner bosonic subalgebras and the newly added gluing operators have sensible, and mutually consistent, actions on the set of twin-plane partitions. We derive all OPE relations by following a similar procedure as the one used in the N = 2 case in [23].…”

“…Relative orientations of the two corners: p = 0 on the left, p = −1 on the right. N = 2 case of [23], the nilpotency-on-vacuum property could have also been derived from the map to the W N =2 ∞ algebra. Here we extend this statement to all values of ρ.…”

“…the triangle relation (1.1) between W 1+∞ , the affine Yangian of gl 1 , and plane partitions. Then we review the construction of the N = 2 version of this triangle [22,23] via gluing. The gluing construction in the current paper is a two-parameter generalization of the N = 2 gluing.…”