Affine Yangian Research Articles

Quantum crystals, Kagome lattice, and plane partitions fermion-boson duality

In this work, we study quantum crystal melting in three space dimensions. Using an equivalent description in terms of dimers in a hexagonal lattice, we recast the crystal melting Hamiltonian as an occupancy problem in a Kagome lattice. The Hilbert space is spanned by states labeled by plane partitions and writing them as a product of interlaced integer partitions, we define a fermion-boson duality for plane partitions. Finally, based upon the latter result we conjecture that the growth operators for the quantum Hamiltonian can be represented in terms of the affine Yangian ${\cal Y}[\widehat{\mathfrak{gl}}(1)]$.

Liouville reflection operator, affine Yangian and Bethe ansatz

In these notes we study integrable structure of conformal field theory by means of Liouville reflection operator/Maulik Okounkov R-matrix. We discuss relation between RLL and current realization of the affine Yangian of mathfrak{gl}(1) . We construct the family of commuting transfer matrices related to the Intermediate Long Wave hierarchy and derive Bethe ansatz equations for their spectra discovered by Nekrasov and Okounkov and independently by one of the authors. Our derivation mostly follows the one by Feigin, Jimbo, Miwa and Mukhin, but is adapted to the conformal case.

Affine Yangian and 3-Schur functions

3D (3 dimensional) Young diagram is a generalization of 2D Young diagram. In this paper, from the orthogonality of 3D Young diagrams and the properties in affine Yangian and its MacMahon representation, we obtain the Schur functions corresponding to 3D Young diagrams, which are called 3-Schur functions. 3-Schur functions are a generalization of Schur functions in the sense that when h1=1,h2=−1,h3=0, the 3-Schur functions of 3D Young diagrams become Schur functions of 2D Young diagrams, which is a special case of h1=h,h2=−1h,h3=1h−h. When h1=h,h2=−1h,h3=1h−h, the 3-Schur functions turn into the Jack symmetric polynomials of 2D Young diagrams by multiplying a coefficient. We will see that 3-Schur functions are symmetric about three coordinate axes.

Gluing affine Yangians with bi-fundamentals

The affine Yangian of {mathfrak{gl}}_1 is isomorphic to the universal enveloping algebra of {mathcal{W}}_{1+infty } and can serve as a building block in the construction of new vertex operator algebras. In [1], a two-parameter family generalization of mathcal{N} = 2 supersymmetric {mathcal{W}}_{infty } algebra was constructed by “gluing” two affine Yangians of {mathfrak{gl}}_1 using operators that transform as (□, overline{square} ) and ( overline{square} , □) w.r.t. the two affine Yangians. In this paper we realize a similar (but non-isomorphic) two-parameter gluing construction where the gluing operators transform as (□, □) and ( overline{square} , overline{square} ) w.r.t. the two affine Yangians. The corresponding representation space consists of pairs of plane partitions connected by a common leg whose cross-section takes the shape of Young diagrams, offering a more transparent geometric picture than the previous construction.

Instanton R-matrix and mathcal{W} -symmetry

We study the relation between {mathcal{W}}_{1+infty } algebra and Arbesfeld-Schiffmann­ Tsymbaliuk Yangian using the Maulik-Okounkov R-matrix. The central object linking these two pictures is the Miura transformation. Using the results of Nazarov and Sklyanin we find an explicit formula for the mixed R-matrix acting on two Fock spaces associated to two different asymptotic directions of the affine Yangian. Using the free field representation we propose an explicit identification of Arbesfeld-Schiffmann-Tsymbaliuk generators with the generators of Maulik-Okounkov Yangian. In the last part we use the Miura transformation to give a conformal field theoretic construction of conserved quantities and ladder operators in the quantum mechanical rational and trigonometric Calogero-Sutherland models on which a vector representation of the Yangian acts.

Gluing two affine Yangians of \U0001d524\U0001d5291

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.

Cohomological Hall Algebras, Vertex Algebras and Instantons

We define an action of the (double of) Cohomological Hall algebra of Kontsevich and Soibelman on the cohomology of the moduli space of spiked instantons of Nekrasov. We identify this action with the one of the affine Yangian of $\mathfrak{gl}(1)$. Based on that we derive the vertex algebra at the corner $\mathcal{W}_{r_1,r_2,r_3}$ of Gaiotto and Rapcak. We conjecture that our approach works for a big class of Calabi-Yau categories, including those associated with toric Calabi-Yau $3$-folds.

Braid Group Action on Affine Yangian

We study braid group actions on Yangians associated with symmetrizable Kac-Moody Lie algebras. As an application, we focus on the affine Yangian of type A and use the action to prove that the image of the evaluation map contains the diagonal Heisenberg algebra inside $\hat{\mathfrak{gl}}_N$.

Plane partition realization of (web of) mathcal{W} -algebra minimal models

Recently, Gaiotto and Rapčák (GR) proposed a new family of the vertex operator algebra (VOA) as the symmetry appearing at an intersection of five-branes to which they refer as Y algebra. Procházka and Rapčák, then proposed to interpret Y algebra as a truncation of affine Yangian whose module is directly connected to plane partitions (PP). They also developed GR’s idea to generate a new VOA by connecting plane partitions through an infinite leg shared by them and referred it as the web of W-algebra (WoW). In this paper, we demonstrate that double truncation of PP gives the minimal models of such VOAs. For a single PP, it generates all the minimal model irreducible representations of W-algebra. We find that the rule connecting two PPs is more involved than those in the literature when the U(1) charge connecting two PPs is negative. For the simplest nontrivial WoW, mathcal{N} = 2 superconformal algebra, we demonstrate that the improved rule precisely reproduces the known character of the minimal models.

Affine Yangian Action on the Fock Space

The localized equivariant homology of the quiver variety of type A_{N-1}^{(1)} can be identified with the level one Fock space by assigning a normalized torus fixed point basis to certain symmetric functions, Jack( \mathfrak{gl}_N ) symmetric functions introduced by Uglov. We show that this correspondence is compatible with actions of two algebras, the Yangian for \mathfrak{sl}_N and the affine Lie algebra \hat{\mathfrak{sl}}_N , on both sides. Consequently we obtain affine Yangian action on the Fock space.

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

The universal enveloping algebra of mathcal{W} 1+∞ is isomorphic to the affine Yangian of mathfrak{g}{mathfrak{l}}_1 . We study the mathcal{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.

HIGHER LEVEL FOCK SPACES AND AFFINE YANGIAN

We construct actions of the affine Yangian of type A on higher level Fock spaces by extending known actions of the Yangian of finite type A due to Uglov. This is a degenerate analog of a result by Takemura-Uglov, which constructed actions of the quantum toroidal algebra on higher level $q$-deformed Fock spaces.

Homomorphisms between different quantum toroidal and affine Yangian algebras

This paper concerns the relation between the quantum toroidal algebras and the affine Yangians of sln, denoted by Uq1,q2,q3(n) and Yh1,h2,h3(n), respectively. Our motivation arises from the milestone work [11], where a similar relation between the quantum loop algebra Uq(Lg) and the Yangian Yh(g) has been established by constructing an isomorphism of C[[ħ]]-algebras Φ:Uˆexp⁡(ħ)(Lg)⟶∼Yˆħ(g) (with ˆ standing for the appropriate completions). These two completions model the behavior of the algebras in the formal neighborhood of h=0. The same construction can be applied to the toroidal setting with qi=exp⁡(ħi) for i=1,2,3 (see [11,22]). In the current paper, we are interested in the more general relation: q1=ωmneh1/m,q2=eh2/m,q3=ωmn−1eh3/m, where m,n≥1 and ωmn is an mn-th root of 1. Assuming ωmnm is a primitive n-th root of unity, we construct a homomorphism Φm,nωmn between the completions of the formal versions of Uq1,q2,q3(m) and Yh1/mn,h2/mn,h3/mn(mn).

The supersymmetric affine Yangian

The affine Yangian of gl1 is known to be isomorphic to {mathcal{W}}_{1+infty } , the W-algebra that characterizes the bosonic higher spin — CFT duality. In this paper we propose some of the defining relations of the Yangian that are relevant for the mathcal{N}=2 superconformal version of {mathcal{W}}_{1+infty } . Our construction is based on the observation that the mathcal{N}=2 superconformal {mathcal{W}}_{1+infty } algebra contains two commuting bosonic W1+∞ algebras, and that the additional generators transform in bi-minimal representations with respect to these two algebras. The corresponding affine Yangian can therefore be built up from two affine Yangians of mathfrak{g}{mathfrak{l}}_1 by adding in generators that transform appropriately.

Higher spins and Yangian symmetries

The relation between the bosonic higher spin {mathcal{W}}_{infty}left[lambda right] algebra, the affine Yangian of mathfrak{g}{mathfrak{l}}_1 , and the SHc algebra is established in detail. For generic λ we find explicit expressions for the low-lying {mathcal{W}}_{infty}left[lambda right] modes in terms of the affine Yangian generators, and deduce from this the precise identification between λ and the parameters of the affine Yangian. Furthermore, for the free field cases corresponding to λ = 0 and λ = 1 we give closed-form expressions for the affine Yangian generators in terms of the free fields. Interestingly, the relation between the {mathcal{W}}_{infty } modes and those of the affine Yangian is a non-local one, in general. We also establish the explicit dictionary between the affine Yangian and the SHc generators. Given that Yangian algebras are the hallmark of integrability, these identifications should pave the way towards uncovering the relation between the integrable and the higher spin symmetries.

Anomaly in [formula omitted] relation for DIM algebra and network matrix models

We discuss the recent proposal of arXiv:1608.05351 about generalization of the RTT relation to network matrix models. We show that the RTT relation in these models is modified by a nontrivial, but essentially abelian anomaly cocycle, which we explicitly evaluate for the free field representations of the quantum toroidal algebra. This cocycle is responsible for the braiding, which permutes the external legs in the q-deformed conformal block and its 5d/6d gauge theory counterpart, i.e. the non-perturbative Nekrasov functions. Thus, it defines their modular properties and symmetry. We show how to cancel the anomaly using a construction somewhat similar to the anomaly matching condition in gauge theory. We also describe the singular limit to the affine Yangian (4d Nekrasov functions), which breaks the spectral duality.

Classical limits of quantum toroidal and affine Yangian algebras

In this short article, we compute the classical limits of the quantum toroidal and affine Yangian algebras of sln by generalizing our arguments for gl1 from [7] (an alternative proof for n>2 is given in [10]). We also discuss some consequences of these results.

W $$ \mathcal{W} $$ -symmetry, topological vertex and affine Yangian

We discuss the representation theory of non-linear chiral algebra $\mathcal{W}_{1+\infty}$ of Gaberdiel and Gopakumar and its connection to Yangian of $\hat{\mathfrak{u}(1)}$ whose presentation was given by Tsymbaliuk. The characters of completely degenerate representations of $\mathcal{W}_{1+\infty}$ are for generic values of parameters given by the topological vertex. The Yangian picture provides an infinite number of commuting charges which can be explicitly diagonalized in $\mathcal{W}_{1+\infty}$ highest weight representations. Many properties that are difficult to study in $\mathcal{W}_{1+\infty}$ picture turn out to have a simple combinatorial interpretation.

The affine Yangian of [formula omitted] revisited

The affine Yangian of gl1 has recently appeared simultaneously in the work of Maulik–Okounkov [11] and Schiffmann–Vasserot [20] in connection with the Alday–Gaiotto–Tachikawa conjecture. While the presentation from [11] is purely geometric, the algebraic presentation in [20] is quite involved. In this article, we provide a simple loop realization of this algebra which can be viewed as an “additivization” of the quantum toroidal algebra of gl1 in the same way as the Yangian Yh(g) is an “additivization” of the quantum loop algebra Uq(Lg) for a simple Lie algebra g. We also explain the similarity between the representation theories of the affine Yangian and the quantum toroidal algebras of gl1 by generalizing the main result of [10] to the current settings.

Yangians and cohomology rings of Laumon spaces

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GL n. We construct the action of the Yangian of sln in the cohomology of Laumon spaces by certain natural correspon- dences. We construct the action of the affine Yangian (two-parametric deformation of the universal enveloping algebra of the universal central extension of sln(s ±1 , t)) in the cohomology of the affine version of Laumon spaces. We compute the matrix coefficients of the generators of the affine Yangian in the fixed point basis of coho- mology. This basis is an affine analog of the Gelfand-Tsetlin basis. The affine analog of the Gelfand-Tsetlin algebra surjects onto the equivariant cohomology rings of the affine Laumon spaces. The cohomology ring of the moduli space Mn,d of torsion free sheaves on the plane, of rank n and second Chern class d, trivialized at infinity, is