Quantum Toroidal Algebra Research Articles

(q, t)-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces

We describe the general strategy for lifting the Wess-Zumino-Witten model from the level of one-loop Kac-Moody {U}_q{left(widehat{mathfrak{g}}right)}_k to generic quantum toroidal algebras. A nearly exhaustive presentation is given for both {U}_{q,t}left({widehat{widehat{mathfrak{gl}}}}_1right) and {U}_{q,t}left({widehat{widehat{mathfrak{gl}}}}_nright) when the screenings do not exist and thus all the correlators are purely algebraic, i.e. do not include additional hypergeometric type integrations/summations.Generalizing the construction of the intertwiner (refined topological vertex) of the Ding-Iohara-Miki (DIM) algebra, we obtain the intertwining operators of the Fock representations of the quantum toroidal algebra of type An. The correlation functions of these operators satisfy the (q, t)-Knizhnik-Zamolodchikov (KZ) equation, which features the ℛ-matrix. The matching with the Nekrasov function for the instanton counting on the ALE space is worked out explicitly.We also present an important application of the DIM formalism to the study of 6d gauge theories described by the double elliptic integrable systems. We show that the modular and periodicity properties of the gauge theories are neatly explained by the network matrix models providing solutions to the elliptic (q, t)-KZ equations.

Extremal loop weight modules and tensor products for quantum toroidal algebras

ABSTRACTWe define integrable representations of quantum toroidal algebras of type A by tensor product, using the Drinfeld “coproduct.” This allows us to recover the vector representations recently introduced by Feigin–Jimbo–Miwa–Mukhin [7] and constructed by the author [21] as a subfamily of extremal loop weight modules. In addition we get new extremal loop weight modules as subquotients of tensor powers of vector representations. As an application we obtain finite-dimensional representations of quantum toroidal algebras by specializing the quantum parameter at roots of unity.

Several realizations of Fock modules for toroidal $\ddot {U}_{q,d}(\mathfrak {sl}_{n})$

In this paper, we relate the well-known Fock representations of the quantum toroidal algebra of sl(n) to the vertex, shuffle, and L-operator representations of the same algebra. These identifications generalize those for the quantum toroidal algebra of gl(1), which were recently established by Feigin-Jimbo-Miwa-Mukhin.

Integrals of motion from quantum toroidal algebras**Dedicated to the memory of Petr Petrovich Kulish.

We identify the Taylor coefficients of the transfer matrices corresponding to quantum toroidal algebras with the elliptic local and non-local integrals of motion introduced by Kojima, Shiraishi, Watanabe, and one of the authors.That allows us to prove the Litvinov conjectures on the Intermediate Long Wave model.We also discuss the duality of XXZ models in quantum toroidal setting and the implications for the quantum KdV model. In particular, we conjecture that the spectrum of non-local integrals of motion of Bazhanov, Lukyanov, and Zamolodchikov is described by Gaudin Bethe ansatz equations associated to affine .

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.

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.

Branching rules for quantum toroidal [formula omitted

We construct an analog of the subalgebra Ugl(n)⊗Ugl(m)⊂Ugl(m+n) in the setting of quantum toroidal algebras and study the restrictions of various representations to this subalgebra.

Actions of the quantum toroidal algebra of type sl2 on the space of vertex operators for Uq(gl2̂) modules

Highest weight modules for Uq(gl2̂) are endowed with a structure of modules for the quantum toriodal algebra Uκ of type sl2. Using this, we define Uκ actions on the space of vertex operators for irreducible highest weight Uq(gl2̂) modules. Highest or lowest weight vectors of the thus obtained Uκ modules are expressed in terms of an intertwiner for Uq(sl2̂) modules and an extra boson. The submodules generated by these vectors are investigated.

Extremal Loop Weight Modules for U q ( sl ̂ ∞ ) $\mathcal {U}_{q}(\hat {sl}_{\infty })$

We construct by fusion product new families of irreducible representations of the quantum affinization $\mathcal {U}_{q}(\hat {sl}_{\infty })$ . The action is defined via the Drinfeld coproduct and is related to the crystal structure of semi-standard tableaux of type A ∞ . We call these representations extremal loop weight modules. The main motivations are applications to quantum toroidal algebras $\mathcal {U}_{q}(sl_{n+1}^{tor})$ : we prove the conjectural link between $\mathcal {U}_{q}(\hat {sl}_{\infty })$ and $\mathcal {U}_{q}(sl_{n+1}^{tor})$ stated in Hernandez (J. Algebra 329, 147–162, 2011) for these families of representations. We recover in this way the extremal loop weight modules obtained in Mansuy (arXiv: 1305.3481 ).

The Fermion Representation of Quantum Toroidal Algebra on 3D Young Diagrams

We develop an equivalence between the diagonal slices and the perpendicular slices of 3D Young diagrams via Maya diagrams. Furthermore, we construct the fermion representation of quantum toroidal algebra on the 3D Young diagrams perpendicularly sliced.

Twisted Quantum Toroidal Algebras $${T_q^-(\mathfrak g)}$$ T q - ( g )

We construct a principally graded quantum loop algebra for the Kac-Moody algebra. As a special case a twisted analog of the quantum toroidal algebra is obtained together with the quantum Serre relations.

Fermion Representation of Quantum Toroidal Algebra

We construct a fermion analogue of the Fock representation of quantum toroidal algebra and construct the fermion representation of quantum toroidal algebra on the K-theory of Hilbert scheme.

Quantum extremal loop weight modules and monomial crystals

In this paper we construct a new family of representations for the quantum toroidal algebras of type $A_n$, which are $\ell$-extremal in the sense of Hernandez [24]. We construct extremal loop weight modules associated to level 0 fundamental weights $\varpi_\ell$ when $n=2r+1$ is odd and $\ell=1, r+1$ or $n$. To do it, we relate monomial realizations of level 0 extremal fundamental weight crystals with integrable representations of $\mathcal{U}_q(sl_{n+1}^{tor})$, and we introduce promotion operators for the level 0 extremal fundamental weight crystals. By specializing the quantum parameter, we get finite-dimensional modules of quantum toroidal algebras at roots of unity. In general, we give a conjectural process to construct extremal loop weight modules from monomial realizations of crystals.

Characters of representations of the quantum toroidal algebra : Plane partitions with “stands”

An expression for the generating function of plane partitions a i,j subject to the constraints a m,n = 0 and a i,j ⩾ k j , 1 ⩽ j ⩽ n, which is the character of an irreducible representation of the quantum toroidal algebra Open image in new window , is obtained.

A twisted quantum toroidal algebra

As an analog of the quantum TKK algebra, a twisted quantum toroidal algebra of type A_1 is introduced. Explicit realization of the new quantum TKK algebra is constructed with the help of twisted quantum vertex operators over a Fock space.

The quantum toroidal algebra $$\widehat {\widehat {\mathfrak{g}{\mathfrak{l}_1}}}$$ : Calculation of characters of some representations as generating functions of plane partitions

The generating function of plane partitions {ai,j} subject to the constraint am,n = 0 is expressed and calculated as the character of an irreducible representation of the quantum toroidal algebra \(\widehat {\widehat {\mathfrak{g}{\mathfrak{l}_1}}}\) in the case K = q1mq2n.

Two-parameter quantum vertex representations via finite groups and the McKay correspondence

We introduce two-parameter quantum toroidal algebras of simply laced types and provide their group theoretic realization using finite subgroups of SL2(C) via McKay correspondence. In particular our construction contains a realization of the vertex representation of the two-parameter quantum affine algebras of ADE types.

The algebra [formula omitted] and applications

In this note we consider the algebra U q ( sl ˆ ∞ ) and we study the category O of its integrable representations. The main motivations are applications to quantum toroidal algebras U q ( sl n + 1 tor ) , more precisely predictions of character formulae for representations of U q ( sl n + 1 tor ) . In this context, we state a general positivity conjecture for representations of U q ( sl ˆ ∞ ) viewed as representations of U q ( sl n + 1 tor ) , that we prove for Kirillov–Reshetikhin modules.

Quantum affine Gelfand–Tsetlin bases and quantum toroidal algebra via K-theory of affine 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 quantum loop algebra U_v(Lsl_n) in the equivariant K-theory of Laumon spaces by certain natural correspondences. Also we construct the action of the quantum toroidal algebra U^{tor}_v(Lsl}_n) in the equivariant K-theory of the affine version of Laumon spaces. We write down explicit formulae for this action in the affine Gelfand-Tsetlin base, corresponding to the fixed point base in the localized equivariant K-theory.