$\mathfrak{gl}_N$ Higgsed networks

We generalize the framework of Higgsed networks of intertwiners to the quantum toroidal algebra associated to Lie algebra $\mathfrak{gl}_N$. Using our formalism we obtain a systems of screening operators corresponding to W-algebras associated to toric strip geometries and reproduce partition functions of 3d theories on orbifolded backgrounds.


Introduction
Quantum toroidal algebras [1], [2] are quantum deformations of double loop algebras. They are natural from several points of view and have intricate representation theory. Most importantly for us, they play a crucial role in different versions of the AGT correspondence [3], which in its abstract form [4] is a statement about actions of certain algebras on the spaces of BPS objects in gauge theories. In many cases the algebras featuring in the correspondence are quantum toroidal algebras and the spaces of BPS states are K-theories of various moduli spaces. Here we will not be concerned with the mechanism of the action itself, which belongs to the geometric representation theory (see e.g. [5]). We will use the representation theory of the algebras to build certain intertwining operators and screening currents and with their help compute partition functions of 3d gauge theories. In doing so we follow closely our previous work [6], where we have considered similar problem for the algebra U q1,q2,q3 ( gl 1 ).
The intertwiners we are going to construct are between horizontal Fock representations and vertical vector representations of the quantum toroidal algebra U q1,q2,q3 ( gl N ). We will see that, just as in the "abelian" U q1,q2,q3 ( gl 1 ) case, these intertwiners are convenient building blocks which can be glued together into what we have called a Higgsed network. The idea of the construction is as follows. In [7] the intertwining operators between Fock representations of U q1,q2,q3 ( gl 1 ) have been shown to reproduce refined topological vertex computations, which in turn can be used to compute Nekrasov partition functions [8] of 5d gauge theories on C 2 × S 1 . In these 5d gauge theories one can tune the parameters so that the theory becomes dual to a 3d theory living on the worldvolume of vortices in the Higgs phase. This tuning procedure is known as Higgsing. The Higgsed intertwiners provide an algebraic framework, in which correlators automatically reproduce Higgsed theory partition functions. Since a lot of degrees of freedom decouple during the Higgsing, the representations featuring in the Higgsed intertwiners are "smaller" than the Fock representations from [7]. In these smaller vector representations the states are labelled by integers (cf. with the Fock representations where they are labelled by Young diagrams, or tuples of integers). Since the representations are smaller, the intertwiners are easier to handle and the computations become more transparent. Besides, vector representations allow one to better use the full power of the large automorphism group of the quantum toroidal algebra (for details see [6]).
In the present short note we consider the following variation of the Higgsing procedure. Let the original 5d gauge theory live not on C 2 × S 1 , but on an ALE space C 2 /Z N × S 1 . In the algebraic language the partition function of the 5d theory corresponds to a network of Fock space intertwiners for the algebra U q1,q2,q3 ( gl N ) [9] 1 . The tuning of the parameters corresponding to Higgsing can still be performed, and one arrives at a 3d theory on the vortex worldvolume. However, now the theory is "orbifolded": it lives on C/Z N × S 1 and there is an additional Z N R-symmetry twist when going around the orbifold point.
We will obtain the partition function of the orbifolded theory from the network of Higgsed U q1,q2,q3 ( gl N ) intertwiners. It turns out that the form of the partition function and the its algebraic properties are quite similar to those of the original theory. The main difference is that the partition function is not just a function, but a vector-valued function, i.e. there are several partition functions labelled by the indices associated with fundamental hypermultiplets (external lines in the intertwiner picture).
The plan of the paper is as follows. We construct the intertwiners in sec. 2, compute their correlators in sec. 3 and show how to interpret them in the gauge theory and algebraic language in sec. 4. Since, the main ideas have already been laid out in [6], we allow ourselves to be brief and present only the main features of the gl N generalization.
In order not to clutter the main part of the paper with technical details, we move the notations and definitions to the appendices.

The intertwiners
Similarly to [6] we would like to build an intertwining operator Φ : where g is any element of U q1,q2,q3 ( gl N ) and ∆ is the coproduct given in Appendix A.1. Representations V q1 and F (1,0),k q1,q3 (u) featuring in the intertwiner are defined in Appendices A.4 and A.5 respectively. Since there will be only one Fock representation in our story, we omit the indices q1,q3 henceforth.
We represent the intertwiner graphically as a trivalent junction: We denote the vector representation V q1 as blue dashed line, while the Fock space is represented by a solid black line. The essential new ingredients compared to the U q1,q2,q3 ( gl 1 ) case are the C N index j of the intertwiner and the color k of the horizontal Fock representation. The explicit expression for the intertwiner is where l − j = (l − j) mod N . The definitions of the bosons a j,n and the zero modes Q j , P j are given in Appendix A.5. Notice that in the "unrefined limit" q → 1 the second line simplifies, so that Φ j q1 (w) becomes the conventional undeformed free boson vertex operator.
The dual intertwiner Φ * ,q1,k where β = ln q3 ln q1 . Due to the reflection symmetry ρ of the algebra (see Appendix A.2) there is another pair of intertwiners with the vector representation V q3 (denoted by the dashed red lines): We will omit some zero modes henceforth since their contributions to the correlators are just powers of the spectral parameters.

Correlators
Having the explicit form of the intertwining operators (3)-(6) we can straightforwardly compute their correlators. Moreover, since the intertwiners are built free boson vertex operators Wick theorem applies, so it is sufficient to compute only the pairwise correlators. They are given by The proportionality signs in Eqs. (7)- (12) are to remind us that we have omitted some simple factors originating from the zero modes. We don't list the correlators of V q3 intertwiner with another V q3 intertwiner, because they are obtained from Eqs. (7)-(9) by the exchange q 1 ↔ q 3 and j ↔ (−j).

R-matrix
Having computed the correlators, we can deduce the commutation relations for the intertwiners. They feature the diagonal R-matrix of U q1,q2,q3 ( gl N ) acting in the tensor product of two vector representations: where A(x) is a certain q N 1 -periodic combination of theta-functions and the R-matrix is given by Similarly, we learn that where B(x) is another q N 1 -periodic combination. The dual intertwiners commute (up to a q N 1 -periodic function C(x)) We don't pay attention to the q N 1 -periodic functions since in the partition function we are going to study in the next section the intertwiners are going to be under the q N 1 -Jackson integral (or, equivalently, under contour integral with q N 1 -periodic strings of poles).

Screenings and networks
Similarly to [6] we can build a network of intertwiners by combining them either vertically or horizontally.
We have considered all possible horizontal combinations in the previous section. In this section we combine the intertwiners vertically to tie together several horizontal solid lines.

Screening operators
Two possible vertical combinations of intertwiners give rise to two screening currents acting in the tensor product of two Fock spaces: where a (1) i,n and a (2) i,n act on the first and the second Fock spaces respectively. If we sum over a complete basis of states in the intermediate vector representations in Eqs. (17), (18) we will get intertwining operators acting in the tensor product of two Fock spaces: In other words, integrated combinations like Q q1 and Q q3 are screening charges commuting with the action of the whole U q1,q2,q3 ( gl N ) algebra. We can use these screening operators to build "gl N -extended" W k algebras acting on the tensor product of kN Fock spaces 2 . By Fock spaces here we mean representations of a single boson, and denote them by F to distinguish from the Fock representation F of U q1,q2,q3 ( gl N ) which is generated by N bosons. To build the W k -algebra one needs to cluster together N -tuples of Fock spaces: There are k tuples, on which the screening charges Q q3 act between a-th and (a + 1)-th N -tuple and the W k -algebra is generated by the operators commuting with all the screenings. There is a constructive way to obtain the generators: one can simply take a coproduct of any element of U q1,q2,q3 ( gl N ) and evaluate it on the tensor product of k Fock spaces. It will automatically commute with the screening charges. As an example of the "gl N -extended" q-deformed W 2 generators we take ∆(E i (z)) (see Appendices A.1 and A.5 for the definition of the coproduct and the Fock representation respectively): The operators (22) should reproduce the web W -algebras [21] associated with brane diagram of the form shown in Fig. 1.

Networks and partition functions
The network partition functions are built from the screening charges in the same way as in the U q1,q2,q3 ( gl 1 ) case: one draws the solid "warp threads" and then connects them with arbitrary number of blue and red "weft threads". Let us give a simple example of the construction: The integral (23) reproduces the partition function of the orbifolded 3d N = 2 * U (1) theory with two hypermultiplets. The logic of this identification is similar to [6]. One views the picture (23) as a brane diagram of Type IIB string theory, so that the horizontal lines are identified with NS5 branes and the vertical dashed lines are D3 branes. The finite D3 brane segment gives rise to a U (1) gauge theory, while the semi-infinite D3 branes represent the hypermultiplets. The remaining directions of the D3 and NS5 branes are then subject to the Z N orbifolding. The FI, mass and equivariant parameters of the 3d theory are identified with the spectral parameters of the branes: log q1 u1 u2 is the FI parameter, w 1,2 are exponentiated masses of the fundamental hypermultiplets, q 2 is the real adjoint mass, and q 1 is the equivariant parameter of the 3d Ω-background.
A more general example is that of U (n) theory with m fundamental hypermultiplets. To get the partition function we just add more dashed lines to the diagram in (23): where the "orbifolded" (q 1 , q 2 )-deformed Vandermonde determinant is given by Of course, one can add the blue and red dashed lines together to the picture and obtain partition functions of two orbifolded 3d theories living respectively on and and coupled through a 1d theory on S 1 . We limit ourselves to a simple example of two coupled U (1) theories each with a single hypermultiplet:

Conclusions
In this short note we have presented the natural generalization of the Higgsed network formalism to "nonabelian" quantum toroidal algebras U q1,q2,q3 ( gl N ). We have found the intertwiners ("Higgsed vertices"), computed their correlators, and built screening operators from them. The correlators of intertwiners are related to partition functions of 3d gauge theories on orbifolded spacetimes. There are several further questions about the models we have considered. The partition functions we have obtained should satisfy a version of (q, t)-KZ equation [9], and they should also be related to the eignefunctions of a spin generalization of the Ruijsenaars-Schneider integrable system. It would be interesting to clarify the interplay between these two systems.
An even more interesting, but also apparently more difficult task, is to obtain general bosonization formulas for toroidal algebras. The bosonization currently known for U q1,q2,q3 ( gl N ) is the Fock representation [11,12], which is written in terms of N free bosons and therefore has fixed central charge C 1 = q. It is thus the toroidal analogue of the Frenkel-Kac construction [17] for affine Lie algebras. This bosonization also follows naturally from the geometric action of U q1,q2,q3 ( gl N ) on the moduli space of instantons [18].
A general bosonization formula for U q1,q2,q3 ( gl N ), if it exists, should involve N 2 free bosons and produce arbitrary central charges. This would be an analogue of the Wakimoto bosonization [19] for affine Lie algebras.
As we briefly explain in Appendix A.6, quantum toroidal algebras U q1,q2,q3 ( gl N ) are related to toric Calabi-Yau three-folds of the form shown in Fig. 1. More generally, one can consider an arbitrary toric strip geometry, i.e. a manifold whose toric diagram has N semi-infinite horizontal lines attached from the right and M lines attached from the left with no loops. This geometry corresponds to a quantum toroidal superalgebra U q1,q2,q3 ( gl N |M ) recently introduced in [20]. It should be straightforward to generalize our results to this case. An interesting conjecture is that any toric CY threefold corresponds to a quantum algebra, which is a "two-dimensional" generalization of quantum toroidal algebras. The gluing approach recently popularized in [21] could probably help to define these general algebras.
Acknowledgements. The author is partly supported by the RSF grant 18-71-10073.
A The algebra U q 1 ,q 2 ,q 3 ( gl N ) In this Appendix we collect the definitions related to the algebra U q1,q2,q3 ( gl N ). We mainly follow [13]. To simplify the presentation we limit ourselves to the cases when N ≥ 3, since the N = 2 case requires special treatment. We use the standard notations for the quantum parameters of the algebra 3 The algebra U q1,q2,q3 ( gl N ) is generated by modes of the currents: with i = 0, . . . , N − 1 and the central element C 1 satisfying the following commutation relations 4 : 3 In [6] we have used a less symmetric notation Uq,t( gl 1 ) for Uq 1 ,q 2 ,q 3 ( gl 1 ), so that q [6] = (q 1 ) here and t [6] = (q 2 ) −1 here . 4 We conform with the conventions of [13], [9]. The relations given in [15], [14] are obtained from the convention we use by the redefinition (K ± i (z)) [15], [14] = (K ± i (C The second central element of the algebra is given by the product of the zero modes K i : Sometimes it will be more convenient for us to write the relations directly for the modes of the generating currents: where θ n>0 gives 1 if n > 0 or 0 if n ≤ 0. The algebra is doubly graded with the first grading d 1 counting the number of the mode of a current: The second grading d 2 is given by There are two natural quantum affine subalgebras U q ( gl N ) inside U q1,q2,q3 ( gl N ), the "horizontal" and the "vertical" ones. The horizontal subalgebra U q ( gl N ) hor is generated by the zero modes of the currents E i,0 , F i,0 , K i . Notice that i = 0, . . . , N − 1 labels the roots of the affine root system A N . This gives the "traceless" subalgebra U q ( sl N ) hor ⊂ U q ( gl N ) hor . An additional horizontal Heisenberg subalgebra is generated by the sequential commutators of non-zero modes, e.g. E 0,1 and E i,−1 or E 0,−1 and E i,1 . The horizontal subalgebra has vanishing d 1 grading, while d 2 counts the modes of the quantum affine currents.
The vertical quantum affine subalgebra U q ( gl N ) vert is obtained by forgetting about the currents E 0 (z), F 0 (z). The remaining currents E i (z), F i (z), K ± i (z) with i = 1, . . . , N − 1 are Drinfeld currents for the "traceless part" U q ( sl N ) vert ⊂ U q ( gl N ) vert of the vertical subalgebra. Adding K ± 0 (z) gives an extra Heisenberg subalgebra. The role of the gradings d 1 and d 2 is reversed in the vertical subalgebra compared to the horizontal one.

A.1 Coalgebra structure
The algebra U q1,q2,q3 ( gl N ) can be given a coalgebra structure with the coproduct structure given by where C (1) As in the U q1,q2,q3 ( gl 1 ) case, there is an infinite number of equivalent but not identical coalgebra structures, parametrized by splittings of the algebra into pairs of Borel subalgebras. The coproducts corresponding to different coalgebra structures are related by Drinfeld twists. For definiteness we call the coproduct (59)-(63) vertical, since it acts (almost) diagonally on the vertical currents K ± i (z) (61), (62).

A.2 Symmetries of the algebra
The algebra U q1,q2,q3 ( gl N ) has a large group of symmetries including: 1. SL(2, Z) automorphisms. The generator S of this group is also known as the Miki's automorphism, which, in particular, transforms the horizontal quantum affine subalgebra into the vertical one. The central charges C 1 and C 2 and the gradings d 1 and d 2 are also exchanged by the automorphism S. The SL(2, Z) duality group is shared by all known quantum toroidal algebras including the simplest one, U q1,q2,q3 ( gl 1 ), because it originates from the mapping class group of the torus.
2. Reflection symmetry. This transformation, which we call ρ, exchanges the deformation parameters q 1 and q 3 , and reflects the A N indices of the generating currents: Evidently ρ 2 = 1, so the reflection symmetry generates S 2 . This symmetry is smaller, than the corresponding symmetry of the U q1,q2,q3 ( gl 1 ), in which case any the permutation of (q 1 , q 2 , q 3 ) was allowed (besides, no relabelling of the generators was needed).
3. The symmetries listed so far were already present in the U q1,q2,q3 ( gl 1 ) algebra. In the gl N case there is one more important automorphism σ, rotating the roots of A N −1 root system: Evidently, σ N = 1. The central charges C 1 and C 2 are invariant under the symmetry, and so is the d 1 grading. The d 2 grading is modified by the symmetry, but in a controlled way. Let δ i be the grading associated with root i: A.3 The limit of U q 1 ,q 2 ,q 3 ( gl N ) to W 1+∞ [gl N ] and the quantum torus In the limit q 2 → 1 the algebra U q1,q2,q3 ( gl N ) can be described very explicitly as the central extension of the algebra of matrix-valued functions on the quantum torus [13]. The quantum torus T 2 d N is a non-commutative space with coordinates 5 x and p satisfying with d = q 1 = q −1 3 playing the role of the parameter of the noncommutativity. The algebra of matrix-valued functions is generated by where e i,j is a basis in the space of N × N matrices: (e i,j ) k,l = δ i,k δ j,l i, j, k, l = 1, . . . , N.
After the central extension with two central charges c 1 and c 2 , the generators W The generating currents of U q1,q2,q3 ( gl N ) are given by the formulas A.4 Vector representations of U q 1 ,q 2 ,q 3 ( gl N ) Vector representation V q1 is a representation of U q1,q2,q3 ( gl N ) on the space of C N -valued functions of a single variable x. It has trivial central charges, C 1 = C 2 = 1. We can take as basis states of this representation tensor products of Dirac delta functions and basis vectors e j in C N . We write them as 6 |u, j = δ q j 1 x u e j , j = 1, . . . , N.
In Eq. (79) we view the spectral parameter u as the parameter of the state, not of the representations (the latter convention was adopted in [13], [15]). These two views are in fact completely equivalent. The 5 We always assume that the coordinates are complexified. 6 Our notation for the states of the vector representation is related to that of [13] by the formula so that algebra U q1,q2,q3 ( gl N ) acts on the states (79) as matrices valued in difference operators 7 in x: where and e i,j are basis matrices defined in Eq. (70). The eigenvalues of the currents K + (z) and K − (z) are given by the same rational function understood as a series expansion around z = 0 and z = ∞ respectively. Let us also write the zero modes K i explicitly: so that indeed C 2 = N −1 i=0 K i = 1. Due to the symmetry ρ of the algebra (Eq. (64)) there exists a representation V q3 defined similarly to Eq. (80)-(82) with q 1 ↔ q 3 and the labels of the generators reversed.
In [13] a set of N different vector representations V (k) q1 related to each other by the action of σ has been defined. Due to σ being the symmetry of the algebra, all the representations in the family are isomorphic. Here we make do with just one representation V q1 . We do so by noticing that σ can be represented on the space of vector-valued functions as a cyclic permutation matrix s: where In this way all N representations V A.5 Horizontal Fock representation of U q 1 ,q 2 ,q 3 ( gl N ) In the horizontal Fock representation representation [11,12] F (1,0),k q1,q3 (u) all currents of the U q1,q2,q3 ( gl N ) algebra are expressed through vertex operators built from the generators H i,n constituting N copies of the Heisenberg algebra (see Eq. (49)). The central charges of F (1,0),k q1,q3 (u) are C 1 = q = √ q 2 and C 2 = 1.
We define the fundamental vertex operators 7 Note that the action of the shift operator in x on a state |u, j can be understood as an inverse shift of the parameter u: Notice that V ± (z) contain only positive or negative modes respectively and are therefore automatically normal ordered. The product of V − (z)V + (w) is normal ordered too, but V + (z)V − (w) is not, and for it we have the following identity: where whereδ i,j gives one if i = j mod N and zero otherwise. Using V ± (z) we define the vertex operator representation of U q1,q2,q3 ( gl N ): F i (z) = ξ i (z) e −ᾱi z −Hi,0+1 e −δ i,k P , where P is the momentum operator giving ln u on the whole space F (1,0),k q1,q3 (u) (this definition seems superfluous when we are dealing with a single Fock representation, but will be convenient when we will study tensor products of Fock representations). The vertex operators η i (z), ξ i (z) and φ ± i (z) are given by: z Hi,0 eᾱ j = z ai,j eᾱ j z Hi,0 , [H i,0 , H j,0 ] = 0.
The relation (99) can be understood as a C × central extension of the group algebra of the A N −1 root lattice 9 . Notice that We also set Due to Eqs. (102), (103) we can understand H i,0 as measuring discrete "momenta" taking values in the weight lattice of A N −1 and eᾱ i as the generators shifting these momenta by simple roots of A N −1 .
A.6 Branching into U q 1 ,q 2 ,q 3 ( gl 1 ) algebras and toric geometry There is a geometric way of viewing the algebra U q1,q2,q3 ( gl N ), which can be helpful for understanding its representations and subalgebras. The algebra U q1,q2,q3 ( gl N ) is associated with a toric Calabi-Yau three-fold of the form X N × C, where X N = C 2 /Z N is the blowup of the A N singularity. There is a C 3 action on the toric CY; denote its weights by q 1 , q 2 , q 3 . We turn off the C × part of C 3 which scales the canonical bundle, so the weights satisfy q 1 q 2 q 3 = 1. We choose the weights so that q 2 scales the trivial C fiber, while the choice of q 1 is easier to shown on the toric diagram (Fig. 1). ( gl 1 ) subalgebra of U q1,q2,q3 ( gl N ).
Let us notice that for N = 1 the threefold is simply C 3 with weights q i rotating the coordinate planes C. It is therefore logical to assume that the neighbourhoods of the fixed points P a of the (C × ) 3 action inside X N × C might be related to U q (a)