New quantum toroidal algebras from 5D $\mathcal{N}=1$ instantons on orbifolds

Quantum toroidal algebras are obtained from quantum affine algebras by a further affinization, and, like the latter, can be used to construct integrable systems. These algebras also describe the symmetries of instanton partition functions for 5D $\mathcal{N}=1$ supersymmetric quiver gauge theories. We consider here the gauge theories defined on an orbifold $S^1\times\mathbb{C}^2/\mathbb{Z}_p$ where the action of $\mathbb{Z}_p$ is determined by two integer parameters $(\nu_1,\nu_2)$. The corresponding quantum toroidal algebra is introduced as a deformation of the quantum toroidal algebra of $\mathfrak{gl}(p)$. We show that it has the structure of a Hopf algebra, and present two representations, called vertical and horizontal, obtained by deforming respectively the Fock representation and Saito's vertex representations of the quantum toroidal algebra of $\mathfrak{gl}(p)$. We construct the vertex operator intertwining between these two types of representations. This object is identified with a $(\nu_1,\nu_2)$-deformation of the refined topological vertex, allowing us to reconstruct the Nekrasov partition function and the $qq$-characters of the quiver gauge theories.


Introduction
Non-perturbative dynamics of supersymmetric gauge theories is a prolific research subject in theoretical physics. Since the innovative work [1], direct microscopic studies on four-dimensional gauge theories with N = 2 supersymmetry became largely accessible, from exact computations of their partition functions on the (non-commutative) C 2 .
The divergence in the partition function coming from the non-compactness of C 2 is properly regularized by introducing the Ω-background [1,2], effectively localizing the four-dimensional theory to the origin. In turn, the path integral reduces to an equivariant integration on the finite dimensional framed moduli space of non-commutative instantons, for which equivariant localization can be applied for its exact evaluation. The Nekrasov partition function has been a powerful tool to investigate the correspondences of four-dimensional N = 2 supersymmetric quiver gauge theories with other objects in mathematical physics, i.e., quantum integrable systems [3,4,5], two-dimensional CFTs [6,7,8,9,10,11,12], flat connections on Riemann surfaces [13,14], and isomonodromic deformations of Fuchsian systems [15,16,17].
Very rich algebraic structures lie at the heart of these correspondences [18]. For instance, the AGT correspondence [6,7] between Nekrasov partition functions and conformal blocks of Liouville/Toda 2D CFTs can be understood algebraically as the action of W-algebras on the cohomology of instantons moduli space [19,20,21]. In this context, the W-algebra currents are coupled to an infinite Heisenberg algebra, and the total action is formulated in terms of a quantum algebra, namely the Spherical Hecke central algebra [21] (isomorphic to the affine Yangian of gl(1) [22,23]). The coupling to an Heisenberg algebra is essential for the definition of a coalgebraic structure, thus emphasizing the underlying quantum integrability since the coproduct provides the R-matrix satisfying the celebrated quantum Yang-Baxter equation.
A closely related but different connection with quantum algebras arises from the type IIB strings theory realization of the five-dimensional uplifts of 4D N = 2 gauge theories, that is the 5D N = 1 quiver gauge theories compactified on S 1 . In this construction, N = 1 gauge theories emerge as the low-energy description of the dynamics of 5-branes webs [24,25]. Here, each brane carries the charges (p, q), generalizing D5-branes (charge (1, 0)) and NS5-branes (charge (0, 1)). Their world-volume include the five-dimensional gauge theory spacetime, together with an extra line segment in the 56-plane. Individual branes' line segments are joined by trivalent vertices, and form the (p, q)-branes web. Alternatively, the (p, q)-brane web can be seen as the toric diagram of a Calabi-Yau threefold on which topological strings can be compactified [26]. The trivalent vertices are then identified with the (refined) topological vertex, thereby leading to a very efficient method of computing 5D Nekrasov partition functions as topological strings amplitudes [27,28].
Awata, Feigin and Shiraishi observed in [29] that a specific representation of the quantum toroidal gl(1) algebra (or Ding-Iohara-Miki algebra [30,31]) can be associated to each edge of the (p, q)-branes web. The charges (p, q) are identified with the values of the two central charges while the brane position define the weight of the representation. As such, the D5-branes correspond to the so-called vertical representation while an horizontal representation is associated to NS5-branes (possibly dressed by extra D5-branes). 1 The refined topological vertex is then identified with an intertwiner between vertical and horizontal representations, that is in fact the toroidal version of the vertex operator introduced in [34] for the quantum group U q ( sl(2)). In this way, the Nekrasov partition function is written as a purely algebraic object using the quantum toroidal algebra, just like conformal blocks with W-algebras [35,36]. This algebraic construction turns out to be useful in probing various properties of the partition function, e.g. in addressing the (q-deformed) AGT correspondence [37,38], or in studying strings' S-duality [39,40].
In [8], an important class of half-BPS observables, called qq-characters, were defined, whose characteristic property is the regularity of their gauge theory expectation values. This regularity property encodes efficiently an infinite set of constraints on the partition function called non-perturbative Dyson-Schwinger equations [8]. The algebraic nature of these constraints was observed in [41,42]. Actually, the constraints take an even more elegant form in the algebraic construction described above as they express the invariance of an operator T under the adjoint action of the quantum toroidal algebra [43]. This operator is obtained by gluing intertwiners along the edges of the (p, q)-branes web, and its vacuum expectation value reproduces the 5D Nekrasov instanton partition. 1 In fact, the vertical representation is simply the q-deformation of the affine Yangian action mentioned previously, it is expected to describe a quantum toroidal action on the K-equivariant cohomology of the quiver variety describing the instanton moduli space. The equivalent of the horizontal representation can also be defined for 4D N = 2 theories, thus extending the whole algebraic construction of the Nekrasov partition functions [32]. However, for this purpose, it is necessary to consider the central extension of the Drinfeld double of the affine Yangian following from the construction given in [33].
A natural question is how to generalize the algebraic construction to gauge theories on more complicated manifolds. Among other manifolds, the Z p -orbifolded C 2 are of a particular interest, since the partition functions on these spaces can be computed by simply projecting out the contributions which are not invariant under the Z p -action [44,45,10]. The generalization of the algebraic construction is not entirely straightforward since it is necessary to introduce the information of the coloring corresponding to the Z p -action of the orbifolding. In this scope, deformations of the quantum toroidal gl(1) algebra must be considered. A special case of the Z p -orbifolded C 2 is the (un-resolved) A p -type ALE spaces. The ALE instantons were introduced by Kronheimer in [46], and the ALE instanton moduli spaces were constructed as quiver varieties in [47,48]. The algebraic construction of the corresponding 5D Nekrasov partition functions has been realized recently using an underlying quantum toroidal gl(p) algebra [49]. There, the index carried by the Drinfeld currents renders the Z p -coloring due to the orbifolding. Incidentally, the vertical representation of this quantum toroidal algebra should coincide with the q-deformation of the affine Yangian of gl(p) acting on the cohomology of the moduli space of ALE instantons, extending by further affinization the algebraic actions discovered in [48,50].
In this work, we extend the algebraic construction of 5D Nekrasov partition functions to a more general Z porbifolding depending on two integer parameters (ν 1 , ν 2 ). We propose an extended quantum toroidal algebra relevant to the construction, and prove its Hopf algebra structure. We define both horizontal and vertical representations, and derive the vertex operator which intertwines between these representations. Finally, using these ingredients, we give an algebraic construction of Nekrasov partition functions and qq-characters. The orbifolds considered in this work incorporate the case of codimension-two defect insertion, whose applications to BPS/CFT correspondence, Bethe/gauge correspondence, and Nekrasov-Rosly-Shatashvili correspondence have been largely investigated [11,12,51,52,14].
This paper is written in such a way that mathematicians interested only in the formulation of the extended algebra can focus on the reading of section three, together with the appendices A (quantum toroidal gl(p)), C (representations) and D (automorphisms and gradings) for more details. Instead, the section two provides a brief description of the physical context in which the algebra emerges, i.e. instantons of 5D N = 1 gauge theories on the spacetime C 2 /Z p . Finally, the section four is dedicated to the algebraic construction of gauge theories observables, giving the expression of the (ν 1 , ν 2 )-colored refined topological vertex and a few examples of application.
2 Instantons on orbifolds 2.1 Action of the abelian group Z p on the ADHM data In order to derive the group action on the instanton moduli space, we focus first on the case of a pure U (m) gauge theory. In this case, the ADHM construction of the moduli space [53] involves only two vector spaces M and K of dimension m and k respectively, where k is the instanton number. Introducing the four matrices B 1 , B 2 : K → K I : M → K and J : K → M , the instanton moduli space is identified with the quiver variety (see, for instance, [54]) The complexified global symmetry group GL(M ) × SL(2, C) 2 acts on the ADHM matrices, preserving the quiver variety M k . It contains an (m + 2)-dimensional torus that acts follows, The fixed points of this action parameterize the configurations of instantons with total charge k. They are in oneto-one correspondence with the m-tuple partitions λ = (λ (1) , · · · λ (m) ) of the integer k, here identified with the m-tuple Young diagrams with |λ| = k boxes. At the fixed point, the vector space K is decomposed into where M α denotes the one-dimensional vector spaces generated by the basis vectors of M . Thus, each box = (α, i, j) of the m-tuple partition λ with coordinate (i, j) ∈ λ (α) corresponds to a one-dimensional vector space We further associate to the box the complex variable φ = a α + (i − 1)ε 1 + (j − 1)ε 2 called instanton position or, sometimes, the box content of . The parameters a 1 , · · · , a m are the Coulomb branch vevs of the gauge theory. We also define the exponentiated quantities v α = e Raα , (q 1 , q 2 ) = (e Rε 1 , e Rε 2 ) and χ = e Rφ = v α q i−1 1 q j−1 2 . In this paper, gauge theories are considered on the 5D orbifolded omega-background S 1 (4). The action of the group Z p on the spacetime is parameterized by two integers ν 1 , ν 2 , Furthermore, it is possible to combine it with a global gauge transformation in the subgroup U (1) m ⊂ U (m). As a result, we obtain an action of Z p on the ADHM data by specialization of the (m + 2)-torus action 2.2, taking This action of the abelian group Z p is parameterized by the m + 2 integers (c α , ν 1 , ν 2 ) considered modulo p. The transformation of the vector spaces in the decomposition 2.3 of K leads to associate to each box = (α, i, j) ∈ λ, in addition to the complex variables φ and χ , the integer c( ) such that We call color any integer parameter defined modulo p. For short, we also say that c α and ν 1 , ν 2 are respectively color of the Coulomb branch vevs, and of the parameters q 1 , q 2 . The map c : λ → Z p defines a coloring of the m-tuple partitions λ, and K has a natural decomposition into sectors of a given color c( ) = ω, Notations We denote C ω (m) the subset of [ [1, m]] such that the Coulomb branch vevs a α (or v α ) with α ∈ C ω (m) have color c α = ω (or, equivalently, that the box (1, 1) ∈ λ (α) with α ∈ C ω (m) is of color c(α, 1, 1) = c α = ω). Similarly, K ω (λ) denotes the set of boxes ∈ λ of the m-tuple colored partition λ that carry the color c( ) = ω. Besides, in the generic case ν 1 + ν 2 = 0, the shift of color indices ω by the quantity ν 1 + ν 2 appears in many formulas. To simplify these expressions, we introduce the notationω = ω + ν 1 + ν 2 for the shifted indices, along with the mapc( ) = c( ) + ν 1 + ν 2 . Finally, we also introduce the extra variables q 3 and ν 3 such that q 1 q 2 q 3 = 1 and ν 1 + ν 2 + ν 3 = 0. Due to the fact that the Z p -action coincides with a subgroup of the torus action, in all formulas the shift of color indices ω + ν i coincide with a factor q i multiplying the parameters associated to instanton positions in the moduli space.
McKay subgroups in SO(4) Although we are considering here a different problem, it is interesting to make a short parallel with the action of SU (2) L × SU (2) R ⊂ SO(4) on the omega-background (see, for instance, [55]). This action takes a simpler form if we employ the quaternionic coordinates Then the 2 × 2 matrices (G L , G R ) ∈ SU (2) L × SU (2) R act on the quaternions as Z → G L ZG R . The McKay subgroups of SU (2) possess an ADE-classification. For instance, the A p−1 series corresponds to the action of Z p , it is generated (multiplicatively) by the diagonal matrices Considering only the action of the A p−1 subgroup on the left, the background coordinates transform as (z 1 , z 2 ) → (e 2iπ/p z 1 , e −2iπ/p z 2 ). This transformation can be recovered from the action 2.4 of Z p by choosing ν 1 = −ν 2 = 1.
The orbifold of the spacetime under this action of Z p reproduces the ALE space constructed in [46]. Instantons of N = 1 gauge theories defined on ALE spacetimes have been extensively studied [46,47,48,50]. In [49], their contributions to the gauge theory partition functions have been reproduced using algebraic techniques based on the quantum toroidal algebra of gl(p). The generalization to DE-type McKay subgroups with only left action is expected to involve quantum toroidal algebras based on either so(p) or sp(p) Lie algebras [50]. It is also possible to consider simultaneously the action of two McKay subgroups A p 1 −1 and A p 2 −1 , with one acting on the left, the other on the right. As a result, coordinates now transform as (2.10) We recognize here another particular case of the Z p -action defined in 2.4, albeit more general than before. It is simply obtained by the specialization ν 1 = p 1 + p 2 , ν 2 = p 1 − p 2 and p = p 1 p 2 . Thus, the action 2.4 leads to a particularly rich context. Moreover, taking ν 1 = 0, the first coordinate z 1 is invariant and the orbifolded spacetime can be reinterpreted as the insertion of a codimension-two defect in a 5D omega background with no orbifold [11,44]. We build here a general algebraic framework to address this kind of problems. It may be possible to further generalize our approach to the action of DE-type McKay subgroups with both left and right actions, but this is beyond the scope of this paper.

Instantons partition function
The computation of the Nekrasov instanton partition function on such Z p -orbifolds has been performed in [44,45,10]. 2 For simplicity, we do not introduce fundamental matter multiplets, those being obtained in the limit q → 0 of the gauge coupling parameters. Furthermore, we only discuss the case of linear quiver gauge theories A r , with U (m (i) ) gauge groups at each node i = 1 · · · r. Thus, the node i carries the following parameters: • a set of colored exponentiated gauge couplings q ω,i , • a p-vector of colored Chern-Simons levels κ (i) = (κ • an m (i) -vector of Coulomb branch vevs a (i) = (a • an associated vector of colors c (i) = (c In addition, each link i → j between two nodes i and j represent a chiral multiplet of matter fields in the bifundamental representation of the gauge group U (m (i) ) × U (m (j) ), with mass µ ij ∈ C. For linear quivers, all bifundamental masses can be set to q −1 3 by a rescaling of the Coulomb branch vevs. The instantons contribution to the gauge theories partition function is expressed as a sum over the content of r m (i) -tuple Young diagrams λ (i) describing the configuration of instantons at the ith node. Each term can be further decomposed into the contributions of vector (gauge) multiplets, bifundamental chiral (matter) multiplets, and Chern-Simons factors: (2.11) Vector and bifundamental contributions are written in terms of the Nekrasov factor N (v, λ|µv ′ , λ ′ ). For a better readability, we drop the node indices in the following, and simply distinguish the two nodes involved in the definition of the Nekrasov factor with a prime. In order to write down the expression of N (v, λ|µv ′ , λ ′ ) given in [10], we need to introduce the equivariant character M v and K λ of the vector spaces M and K associated to each node, A linear involutive operation * acts on such characters by flipping the sign of R: (e Raα ) * = e −Raα , (q * 1 , q * 2 ) = (q −1 1 , q −1 2 ) and thus (e Rφ ) * = e −Rφ (see [8,9,10] for more details on these notations). Introducing S λ = M − P 12 K λ with P 12 = (1 − q 1 )(1 − q 2 ), the Nekrasov factor writes where the I-symbol is the equivariant index functor, and [· · · ] Zp denotes the operation of keeping only the Z p -invariant parts. In particular, the RHS of (2.13) involves a coloring function c : Z[a α , ε 1 , ε 2 ] → Z p defined on weights w i as the linear map taking the values c(a α ) = c α , c(ε 1 ) = ν 1 and c(ε 2 ) = ν 2 so that c(φ ) = c( ) (justifying our slight abuse of notations). The [· · · ] Zp projects on Z p -invariant factors. Replacing the equivariant characters by their expressions 2.12, the Nekrasov factor can be written in a more explicit form, The function S ωω ′ (z) is sometimes called the scattering function, it carries two color indices ω, ω ′ : In this expression, the non-zero matrix elements have been expressed in a compact way using the delta function δ ω,ω ′ defined modulo p (i.e. δ ω,ω ′ = 1 iff ω = ω ′ modulo p, zero otherwise). In fact, S ωω ′ (z), and more generally all the matrices of size p × p with indices ω, ω ′ appearing in this paper, are circulant matrices: their matrix elements only depend on the difference ω − ω ′ of row and column indices. In particular, S ω+ν ω ′ (z) = S ω ω ′ −ν (z) for all ν ∈ Z p . Finally, the function S ωω ′ (z) satisfies a sort of crossing symmetry, with the function f ωω ′ (z) = F ωω ′ z β ωω ′ defined by 3

Y-observables
A new class of BPS-observables for supersymmetric gauge theories was introduced in [8], they are called qqcharacters. As the name suggests, they correspond to a natural deformation of the q-characters of Frenkel-Reshetikhin [57] from the gauge theory point of view [18]. They were defined in [8] as particular combinations of chiral ring observables in such a way that their expectation values exhibit an important regularity property [8,9]. This regularity property encodes an infinite set of constraints called non-perturbative Dyson-Schwinger equations. From a different viewpoint, qq-characters in 5D gauge theories can also be defined in terms of Wilson loops [58] (see also [59] for a string theory perspective). 4 The qq-characters are half-BPS observables written as combinations of Y-observables. In the case of a Z porbifold, it is natural to introduce two inequivalent Y-observables Y Replacing the equivariant characters with the expressions 2.12, we find the explicit formulas Due to the crossing symmetry 2.17, these two Y-observables satisfy the relation It will allow us to express all the equations below in terms of Y ω (z) only. 5 Furthermore, the Y-observables possess an alternative expression following from the shell formula derived in appendix B, (2.23) 3 The function f ωω ′ (z) also controls the asymptotics of the scattering function since and βω′ ω = β ωω ′ . 4 In [11], the qq-characters of 4D N = 2 gauge theories with the insertion of surface defects were considered. In this case, the nonperturbative Dyson-Schwinger equations produce either Knizhnik -Zamolodchikov equations or BPZ equations that are satisfied by the surface defect partition functions [12,52]. These surface defect partition functions were investigated in the context of Bethe/gauge correspondence in [51], and in their relation to the oper submanifold of the moduli space of flat connections on Riemann surfaces in [14]. 5 The presence of the function f ω (z) can be interpreted as follows. Note that I(X * ) = (−1) rkX * det X * I(X), for X = i∈I + e Rw i − i∈I − e Rw i , rkX = |I+| − |I−|, and det X = i∈I + e Rw i / i∈I − e Rw i . Applying this reflection relation to X = e −Rζ S λ Zp , we recover the relation (2.21) with f Here, the sets A ω (λ) and R ω (λ) denote respectively the set of boxes of color ω that can be added to or removed from the m-tuple Young diagram λ. This expression arises from the cancellations of contributions by neighboring boxes, it plays an essential role in the definition of the vertical representation of the algebra.

New quantum toroidal algebras
In order to reconstruct the instanton partition functions on the general orbifold 2.4, the definition of a new quantum toroidal algebra is necessary. In addition to the complex parameters q 1 , q 2 and the rank p ∈ Z >0 , this algebra will depend on the integers (ν 1 , ν 2 ) modulo Z p . Taking ν 1 = −ν 2 = 1, the Z p -action 2.4 reduces to the standard action defining ALE spaces. Thus, in this limit the (ν 1 , ν 2 )-deformed algebra should reduce to the quantum toroidal algebra of gl(p). In fact, this is true only up to a twist in the definition of the Drinfeld currents (see the subsection A.4 of the appendix). A brief reminder on the quantum toroidal algebra of gl(p) is given in appendix A, it includes its two main representations called, in the gauge theory context, vertical and horizontal representations.
The key ingredient to define the deformation of the quantum toroidal algebra of gl(p) is the scattering function S ωω ′ (z) defined in 2.16. Indeed, this function plays an essential role in the two elementary representations involved in the algebraic engineering of partition functions and qq-characters. In the vertical representation, it enters through the definition 2.20 of the Y-observables that describe the recursion relations among Nekrasov factors. Instead, in the horizontal representation, it expresses the normal-ordering relations between vertex operators. Thus, from the physics perspective, the scattering function is the natural object to consider for the deformation of the algebra. Moreover, through the crossing symmetry relation 2.17, this function defines the p × p matrix β ωω ′ that could be identified with the underlying Cartan matrix of the deformed quantum toroidal algebra (see the subsection A.4). Note that the matrix β ωω ′ naturally reduces to the generalized Cartan matrix of the Kac-Moody algebra gl p when ) are written in terms of the pth root of unity Ω j = e 2iπj/p , and the corresponding eigenvalues read λ j = −4e iπν 3 j/p sin(πν 1 j/p) sin(πν 2 j/p). (3.1) In particular, the eigenvector v 0 = (1, 1, · · · , 1) has the eigenvalue zero which relates β ωω ′ to the Cartan matrix of affine Lie algebras, and thus justifies the designation toroidal of the deformed algebra.

Definition of the algebra
Like in the case of gl(p), the (ν 1 , ν 2 )-deformed quantum toroidal algebra is defined in terms of a central element c and 4p Drinfeld currents, denoted x ± ω (z) and ψ ± ω (z), with ω ∈ Z p . The currents ψ ± ω (z) (together with c) generate the Cartan subalgebra, while the currents x ± ω (z) deform the notion of Chevalley generators e ω , f ω . The algebraic relations obeyed by the currents resemble those defining the quantum toroidal algebra of gl(p) in A.3, the main difference being the presence of shifts in the indices ω by the product ν 3 c: 6 (3.2) 6 Comparing with the standard definition of quantum toroidal algebras, the Drinfeld currents have been redefined as follows: . This redefinition makes the coincidence between shifts of indices ω ± ν3c and spectral parameters zq ±c 3 manifest. In fact, this asymmetric form of the algebraic relations appears naturally in the construction of a central extension of the Yangian double [33].
In the last relation δ(z) = k∈Z z k denotes the multiplicative Dirac delta function and we introduced the complex parameter The other relations in 3.2 involve the structure function g ωω ′ (z) defined as a ratio of two scattering functions. This function depends on the variables (q 1 , q 2 ) ∈ C × × C × and the integers (ν 1 , ν 2 ) ∈ Z p × Z p : where the extra variables q 3 and ν 3 obey q 1 q 2 q 3 = 1 and ν 1 + ν 2 + ν 3 = 0. Note that the invariance under the S 3 -permutation of indices (ν i , q i ) is broken to S 2 corresponding to exchange (ν 1 , q 1 ) and (ν 2 , q 2 ). The structure function satisfies the property g ωω ′ (z)g ω ′ ω (z −1 ) = 1 necessary for the definiteness of the algebraic relations. The algebraic relations 3.2 are expected to include additional Serre relations. However, the Drinfeld currents employed here are a twisted version of those used in the formulation of the quantum toroidal algebra of gl(p). This explains why the function g ωω ′ (z) defined in 3.4 does not quite reproduce the gl(p) structure function A.6 as we set ν 1 = −ν 2 = 1. Even in the case of gl(p), the twist of the currents make the derivation of Serre relations difficult. We hope to come back to this question in the near future.
Due to the non-trivial power of z in the asymptotics of the functions g ωω ′ (z), namely the Cartan currents ψ ± ω (z) cannot be expanded in powers of z ∓k with k > 0 as it is usually the case for quantum groups. Instead, it is necessary to introduce a zero modes part using extra operators a ± ω,0 : In the appendix D, the operators ψ ω,0 z ∓a ± ω,0 are constructed as a specific combination of grading operators. The Cartan zero modes ψ ± ω,0 are invertible, they can be used to define another central elementc setting Note that the ordering of the zero modes is important since they do not commute. It is chosen here such that the expression of the coproduct defined below simplifies.
Coalgebraic structure A Hopf algebra A over the field C is a C-module equipped with a unit 1 A , a product ∇, a counit ε, a coproduct ∆ and an antipode S satisfying the following properties [60].
• The counit ε : A → C and the coproduct ∆ : A → A ⊗ A are homomorphisms of algebras. The compatibility with the scalar multiplication and the addition are trivially satisfied. On the other hand, the compatibility with the product requires to verify ε(ee ′ ) = ε(e)ε(e ′ ) and ∆(e)∆(e ′ ) = ∆(ee ′ ) for any two elements e, e ′ ∈ A.
• The antipode S : The algebra 3.2 is a Hopf algebra with the coproduct, counit and antipode given by The proof is a tedious but straightforward calculation that the axioms defining a Hopf algebra hold for any pair of currents. The antipode is an anti-homomorphism of algebra, it satisfies S 2 = (−1) 1+ε Id. Using the coproduct of the Cartan zero modes ψ ± ω,0 , it is possible to compute the coproduct of the central chargec defined in 3.7, we find 7 In order to reconstruct the instanton partition functions, we need to introduce two types of representations: a vertical representation ρ (V ) with level c = 0 and a horizontal representation ρ (H) with level c = 1. Such representations are already known in the case of quantum toroidal algebras of gl(p) (see [61,62], or the brief summary presented in appendix A), but also for the quantum toroidal gl(1) algebra (or Ding-Iohara-Miki algebra [30,31]) [63,64]. In fact, there are two different point of view concerning these representations. In the mathematics literature [63,64,61], one often considers a single module, the Fock module, and present the action of two subalgebras called horizontal and vertical. Miki's automorphism S [65,31] exchanges the two subalgebras, allowing us to define (for instance) ρ (H) = ρ (V ) • S. On the opposite, physicists usually introduce two different types of modules referred as vertical and horizontal modules, somehow fixing the choice of subalgebra. Of course, the modules are isomorphic thanks to Miki's automorphism and the two point of views are equivalent [40]. However, no analogue of Miki's automorphism is known yet for the (ν 1 , ν 2 )-deformed algebra. Thus, at this stage, we have no choice but to follow the second approach and define two distinct representations. This will be done in the next two subsections.

Vertical representation
The vertical representation presented here is a deformation of the Fock representation for the quantum toroidal algebra of gl(p) [61] (see appendix A.3). This representation is similar to the usual finite dimensional representations of quantum groups. Indeed, the Cartan currents ψ ± ω (z) are diagonal on a set of weight vectors. The currents x − ω (z) annihilates the highest weight (or vacuum) |∅ , and x + ω (z) creates excitations. However, the weight vectors are labeled here by the box configurations of an m-tuple Young diagrams λ. Thus, this representation is infinite dimensional, yet it is graded by the total number of boxes |λ|.
From the gauge theory perspective, the vertical representation of the algebra 3.2 describes the relation between sectors of different instanton numbers. Thus, vertical modules are characterized by a basis of states |λ labeled by 7 The extra factor in the RHS comes from the shifts of the currents' arguments in the coproduct that brings instanton configurations. Accordingly, the representation depend on a set of m (highest) weights v = (v α ) α=1···m and a choice of color c α for each weight. This coloring defines the integers m ω = |C ω (m)| corresponding to the number of weights v α of color ω. The integers m ω provide the levels of the vertical representation: ρ (V ) (c) = 0 and ρ (V ) (c) = m with m = ω∈Zp m ω . As mentioned previously, the Cartan currents ψ ± ω (z) are diagonal on the basis |λ . On the other hand, the operators x ± ω (z) relate the sectors of instanton charge |λ| and |λ|±1 by adding/removing a box to the m-tuple Young diagram λ. Their action encodes the recursion relation 2.19 obeyed by Nekrasov factors [66]. The action of the Drinfeld currents on the states |λ is derived in appendix C.1, it reads 8 (3.12) In the first two lines, A ω (λ) and R ω (λ) correspond respectively to the set of boxes of color ω that can be added to or removed from λ. In the last line, the subscript ± denotes the expansion of the function Ψ ω (z) for |z| ±1 → ∞. This function is written as a ratio of the Y-observables defined in 2.20, We notice that the highest weights are still encoded in the form of a Drinfeld polynomial p ω (z): (3.14) When ν 3 = 0, we haveω = ω and the prefactor reduces to the usual expression q mω As a result, the action of the zero-modes of the Cartan currents read The value of the second central charge is obtained by taking the product over ω, we recover ρ (V ) (c) = m. 8 The definition of the vertical representation is not unique due, for instance, to the following invariance of Drinfeld currents at c = 0: Here a particular choice is made to simplify the derivation of intertwiners in section 4 below.

Contragredient representation
The definition of intertwiners in the next section requires the introduction of the dual basis λ|. The algebra 3.2 acts on the dual basis with the contragredient representation ρ (V ) * , defined such that for any element e of the algebra. Thus, the action of the contragredient representation depends on the choice of a scalar product for the vertical states. It turns out that the analysis of intertwining relations simplifies for a particular choice of scalar product for which states are orthogonal but not orthonormal, The norms a λ (v) −1 are chosen so that the contragredient representation of x ± ω (z) acts on λ| in the same way as the original representation ρ (V ) (x ∓ ω (z)) acts on |λ (note that x ± ω becomes x ∓ ω ). As a result, the norms have to obey the two following recursion relations for a box x of color c( ) = ω: The solution is expressed in terms of the vector contribution Z vect. (v, λ) defined in 2.11, (3.20)

Horizontal representation
The horizontal representation of the algebra 3.2 is the equivalent of the vertex representations constructed by Saito in [62] for quantum toroidal algebras of gl(p). It has level ρ (H) (c) = 1 and depends on p weights u ω ∈ C × and p integers n ω ∈ Z. In this representation, Drinfeld currents are constructed as a direct product of two (commuting) algebras. The first algebra is called here the zero modes factor, it is defined in terms the two operators Q ω (z), P ω (z) satisfying the exchange relation In appendix C.2, these operators are constructed explicitly in terms of 2p Heisenberg algebras. As a result, the operator P ω (z) acts on the vacuum state |∅ as P ω (z) |∅ = |∅ , and Q ω (z) acts on the dual vacuum ∅| as ∅| Q ω (z) = ∅|. Accordingly, we define the normal ordering of these operators by writing the Q ω (z)-dependence on the left. The second algebra involved in the horizontal representation is defined upon the modes α ω,k of p coupled free bosons (ω ∈ Z p and k ∈ Z × ) satisfying the commutation relations, 9 The RHS of these commutation relations involves the coefficients σ appearing in the expansion of the scattering function 2.16, As usual, the vacuum state |∅ is annihilated by the positive modes (k > 0), while negative modes create excitations. The dual state ∅| is annihilated by negative modes. Thus, these modes are normal ordered by moving the positive modes to the right. The representation of the Drinfeld currents x ± ω and ψ ± ω is given in terms of the vertex operators Combining the zero modes and vertex operators, the horizontal representation writes It is shown in appendix C.2 that the expressions in the RHS obey the algebraic relations 3.2 at the levels ρ (H) (c) = 1 and ρ (H) (c) = n + p if ν 1 + ν 2 < p and ρ (H) (c) = n otherwise, where n = ω∈Zp n ω . Note that even in the ALE case ν 1 = −ν 2 = 1, the horizontal representation given here is slightly more general than the one proposed in [67]. Indeed, in the latter the Z p -symmetry is broken by a choice of color ω 0 , setting u ω = uδ ω,ω 0 and n ω = nδ ω,ω 0 . Instead, in our construction of the gauge theory partition functions, it is necessary to keep u ω and n ω arbitrary in order to be able to assign a different gauge coupling q ω and Chern-Simons level κ ω for each color ω.

Algebraic engineering
The algebraic engineering of 5D N = 1 quiver gauge theories on C ε 1 × C ε 2 × S 1 R (without orbifold) follows from their correspondence with topological string theories in which the Nekrasov instanton partition function is obtained as a topological strings amplitude [27]. Indeed, these amplitudes are computed using the (refined) topological vertex [28,68,69] that was identified in [38] with an intertwiner between certain modules of the Ding-Iohara-Miki algebra [30,31], also known as the quantum toroidal algebra of gl(1). This intertwiner is in fact the toroidal analogue of the vertex operators introduced in [34] to compute the form factors of the XXZ Heisenberg spin chain. As result, the powerful topological strings computational methods for supersymmetric gauge theories can be reformulated in the language of quantum integrability.
The correspondence between 5D N = 1 gauge theories and quantum toroidal algebras is better formulated using the (p, q)-brane realization of the gauge theories in type IIB string theory [24,25]. In this realization, quiver gauge theories are reproduced by the low energy dynamics of a network of 5-branes with charges (p, q). These branes generalize both NS5-branes (0, 1) and D5-branes (1, 0). They wrap the 5-dimensional spacetime, and define a line segment in the 56-plane of the ten dimensional strings spacetime. These segments meet at trivalent vertices and form a web called the (p, q)-branes web. For instance, in the case of linear quivers, a set of m-D5 branes is associated to each node bearing a U (m) gauge group. These D5-branes are suspended between dressed NS5-branes (i.e. branes of charge (n, 1)). In this context, the relevant quantum toroidal algebra is determined by the spacetime of the gauge theory. Then, each brane of the (p, q)-branes web is associated to a representation of the algebra, identifying the levels with the charges ρ(c) = q, ρ(c) = p and the weights with the (exponentiated) position of the branes [49,36,35]. Thus, to a D5-brane corresponds a vertical representation with m = 1, while horizontal representations are associated to dressed NS-branes of charge (n, 1). It was further noticed in [43] that the set of m D5-branes of a single node (with a U (m) gauge group) can be directly described by a vertical representation with ρ (V ) (c) = m. Following the identification of the (p, q)-branes web with the toric diagram of the Calabi-Yau in topological strings [26], the trivalent junctions of branes coincide with the vertex operator of the algebra acting on the modules determined by the branes charge. Finally, the automorphisms of the algebra renders the various geometrical operations (translations, rotations) applied to the branes web [40].
For each (p, q)-branes web it is possible to write down an operator T constructed by 'gluing' the vertex operators of nodes connected by an edge. The gluing procedure is done by a product of operators in horizontal representations (NS5), and a scalar product in vertical ones (D5). The T -operator obtained in this way acts on the tensor product of representations corresponding to the external branes of the web (i.e. the semi-infinite line segments). These representations are in fact horizontal modules, and the vacuum expectation value of the T -operator reproduces the instantons partition function. The qq-characters are further obtained by introducing algebra elements (in the proper representation) within the vacuum expectation value [43]. We will give several examples below.
This algebraic construction of gauge theories BPS-observables has been generalized in a several directions: Dtype quivers [70], 6D spacetime and elliptic algebras [71], 4D N = 2 gauge theories and the affine Yangian of gl(1) [32], 5D N = 1 gauge theories on ALE spaces [49], and 3D N = 2 * gauge theories [67]. In this section, we present yet another generalization corresponding to deformed ALE spaces with the Z p -action described in section two. However, we do not wish to reproduce the whole construction here as it is a straightforward application of the methods developed earlier [49,36,35,43]. Instead, we will only provide the main ingredient, namely the expression of the vertex operator, and a few selected examples to illustrate the construction.

Vertex operators
We consider two types of vertex operators, denoted Φ and Φ * , and obtained, up to a normalization factor, by solving the following equations where e is any of the currents x ± ω (z), ψ ± ω (z) or the central charge c. 10 Here ∆ ′ denotes the opposite coproduct obtained by permutation ∆ ′ = P∆P. In order to distinguish the two horizontal representations, we denoted them ρ (H) and ρ (H ′ ) , they depend on the parameters u ω , n ω and u ′ ω , n ′ ω respectively. Thus, the vertex operator Φ (and also Φ * ) depend on the set of weights u ω , u ′ ω , v ω and integers n ω , n ′ ω , m ω . A solution to the equations 4.1 is found only if these parameters satisfy the two constraints The first relation expresses a constraint among the position of the branes in the 56-planes. The second equation is the charge conservation at the vertex. Due to the spacetime orbifold, the branes charges p in (p, q) degenerates into charges p ω with ω ∈ Z p identified with the integers n ω and mω of horizontal/vertical representations. 11 Summing over ω, these constraints reproduce the conservation of the levels n ′ = n + m that follows from the application of the intertwining relations 4.1 to the element e =c with the coproduct 3.10. Due to the presence of an algebra automorphism exchanging c andc in the gl(p)-case [65], we expect a similar degeneration of the charge q into q ω . It is not observed here because only a single charge q = 1 flow through the topological vertex. By definition, the vertex operator Φ * is a vector in the vertical module while Φ is a dual vector, Each vertical component Φ λ (or Φ * λ ) is a Fock vertex operator acting on the horizontal module,

(4.4)
A sketch of the derivation can be found in the appendix E, together with the (rather lengthy) expressions of the vacuum components Φ ∅ and Φ * ∅ . The vertex operators Φ and Φ * given here are a generalization of the colored refined topological vertex derived in [49,72] with extra parameters (ν 1 , ν 2 ).
The vertical components 4.4 of the vertex operators obey important normal ordering relations, from which we recover the vector and bifundamental contributions to the partition functions [43], 12 The expression of the one-loop factors G(v|v ′ ) can be found in appendix E, formula E.11. Note also that, following the method presented in [35,49], it is a priori possible to show that Φ λ and Φ * λ are solutions of the double deformed Knizhnik -Zamolodchikov (or (q, t)-KZ) equations.

Partition functions and qq-characters
The simplest example of algebraic engineering is given by the pure U (m) gauge theory with quiver A 1 . In this case, the (p, q)-brane web can be described roughly as a set of m D5-branes suspended between two (dressed) NS5branes. The corresponding T -operator is obtained as a product of vertex operators Φ and Φ * in the vertical channel [43], it acts on the tensor product of two horizontal modules, In order to distinguish the horizontal modules, we added the subscript * to the ones on which Φ * act. Accordingly, we denote the parameters of these representations (n * ω , u * ω ) and (n * ′ ω , u * ′ ω ). Evaluating the vacuum expectation value of this operator, we recover the instanton partition function of the underlying gauge theory: (4.8) 12 To simplify the notations, we have omitted the dependence of the Nekrasov factors in the vectors of colors c = (cα) m α=1 and c ′ . The shortcut notation q −1 should be understood as a shift of the weights q −1 3 v ′ α together with the corresponding shift of indices c ′ α − ν3 =c ′ α . Thus, we have the important relation where we have identified the colored gauge coupling q ω and Chern-Simons level κ ω with By construction, the operator T [U (m)] commutes with the action of the algebra defined by the opposite coproduct ∆ ′ [43], namely, (4.10) For this reason, T [U (m)] plays the role of the screening operator in [35]. The gauge theory expectation value of the fundamental qq-characters is obtained by insertion of ∆ ′ (x − ω+ν 3 (q 3 z)) in the horizontal vacuum expectation value, where the gauge averaging of a chiral ring observable O [λ] is performed over the instanton configurations weighted by the vector (and Chern-Simons) contributions to the partition function, and the qq-character writes .

(4.13)
Note that the first term involve the Y-observable Y ω (z). As shown in [43], it follows from the commutation relations (4.10) that the quantity X . This is in fact due to the radial ordering of operators in the horizontal Fock spaces. Indeed, when x − ω (z) is inserted on the left of T , the correlator as a well-defined expansion around z = ∞. On the other hand, when x − ω (z) in inserted on the right, the expansion around z = 0 is now well-defined. The non-trivial equality between the two expansions (4.10) implies that both series are finite, and thus that the correlator is a finite Laurent series in z. Asymptotically, the Y-observables behave as Y ω becomes independent of λ, β [λ] ω = m ω . As a result, the gauge average of the qq-character X ω (z) is a polynomial of degree m ω when |κ ω | < m ω . Unfortunately, when ν 3 = 0 not much can be said.
Another fundamental qq-character can be obtained using the generator x + ω (z) instead, . (4.14) The presence of two different fundamental qq-characters is a specificity of 5D N = 1 gauge theories on orbifolds: when p = 1, the two qq-characters are equivalent (they only differ by multiplication of a constant times a power of z). Further, as we shall see below, in the 4D limit R → 0, the qq-characters X ω (z) and X [λ] * ω (z) reduce to the same expression. The gauge averages (4.11) and (4.14) for the qq-characters have been computed at the first few orders in the gauge couplings q ω for the gauge groups U (1) and U (2) and various orbifold parameters. In all cases, it has been observed that these quantities are indeed finite Laurent series in the argument z. Finally, it is worth mentioning that higher qq-characters can be obtained by multiple insertions of the coproducts ∆ ′ (x ± ω (z)). We refer to [43] for more details on the computation of qq-characters.
4D limit When the radius R of the background circle S 1 R is sent to zero, the gauge theory reduces to a 4D N = 2 gauge theory. This limit can be performed directly on the partition functions and qq-characters, re-introducing the radius dependences in the parameters (q 1 , q 2 ) = (e Rε 1 , e Rε 2 ), v α = e Raα , χ = e Rφ ,... Sending R → 0 in the expression 2.11 of the instanton partition function, we observe that the Chern-Simons contribution is subdominant while, after setting the spectral variable to z = e Rζ , the scattering function 2.16 becomes This function satisfies a simpler crossing symmetry S ωω ′ = (−1) β ωω ′ in now independent of the spectral variable ζ. As a result, the function f ω (z) reduces to a sign. When ν 3 = 0, this sign is simply (−1) mω , it can be absorbed in the definition of q ω . In this way, both X ω (z) and X [λ] * ω (z) reproduce the expression of the 4D fundamental qq-character given in [10,11].
A 2 quiver Linear quiver gauge theories can be treated along the same lines. For instance, the A 2 quiver gauge theory with gauge group U (m 1 ) × U (m 2 ) is obtained by considering two sets of m 1 and m 2 D5-branes suspended between three dressed NS5-branes. The T -operator is simply the product of the single nodes operators T [U (m 1 )] and T [U (m 2 )] in a common horizontal representation, The vacuum expectation value is computed using the normal ordering relation 4.6 for the product Φ It reproduces the instanton partition function 2.11 for the A 2 quiver gauge theory, with the identification 4.9 of the parameters at each node i = 1, 2. The qq-characters can also be constructed along the lines of [43].

Concluding remarks
In this paper, we have reconstructed algebraically the instanton partition functions for N = 1 linear quiver gauge theories with unitary gauge groups on the five dimensional background S 1 R × (C ε 1 × C ε 2 )/Z p . The action of the abelian group considered here is a generalization by two integers (ν 1 , ν 2 ) of the standard action defining ALE spaces. These extra parameters led us to introduce a deformation of the quantum toroidal algebra of gl(p). This new quantum toroidal algebra appears to be defined upon a non-symmetrizable Cartan matrix β ωω ′ . Yet, we have shown that it still possesses the structure of a Hopf algebra with the deformed Drinfeld coproduct given in 3.8. We have also presented two different representations, called vertical and horizontal, that are respectively the deformation of the Fock module [61] and the vertex representation [62] of the quantum toroidal algebra of gl(p). Other types of representations should exist, like the Macmahon representation obtained for gl(p) as a tensor products of Fock modules in [61]. Although the definition of this new algebra may appear intricate, the physical context in which it emerges is very natural, and its representations are simple generalizations of the usual ones.
Quantum toroidal algebras extend the definition of quantum affine algebras (or quantum groups) by an extra affinization. In fact, the quantum toroidal algebra of gl(p) is generated by two orthogonal quantum affine subalgebra U q ( sl(p)) [73,74]. Then, one may wonder if the (ν 1 , ν 2 )-deformed algebra possesses a similar property. Of course, it is assuming that a quantum affine algebra built upon the Cartan matrix β ωω ′ can be defined properly. In fact, we expect that this is indeed the case, and that such quantum affine algebra retains a quasitriangular Hopf algebra structure, making it suitable for the construction of new quantum integrable systems.
On the gauge theory side, several generalizations of our approach could be implemented. For instance, the abelian group Z p could be replaced by a Mckay subgroup of SU (2) of type DE, with either left, right, or both leftright action. As shown by Nakajima in [48,50], in the first two cases a quantum affine algebra of type so/sp acts on the cohomology of the instanton moduli space. This action is expected to be lifted to a quantum toroidal algebra in K-theory. Accordingly, the algebraic engineering should involve the quantum toroidal so/sp algebras. However, the effective construction requires some new developments in the representation theory of these algebras.
When ν 2 = 0, the orbifold can be interpreted as the presence of a surface defect [11]. In this case, the Cartan matrix β ωω ′ appears to vanish but the algebra remains non-trivial, When ν 1 = 1, the structure function g ωω ′ (z) reproduces the one that defines the quantum toroidal algebra of gl(p) with q 2 and q 3 exchanged (up to a factor q m ωω ′ /2 3 ). However, the function S ωω ′ (z) is different from the one appearing in A.18, and thus horizontal and vertical representations of the (ν 1 , ν 2 )-deformed algebra degenerate into new representations for the quantum toroidal algebra of gl(p). We hope to come back to the study of this problem in a future publication.
Finally, an important question was left behind in our study, namely the correspondence with (q-deformed) Walgebras. This type of correspondences is now well-understood in the case of quantum toroidal gl(1). There, the q-W-algebras appearing in horizontal or vertical representations play different roles. In the horizontal case, a representation of level c = m can be built by tensoring m level one representations. It is thus expressed in terms of m sets of bosonic modes that are coupled through their commutation relations. Diagonalizing these relations, the Drinfeld currents can be expressed in terms of q-W m currents coupled to an infinite Heisenberg algebra. This dual q-W-algebra corresponds to the quiver W-algebra of Kimura and Pestun [75]. Using Miki's automorphism [65,31], vertical representations of levelc = m can be mapped on horizontal ones, and thus expressed in terms of q-W m currents coupled to the Heisenberg algebra. In the vertical case, the dual W-algebra is responsible for the AGT-like correspondence with q-deformed conformal blocks [37]. Alternatively, the AGT correspondence can also be seen directly in the degenerate limit R → 0 in which the vertical representation of the toroidal algebra reduces to a representation of the affine Yangian of gl(1) that is known to contain the action of W m -currents [21,22,23].
A similar type of duality is believed to hold between the degenerate limit of the quantum toroidal algebra of gl(p) and the coset 13 gl(α) m / gl(α − p) m , leading to an AGT correspondence between instantons on ALE spaces and parafermionic conformal field theories [76,77,78,56,79]. This conjecture has been verified for small values of p and m by comparing the conformal blocks of the coset theory with the gauge theories instanton partition functions [76,80,81,82,83,84], or the limit R → 0 of 5D topological strings amplitudes [72]. There are two main strategies to extend this duality to the (ν 1 , ν 2 )-deformed algebra. One possibility is again to compare instanton partition functions with conformal blocks. This approach was taken in [45] where the gauge theory calculations led to conjectural expressions for these conformal blocks. But, unfortunately, the corresponding conformal field theory appears to be unknown. Another possible approach consists in identifying directly the (q-deformed) coset algebra generators acting on the vertical modules of the quantum toroidal algebra. For this purpose, one could diagonalize the commutation relations for the modes α ω,k in the horizontal representations, and then define the analogue of Miki's automorphism to map the horizontal representations to the vertical ones. From the strings theory perspective, the latter is expected to exist since it should describe the fiber-base duality of the topological strings (or, the S-duality in Type IIB string theory) [39,40]. This approach appears very promising and we hope to be able to report soon on this problem.

Acknowledgments
JEB would like to thank Omar Foda for discussions and generous support during his visit of the university of Melbourne. SJ is greatly indebted to Nikita Nekrasov for numerous discussions and supports. SJ is also grateful to Korea Institute for Advanced Study for providing support during his visit. The work of SJ was supported in part by the NSF grant PHY 1404446 and also by the generous support of the Simons Center for Geometry and Physics.

A Quantum toroidal algebra of gl(p)
In this appendix, we remind the definition of the quantum toroidal algebra of gl(p), give its vertical and horizontal representations, and comment on the reduction ν 1 = −ν 2 = 1 of the algebraic relations 3.2.

A.1 Definition
Quantum toroidal algebras were introduced by V. Ginzburg and M. Kapranov and E. Vasserot in [73]. In general, they can be built over an affine Kac-Moody algebraĝ, but we will focus in this appendix on the case of an algebra of type A (1) p−1 , also called quantum toroidal g = gl(p) algebra. This algebra is formulated in terms of the Drinfeld currents Like the Chevalley generators, the operators x ± ω (z) are associated to the simple roots α ω ofĝ. On the other hand, the operators ψ ± ω (z) describe the Cartan sector of the algebra, they are naturally associated to the coroots α ∨ ω . We denote the Cartan matrix β ωω ′ = α ∨ ω , α ω ′ , in the case of gl(p), we have β ωω ′ = 2δ ω,ω ′ − δ ω,ω ′ +1 − δ ω,ω ′ −1 (here δ ω,ω ′ denotes the Kronecker delta with indices taken modulo p). In this case, the original relations can be deformed by an extra central parameter κ, using the antisymmetric matrix m ωω ′ = δ ω,ω ′ −1 − δ ω,ω ′ +1 [62]: 14 14 The generators of this algebra are sometimes denoted More rigorously the x ± − x ± exchange relation should be written This subtlety only affects the colliding points z = q ±β ωω ′ κ −m ωω ′ w. The parameter κ here bears not connection with the Chern-Simons levels κω of the gauge theory. and ψ + ω,0 ψ − ω,0 = ψ − ω,0 ψ + ω,0 = 1. In these relations, q ∈ C × , c is a central element, and the matrix g ωω ′ (z) writes 15 In order to compare with the gauge theory quantities, we should set q = q 1/2 3 , κ = (q 1 /q 2 ) 1/2 , then Modes decomposition The algebraic relations A.3 can also be directly written for the modes of the Drinfeld currents. In particular, introducing we find, where [· · · ] ± denotes the expansion in powers of z ∓1 . In addition to the central charge c, it possible to define a second central charge using the zero modes of the Cartan currents: Finally, the algebra can be supplemented with the following grading operators, (A.12) 15 This matrix is sometimes also written 16 Alternatively, is the mass-deformed Cartan matrix of Kimura and Pestun [75] with the mass µe = q1 associated to each link e : ω → ω + 1 of the necklace quiver.

A.2 Horizontal representation
Representations of this type have central charge c = 1, they have been constructed by Saito in [62] under the name vertex representations. We review here this construction.
For c = 0, the Cartan modes a ω,k define p coupled Heisenberg subalgebras. For later convenience we introduce the rescaled modes The representation of the currents x ± ω (z) and ψ ± ω (z) can be factorized into two commuting parts: a zero mode part (X ± ω (z), Y ± ω (z)), and a vertex operator part (η ± ω (z), ϕ ± ω (z)) built over the Cartan modes α ω,k : We focus first on the vertex operators part, it writes . The Fock vacuum |∅ is annihilated by positive modes α ω,k>0 , and we define accordingly the normal ordering : · · · : by writing positive modes on the right. It is a matter of simple algebra to derive the following normal-ordering relations: with the function In fact, it is possible to resum the infinite series and write the matrix elements S ωω ′ (z) as simple rational functions: We then observe the crossing symmetry, The structure function g ωω ′ (z) can be written as a ratio of functions S ωω ′ (z) with shifted arguments, We now turn to the analysis of the zero-modes. In [62], Saito introduces the symbols e αω associated to the roots α ω , and obeying the commutation relations e αω e α ω ′ = (−1) β ωω ′ e α ω ′ e αω (in particular symbols attached to the same root commute). These symbols, together with the operators a ω,0 and ∂ αω act on states parameterized by a root α = ω∈Zp r ω α ω (r ω ∈ Z) and a fundamental weight Λ ω 0 , In this representation, Thus, introducing X ± ω (z) = e ±αω z 1±a ω,0 and Y ± ω = q ±∂α ω , we find the algebraic relations It is easy to verify that these are indeed the factors needed to reproduce the algebraic relations A.3. The only difficulty appears in the verification of the commutation relation [x + ω , x − ω ′ ] for which we need to use the property z a ω,0 w −a ω,0 = z ∂α ω w −∂α ω to treat the zero mode dependence. The value of the central chargec can be recovered by noticing that This representation has been extended to higher levelc in [49]. Note however that in the definition of the (ν 1 , ν 2 )deformed horizontal representation, a set of 4p Heisenberg algebras will be employed to define to the zero-modes X ± ω and Y ± ω instead of the symbols introduced in A.21.

A.3 Vertical representations
The vertical representations have central charge c = 0 and thus the Cartan currents ψ ± ω (z) commute. They are diagonal in the basis of states |λ labeled by m-tuple Young diagram λ = (λ (1) , · · · , λ (m) ). The representation depends on an m-vector of weights v = (v 1 , · · · , v m ) and a choice of coloring c α for each component v α . We denote m ω = |C ω (m)| the number of weights v α of color c α = ω (obviously, m = ω∈Zp m ω ). The action of the Drinfeld currents on the states |λ reads (A. 25) In the first two lines, summations are performed over the set of boxes of color ω that can be added (A ω (λ)) to or removed from (R ω (λ)) the m-tuple Young diagram λ. The summands are expressed in terms of residues involving the functionsỸ [λ] ω (z), and the action of the Cartan is given as an expansion of the functionsΨ [λ] ω (z) in powers of z ∓1 . These two sets of functions are defined as follows: (A. 26) The zero modes of the Cartan act as .27) and, taking the product over the index ω, we deduce the level ρ (V ) (c) = m.
A.4 Relation with the (ν 1 , ν 2 )-deformed algebra The physical quantity we need to reproduce is the scattering function S ωω ′ (z) defined in 2.16. In this scope, it is easier to compare the horizontal representations, and reproduce the commutation of the Heisenberg subalgebras 3.22 using the gl(p) formula A.13. This leads to identify the Cartan matrix with the matrix β ωω ′ defined in 2.18. Furthermore, the factor κ m ωω ′ has to be replaced with a more general matrix κ ωω ′ that reads 17 The definition of the structure function g ωω ′ (z) is a little more difficult because of the freedom in defining the zero-mode factor. Comparing with the formula A.20 for the case of gl(p), the most natural choice would be Unfortunately, with this definition, the identity g (1) ωω ′ (z)g (1) ω ′ ω (z −1 ) = 1 is NOT satisfied, yet it is necessary for the consistency of the algebraic relations. This prompts us to propose instead the definition given in 3.4 where the factor f ωω ′ (qz) is missing. Unfortunately, this redefinition of the structure function g ωω ′ (z) breaks the natural symmetry between positive and negative currents, and makes the definition of the central chargec more difficult. Note also that another possibility could have been to define but this would require us to modify the definition of the function S ωω ′ (z): (A.32) 17 We could also express the coefficients σ ωω ′ in terms of the mass-deformed Cartan matrix β ] ω (z). Thus, it is possible to focus on the case of a single Young diagram λ (α) corresponding to a weight v α of color c α . The proof will be done by recursion on the number of boxes. We start with an empty Young diagram, for which R ω (∅) = ∅.
There are four possible configurations for adding box in λ (α) , all represented in figure 1. We start with the generic case, for which We employed here the shortcut notation q ±1 1 (q ±1 2 ) to designate the box of coordinate (i ± 1, j) (resp. (i, j ± 1)) next to = (i, j). In this generic case, the factors induced by the variation of the content of the sets A ω and R ω−ν 1 −ν 2 reproduce the extra factor S c( )ω (χ /z) in the RHS of B.1.
We now turn to the first case of the degenerate configurations represented on figure 1. In this case, only one more box can be added to A ω (λ (α) + ). On the other hand, the addition of the box prevents the removal of the box q −1 1 . As a result, Once again, we observe the agreement between the variation of the RHS 2.23 and the recursion relation B.1. The other two cases are treated in the same way.
C Representations of the extended algebra

C.1 Vertical representation
The vertical representation is of the highest weight type. The highest state |∅ , also called vacuum state, is annihilated by the currents x − ω (z), while x + ω (z) create excitations. The excited states |λ are parameterized by an m-tuple Young diagram λ. The weights v = (v 1 , · · · , v m ) parameterize the action of the Cartan ψ ± ω (z) on the vacuum state. The two Cartan currents commute, they are diagonal in the basis |λ , with the eigenvalue [Ψ [λ] ω (z)] ± where ± denotes an expansion in powers of z ∓1 . The action of x ± ω (z) add/remove a box of color ω. In order to produce the Dirac δ-function in the commutator [x + , x − ], it is natural to assume that modes x ± ω,k depends on the index k only through a factor of χ k where is the box that is added/removed. Taking all these assumptions in consideration, we arrive at the following ansatz: are the coefficients to be determined. When the central charge c is vanishing, the algebra 3.2 simplifies drastically, Plugging in the ansatz C.1, and the expression 3.13 for Ψ [λ] ω (z), we find that these relations are satisfied provided that , c( ) = ω ′ , c(y) = ω. (C. 3) The first two relations come from the projection of the commutator [x + , x − ] on the basis |λ , decomposing the RHS as The last equation in C.3 arises from the exchange relations x ± x ± . Then, it is simply a matter of calculation to check that the following coefficients do indeed satisfy the relations C.3, (C.5)

C.2 Horizontal representation
Here the strategy is to start by computing the algebraic relations satisfied by the vertex operators η ± ω and ϕ ± ω , compare them with 3.2, and introduce the zero-modes factors to compensate unwanted factors. Using the definition 3.24, we can compute the normal-ordering relations It remains to compute the central chargec. The zero modes of the Cartan currents write (C.14) We deduce that Since [q ω , P ω ′ (w)] = β ω ′ ω P ω ′ (w), the operatorq commute with P ω (z), thus it is central in this representation. Moreover, since Q ω (z) acts trivially on the dual state ∅|, we haveq = 0. Finally, we also have to take into account the non-commutation of the zero modes which brings the extra factor Since β ωω ′ is circulant, it is easy to compute assuming 0 ≤ ν 1 , ν 2 ≤ p − 1. This gives us the value of the central chargec.

D Automorphisms, gradings and modes expansion D.1 Automorphisms and gradings
The algebraic relations 3.2 can be supplemented with the grading operators d andd ω (ω ∈ Z p ) acting on the currents as e αd x ± ω (z)e −αd = x ± (e α z), e αd ψ ± ω (z)e −αd = ψ ± ω (e α z), for any parameter α ∈ C. The grading operator d reflects the invariance of the algebra under rescaling of the variable z → e α z, it defines the automorphisms τ α acting on an element x of the algebra as τ α (x) = e αd xe −αd . Similarly, the grading operatorsd ω defines the automorphismsτ ω,α (x) = e αdω xe −αdω associated to the invariance under the following rescaling of the currents for a fixed ω: while the currents x ± ω ′ =ω (z), ψ + ω ′ (z) and ψ − ω ′ =ω,ω+ν 3 c (z) remain invariant. In addition to the automorphisms τ α andτ ω,α , the algebraic relations are invariant under a third class of automorphismsτ ω,α (x) = e αdω xe −αdω defined as This transformation is the generalization of the element T of the SL(2, Z) group of automorphisms for the quantum toroidal algebra of gl 1 (or Ding-Iohara-Miki algebra) [43]. With a slight abuse of terminology, we will also calld ω a grading operator.

D.2 Modes expansion
In order to define properly the modes expansion of the currents x ± ω (z) and ψ ± ω (z), we need to remove some part of the zero modes factors. For this purpose, we use a twist by a combination of automorphisms to define the new currentsx ± ω (z) andψ ± ω (z) with proper modes expansion. First, we introduce the following combinations of grading operators, Note that this operator becomes central if ν 3 = 0 or c = 0. Expanding in powers of the spectral parameters, the exchange relationsψ −x andψ −ψ given in D.8 provide the commutation relations between the modes, In particular, when c = 0, the modes a ω,k of the Cartan currents define p Heisenberg subalgebras. This property is used to build the horizontal representation in appendix C.2. The exchange relationsx −x can also be written in terms of modes by projecting the following relations: (D.14) A priori, the commutator [x + ,x − ] could also be written in terms of modes, but the expression is rather cumbersome.

D.3 Coproduct
The Hopf algebra structure can be extended to include the grading operators, provided we define the coproduct, counit and antipode as

(D.24)
Thus, we find the representation for the twisted currents, We can verify thatã ω,0 commutes with the twisted currents, and satisfies the relation D.10 with the grading operator ξ ω (z).

E Derivation of the vertex operators E.1 Definition of the vacuum components
Before sketching the derivation of the solution for the intertwining relations, we would like to provide a bold argument for the definition of the vacuum components Φ ∅ and Φ * ∅ entering in the definition 4.4 of the intertwiners. In fact, the full partition function of the gauge theory, including classical, one-loop and instantons contributions, has a nice description in terms of the melting crystal picture [2,85]. Indeed, the one-loop contribution can be written as a double product over the boxes of completely filled (infinite) Young diagrams λ ∞ = {(α, i, j) α = 1 · · · m, i = 1 · · · ∞, j = 1 · · · ∞}, assuming a ζ 2 -regularization for the infinite product. Then, the instanton correction of order O(q k ) is obtained by removing k boxes to λ ∞ , taking the double product over λ c = λ ∞ \ λ = {(α, i, j) α = 1 · · · m, i = λ (α) j + 1 · · · ∞, j = 1 · · · ∞} and summing over the configurations λ of k = |λ| boxes. The vacuum component Φ ∅ of the intertwiner Φ is associated to this infinite product over boxes in λ ∞ , so that formally and similarly for Φ * λ , replacing η + c( ) (χ ) with η − c( )+ν 3 (q 3 χ ). In order to develop this idea, we may introduce a very crude cut-off N such that λ ∞ is obtained as the limit N → ∞ of m Young diagrams consisting of squares of size (pN ) × (pN ), i.e. λ N = {(α, i, j) α = 1 · · · m, i = 1 · · · pN, j = 1 · · · pN }. Then, we may consider the product over boxes (α, i, j) ∈ λ N and decompose the indices (i, j) as i =ī + 1 + k i p, j =j + 1 + k j p withī,j = 0 · · · p − 1 and k i , k j = 0 · · · N − 1. We end up with At this stage, the limit N → ∞ is ill-defined because the first exponential converges when |q 1 |, |q 2 | < 1 while the second exponential for |q 1 |, |q 2 | > 1. However, we notice that each color can be treated independently, and their contribution written in terms of the vacuum component for the intertwiner describing instantons on a omega-background with no orbifold [38,43], with the replacement ε 1 , ε 2 → pε 1 , pε 2 . Thus, we can borrow the corresponding operator and simply define α cα+īν 1 +jν 2 ,k : .

(E.4)
The appearance of quantities defined on the background C pε 1 × C pε 2 × S 1 R is reminiscent of the surface defect interpretation of the orbifold developed in [11,14]. It may also be related to the abelianization procedure described in the case of gl(p) (unrefined, i.e. q 3 = 1) in [49].
(E. 16) From these relations, we to deduce the normal ordering relations for the currents x ± ω and ψ ± ω . Then, the relation E.12 and E.13 for the Cartan currents ψ ± ω follow directly, provided that the weights and levels satisfy the relation 4.2. This condition is related to the difference between the functions f . (E.17) The relations E.12 and E.13 involving the currents x ± ω are harder to prove. This is done by decomposition of the functions Y ω (z) as sum over poles. We refer the reader to [43] for a more detailed explanation.