Nested coordinate Bethe wavefunctions from the Bethe/gauge correspondence

In [1, 2], Nekrasov applied the Bethe/gauge correspondence to derive the $\mathfrak{su}\, (2)$ XXX spin-chain coordinate Bethe wavefunction from the IR limit of a 2D $\mathcal{N}=(2, 2)$ supersymmetric $A_1$ quiver gauge theory with an orbifold-type codimension-2 defect. Later, Bullimore, Kim and Lukowski implemented Nekrasov's construction at the level of the UV $A_1$ quiver gauge theory, recovered his result, and obtained further extensions of the Bethe/gauge correspondence [3]. In this work, we extend the construction of the defect to $A_M$ quiver gauge theories to obtain the $\mathfrak{su} \, ( M + 1 )$ XXX spin-chain nested coordinate Bethe wavefunctions. The extension to XXZ spin-chain is straightforward. Further, we apply a Higgsing procedure to obtain more general $A_M$ quivers and the corresponding wavefunctions, and interpret this procedure (and the Hanany-Witten moves that it involves) on the spin-chain side in terms of Izergin-Korepin-type specializations (and re-assignments) of the parameters of the coordinate Bethe wavefunctions.

In [1,2], Nekrasov introduced an orbifold-type codimension-2 defect in the IR limit of an A 1 quiver gauge theory on the gauge side of the correspondence, and obtained the coordinate Bethe wavefunction in the su (2) XXX spin- 1 2 chain on the Bethe side [6,7] 2 . In [3], Bullimore, Kim and Lukowski implemented Nekrasov's defect construction at the level of the UV A-twisted N = (2, 2) supersymmetric A 1 quiver gauge theories on S 2 , and used the localization formulae of [8,9] to recover Nekrasov's result, amongst other results that further extend the Bethe/Gauge correspondence.
In this work, we extend the construction of [3] to A M quiver gauge theories with orbifold-type codimension-2 defects to obtain su(M + 1) XXX spin-chain nested coordinate Bethe wavefunctions, with spin states in the fundamental representation.
Further, we apply a Higgsing procedure to obtain more general A M quiver gauge theories with codimension-2 defects, and their corresponding nested coordinate Bethe wavefunctions, and interpret these generalizations on the spin-chain side (and the Hanany-Witten moves that they involve) as Izergin-Korepin-type specializations (and re-assignments of the roles) of the parameters. While we focus on results in XXX spin-chains, we outline their straightforward extension to XXZ spin-chains.
1.1. Outline of contents. In Section 2, we recall the localization formulae of Closset, Cremonesi and Park, for 2D N = (2, 2) quiver gauge theories on S 2 [8] (and that of Benini and Zaffaroni, for N = 2 quiver gauge theories on S 2 × S 1 [9]), used in [3] to construct Nekrasov's orbifold defect in 2D A 1 quiver gauge theory. Generalizing the discussion in [3] for A 1 quiver gauge theory, we formally introduce the equivariant characters and reconstruct the localization formulae in [8,9] from them.
In Section 3, we recall the Bethe/Gauge correspondence [4,5] for A M linear quiver gauge theories, and provide the equivariant characters for them.
In Section 4, after briefly recalling the construction of orbifold defects in A 1 quiver gauge theory, we extend this construction to a simple A M linear quiver gauge theory and obtain the nested coordinate Bethe wavefunctions of the su(M + 1) XXX spin-chain with spin states in the fundamental representation.
In Section 5, using a Higgsing procedure, we generalize the orbifold construction of Section 4 to more general A M linear quiver gauge theories and find the corresponding partition functions (as specializations of coordinate Bethe wavefunctions) of the corresponding su(M + 1) vertex lattice models. In Section 6, we interpret the Higgsing procedure as an Izergin-Korepin-type specialization of the lattice parameters on the Bethe side of the Bethe/Gauge correspondence, and interpret the Hanany-Witten moves that are involved in the Higgsing on the gauge side as a re-assignment of the lattice parameters on the Bethe side, and in Section 7, we include remarks.
In Appendix A, we discuss the partial domain wall partition functions (DWPFs) in the rational su(2) (six-)vertex model and prove Proposition 4.5, and in Appendix B, we define the partition function of the su(M +1) vertex model which corresponds to the orbifold defect in the A M quiver constructed in the present work.

Rewriting the localization formulae
We review the localization formulae of the partition functions of the A-twisted gauged linear sigma models (GLSMs) [10,11], which are 2D A-twisted N = (2, 2) supersymmetric gauge theories on the Ω-deformed S 2 [8] (and 3D twisted N = 2 gauge theories on S 2 × S 1 [9]), where is the Ω-deformation parameter. Following [3], we provide the localization formulae in terms of equivariant characters which are more fundamental objects than the partition functions. The equivariant characters are used to construct orbifold defects in Sections 4 and 5. 3 2.1. Equivariant characters. Consider the 2D N = (2, 2) (or 3D N = 2) supersymmetric gauge theory consisting of a vector multiplet V in a Lie algebra g, of gauge group G, and L chiral matter multiplets Φ r i R i , with representations R i of g, U(1) vector R-charges r i and twisted masses λ i , i = 1, . . . , L. In the present work, r i = 0 or 2. The vector multiplet V contains scalars u = {u 1 , . . . , u rk(g) }, which take values in h ⊗ R C and parametrize the Coulomb branch of the gauge theory, where rk(g) is the rank of g and h is the Cartan subalgebra of g. where P = V or Φ r R , and u a | ± = u a ∓ d a 2 (2.3) where the characters χ P + (u| + ) and χ P − (u| − ) are defined on the north pole and the south pole of S 2 , respectively. Definition 2.2. The total equivariant character is 3 In the present work, we use 'equivariant character', as well as the notation for the Ω-deformation parameter, and γ for a twisted mass parameter. In [3], Bullimore et al. use 'equivariant index' instead of 'equivariant character', as well as the notation ǫ for the Ω-deformation parameter, and for the twisted mass parameter. and the total equivariant character with charges d is where (2.6) defines w I and v J , and we obtain the building blocks of the topologically twisted partition functions of 2D and 3D gauge theories by and and respectively, give the building blocks of 2D and 3D topologically twisted partition functions in [8,9] (see Proposition 2.4). Here U a = e −ua , Λ = e −λ , q = e − , u α = e −α(u) , and u ρ = e −ρ(u) .
In the above proposition, the Pochhammer and q-Pochhammer symbols are, respectively, defined by and In the following, we assume that the gauge group G contains central U(1) c factors, and then one can deform the gauge theory by the associated Fayet-Iliopoulos (FI) parameters ξ a , and theta angles θ a , a = 1, . . . , c.
Proposition 2.4 ( [8,9]). Combining the complexified FI parameters τ a = i ξ a + 1 2π θ a with the building blocks in Proposition 2.3 obtained from (2.5), up to sign factors, the partition function of the A-twisted GLSM on S 2 is given by Here |W| is the order of the Weyl group of G, and the pairing τ (d) = a τ a d a is defined by embedding τ into h * ⊗ R C. The contour integral along Γ is given by the Jeffrey-Kirwan residue operation (JK contour integral) [12,13,14] (see also [15]), which picks relevant poles of the integrand. Similarly, the correlation function of two codimension-2 defects, O N (u) and O S (u), inserted at the north pole and at the south pole of S 2 , respectively, is given by The topologically twisted partition function and correlation functions of the N = 2 gauge theory on S 2 × S 1 are obtained, up to Chern-Simons factors, by replacing Z P d (u; ) with Z K,P d (u; ) in the above formulae.

The Bethe/Gauge correspondence
We recall the basics of the Bethe/Gauge correspondence [4,5], between the supersymmetric vacua of 2D N = (2, 2) (resp. 3D N = 2) A M linear quiver gauge theories as in Figure 1, and the Bethe eigenfunctions of XXX (resp. XXZ) spin-chain Hamiltonians, with spins in the fundamental representation of su(M + 1).
The A M quiver gauge theory which corresponds to the su(M + 1) XXX spin-chain, with spins in the fundamental representation, is described by an A-twisted GLSM, on the Ω-deformed S 2 , Figure 1. A M linear quiver with the matter content in Table 1. Table 1. The matter content of the A M quiver gauge theory in Figure 1 which describes an su(M + 1) spin-chain. For X and U(1) R denotes the U(1) vector R-charge.
with the gauge group U(k 1 ) × · · · × U(k M ), the matter content in Table 1, and the superpotential It contains a set of vector multiplet scalars u M k = {u  L p } and γ, associated with the U(L 1 ) ×· · ·×U(L M ) ×U(1) flavor symmetry. In Table 2, we summarize the Bethe/Gauge dictionary [4,5] which translates the gauge theory language to the spin-chain language. From Definition 2.1, we obtain the equivariant characters 2D/3D gauge with A M quiver ( Figure 1) su(M + 1) XXX/XXZ spin-chain u  for the U(k p ) vector multiplets V (p) and the chiral matter multiplets in Table 1 as where u Combining (3.2) with the complexified FI parameters τ p , p = 1, . . . , M, associated with the central U(1) M ⊂ U(k 1 ) × · · · × U(k M ), the integrand (2.14) of the S 2 partition function is where q p = (−1) L p+kp−1+kp e 2πi τ p are exponentiated FI parameters.
In the limit → 0, with positive FI parameters ξ p > 0, and summing over the GNO charges d M a ≥ 0, the partition function (2.13) can be written in terms of a contour integral around the roots of the vacuum equations [8] (3.5) Here the effective twisted superpotential [4] (in the denominator of the integrand), consists of The contour Γ eff encloses the roots of vacuum equations e where a = 1, . . . , k p , p = 1, . . . , M, k 0 = k M +1 = 0. These vacuum equations are the nested Bethe equations of the su(M + 1) XXX spin-chain (see the Bethe/Gauge dictionary in Table 2), which is the starting point of the Bethe/Gauge correspondence [4,5].
Remark 3.1. The 3D uplift of the above results for XXZ spin-chains is straightforward [4] (see also [16]). Figure 2. The A M linear quiver that corresponds to Figure 1 with the ranks of the flavour groups L i = 0, i = 1, . . . , M − 1, L M = L, and the ranks of the gauge Table 3. The matter content of the A 1 quiver gauge theory that corresponds to the su(2) spin-chain. Here i = 1, . . . , L, k ≤ L.

Nested coordinate Bethe wavefunctions from orbifold defects
We review the construction of orbifold-type codimension-2 defects in the A 1 quiver gauge theory that corresponds to the su(2) spin-chain [1,2,3], then extend that to the A M quiver gauge theory that corresponds to the su(M + 1) spin-chain, described by the A M quiver in Figure 2. 4

4.1.
Orbifold defect for A 1 quiver. Consider the A-twisted U(k) GLSM on S 2 , with matter content as in Table 3, and the superpotential In this case, the equivariant characters (3.1) are given by Now, we recall the construction of orbifold defects for the A 1 quiver in [1,2,3]. The orbifold defects inserted at the north (or south) pole of S 2 are characterized by a discrete holonomy ω n , 4 For a suitable choice of the FI parameters, the GLSM described by the A M quiver in Figure  n = 0, 1, . . . , L − 1, with ω L = 1, associated with a Z L orbifold around the north (or south) pole, such that the gauge symmetry U(k) and the flavor symmetry U(L) are broken to a maximal torus. Firstly, for constructing such orbifold defects, we change the parameters in the total equivariant character χ total is an ordered set that characterizes the orbifold defect. Next, taking the Z L invariant part under → + 2πin, n = 0, 1, . . . , L − 1, of χ total which follows from the following lemma.
the lemma is proved.
Symmetrizing in the variables u k , the contribution of a defect inserted at the north (resp. south) pole of S 2 to the integrand of the JK contour integral 5 is ψ Here Sym u k stands for the symmetrization of a function f (u k ) in the variables u k , and S k is the symmetric group of degree k.
. Consider a normalization of the A 1 quiver orbifold defect as Then, the defect ψ (L) I k (u k ; m L ) gives the coordinate Bethe wavefunction of su(2) XXX spin-1 2 chain.
Note that the defects inserted at the north pole and the south pole of S 2 are given by  I k (u k ; m L ) from the character in (4.4), using (2.7). Using (2.8) instead of (2.7), we obtain the coordinate Bethe wavefunction of the su(2) XXZ spin-1 2 chain, and (4.9) is replaced by where [x] = 2 sinh(x/2).
From the orbifold defect ψ (L) I k (u k ; m L ), one obtains the su(2) six-vertex model partial domain wall partition function (DWPF) [17]. In Appendix A, we prove the following proposition as a corollary of Propositions 4.2 and A.4.
Then, the partition function Z k (u k ; m L ) agrees with the su(2) six-vertex model partial DWPF, which has the determinant expression [17], 12) or equally [18,19], Orbifold defect for a simple A M quiver. We extend the above construction of the A 1 quiver orbifold defects to the simple A M linear quiver in Figure 2, with the matter content in Table 1. The equivariant characters are given in (3.1), with L p = 0, p = 1, . . . , M − 1, and To construct orbifold defects which break the gauge symmetry U(k 1 ) × · · · × U(k M ) and the flavor symmetry U(L) to a maximal torus, as a generalization of the change of parameters (4.2), we consider (4.14) in the total equivariant character χ total Taking the Z L invariant part of the total equivariant character under → + 2πin, n = 0, 1, . . . , L − 1, from Lemma 4.1 one finds where the set I (p) a is defined by the map which can be explained as follows. I Using a normalization similar to that in (4.8), we find the following proposition.
Proposition 4.6. The orbifold defect, for the simple A M quiver in Figure 2, gives the nested coordinate Bethe wavefunction (see e.g. [20]) in the su(M + 1) XXX spin-chain with spins in the fundamental representation, where ω (L)  Remark 4.8. In [21], the orbifold defect (4.19) was geometrically constructed as an element in a stable basis [22] in the cotangent bundle of a partial flag variety (see also [23,24]).
Then, the partition function factorizes into the A 1 quiver orbifold defect in (4.8) and the su (2) six-vertex model partial DWPF (4.11), Further, Proof. From 3 in Proposition A.3 and (A.2), the partition function Z k 1 (u (1) k 2 , and then,  Table 4. On the left is a type-IIA brane configuration, and on the right is the type-IIB brane configuration in [16] that corresponds to it by T-duality along the x 2 -direction (see also Figure 4).

Generalizations by Higgsing
In Section 5.1, we recall the Higgsing procedure in the A M quiver in Figure 1 Table 4 and Figure 4, we describe a type-IIA brane configuration, and a type-IIB brane configuration in [16] that corresponds to it by T-duality along the x 2 -direction. By introducing the twisted mass γ in Table 1, which breaks half the supersymmetry, these configurations describe, respectively, 2D N = (2, 2) quiver gauge theories on the (x 0 , x 1 )-directions, and 3D N = 2 quiver gauge theories on the (x 0 , x 1 , x 2 )-directions. For the purposes of this work, the basic idea, as described in Figure 4, is 1. to fine-tune the quiver data so that two D4/D5 branes are aligned at the same position in x 7,8,9 x 6 x / 2,3,4,5 Figure 4. The Higgsing procedure including Hanany-Witten moves for type-IIA/IIB brane configuration [16], where the vertical (resp. horizontal) lines represent NS5 (resp. D2/D3) branes, and each x represents a D4/D5 brane. The corresponding quiver description is in Figure 5. work, as in [16], the Higgsing procedure, involves the annihilation of D2/D3 branes, as in Step 3 of Figure 4. We thank A Hanany for discussions on this point.
Higgses the n-th gauge node of the A M quiver on the left hand side of Figure 5, and changes the quiver parameters as follows, k p → k p , (for p = n), L p → L p , (for p = n, n ± 1), as depicted in the transition from the left hand side to the right hand side of Figure 5  where the following characters remain unchanged while the following characters change and The sum of the Higgsed characters in (5.5) becomes a and can be decoupled from the quiver gauge theory. One also finds that the extra factors of the first and third (resp. second and fourth) characters in (5.6), agree with the contributions from extra (anti-)fundamental matter with mass parameter µ H at the (n − 1)-th (resp. (n + 1)-th) gauge node. As a result, the transition (5.2), under the specialization (5.1), is confirmed.   Figure 6. Higgsing the A 2 linear quiver.

5.2.1.
Higgsing the A 2 quiver gauge theory in Figure 6. We set the A 2 quiver data k 1 ≤ k 2 ≤ L−1, so that the specialization (5.1) becomes As before, the twisted masses m L−1 and m L decouple after Higgsing, and compared with the discussion in Section 4.2, one only needs to consider the second factors on the right hand side in the Higgsed characters (5.6).

5.2.2.
Construction of A 2 quiver orbifold defects. Instead of the change of parameters (4.14), we consider which is consistent with Higgsing, where a+1 , (5.11) and the symbol ⊎ denotes the pairwise disjoint union. Then, as a generalization of (4.19) for M = 2, we obtain an orbifold defect where ω (L) I k (u k ; m L ) is defined in (4.9), and I Note that, instead of the change of parameters (5.10), by considering with I (1) a+1 , (5.14) we obtain an another orbifold defect

5.2.3.
More general A 2 quiver orbifold defects. One can apply the above Higgsing procedure repeatedly. Consider the A 2 quiver in Figure 1 with M = 2, set k 1 ≤ L 1 + k 2 ≤ L 1 + L 2 , and apply the change of parameters As a result, we find an orbifold defect for the A 2 quiver, which generalizes the defect (5.12). Similarly, instead of (5.16), by considering we find an another orbifold defect for the A 2 quiver,  Table 1 for the matter content), where we set The orbifold defects for the A 2 quiver are composed of the orbifold defect ω (L) I k (u k ; m L ) in (4.9) for the A 1 quiver, and it is useful to define Now, from the equivariant characters (3.1), we find orbifold defects for the A M quiver, which generalize the simple A M quiver orbifold defect (4.19), (1) a+1 . Our claim is that the orbifold defect   Figure 10. In this case, before taking the limit x → ∞, we split this variable into three pieces x, x ′ and x ′′ . As we will discuss in Section 6.3, the splitting of the variable x is interpreted as the lattice version of a Hanany-Witten move. Further, we deform the lattice configuration by taking the decoupling limit x ′′ → ∞ to remove the horizontal line, with the variable x ′′ and colour i at the two end boundaries, which uniquely results in no c vertex 7 .
7 On the gauge side, we can interpret taking the limit x ′′ → ∞ as the decoupling of a pair of fundamental and anti-fundamental matter, with twisted masses x ′′ , at the (n + 1)-th node after the Hanany-Witten move in Repeating this process, one can construct the su(M + 1) lattice configurations in Appendix B (e.g. see Figure 7) from the lattice configuration associated with the simple A M linear quiver in in the su(2) spin-1 2 six-vertex model [27]. As explained in Appendix A, Korepin proposed a specialization of the parameters (the rapidities and the inhomogeneities) of the DWPF that leads to a recursion relation (and an initial condition) that completely determine it [27]. In [28], The specialization (5.1) of the parameters, used in the Higgsing procedure by Gaiotto and Koroteev, is a variation on Korepin's, in the sense that it is essentially the same with two differences between them.
6.2.1. The first difference. Korepin's derivation of the recursion relation requires one type of conditions, and can either, while the Higgsing procedure requires both as in (5.31).
In the case of domain wall boundary conditions, the DWPF is symmetric in the horizontal-line variables and also in the vertical-line variables, and one can choose the variables that one wishes to specialize to be those on lines on the boundaries of the (finite) lattice configuration on which the DWPF is defined. Once we do that, we have more information about the colours of the state variables on these lines, and only one condition is needed to derive the recursion relation.
In the case of Higgsing, when translated to the lattice, one deals with lattice configurations that correspond to the coordinate Bethe wavefunction which is not a symmetric function in the vertical-line (inhomogeneity) variables, one cannot (for general choices of the Higgsing parameters) associate the variables that one wishes to specialize to boundary lattice lines, and one requires (in general) two independent conditions. This makes the specialization of Gaiotto and Koroteev a more general version of Korepin's. On the other hand, in the lattice version of Higgsing, one wishes to trivialize the contributions of a single horizontal and two vertical lattice lines, and this is achieved by identifying three parameters (using two conditions), then taking that parameter to infinity and normalizing appropriately. In the type-IIA/IIB brane realizations in Figure 4, this Higgsing procedure corresponds to Steps 1 and 2. This makes the specialization of Gaiotto and Koroteev a limiting case of Korepin's. 7. Remarks 7.1. Affinization. In [29], Bonelli, Sciarappa, Tanzini and Vasko studied connections between 4D N = 2 supersymmetric gauge theories and quantum integrable systems of the hydrodynamic type. In particular, the A M quiver gauge theory that plays a central role, and appears in Figure   1, in [29], is the affine version of the A M quiver gauge theory that plays a central role, and appears in Figure 1, in the present work. This leads us to expect that the present work has an affine extension along the lines of [29]. 7.2. Quiver with orbifold defects. In Section 4, we considered a Z L orbifold of the 2D gauge theory described by the A M quiver in Figure 2 and obtained the orbifold defect (4.19) labeled by the nested sequences (4.15). In [30], Bonelli, Fasola and Tanzini studied a class of 4D A 1 quiver gauge theories, with a codimension-2 surface defect, that supports nested instantons obtained by an orbifold and labeled by nested partitions. They discussed a 2D gauge theory described by a quiver that is different from that used in the present work, and that corresponds to the moduli space of nested instantons. It would be interesting to find the relation, if any, between the two constructions.
Appendix A. The su(2) six-vertex model partial domain wall partition function We prove Proposition 4.5, to the effect that the partition function Z k (u k ; m L ) in (4.11), constructed from the orbifold defect ψ Proof. We show that Z k (u k ; m L ) is regular at u A = u B . Let σ A,B be an operator acting on functions of u k which exchanges u A and u B , and consider the defect ω I k (u k ) := ω (L) (4.9). Symmetrizing in the variables u k , the pole at u A = u B , A < B, in ω I k (u k ) cancels the corresponding pole in σ A,B ·ω I k (u k ). Therefore, ω I k (u k )+ σ A,B ·ω I k (u k ) has no poles at u A = u B , thus ψ (L) I k (u k ; m L ) and Z k (u k ; m L ) are regular at u a = u b , a, b = 1, . . . , L, and polynomials of degree L − 1 in each u a .
Lemma A.2. One can decouple the vector multiplet scalars by and further, Proof. The orbifold defect ψ Proof. Condition 1 follows from the definition of Z L (u L ; m L ), and Condition 2 follows from Lemma A.1. To prove Condition 3, it is sufficient to show that ω I L (u L ) + σ A,A+1 · ω I L (u L ), in Z L (u L ; m L ), is invariant under the permutation of m A and m A+1 , where ω I L (u L ) = ω (L) (4.9). The point is that, once this is shown, then by symmetrizing u A , u A+1 , and u A+2 in ω I L (u L ), the symmetrized function becomes invariant under permuting m A , m A+1 , and m A+2 . In ω I L (u L ), the factor that contains u A and u A+1 is Because the factor that contains m A and m A+1 , but does not contain u A and u A+1 , is manifestly symmetric under the exchange of m A and m A+1 , in ω I L (u L ) + σ A,A+1 · ω I L (u L ), it is sufficient to consider the following factor in (A.4) Since this factor is symmetric under permuting m A and m A+1 , Condition 3 is proved. To prove Condition 4, we assign u L = m L − γ 2 in Z L (u L ; m L ), the non-zero terms only come from Sym u L−1 ω I L (u L ), and by By induction in L, any function which satisfies the four conditions in Proposition A.3 is uniquely determined 9 , and the partition function Z L (u L ; m L ) agrees with Izergin's determinant expression for the su(2) DWPF [28] (see [31] for a review), which satisfies the same four conditions, as well as Kostov's determinant expression [18,19], which is equivalent to Izergin's.   a are the rapidities and inhomogeneities, respectively. The translation of the lattice parameters to the gauge theory parameters in Table 1 is We associate lattice configurations to the A M linear quiver in Figure 1, where k p ≤ L p + k p+1 , k M ≤ L M , p = 1, . . . , M − 1, as follows. For the first node with U(k 1 ) gauge group, we associate one of the lattice configurations in Figure 12. i (1) x (1) i (1) x (1) Figure 12. Two possible lattice configurations that can be associated with the first quiver node.
All bonds on the left boundary are assigned the (fixed and same) colour 1, all bonds on the right and lower boundaries are assigned the (fixed and same) colour 2, and the L 1 bonds on the top boundary are assigned (fixed but varying) colours i (1) ℓ ∈ {1, 2}, ℓ = 1, . . . , L 1 . Each dashed line that carries a rapidity variable x (2) a is connected with another dashed line that carries a rapidity variable x (2) a associated with the second quiver node. For the p-th quiver node with a U(k p ) gauge group, p = 2, . . . , M − 1, we associate one of the lattice configurations in Figure 13, where, each dashed line that carries a rapidity x (p) a is connected with another dashed line that carries a rapidity variable x (p) a . All bonds on the right boundary and on the lower boundary are assigned the (fixed and same) colour p + 1, the L p bonds on the top boundary are assigned (fixed but varying) colours i (L p ) ℓ ∈ {1, . . . , p + 1}, ℓ = 1, . . . , L p , and we label the above left (resp. right) associated lattice configuration by s p = 1 (resp. s p = 2). For the M-th quiver node with U(k M ) gauge group, we associate the unique lattice configuration in Figure 14.