A Note on the Algebraic Engineering of 4D N=2 Super Yang–Mills Theories

Physics Letters B 789 (2019) 610–619

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

A note on the algebraic engineering of 4D N = 2 super Yang–Mills theories ∗ J.-E. Bourgine a, , Kilar Zhang b,1 a Korea Institute for Advanced Study (KIAS), Quantum Universe Center (QUC), 85 Hoegiro, Dongdaemun-gu, Seoul, South Korea b Department of Physics, National Taiwan Normal University, No. 88, Sec. 4, Ting-Chou Road, Taipei 11677, Taiwan a r t i c l e i n f o a b s t r a c t

Article history: Some BPS quantities of N = 15D quiver gauge theories, like instanton partition functions or qq- Received 11 November 2018 characters, can be constructed as algebraic objects of the Ding–Iohara–Miki (DIM) algebra. This construc- Accepted 12 November 2018 tion is applied here to N = 2super Yang–Mills theories in four dimensions using a degenerate version Available online 2 January 2019 of the DIM algebra. We build up the equivalent of horizontal and vertical representations, the ﬁrst one Editor: N. Lambert being deﬁned using vertex operators acting on a free boson’s Fock space, while the second one is essentially equivalent to the action of Vasserot–Shiffmann’s Spherical Hecke central algebra. Using intertwiners, the algebraic equivalent of the topological vertex, we construct a set of T -operators acting on the tensor product of horizontal modules, and the vacuum expectation values of which reproduce the instanton partition functions of linear quivers. Analysing the action of the degenerate DIM algebra on the T -operator in the case of a pure U (m) gauge theory, we further identify the degenerate version of Kimura–Pestun’s quiver W-algebra as a certain limit of q-Virasoro algebra. Remarkably, as previously noticed by Lukyanov, this particular limit reproduces the Zamolodchikov–Faddeev algebra of the sine-Gordon model. © 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction (formally equivalent to the affine Yangian of gl1 [12,13]), for which representations of level m contains the action of the Wm-algebra Since the pioneer work of Seiberg and Witten [1,2], 4D N = 2 involved in the AGT correspondence. However, the complete de- super Yang–Mills (SYM) theories have manifested deep connections scription of algebraic structures sustaining all the observed inte- with integrability. Thus, in the Euclidean background R4, their BPS grable properties is still lacking. sector is described by a classical integrable system [3,4]. This sys- As it turns out, the algebraic description is better understood tem becomes quantized after the (partial) omega-deformation of when the gauge theory is lifted to five dimensions by an ex- 1 R2 × R2 tra S . The corresponding gauge theories have N = 1 supersym- the background ε , the parameter ε1 playing the role of 1 metry, and describe the low-energy dynamics of (p, q)-branes in a Planck constant [5]. Furthermore, the partition function of the type IIB string theory [14,15]. Alternatively, they can be obtained (fully) omega-deformed theories on R2 × R2 has been computed ε1 ε2 as the topological string amplitude on a toric Calabi–Yau three- exactly by Nekrasov using a localization technique [6]. These quan- fold [16–18], and the toric diagram coincides with the (p, q)-brane tities receive non-perturbative (instanton) corrections. They were web [19]. This string theory realization is the basis of an algebraic further shown to be equal to the W-algebras’ conformal blocks construction, first proposed in [20,21], based on the Ding–Iohara– describing correlation functions of Liouville/Toda integrable con- Miki (DIM) algebra [22,23], which we call here algebraic engineer- formal field theories under the celebrated AGT-correspondence ing. Representations of the DIM algebra are labeled by two integer [7,8]. Some of these integrable aspects follow from an action of levels, and are essentially made from two basic ones called ver- Vasserot–Schiffmann’s Spherical Hecke central (SHc) algebra [9–11] tical (0, m) [24] and horizontal (1, n) [25]. In engineering N = 1 gauge theories, a representation of level (q, p) is associated to each brane of charge (p, q). Thus, D5 branes are associated to vertical Corresponding author. * representations (0, 1), and NS5-branes (possibly dressed by n D5s) E-mail addresses: [email protected] (J.-E. Bourgine), [email protected] (K. Zhang). to horizontal ones. A refinement of this technique has been de- 1 On leave from the Institute of Theoretical Physics, Chinese Academy of Sciences, veloped in which stacks of m D5-branes (with different positions) Beijing 100190, Mainland China. correspond to vertical representations of higher levels (0, m) [26]. https://doi.org/10.1016/j.physletb.2018.11.066 0370-2693/© 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. J.-E. Bourgine, K. Zhang / Physics Letters B 789 (2019) 610–619 611 z + εα The junctions between branes are described in string theory us- g(z) = , (2.2) z − εα ing the refined topological vertex, which has also been written in α=1,2,3 the form of an intertwiner for the DIM algebra [27]. In this way, it has been possible to associate to the brane web of linear (and plays the role of a scattering factor. It obeys the unitarity property − = also D-type, see [28]) quivers a certain T -operator acting on the g(z)g( z) 1. In contrast with the case of DIM, the delta function tensor product of horizontal representations associated to external δ(z) here is not the multiplicative one, but it is simply defined as (dressed) NS5-branes [26]. The vacuum expectation value (vev) of 2iπ times the Dirac delta function. this operator reproduces Nekrasov’s instanton partition functions, We will not discuss in details the mathematical definition of while further insertions of DIM currents define the qq-characters the algebra (2.1) that should correspond to a Drinfeld double of 2 [29–33]. the affine Yangian of gl1 [13]. Instead, we will present two types In this note, we extend this algebraic construction to the case of representations: an horizontal representation of weight u, de- (H) (H) = of 4D N = 2theories. For this purpose, we introduce a formal cur- noted ρu and having level ρu (c) 1, and a set of vertical (m) rent algebra seen as a degenerate limit of the DIM algebra. This representations of rank m and weight a, denoted and such ρa algebra has a Hopf algebra structure with a Drinfeld-like coprod- (m) that ρ (c) = 0. The 4D N = 2SYM theories can also be obtained uct, an important feature to define intertwiners. We proceed by a from a brane construction, this time using type IIA string the- constructing two different types of representations, which we call ory [37]. In this construction, D5 and NS5 branes are replaced by again vertical and horizontal, by analogy with the DIM algebra. The D4 and solitonic 5-branes respectively. The horizontal representa- vertical representation is roughly equivalent to the action of SHc tion will be associated to the solitonic 5-branes, while stacks of m in which generators act on states labeled by a set of Young diagrams. On the other hand, the horizontal representation appears D4-branes, on which act a U (m) gauge group, will be associated to to be new. It is defined in terms of twisted vertex operators built the rank m vertical representation. The weights of representations upon the modes of a free boson. Then, we solve the intertwin- will be identified with the branes positions. In addition, automor- ing relation coupling these two representations, thus defining a phisms of the DIM algebra encode the invariance of the brane-web 4D equivalent of the topological vertex. From this point, the 5D under geometric transformations [38]. These automorphisms can method can be readily adapted to N = 2theories, and, as an also be defined on the algebra (2.1), except for Miki’s automor- ◦ example, we construct the T -operator for the pure U (m) gauge phism S [23] corresponding to a 90 rotation of the brane-web theory, and re-derive both the instanton partition function and that maps D5 on NS5 and vice versa, thus realizing the S-duality ◦ the fundamental qq-character. This T -operator commutes with the of type IIB string theory. On the other hand, the 180 rotation S2 action of the degenerate DIM algebra in a certain tensored hori- is preserved: zontal representation. In the 5D case, the corresponding action of 2 2 ± ∓ 2 ± ∓ DIM can be decomposed into q-Heisenberg and q-Virasoro actions S · c =−c, S · x (z) =−x (−z), S · ψ (z) = ψ (−z), [20], the latter being identified with Kimura–Pestun (KP) quiver ± ± ± ± τω · c = c, τω · x (z) = x (z + ω), τω · ψ (z) = ψ (z + ω), W -algebra for the A1-quiver [34]. A similar decomposition is ob- ± ±1 ± ± ± served in the degenerate case, leading to define the equivalent of τ¯ω¯ · c = c, τ¯ω¯ · (x (z)) = ω¯ x (z), τ¯ω¯ · ψ (z) = ψ (z). KP’s algebra for 4D theories. In this simple case, it is identified (2.3) with a certain limit of the q-Virasoro algebra [35] that, surpris- Here and ¯ ¯ define translations in the directions correspond- ingly, also coincides with the Zamolodchikov–Faddeev (ZF) algebra τω τω for the sine-Gordon model [36]. ing to the solitonic 5-branes and the D4-branes respectively. In addition, two reflection symmetries σV and σH (resp. along the 2. Degenerate DIM algebra and representations directions of D5/D4-branes and NS5/solitonic 5-branes) were introduced in [26,28]as involutive isomorphisms of algebra. Their 2.1. Definition of the algebra definition extend to the algebra (2.1), mapping it to the same algebra but with the opposite parameters εα →−εα, and acting on Following the analysis of the degenerate limit of the DIM alge- the generators as bra presented in Appendix A, we introduce the following algebra ± ∓ ± ± ± ± c = c · x z =−x z · z = z obeyed by four currents denoted x (z), ψ (z), and a central ele- σV ( ) , σV ( ) ( ), σV ψ ( ) ψ ( ), ± ± ± ∓ ment c, σH (c) =−c, σH · x (z) = x (−z), σH · ψ (z) = ψ (−z). ± ± [ψ (z), ψ (w)]=0, (2.4) ± ∓ g(z − w − cε+/2) ∓ ± Note that S2 is also involutive, and σ σ = S2. ψ (z)ψ (w) = ψ (w)ψ (z), V H g(z − w + cε+/2) The main difference between (2.1) and the DIM algebra is the ± ± + ± ±1 ± + absence of a mode expansion for the currents x (z) and ψ (z). ψ (z)x (w) = g(z − w ∓ cε+/4) x (w)ψ (z), − ± ± ± − This is due to the presence of zero modes in the horizontal repre- ψ (z)x (w) = g(z − w ± cε+/4) 1x (w)ψ (z), (2.1) sentation, but also to the fact that we use the Dirac delta function ± ± ± ± ± x (z)x (w) = g(z − w) 1x (w)x (z), instead of the multiplicative one. In practice, the algebra (2.1)is + − simply a convenient way to combine the algebraic relations satis- [x (z), x (w)] fied by the currents in the two types of representations. Yet, its ε1ε2 + =− δ(z − w + cε+/2)ψ (z + cε+/4) main interest lies in the possibility to define a co-algebraic struc- ε+ ture and, later on, the corresponding intertwiners. The coproduct − − δ(z − w − cε+/2)ψ (z − cε+/4) . can be seen as a degenerate limit of the Drinfeld coproduct for the DIM algebra, This algebra depends on two parameters ε1 and ε2, identified with the omega-background equivariant parameters. It has been con- =− = + venient to introduce the notation ε+ ε3 ε1 ε2, and the 2 We would like to thank Anton Nedelin, Sara Pasquetti and Yegor Zenkevich for function communicating this observation to us. 612 J.-E. Bourgine, K. Zhang / Physics Letters B 789 (2019) 610–619

+ + − + (x (z)) = x (z) ⊗ 1 + ψ (z − +c /4) ⊗ x (z − +c /2), ε (1) ε (1) ε1, ε2 are assumed to be real. The only non-trivial relation to be − − − + [ + −] (x (z)) = 1 ⊗ x (z) + x (z − ε+c /2) ⊗ ψ (z − ε+c /4), checked is the commutator x , x . It follows from the identities (2) (2) ± + − ± ± ± ξ (z) =: η (z ∓ ε+/4)η (z ± ε+/4):, together with pole decompo- z = z ∓ +c 4 ⊗ z ± +c 4 (ψ ( )) ψ ( ε (2)/ ) ψ ( ε (1)/ ), sition of the function (2.5) = + ε1ε2 1 − 1 ⇒ (c) = c(1) + c(2) with c(1) = c ⊗ 1 and c(2) = 1 ⊗ c. In fact, the S(z) 1 ε+ z z + ε+ whole Hopf algebra structure survives the limit, the co-unit taking (2.11) ± ± ε1ε2 the usual form (x (z)) = (c) = 0, (ψ (z)) = (1) = 1, and the [S(z)]+ −[S(z)]− = (δ(z) − δ(z + ε+)) , ± ± − antipode θ(c) =−c, θ(ψ (z)) = ψ (z) 1 and ε+ + − − + where [S(z)]± = S(z ∓ i0) defines the poles prescription (see Ap- θ(x (z)) =−ψ (z + ε+c/4) 1x (z + ε+c/2), pendix A). − − + − (2.6) θ(x (z)) =−x (z + ε+c/2)ψ (z + ε+c/4) 1. 2.3. Vertical representations and SHc algebra As in [38], the opposite (or permuted) coproduct can also be obtained as a twist of by the automorphism S2. Vertical representations of rank m and weight a = a1, ··· , am | 2.2. Horizontal representation act on a basis of states a, λ parameterized by a set of m Young diagrams λ = λ(1), ··· , λ(m), The horizontal representation is built using vertex operators (m) + −1 ρ (x (z)) |a, λ = δ(z − φ ) Res Y (z) |a, λ + x , acting on the Fock space of a free boson. The modes αn and P , Q a x λ z=φx [ ] = ∈ satisfy the standard Heisenberg commutation relations αn, αm x A(λ) nδ + and [P, Q ] = 1. The vacuum |∅ is annihilated by the (m) − n m x z |a = z − Res Y z + + |a − x ρa ( ( )) , λ δ( φx) λ( ε ) , λ , positive modes αn>0 and P , while negative modes αn<0 and z=φx x∈R(λ) Q create excitations. As usual the normal ordering : ···:is de- (m) ± | = | fined by moving the positive modes to the right. We also intro- ρ (ψ (z)) a, λ λ (z) ± a, λ . ∅| a duce the dual state annihilated by negative modes, so that (2.12) ∅|:···:|∅ = 1. In our construction, and just like in DIM’s horizontal representations [25], twists between positive and nega- We have denoted A(λ) (resp. R(λ) ) the set of boxes that can be tive modes have to be introduced, and it is convenient to de- added to (removed from) the Young diagrams composing λ. In fine addition, to each box x = (l, i, j) ∈ λ with (i, j) ∈ λ(l), we have − = + − + − z n zn associated the complex number φx al (i 1)ε1 ( j 1)ε2 ϕ+(z) = P log z − αn, ϕ−(z) = Q + α−n ⇒ that is sometimes called instanton position (in the moduli space). n n Y n>0 n>0 Finally, the functions λ (z) and λ (z) appearing in (2.12)are de- eϕ+(z)eϕ−(w) = (z − w) : eϕ+(z)eϕ−(w) : . (2.7) fined by Y + − In fact, the bosonic modes enter in the representation only (z ε+) x∈A(λ) z φx − − + = λ Y = = ϕ−(z ε2/2) ϕ−(z ε2/2) λ (z) , λ (z) . (2.13) through the combinations V −(z) e and Y − + − + − − λ(z) x∈R(λ) z ε φx V +(z) = eϕ+(z ε1/2) ϕ+(z ε1/2) that obey the normal-ordering property Parameters al and ε1, ε2 are chosen real, so that the poles lie on + − the real axis. The commutator [x , x ] in (2.1) follows from the −1 V +(z)V −(w) = S(z − w − ε+/2) : V +(z)V −(w) :, pole decomposition of λ (z). The other relations are derived using (z + ε1)(z + ε2) the shell formulas with S(z) = . (2.8) z(z + ε+) m Y = − − The function S(z) is related to the scattering function g(z) ap- λ (z) (z al) S(φx z), = pearing in (2.1)by g(z) = S(z)/S(−z), it satisfies the crossing l 1 x∈λ symmetry S(−ε+ − z) = S(z). Then, the horizontal representa- m + − = z ε+ al − tion of weight u can be defined in terms of the vertex opera- λ (z) g(z φx). (2.14) z − al tors l=1 x∈λ ± ± ± η (z) = V −(z ± ε+/4) 1 V +(z ∓ ε+/4) 1, In order to define the dual intertwiner, we need to introduce a | ± −1 dual basis a, λ on which acts the contragredient representation ξ (z) = V ±(z ∓ ε+/2)V ±(z ± ε+/2) , (2.9) (m) ˆ such that ρa it reads (m) ˆ(m) a, λ| ρ (e) |a, μ = a, λ| ρ (e) |a, μ . (2.15) (H) ± ±1 ± (H) ± ± (H) a a ρu (x (z)) = u η (z), ρu (ψ (z)) = ξ (z), ρu (c) = 1. The form of the contragredient representation depends on the def- (2.10) inition of the scalar product between vertical states. Like in the + − Note that ξ (z) (resp. ξ (z)) is expressed only in terms of pos- case of DIM’s vertical representations [28,26], and in contrast with itive (negative) modes. The algebraic relations (2.1) follow from our previous choice [32], it is convenient to introduce a non-trivial the normal-ordering properties of the vertex operators given in norm for these states, namely (B.1). Here, products of operators are ordered along the imagi- O O | = Z −1 nary axis: 1(z) 2(w) implies Im(z) < Im(w), and the parameters a, λ a, μ vect.(a, λ) δλ, μ , (2.16) J.-E. Bourgine, K. Zhang / Physics Letters B 789 (2019) 610–619 613

where Zvect.(a, λ) denotes the vector contribution to the N = 2 states |v, λ coincide with the one presented in [32], up to a instanton partition function.3 With this choice, the contragredi- change of states normalization.4 ent representation reads exactly as in (2.12), but with the action ± ∓ of x (z) and −x (z) exchanged. At first encounter, this choice 3. Reconstructing the gauge theories’ BPS quantities might look like an artificial way to introduce the vector contribution. However, the underlying algebraic structure is in fact very 3.1. Intertwiners rigid, and had we chosen a trivial norm, we would have recovered ∗ the vector contribution Zvect.(a, λ) in the normalization factor t In the algebraic engineering of 5D N = 1gauge theories, the ∗ λ of the dual intertwiner defined below. Indirectly here, we have role of the topological vertex is devoted to the intertwiners of the exploited the reflection symmetry σV in order to simplify the ex- DIM algebra, first introduced by Awata, Feigin and Shiraishi (AFS) ∗ pression of . in [27], and further generalized in [26,38]. In the 4D case, the ∗ The connection between the vertical representation (2.12) and intertwiners, denoted (m)[u, a] and (m)[u, a], are obtained as the action of SHc algebra on the basis |a, λ follows from the solution to the following equations, (m) ± ± ± ± decomposition ρ (x (z)) =[X (z)]+ −[X (z)]− and ψ (z) = a (H) (m) (m) (m) (H) [ ]± e [u a]= [u a] ⊗ e (z) with ρu ( ) , , ρa ρu ( ) , (3.1) (m) (H) ∗(m) ∗(m) (H) + 1 − ⊗ e [u a]= [u a] e | = Y 1 | + ρa ρu ( ) , , ρu ( ), X (z) a, λ Res λ (z) a, λ x , − z=φ z φx x x∈A(λ) where e is any of the four currents defining the algebra (2.1), is (2.18) 1 the coproduct given in (2.5), and the opposite coproduct ob- − | = Y + | − X (z) a, λ Res λ (z ε+) a, λ x , − z=φ tained by permutation. Note that the charge c is conserved in these z φx x x∈R(λ) relations. For each equation, the solution can be expanded on the vertical basis, their components are operators acting on the hori- and (z) |a, λ = (z) |a, λ . This operator can be expressed us- λ zontal Fock space: ing the half-boson (i.e. containing only commuting positive modes) (m) (z), (m)[u,a]= Z (a, λ) [u,a] a, λ| , vect. λ m λ z + ε+ − al + − − (z) = e(z εα) (z εα), m (m) (m) + z − al [ ]= [ ]: : l=1 α=1,2,3 u,a t u,a ∅(al) η (φx) , (2.19) λ λ ∞ l=1 ∈ 1 x λ = − ∗ (3.2) (z) log(z)0 n, ∗(m)[ ]= Z (m)[ ]| nzn u,a vect.(a, λ) u,a a, λ , n=1 λ λ which acts diagonally on the states |v, λ , m ∗(m) ∗(m) ∗ − [u,a]=t [u,a]: (a ) η (φ ) :, ⎛ ⎞ λ λ ∅ l x l=1 x∈λ | = ⎝ n⎠ | n v, λ φx v, λ , (m) m|λ| |λ| ∗(m) −|λ| where t [u, a] = (−1) u and t [u, a] = u are normal- x∈λ λ λ ⎛ ⎞ (2.20) ization coefficients, and |λ| denotes the total number of boxes con- | = ⎝ 1 ⎠ | tained in λ. In addition, the horizontal weights must also obey the ∂z(z) v, λ v, λ . m z − φx constraint u = (−1) u. The intertwiners (3.2)taken at m = 1de- ∈ x λ fine a degenerate version of the refined topological vertex, in the ± AFS free field presentation [27], that can be employed to construct The new operators X (z), (z) and ∂z(z) obey the algebraic re- directly 4D N = 2gauge theories. lations ∗ The definition of the vacuum components ∅(a) and ∅(a) − + − ε1ε2 (z) (w) in (3.2)requires some explanation. Physically, these operators de- [X (z), X (w)]= , ± ε+ z − w fine a sort of Fermi sea for which η creates excitations. Thus, (2.21) ± 1 ± and by analogy with AFS intertwiners, we would like to de- [∂z(z), X (w)]=± X (w), z − w fine them as a (normal-ordered) infinite product of the opera- ± −1 √ tors η (φx) over the boxes of a fully filled Young diagram, i.e. leading to identify them with the SHc currents − D± z , ε1ε2 1( ) φx = a + (i − 1)ε1 + ( j − 1)ε2 with i, j = 1 ···∞. However, in E(z) and D0(z) (respectively) in the holomorphic presentation of contrast with the DIM scenario, it is not possible to express the reference [32]. Furthermore, the actions (2.18) and (2.20)on the resulting operators as vertex operators built upon the modes αn.

3 We remind here the formula for the contributions to the instanton partition 4 Alternatively, it is possible to expand the currents at inﬁnity, thus deﬁning the Z functions of the multiplets vector vect.(a, λ) and bifundamental (with mass m) modes − Z (a, λ; b, μ|m): Z (a, λ) = Z (a, λ; a, λ|0) 1 and bfd. vect. bfd. ∞ ∞ ± = −n−1 ± = + −n−1 X (z) z Xn , (z) 1 ε+ z n, (2.22) Zbfd.(a, λ; b, μ|m) = (φx − bl + ε+ − m) × (φx − al + m) n=0 n=0 x∈λ l x∈μ l ± and identify√ Xn , n and n with, respectively, the representation of the SHc generators −ε ε D± , E and D + . The latter satisfy the algebraic relations × S(φx − φy − m). (2.17) 1 2 1,n n 0,n 1 [D , D ] = 0, [D + , D± ] =±D± + , [D− , D ] = E + and a certain x∈λ 0,n 0,m 0,n 1 1,m 1,n m 1,n 1,m n m ∈ y μ exponential relation between En and D0,n deriving from (2.19). 614 J.-E. Bourgine, K. Zhang / Physics Letters B 789 (2019) 610–619

We propose two different approaches to deal with this issue. The under the identification of a with the Coulomb branch vevs, and ∗ first one is to cut-off the size of the infinite Young diagram, effec- q = (−1)mu/u with the exponentiated gauge coupling. The parti- tively regularizing the double product to i, j = 1 ···N, and sending tion function can also be recovered as a scalar product of Gaiotto N to infinity at the end of the calculation. This approach is a lit- states, corresponding here to the vevs of the intertwiners: tle cumbersome as it introduces extra factors that are eventually ∗ ∗ |G,a = (1 ⊗ ∅|) (m)[u ,a]|∅ canceled. Alternatively, it is possible to introduce an auxiliary free ˜ = ˜ − − ˜ + ∗(m) ∗ boson ϕ(z) such that ϕ−(z) ϕ−(z ε1/2) ϕ−(z ε1/2) and = Z (a, λ) ∅| [u ,a]|∅ |a, λ , vect. λ ϕ+(z) = ϕ˜+(z + ε2/2) − ϕ˜+(z − ε2/2). Assuming that the propa- λ gator of the boson ϕ˜(z) is given by (minus) the log of a function (3.8) G,a|= ∅|(m)[u,a] (1 ⊗|∅ ) F (z) (conveniently shifted here by +ε+/2), the relations between the positive/negative modes of the bosons ϕ and ϕ˜ imply a certain (m) = Zvect.(a, λ) ∅| [u,a]|∅ a, λ| , functional relation for the propagator5: λ λ ϕ˜(z) ϕ˜(w) −1 ϕ˜(z) ϕ˜(w) = − + + : : e e F (z w ε /2) e e , so that TU (m) = G,a|G, a . In fact, these Gaiotto states coincide F (z + ε1)F (z + ε2) (3.4) with those defined in [32]if we take into account the different = z. ± F (z)F (z + ε+) normalization of the basis. The action of SHc currents X (z) on these states has been derived in [32]. It can be recovered here us- This relation can be solved using the double-gamma function, ing the intertwining property (3.1). For instance, F (z) = 2(z) with pseudo-periods ε1 and ε2 (see [39,41]). Defining (m) + ∗ −ϕ˜−(a−ε+/4) −ϕ˜+(a−3ε+/4) ρ (x (z)) |G,a = − u ([Y(z + ε+)]+ −[Y(z + ε+)]−) |G,a , ∅(a) = e e , a (3.5) (m) − | = − ∗ −1 [Y −1] −[Y −1] | ∗ ϕ˜−(a−3ε+/4) ϕ˜+(a−ε+/4) ρ (x (z)) G,a (u ) (z) + (z) − G,a , ∅(a) = e e , a (3.9) all the necessary normal-ordering relation can now be derived, they have been gathered in Appendix B.2. Adapting the method where Y(z) is a diagonal operator in the basis |a, λ with eigen- Y described in the Appendix C of [26], they can be used to show that values λ (z). the expressions given in (3.2)indeed solve the equations (3.1). By construction, the operator TU (m) commutes with the action of any element e of the algebra (2.1)in the appropriate represen- 3.2. Instanton partition functions and qq-characters tation, (H) (H) (H) (H) From this point, we can repeat the construction presented in ⊗ ∗ T = T ⊗ ρu ρu (e) U (m) U (m) ρu ρu∗ (e) . (3.10) [26]for the instanton partition functions and qq-characters of linear quivers gauge theory. For the purpose of illustration, we The fundamental qq-character can be obtained by insertion of the + 6 present here the main results concerning (A1 quiver) pure U (m) current x (z) before taking the vev, gauge theories. The corresponding T -operators are obtained by (m) ∗(m) ∗ coupling an intertwiner [u, a] to its dual [u , a] through χ(z − ε+/2) · a scalar product in their common vertical representation, (H) (H) + ∅| ⊗ ∅| ⊗ ∗ T |∅ ⊗ |∅ ∗ −1 ( ) ρu ρu (x (z)) U (m) ( ) ∗ ∗ = (u ) . T = (m)[u,a]· (m)[u ,a] ( ∅| ⊗ ∅|) T (|∅ ⊗ |∅ ) U (m) U (m) (m) ∗(m) ∗ (3.11) = Z (a, λ) [u,a]⊗ [u ,a]. (3.6) vect. λ λ λ After evaluation of the vev using the normal-ordering relations (B.3)of Appendix B, we recover the well-known expression This operator acts on the tensor product of two bosonic Fock spaces, it is easy to check that its vev reproduces the instanton −1 |λ| q χ(z) = Z [U(m)] q Z (a, λ) Y (z + ε+) + . partition function of the gauge theory, inst. vect. λ Y λ (z) λ Z [ ]= ∅| ⊗ ∅| T |∅ ⊗ |∅ = |λ|Z inst. U(m) ( ) U (m) ( ) q vect.(a, λ), (3.12) λ The covariance Property (3.10)of the operator T implies the (3.7) U (m) cancellation of poles between the two terms in (3.12), and so χ(z) is a polynomial of degree m. Higher qq-characters are constructed + 5 by insertion of multiple operators x (zi ) in the vevs. Note however, that the asymptotic expansion at z =∞ of the function 2(z) (see, for instance, reference [39]), These results can be extended to linear quivers of higher rank. The corresponding T -operator is obtained by coupling several of 2 z 3 ε+ T log 2(z) ∼− log(z) − + z (log(z) − 1) + O (log(z)) (3.3) the operators U (m) defined in (3.6)in the horizontal channel. Bi- 2ε1ε2 2 2ε1ε2 fundamental contributions Zbfd. to the partition function appear as is not compatible with the usual mode expansion (2.7)of a free boson. To re- the product of intertwiners are normal-ordered following the two solve this, it is possible to modify the expression of the propagator, defining relations F (z) = θ2(z)2(z) where θ2(z) obeys the same functional equation as 2(z) but with 1instead of z in the RHS, and possesses an asymptotic expansion that cancels the unwanted terms in (3.3). However, it turns out that the one-loop determinant contributing to the partition function [40]is usually regularized using the function 6 Note that, as consequence of the invariance of the brane web under the reflec- + − 2(z) [41]. Another possibility is to add extra zero-modes to the boson ϕ˜(z) to gen- tion encoded by σV , we find the same quantity if we replace x (z) by x (z) (and ∗ − erate the terms z2 log z, z log z, .... (u ) 1 by u ) in this formula. J.-E. Bourgine, K. Zhang / Physics Letters B 789 (2019) 610–619 615

∗ (m)[ ] ∗(m )[ ∗ ∗] u,a ∗ u ,a of KP’s q-Virasoro algebra and a q-Heisenberg algebra [20,26]. The λ λ ¯ ± ∗ latter encodes the action of the currents ψ (Z), it can be factor- = − m|λ |Z ; ∗ ∗| ± ( 1) bfd.(a, λ a , λ ε+/2) ized out of the action of x¯ (Z), thereby providing the action of ∗ m,m q-Virasoro on the T -operator. Here, we would like to repeat this ∗ ∗ (m) ∗(m ) ∗ ∗ × F (a − a ∗ + ε+/2) : [u,a] [u ,a ]:, construction for the degenerate DIM algebra (2.1). l l λ λ∗ l,l∗=1 The first step is to isolate the q-Heisenberg action. For this pur- ∗ ∗(m ) ∗ ∗ (m) pose we need to introduce an auxiliary bosonic field ϕW (z), with [u ,a ] [u,a] λ∗ λ propagator log h(z), such that the (half) vertex operators V ±(z) are ∗ = ϕW ±(z)7 m |λ| ∗ ∗ decomposed into products of W±(z) e : = (−1) Zbfd.(a, λ; a , λ |ε+/2) ∗ m,m V ±(z) = W±(z + ε+/2)W±(z − ε+/2), ∗ ∗ (m) ∗(m ) ∗ ∗ (3.14) × F (a ∗ − a + ε+/2) : [u,a] [u ,a ]:. = − : : l l λ λ∗ W+(z)W−(w) h(z w) W+(z)W−(w) . l,l∗=1 It turns out convenient to introduce also the function f (z) = (3.13) − − h(z − ε+/2) 1h(z + ε+/2) 1. Since the decomposition (3.14)has In these formulas, the F -dependent factor can be interpreted as to reproduce the normal-ordering property (2.8)for the vertex op- one-loop bifundamental contributions with the 2-regularization erators V ±(z), we find the functional relation f (z) f (z +ε+) = S(z) of infinite products. The fundamental qq-characters attached to constraining f (z) in terms of the function S(z). The exact formula the two nodes at the extremity of the quiver are obtained by in- ± for the function f (z) will be specified later. Using the decomposi- sertion of x (z) (with repeated coproduct). On the other hand, tion (3.14), it is easy to observe that the representation ρ(H) ⊗ρ(H) qq-characters corresponding to middle nodes require the introduc- ± ± of the currents x (z), ψ (z) can be written as8 tion of more involved operators [26]. (H) ⊗ (H) ± ρu ρu (x (z)) 3.3. Degenerate limit of Kimura–Pestun’s quiver W-algebra 1 √ 2 ±1 ±1 ±1 = u1u2 U−(z ± ε+/2) T (z)U+(z ∓ ε+/2) , (3.15) Instanton partition functions of 5D N = 1quiver gauge the- (H) ⊗ (H) ± = ∓ ± −1 ρu ρu (ψ (z)) U±(z ε+)U±(z ε+) , ories are covariant under the action of two distinct q-deformed 1 2 W-algebras. The first one is involved in the q-deformed AGT cor- where we have introduced the following quantities: respondence [42], it appears at each individual node of the quiver. U±(z) = W±(z ± ε+/4) ⊗ W±(z ∓ ε+/4), For instance, the partition function of a linear quiver theory bear- − ing a U (m) gauge group at each node coincides with a conformal T (z) = R(z)+:R(z − ε+) 1 :, block of the q-Wm-algebra. A second form of q-W-algebra covari- u2 − − (3.16) ance has appeared in the work of Kimura and Pestun (KP) [34]. R(z) = W−(z + ε+/4) 1 W+(z − ε+/4) 1 In their construction, the Dynkin diagram of the algebra is identi- u1 fied with the quiver of the gauge theory. Thus, the partition func- ⊗ W−(z − ε+/4)W+(z + ε+/4). tion of a linear quiver with k nodes is covariant under the action It is possible to refine the decomposition (3.15), observing that of a q-Wk+1-algebra. The DIM algebra offers a third perspective which is somehow more elementary as it allows us to understand the vertex operators U±(z) and R(w) commute as a result of the the relation between these two q-W-algebras (at least in the case cancellation between the factors obtained in each channel when of linear quivers). For instance, each node of a quiver bearing a normal-ordering. Commuting the factors of U±(z) with T (z), we ± ± U (m) gauge group corresponds to a stack of m D5-branes in the can write the representation of the currents x (z) and ψ (z) in (p, q)-brane construction. Accordingly, the node carries a vertical the form representation (0, m) that contains the q-W action involved in m (H) ⊗ (H) ± = ± the q-AGT correspondence. On the other hand, the DIM algebra ρu1 ρu2 (x (z)) V B (z)T (z), acts on the T -operators in a different representation, obtained as (H) (H) ± + − ρu ⊗ ρu (ψ (z)) =: V (z ∓ ε+/2)V (z ± ε+/2) :, a tensor product of horizontal representations (1, n) attached to 1 2 B B the external NS5-branes (possibly dressed by D5s). This tensored (3.17) √ horizontal representation factorizes into a q-Heisenberg contribu- ± ±1 ±1 ±1 where V (z) = u1u2 U−(z ± ε+/2) U+(z ∓ ε+/2) com- tion, and the action of KP’s q-W-algebra [20,26]. Furthermore, the B mutes with T (w). As in the case of the DIM algebra, we observe automorphism S discovered by Miki [23]maps vertical to hori- ± that the action of the currents ψ (z) is given only in terms of zontal representations (and vice-versa), effectively implementing a ± the vertex operators V B (z). These operators can be factorized out rotation of the (p, q)-brane web, and thereby exchanging the role ± of the action of x (z), leaving only the operator T (z). The latter of the two different q-W-algebras. We refer to the upcoming paper 9 [38]for a deeper discussion of this automorphism in the formalism obeys the following weighted commutation relation, of algebraic engineering.

In this section, we would like to define the degenerate ver- 7 In terms of bosonic fields, we have the relations ϕW +(z + ε+/2) + ϕW +(z − sion of KP’s quiver W-algebra that acts on partition functions of ε+/2) = ϕ+(z+ε /2) −ϕ+(z−ε /2) and ϕ −(z+ε+/2) +ϕ −(z−ε+/2) = ϕ−(z− N = 1 1 W W 4D 2gauge theories. We will restrict ourselves to the case of ε2/2) − ϕ−(z + ε2/2) for positive/negative modes. a single node quiver with U (m) gauge group. For the 5D version 8 It is possible to work with either coproduct or by simply permuting the of the gauge theory, the relevant q-W-algebra is the q-Virasoro two Fock spaces. Here we chose in agreement with the previous section. 9 This relation is obtained by first computing the normal-ordering algebra, constructed in [35]as a q-deformation of the usual Vi- − − − rasoro algebra. In the DIM description, the representation acting T (z)T (w) = f (z − w) 1 : R(z)R(w) :+:R(z − ε+) 1 R(w − ε+) 1 : on the T -operator has levels (2, 0), it is obtained by coproduct − + f (z − w + ε+) : R(z)R(w − ε+) 1 : of the two representations (1, n) ⊗ (1, −n) associated to the two − (dressed) NS5-branes. It can be decomposed as a tensor product + f (z − w − ε+) : R(z − ε+) 1 R(w) :, (3.18) 616 J.-E. Bourgine, K. Zhang / Physics Letters B 789 (2019) 610–619 f (z − w)T (z)T (w) − f (w − z)T (w)T (z) for 3D T [U (N)] theories, it provides an additive analogue of the q-Virasoro algebra called d-Virasoro [43]. Screening currents, ver- =−ε1ε2 − + − − − (δ(z w ε+) δ(z w ε+)) , (3.19) tex operators and Dotsenko–Fateev representations of conformal ε+ blocks were also defined in this paper. Furthermore, a correspon- that will be identified with a certain degenerate limit of the q- dence with conformal blocks of ordinary Virasoro algebra was also Virasoro algebra. We claim that this algebra is the degenerate observed, it suggests the presence of an analogue of Miki’s au- version of KP’s quiver W-algebra in the case of a single node. Re- tomorphism in the degenerate case. Indeed, in the self-dual pure peating this procedure for an arbitrary quiver, it should be possible U (2) gauge theory, the exchange of vertical/horizontal representa- to formulate the general degenerate KP algebras. tions induced by the automorphism should reduce to an exchange The q-Virasoro algebra is defined in [35]in terms of its stress– of the role played by the degenerate KP algebra (d-Virasoro) and energy tensor T¯ (Z), a current that obeys the algebraic relation the ordinary Virasoro. We hope to come back to this problem in a near future. ¯ ¯ ¯ ¯ ¯ ¯ f (W /Z)T (Z)T (W ) − f (Z/W )T (W )T (Z) Interestingly, the d-Virasoro algebra is also related to the 2D − − (massive) sine-Gordon model describing the scaling limit of the = (1 q1)(1 q2) ¯ − ¯ δ(Z/q3 W ) δ(q3 Z/W ) , (3.20) XYZ model.11 Indeed, in the short note [36], Lukyanov noticed that (1 − q1q2) the ratio12 − where q = q and q = t 1 are two complex parameters, q = 1 2 3 f (z) tan (π(z + ε2)/2ε+) −1 −1 ¯ = (3.26) q1 q2 , δ(z) is the multiplicative delta function (A.4), and the − − ¯ f ( z) tan (π(z ε2)/2ε+) function f (Z) writes (the second equality is valid when |q3| < 1): reproduces the scattering factor of the sine-Gordon model ∞ − n − n ¯ 1 (1 q1)(1 q2) n + f (Z) = exp − Z sinh β i sin πξ n (1 + q n) (3.27) n=1 3 sinh β − i sin πξ ∞ −1 2n −1 2n 1 (1 − q q Z)(1 − q q Z) = + =− + = 1 3 2 3 setting β iπ z/ε , and ξ ε2/ε (note that the ordering − + − + . (3.21) − − 1 2n 1 − 1 2n 1 Im z1 < Im z2 becomes Re β1 > Re β2). This observation, together 1 Z = (1 q q Z)(1 q q Z) n 0 1 3 2 3 with the existence of simple poles at β =±iπ (corresponding to ¯ This function f (Z) satisfies the functional relation the delta functions in (3.19)), led him to identify T (z) with the generator of the corresponding Zamolodchikov–Faddeev (ZF) alge- − − bra [46,47]. In this context, the operator T creates a breather ¯ ¯ −1 (1 q1 Z)(1 q2 Z) (β) = ¯ ¯ = 13 f (Z) f (q3 Z) S(Z), with S(Z) . with rapidity β, the basic particle of the theory. (1 − Z)(1 − q1q2 Z) (3.22) 4. Perspectives In order to recover the relation (3.19)for (3.20), we need to take a degenerate limit. Just like in the case of DIM (see Appendix A), We have presented here an algebraic construction for the BPS Rε Rz Rw ¯ quantities of 4D N = 2SYM theories based on a degenerate ver- we set qα = e α , Z = e , W = e , and send R → 0. Then, S(Z) sion of the DIM algebra. The two essential ingredients are the tends to S(z) and the functional relation (3.22)reproduces the one action of the SHc algebra (vertical representation), and an algebra satisfied by the function f (z) (which is in fact also satisfied by of twisted vertex operators (horizontal representation). It would be f (−z) due to the crossing symmetry S(−ε+ − z) = S(z)). Since this interesting to understand how the latter fits into the formalism of functional equation is the only constraint for the (entire) function Maulik–Okounkov [50], and to relate the intertwiners to Smirnov’s ¯ −1 f (z), we choose to identify f (z) with the limit of f (Z ): instanton R-matrix [51]. For linear quivers, the T -operator con- ¯ −Rz structed here is expected to be the Baxter T-operator of a 1D f (z) = lim f (e ) R→0 quantum integrable system, with two spectral parameters associ- ∞ ated to vertical and horizontal weights, that produces the set of z − ε2 (z + ε2 + 2nε+)(z − ε2 + (2n + 1)ε+) = . (3.23) commuting Hamiltonians by expansion. Given the rich structure z (z − ε2 + 2nε+)(z + ε2 + (2n + 1)ε+) of this algebra, there is little doubt that the underlying statistical n=0 model will reveal itself of great interest. Applying the limiting procedure of Appendix A to treat the delta Following the program initiated in [52], it should be possible functions in the RHS of (3.20), it is easy to show that the limit T (z) to take the Nekrasov–Shatashvili limit ε → 0of the algebra (2.1). ¯ 2 of T (Z) obeys (3.19). Note that this degenerate limit is different In this way, one hopes to recover the quantum algebra behind the from the usual one that sends q-Virasoro to Virasoro, and Macdon- integrable system of the Bethe/gauge correspondence [5]. This con- 10 ald to Jack polynomials. This particular limit of the q-Virasoro struction should provide us with a proper interpretation for the algebra has already been encountered in the study of dualities ε2-deformation of quantum integrable systems, and reproduce the

using (3.14). Multiplying this relation with f (z − w), the function S(z − w) appears 11 This connection is a priori unrelated with the one observed in [44] between in the RHS. In order to recover (3.19), we take the difference between the two pure SU(2) gauge theory and the 1D sine-Gordon model. normal-ordering relations with z and w exchanged. The ﬁrst terms cancel, while 12 We use here the inﬁnite product formula GR.1.431.1 and GR.1431.3 of [45]for the remaining ones produce the δ-function through the pole decomposition (2.11) the trigonometric functions, of the function S(z). ∞ ∞ 10 Rxi 2 2 Presumably, in our limit the variables of the polynomial Xi = e are also sent z 4z sin z = z 1 − , cos z = 1 − . (3.25) to one, so that the Macdonald operator does not reduce to the Calogero–Sutherland 2 2 + 2 2 = k π = (2k 1) π Hamiltonian, but instead produces a difference operator k 1 k 0

−1 13 q X − X X ∂ − − See also [48,49]forthe derivation of the q-Virasoro algebra as the ZF algebra of 2 i j i Xi xi x j ε2 ε ∂ D = q → e 1 xi (3.24) q,t 1 . the ﬁrst breather in the scaling limit of the spin 1/2 XYZ model using Bethe Ansatz Xi − X j xi − x j i j=i i j=i techniques. J.-E. Bourgine, K. Zhang / Physics Letters B 789 (2019) 610–619 617

subleading order terms in ε2 derived in [53,54]by expansion of in the last relation have to be treated carefully. In the DIM alge- the partition function. bra, these delta functions are multiplicative ones, they arise from Finally, the second main result of this note is the identification the analytical continuation around a pole at Z = 1in the complex of the degenerate version of KP’s quiver W-algebra. In the A1 case, plane we observed an unsuspected connection with the ZF algebra of the sine-Gordon model. This is particularly intriguing given the simi- 1 1 δ(¯ Z) = Zk = − , (A.4) larity between the algebraic engineering and the method devel- − − ∈Z Z 1 +¯ Z 1 −¯ oped by Lukyanov to compute form factors of massive integrable k models [55]. Indeed, the vertical representation is reminiscent of where [ f (Z)]+¯ denotes the Laurent series in Z obtained after ex- the ZF algebra’s representation πA associated to asymptotic states, pansion of the function f (Z) for |Z| > 1 and, [ f (Z)]−¯ the series while both horizontal representation and Lukyanov’s πZ are built obtained by expansion for |Z| < 1. Taking the limit R → 0, we find upon free fields. It appears essential to understand better the con- that nection between the two methods. In this perspective, one could R 1 try to extend our algebraic construction to more general scatter- lim = , (A.5) Rz ing functions g(z). We hope to come back to this issue in a future R→0 e − 1 ±¯ z ± publication. where the subscript ± now has a different meaning which we ex- plain now. For definiteness, we assume that R is a purely positive Acknowledgements imaginary number, i.e. R ∈ iR>0. This corresponds to the choice of JEB would like to thank Davide Fioravanti and Vincent Pasquier an axis for ordering the variables z, and we take here the imagi- for discussions, and the IPhT CEA-Saclay, DESY, and the Univer- nary axis. Then, the condition |Z| > 1 corresponds to Im z < 0 (and sity of Bologna for hospitality at various stages of this project. KZ obviously |Z| < 1to Im z > 0). Thus, under the inverse conformal (Hong Zhang) thanks Wei Li for guidance, and Shuichi Yokoyama map z(Z), the complex plane is sent to an infinite cylinder of ra- for discussions. The work of KZ is partially supported by the Gen- dius i/R, and the radial ordering is mapped to a “chronological” eral Financial Grant from the China Postdoctoral Science Founda- ordering along the imaginary axis identified with the axis of the tion, with Grant No. 2017M611009. cylinder. In the limit R → 0, the pole at Z = 1 becomes a pole at z = 0, and the prescription [···]± should be defined along the Appendix A. Degenerate limit of DIM algebra imaginary axis, In the second Drinfeld presentation, the DIM algebra is defined 1 1 ¯± ¯ ± = ⇒ in terms of four currents x (Z) and ψ (Z), and a central ele- z ± z ∓ i0 ment c, that obey the algebraic relations, (A.6) ¯ Rz ε1ε2 1 1 ε1ε2 [ ¯ ± ¯ ± ]= lim κδ(e ) =− − =− δ(z). ψ (Z), ψ (W ) 0, R→0 ε+ z − i0 z + i0 ε+ c/2 + − g¯(q Z/W ) − + ψ¯ (Z)ψ¯ (W ) = 3 ψ¯ (W )ψ¯ (Z), Appendix B. Formulas for operators’ normal-ordering ¯ −c/2 g(q3 Z/W ) ± ± ± ± ± x¯ (Z)x¯ (W ) = g¯(Z/W ) 1x¯ (W )x¯ (Z), In this section, we gather several useful identities for the com- putation of normal-orderings. In order to write shorter formulas, + ± ±c/4 ± ± + ψ¯ (Z)x¯ (W ) = g¯(q Z/W ) 1x¯ (W )ψ¯ (Z), 3 we will abbreviate the expressions of the type O1(z)O1(w) = − ± ∓c/4 ±1 ± − − : O O O O :: − ψ¯ (Z)x¯ (W ) = g¯(q Z/W ) x¯ (W )ψ¯ (Z), S12(z w) 1(z) 2(w):into 1(z) 1(w) S12(z w). Start- 3 ± ± + − ing with the vertex operators η (z) and ξ (z) defined in (2.9), we [x¯ (Z), x¯ (W )] find the following non-trivial normal-ordering relations: ¯ −c/2 ¯ + −c/4 ¯ c/2 ¯ − c/4 = κ δ(q Z/W )ψ (q Z) − δ(q Z/W )ψ (q Z) . ± ± −1 3 3 3 3 η (z)η (w) :: S(∓z ± w) , (A.1) ± ∓ η (z)η (w) :: S(z − w − ε+/2), The algebra depends on the complex parameters q1, q2, q3, + ± ± ξ (z)η (w) :: g(z − w ∓ ε+/4) 1, with the constraint q1q2q3 = 1, entering in the definition of the (B.1) ± − ± C-number κ and the function g¯(z): η (z)ξ (w) :: g(z − w ∓ ε+/4) 1, − − (1 − q1)(1 − q2) 1 − qα Z + − g(z w ε+/2) κ = , g¯(Z) = . (A.2) ξ (z)ξ (w) :: . −1 − + (1 − q1q2) 1 − q Z g(z w ε+/2) α=1,2,3 α Next, using the Definition (3.5)of the intertwiners’ vacuum com- In order to define the degenerate limit of the algebra, we in- ponents, we find troduce a parameter R, interpreted as the radius of the com- Rε pact dimension for the 5D background, and write qα = e α , Z = + −1 η (z)∅(a) :: (z − a) , eRz and W = eRw. The algebra is invariant under the rescaling + − ± ±1 ± :: − − 1 x¯ (Z) → ω¯ x¯ (Z) (the equivalent of the automorphism τ¯ω¯ in ∅(a)η (z) (a z ε+) , (2.3)), which leads to some freedom for the definition of the cur- − η (z)∅(a) :: (z − a + ε+/2), rents’ limit: − ∅(a)η (z) :: (a − z − ε+/2), ± ±α ¯± Rz ± ¯ ± Rz x (z) = lim R x (e ), ψ (z) = lim ψ (e ). (A.3) − + R→0 R→0 + z a 3ε+/4 ξ (z)∅(a) :: , − − As R is sent to zero, the function g¯ (eRz) tends to g(z), and the re- z a ε+/4 + − lations (A.1)reduce to (2.1). However, the delta functions entering ∅(a)ξ (z) :: 1,ξ(z)∅(a) :: 1, (B.2) 618 J.-E. Bourgine, K. Zhang / Physics Letters B 789 (2019) 610–619

− z − a + ε+/4 ∅(a)ξ (z) :: , [11] S. Kanno, Y. Matsuo, H. Zhang, Extended conformal symmetry and recursion z − a + 5ε+/4 formulae for Nekrasov partition function, J. High Energy Phys. 08 (2013) 028, + ∗ :: − + arXiv:1306 .1523 [hep -th]. η (z)∅(a) (z a ε+/2), [12] T. Procházka, W-symmetry, topological vertex and affine Yangian, J. High En- ∗ + ergy Phys. 10 (2016) 077, arXiv:1512 .07178 [hep -th]. ∅(a)η (z) :: (a − z − ε+/2), [13] A. Tsymbaliuk, The affine Yangian of gl1 revisited, Adv. Math. 304 (2017) − ∗ −1 583–645, arXiv:1404 .5240 [math .RT]. η (z) (a) :: (z − a + ε+) , ∅ [14] O. Aharony, A. Hanany, Branes, superpotentials and superconformal fixed ∗ − −1 points, Nucl. Phys. B 504 (1997) 239–271, arXiv:hep -th /9704170 [hep -th]. ∅(a)η (z) :: (a − z) , [15] O. Aharony, A. Hanany, B. Kol, Webs of (p, q) 5-branes, five dimensional field − + + theories and grid diagrams, J. High Energy Phys. 9801 (1998) 002, arXiv:hep - + ∗ :: z a ε /4 ξ (z)∅(a) , th /9710116 [hep -th]. z − a + 5 + 4 ε / [16] M. Aganagic, A. Klemm, M. Marino, C. Vafa, The topological vertex, Commun. ∗ + − ∗ ∅(a)ξ (z) :: 1,ξ(z)∅(a) :: 1, Math. Phys. 254 (2) (2005) 425–478. [17] A. Iqbal, C. Kozcaz, C. Vafa, The refined topological vertex, J. High Energy Phys. − + ∗ − :: z a 3ε+/4 2009 (0910) (2009) 069, arXiv:hep -th /0701156 [hep -th]. ∅(a)ξ (z) . [18] H. Awata, H. Kanno, Instanton counting, Macdonald functions and the moduli z − a − ε+/4 space of D-branes, J. High Energy Phys. 05 (2005) 039, arXiv:hep -th /0502061 Finally, we present the relations involving intertwiners: [hep -th]. [19] N. Leung, C. Vafa, Branes and toric geometry, Adv. Theor. Math. Phys. 2 (1998) + (m) −1 91–118, arXiv:hep -th /9711013 [hep -th]. η (z) [u,a]::Y (z) , λ λ [20] A. Mironov, A. Morozov, Y. Zenkevich, Ding–Iohara–Miki symmetry of network − (m) matrix models, Phys. Lett. B 762 (2016) 196–208. η (z) [u,a]::Y (z + ε+/2), λ λ [21] H. Awata, H. Kanno, T. Matsumoto, A. Mironov, A. Morozov, A. Morozov, Y. + (m) Ohkubo, Y. Zenkevich, Explicit examples of DIM constraints for network ma- ξ (z) [u,a]:: (z − ε+/4), trix models, J. High Energy Phys. 07 (2016) 103. λ λ [22] J. Ding, K. Iohara, Generalization of Drinfeld quantum affine algebras, Lett. + (m)∗ η (z) [u,a]::Y (z + ε+/2), Math. Phys. 41 (2) (1997) 181–193. λ λ [23] K. Miki, A (q, γ ) analog of the W 1+∞ algebra, J. Math. Phys. 48 (12) (2007) − ∗ − (m) [ ]::Y + 1 3520. η (z) u,a λ (z ε+) , λ [24] B. Feigin, E. Feigin, M. Jimbo, T. Miwa, E. Mukhin, Quantum continuous gl∞: + (m)∗[ ]:: + −1 semi-infinite construction of representations, Kyoto J. Math. 51 (2) (2011) ξ (z) u,a λ (z ε+/4) , λ (B.3) 337–364. (m) + m −1 [25] B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi, S. Yanagida, A commutative [u,a]η (z) :: (−1) Y (z + ε+) , 1 λ λ algebra on degenerate CP and Macdonald polynomials, J. Math. Phys. 50 (9) (m) − m (Sept. 2009) 095215, arXiv:0904 .2291 [math .CO]. [u,a]η (z) :: (−1) Y (z + ε+/2), λ λ [26] J.-E. Bourgine, M. Fukuda, K. Harada, Y. Matsuo, R.-D. Zhu, (p, q)-webs of DIM representations, 5d N = 1 instanton partition functions and qq-characters, (m) − −1 [u,a]ξ (z) :: (z + ε+/4) J. High Energy Phys. 11 (2017) 034, arXiv:1703 .10759 [hep -th]. λ λ [27] H. Awata, B. Feigin, J. Shiraishi, Quantum algebraic approach to refined topo- (m)∗ + [ ] :: − mY + logical vertex, J. High Energy Phys. 03 (2012) 041, arXiv:1112 .6074 [hep -th]. u,a η (z) ( 1) λ (z ε+/2), λ [28] J.-E. Bourgine, M. Fukuda, Y. Matsuo, R.-D. Zhu, Reflection states in Ding– ∗ (m) − m −1 Iohara–Miki algebra and brane-web for D-type quiver, J. High Energy Phys. 12 [u,a]η (z) :: (−1) Y (z) , λ λ (2017) 015, arXiv:1709 .01954 [hep -th]. (m)∗ − [29] N. Nekrasov, V. Pestun, S. Shatashvili, Quantum geometry and quiver gauge the- [u,a]ξ (z) :: (z − ε+/4). λ λ ories, arXiv:1312 .6689 [hep -th]. [30] N. Nekrasov, BPS/CFT correspondence: non-perturbative Dyson–Schwinger equations and qq-characters, arXiv:1512 .05388. References [31] H.-C. Kim, Line defects and 5d instanton partition functions, J. High Energy Phys. 03 (2016) 199, arXiv:1601.06841 [hep -th]. [32] J.-E. Bourgine, Y. Matsuo, H. Zhang, Holomorphic field realization of SHc and [1] N. Seiberg, E. Witten, Monopoles, duality and chiral symmetry breaking in quantum geometry of quiver gauge theories, J. High Energy Phys. 04 (2016) = N 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484–550, arXiv:hep -th / 167, arXiv:1512 .02492 [hep -th]. 9408099 [hep -th]. [33] J.-E. Bourgine, M. Fukuda, Y. Matsuo, H. Zhang, R.-D. Zhu, Coherent states [2] N. Seiberg, E. Witten, Electric–magnetic duality, monopole condensation, and in quantum W +∞ algebra and qq-character for 5d Super Yang–Mills, PTEP = 1 confinement in N 2 supersymmetric Yang–Mills theory, Nucl. Phys. B 426 2016 (12) (2016) 123B05, arXiv:1606 .08020 [hep -th]. (1994) 19–52, arXiv:hep -th /9407087 [hep -th], Erratum: Nucl. Phys. B 430 [34] T. Kimura, V. Pestun, Quiver W-algebras, Lett. Math. Phys. 108 (6) (2018) (1994) 485. 1351–1381, arXiv:1512 .08533 [hep -th]. [3] A. Gorsky, I. Krichever, A. Marshakov, A. Mironov, A. Morozov, Integrability and [35] J. Shiraishi, H. Kubo, H. Awata, S. Odake, A quantum deformation of the Vi- Seiberg–Witten exact solution, Phys. Lett. B 355 (1995) 466–474, arXiv:hep -th / rasoro algebra and the Macdonald symmetric functions, Lett. Math. Phys. 38 9505035 [hep -th]. (1996) 33–51, arXiv:q -alg /9507034 [q -alg]. [4] R. Donagi, E. Witten, Supersymmetric Yang–Mills theory and integrable sys- [36] S.L. Lukyanov, A note on the deformed Virasoro algebra, Phys. Lett. B 367 tems, Nucl. Phys. B 460 (1996) 299–334, arXiv:hep -th /9510101 [hep -th]. (1996) 121–125, arXiv:hep -th /9509037 [hep -th]. [5]. N.A Nekrasov, S.L. Shatashvili, Quantization of integrable systems and four [37] E. Witten, Solutions of four-dimensional field theories via M theory, Nucl. Phys. dimensional gauge theories, in: Proceedings, 16th International Congress on B 500 (1997) 3–42, arXiv:hep -th /9703166 [hep -th], [452 (1997)]. Mathematical Physics, ICMP09, Prague, Czech Republic, August 3–8, 2009, [38] J.-E. Bourgine, Fiber-base duality from the algebraic perspective, arXiv:1810 . pp. 265–289, arXiv:0908 .4052 [hep -th], 2009. 00301 [hep -th]. [6] N. Nekrasov, Seiberg–Witten prepotential from instanton counting, Adv. Theor. [39] M. Spreafico, On the Barnes double zeta and Gamma functions, J. Number The- Math. Phys. 7 (2004) 831, arXiv:hep -th /0306211. ory 129 (9) (2009) 2035–2063. .[7] L.F Alday, D. Gaiotto, Y. Tachikawa, Liouville correlation functions from four- [40] A.S. Losev, A. Marshakov, N.A. Nekrasov, Small instantons, little strings and free dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167–197, arXiv:0906 . fermions, in: M. Shifman, et al. (Eds.), From Fields to Strings, Vol. 1, 2003, 3219 [hep -th]. pp. 581–621, arXiv:hep -th /0302191 [hep -th]. .[8] V.A Alba, V.A. Fateev, A.V. Litvinov, G.M. Tarnopolsky, On combinatorial expan- [41] J.-E. Bourgine, D. Fioravanti, Seiberg–Witten period relations in Omega back- sion of the conformal blocks arising from AGT conjecture, Lett. Math. Phys. 98 ground, J. High Energy Phys. 08 (2018) 124, arXiv:1711.07570 [hep -th]. (2011) 33–64, arXiv:1012 .1312v4 [hep -th]. [42] H. Awata, Y. Yamada, Five-dimensional AGT conjecture and the deformed Vira- [9] O. Schiffmann, E. Vasserot, Cherednik algebras, W algebras and the equivariant soro algebra, J. High Energy Phys. 01 (2010) 125, arXiv:0910 .4431 [hep -th]. cohomology of the moduli space of instantons on A2, Publ. Math. IHÉS 118 (1) [43] A. Nedelin, S. Pasquetti, Y. Zenkevich, T [U (N)] duality webs: mirror symmetry, (2013) 213–342, arXiv:1202 .2756v2 [math .QA]. spectral duality and gauge/CFT correspondences, arXiv:1712 .08140 [hep -th]. [10] S. Kanno, Y. Matsuo, H. Zhang, Virasoro constraint for Nekrasov instanton par- [44] A. Mironov, A. Morozov, Nekrasov functions and exact Bohr–Sommerfeld inte- tition function, J. High Energy Phys. 10 (2012) 097, arXiv:1207.5658 [hep -th]. grals, J. High Energy Phys. 04 (2010) 040, arXiv:0910 .5670 [hep -th]. J.-E. Bourgine, K. Zhang / Physics Letters B 789 (2019) 610–619 619

[45] I. Gradshteyn, A. Jeffrey, I. Ryzhik, Table of Integrals, Series, and Products, Aca- [51] A. Smirnov, On the instanton R-matrix, Commun. Math. Phys. 345 (3) (2016) demic Press, 1996. 703–740, arXiv:1302 .0799 [math .AG]. [46] A.B. Zamolodchikov, A.B. Zamolodchikov, Factorized S-matrices in two- [52] J.-E. Bourgine, Spherical Hecke algebra in the Nekrasov–Shatashvili limit, dimensions as the exact solutions of certain relativistic quantum field models, J. High Energy Phys. 01 (2015) 114, arXiv:1407.8341 [hep -th]. Ann. Geophys. 120 (1979) 253–291, [559 (1978)]. [53] J.-E. Bourgine, D. Fioravanti, Finite 2-corrections to the N = 2SYM prepoten- [47] L.D. Faddeev, Quantum completely integral models of field theory, Sov. Sci. Rev. tial, Phys. Lett. B 750 (2015) 139–146, arXiv:1506 .01340 [hep -th]. C 1 (1980) 107–155. [54] J.-E. Bourgine, D. Fioravanti, Mayer expansion of the Nekrasov prepotential: the [48] D. Fioravanti, M. Rossi, From finite geometry exact quantities to (elliptic) scat- subleading 2-order, Nucl. Phys. B 906 (2016) 408–440, arXiv:1511.02672 [hep - tering amplitudes for spin chains: the 1/2-XYZ, J. High Energy Phys. 08 (2005) th]. 010, arXiv:hep -th /0504122 [hep -th]. [55] S.L. Lukyanov, Free field representation for massive integrable models, Com- [49] D. Fioravanti, M. Rossi, The elliptic scattering theory of the 1/2-XYZ and higher mun. Math. Phys. 167 (1995) 183–226, arXiv:hep -th /9307196 [hep -th]. order Deformed Virasoro Algebras, Ann. Henri Poincaré 7 (2006) 1449–1462, arXiv:hep -th /0602080 [hep -th]. [50] D. Maulik, A. Okounkov, Quantum groups and quantum cohomology, arXiv: 1211.1287v1 [math .AG], 2012.