Explicit examples of DIM constraints for network matrix models

Journal of High Energy Physics, Jul 2016

Dotsenko-Fateev and Chern-Simons matrix models, which describe Nekrasov functions for SYM theories in different dimensions, are all incorporated into network matrix models with the hidden Ding-Iohara-Miki (DIM) symmetry. This lifting is especially simple for what we call balanced networks. Then, the Ward identities (known under the names of Virasoro/\( \mathcal{W} \)-constraints or loop equations or regularity condition for qq-characters) are also promoted to the DIM level, where they all become corollaries of a single identity.

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Explicit examples of DIM constraints for network matrix models

Received: May Explicit examples of DIM constraints for network matrix models Hidetoshi Awata 1 2 6 7 8 9 10 11 Hiroaki Kanno 1 2 4 6 7 8 9 10 11 Takuya Matsumoto 1 2 6 7 8 9 10 11 Andrei Mironov 0 1 2 3 5 7 8 9 10 11 Alexei Morozov 0 1 2 3 7 8 9 10 11 Andrey Morozov 0 1 2 3 7 8 9 10 11 g Yusuke Ohkubo 1 2 6 7 8 9 10 11 Yegor Zenkevich 1 2 7 8 9 10 11 Field Theories, Topological Strings 0 National Research Nuclear University MEPhI 1 Leninsky pr. , 53, Moscow 119991 , Russia 2 Nagoya , 464-8602 , Japan 3 Institute for Information Transmission Problems 4 KMI, Nagoya University 5 Theory Department, Lebedev Physics Institute 6 Graduate School of Mathematics, Nagoya University 7 60-letiya Oktyabrya pr. , 7a, Moscow 117312 , Russia 8 Bratiev Kashirinyh , 129, Chelyabinsk 454001 , Russia 9 Kashirskoe sh. , 31, Moscow 115409 , Russia 10 Bol.Karetny , 19 (1), Moscow 127994 , Russia 11 Bol.Cheremushkinskaya , 25, Moscow 117218 , Russia Dotsenko-Fateev and Chern-Simons matrix models, which describe Nekrasov functions for SYM theories in di erent dimensions, are all incorporated into network matrix models with the hidden Ding-Iohara-Miki (DIM) symmetry. This lifting is especially simple for what we call balanced networks. Then, the Ward identities (known under the names of Virasoro/W-constraints or loop equations or regularity condition for qq-characters) are also promoted to the DIM level, where they all become corollaries of a single identity. Conformal and W Symmetry; Supersymmetric gauge theory; Topological 1 Introduction 2 Basic example: theme with variations The main theme Variation I: matrix elements in the free- eld theory Variation II: generating functions Variation III: DF model Variation IV: multi- eld case Variation V: Chern-Simons (CS) model Variation VI: correlators with vertex operators Variation VII: Nekrasov functions 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Variation VIII: network model level. Network as a Feynman diagram Variation IX: balanced network model 2.10 Variation X: q-deformation 2.11 Variation XI: q; t; : : :-deformations 2.12 Variation XII: -deformation to non-unit Virasoro central charge 3 DIM calculus for balanced network model DIM algebra Bosonization in the case of special slopes Relation to topological vertex Building screening charges and vertex operators Network partition function Examples of conformal blocks Compacti ed network and the a ne screening operator 4 5 6 The action of Virasoro and DIM(gl1) Vertical action of DIM Conclusion A Properties of the DIM algebras and their limits A.1 Constructing DIM(gl1) from W1+1 algebra A.2 Elliptic DIM(gl1) algebra A.3 Rank > 1: DIM(gln) = quantum toroidal algebra of type gln A.4 A ne Yangian of gl1 [139] { 1 { u sitting on the horizontal leg of the topological vertex (denoted by the dashed line) is the same as its action on the product of two representations | the \vertical" jv and \diagonal" = uv. b) Appropriate contraction of two intertwiners is also an intertwiner. This gives the vertex operator of the corresponding conformal eld theory with deformed Virasoro symmetry, corresponding to a single vertical brane in gure 2. 1 Introduction Nekrasov functions, describing instanton corrections in supersymmetric Yang-Mills theories [1]{[11], and AGT related conformal blocks [12{16] possess rich symmetries that can be separated into large and in nitesimal. The former describe dualities between di erent models, while the latter de ne equations on the partition functions in each particular case. They are also known as \Virasoro constraints" [17, 18] for associated conformal or Dotsenko-Fateev (DF) matrix models [19{30], which are further promoted to network matrix models [31{34], looking like convolutions of re ned topological vertices [35{37] and possessing direct topological string interpretation. As conjectured in a number of papers throughout recent years [38]{[49] and recently summarized in [50], in full generality the symmetry underlying the AGT correspondence [51{53], is the Ding-Iohara-Miki algebra (DIM) [54]{[69], in particular, the innitesimal Ward identities are controlled by DIM from which the (deformed) Virasoro and WK emerge as subalgebras in particular representations. In other words, the full symmetry of the Seiberg-Witten theory seems to be the Pagoda triple-a ne elliptic DIM algebra (not yet fully studied and even de ned), and particular models (brane patterns or Calabi-Yau toric varieties labeled by integrable systems a la [3, 4]) are associated with its particular representations. The ordinary DF matrix models arise when one speci es \vertical" and \horizontal" directions, then convolutions of topological vertices can be split into vertex operators and screening charges, and the DIM algebra constraints can be attributed in the usual way [70{79] to commutativity of screening charges with the action of the algebra in the given representation. Dualities are associated with the change of the vertical/horizontal splitting, or, more general, with the choice of the section, where the algebra acts [80{82]. All this is illustrated in pictures 1 and 2, which we borrowed from [50], and our purpose in this paper is to provide very explicit examples of how these pictures are converted into formulas. A great deal of these formulas already appeared in the literature. Putting them together, we hope to illustrate their general origin and better formulate the remaining open problems. { 2 { Qm;1 QF;1 Qm;2 mf;2 mf;1 mf;2+mf;1 2 D4 D4 QF;2 QB;1 NS5 NS5 QB;2 mbif a(2) a( 1 ) 1 NS5 Qm;3 QF;3 Qm;4 mf;2+mf;1 2 mf;2 mf;1 2 2 SU (2) SU (2) 2 2 c) i are exponentiated complexi ed gauge couplings, a(a) are Coulomb moduli and ma are the hypermultiplet masses. b) The toric diagram of the Calabi-Yau threefold, corresponding to the 5d gauge theory with the same matter content. Edges represent twocycles with complexi ed Kahler parameters Qi, which play the same role as the distances between the branes in a). c) The quiver encoding the matter content of the gauge theory. SU(2) gauge groups live on each node and bifundamental matter on each edge. The squares represent pairs of (anti)fundamental matter hypermultiplets. The main scheme could be formulated as follows: To build a functor ! rank-r Lie algebra G quantized double-center double-loop DIM(G); (1.1) perhaps, q123:::-dependent and elliptic To obtain a non-linear Sugawara construction of stress tensor and other symmetry generators from a comultiplication DIM . To clarify the interplay between two \orthogonal" (\horizontal" and \vertical") comultipilcations. To apply the functor (1.1) to central-extended loop algebras G, starting from G = (\gl( 1 ), to obtain triple-a ne Pagoda DIM algebras. One of the immediate problems { 3 { is that the known construction of DIM(G) for non-a ne glN algebras [54, 83{85] already involves the a something more sophisticated. ne Dynkin diagrams, thus, for an a ne gdlN one can need An additional light on the problem can be shed by comparative analysis of DIM(gl2), DIM(gl3), DIM(so5), DIM(g2) and DIM(gcl1), rst four of them being explicitly constructed, and by studying their various limits including the one to the a ne Yangian and further to the standard conformal algebras (coset constructions of conformal eld theories, [86]). Actually, the rst three issues are actively studied by various authors (and there has been already achieved a serious progress), and we do not achieve too much in the two last challenging directions in the present paper, which can be considered as an introduction to the problem. What we actually do, is search for a q; t-deformed network analogue of the CFT Ward identity [12] HJEP07(216)3 * a Y V^ a (za) T^+(z) Q^r + where < : : : > denotes the matrix element < vacj : : : jvac > between two vacua of operators in the xed chronological order and in the chiral sector [87, 88]. Here V (z) is a primary eld (vertex operator) in the free eld c = 1 CFT, T (z) is its stress-energy tensor and Q is the corresponding screening charge [70{72], which is the integral Q = Hx S(x) of the screening current S(x). The order of operators in (1.2) means that in the conformal correlator ** Y V a (za)T+(z)Qr a ++ (where << : : : >> denotes the chiral part of the CFT correlator) all jzaj > jzj and jzj > jxij, where xi's lies on the integration contours of the screening currents. The Ward identity (1.2) can be manifestly written as z 2 a b za)(z r i=1 zb) S(xi) + X ++ a;i (z = Pol(z) a za)(z xi) + r X and the notation Pol(z) means a power series, i.e. any positive powers of z are allowed. The underlined terms just contribute to Pol(z) (since jzaj > jzj) and can be omitted giving (1.2) (1.3) (1.4) nally z 2 X eld theory correlators, it is dictated by operator expansions and is especially simple because a free eld formalism is available for conformal theories. The rst one is actually about matrix elements, and the di erence is that it depends on the ordering of operators, while correlators do not. Another way to say this is that the projected stress tensor T+(z) does not have a simple operator product expansion (OPE) with other operators, the projection is a non-local operation and actually depends on the position: if T+(z) was placed to the left of vertex operators V (za), the matrix element would no longer vanish. At the same time, in this case the underlined terms in (1.4) also contribute (since jzj > jzaj), and they exactly cancel non-zero matrix element leading to the same Ward identity (1.5). These are trivial remarks for the old-fashioned eld theory, where the Ward identities were discovered and treated as sophisticated recurrence relations between Feynman diagrams, but in modern CFT we got used to the formalism based on the operator product expansion and moving the integration contours, which provides a shortcut for the derivations. Unfortunately, in the network models, only the operator approach is currently available, and this is the reason why we need to develop the formalism from this starting point. Still, some elements of the free eld formalism are already worked out in particular representations of DIM, and for a special class of balanced network models, drawn as a set of horizontal lines with vertical segments in between, see gure 3, a), one has a direct counterpart of (1.2). In (extremely) condensed notation it looks like * Y a a [za] a [za] T^+(z; uj ) Y b X !+ and involves operators like Y I I [zI ] Y J J [zJ ] ! 0 I;J where [ ; z]n sign(n) q i 1=2t1=2 iz n are the Miwa variables associated with the Young diagram , and the Drinfeld-Sokolov operator (generalized stress energy tensor = Miura transformation from i(z)) T^ (z; uj ) = z1=2 log! z 1=2 log! = K k K X k=1 ui i(z!2(i 1)) : k Y uia : ia (z!2(a 1)) : X i K : Y i=1 X i1<:::<iM a=1 { 5 { de ning numerous ows, is a linear combination of all W are also made from the annihilation and creation operators ^ n, ! = pq=t and T^ (z; uj ) K. Here i(z) (m) with m depends on an additional parameter rameters of DIM representation ui. generating di erent W (m)(z) and on spectral pa(1.6) (1.7) (1.8) (1.9) ( 1 ) 1 z(2) 2 u1 u2 u3 ( 1 ) 3 z(2) 4 z( 1 ) 2 c) z( 1 ) 4 ( 1 ) 4 (2) 4 y4 u1 to the left, through external vertical legs, which do not commute with T^ (z). Moreover, now we can also consider deformations of the section which do not preserve verticality, like the dotted one in gure 3, c), and everything can still be calculated. This should provide a qualitatively new insight into spectral dualities [89{94] associated with global rotations of the network graph. Non-balanced networks, where the right-most and left-most branes in gure 4 are tilted and the number of operators di ers from that of , can be considered as certain limits of the balanced ones, but these limits are non-trivial and singular when, say, q; t the point of view of representation theory these limits should have independent description, making use of more complicated intertwiners. A full- edged free eld description for them comparable to the one in [95{97] for ordinary a ne case still needs to be worked out. ! 1. From { 6 { u1 u2 u3 Restriction to the balanced networks is a great technical simpli cation, but it requires a somewhat lengthy comments on what this means and whether this really restricts the set of handy physical models. DIM is a quantization of double loop (double a ne) algebras, and the existing free eld formalism, which we are going to expose and exploit in the present paper, explicitly breaks the symmetry between the two loops. Bosonized/fermionized are only the Chevalley generators, in the case of DIM there are many, still they depend on one of the two loop parameters, while the other loop is associated with their multiple commutators and is described very di erently: in terms of Young diagrams parameterizing states in the Fock space. This breaks the symmetry of the DIM algebra: the SL(2; Z)-automorphisms acting on the square lattice of the generators and introduces asymmetry between horizontal and vertical directions in the planar graphs which are used to de ne the network models, and makes the spectral dualities interchanging these two directions highly non-trivial. In particular, allowed networks look like in nite \horizontal" lines, connected by vertical segments, see gure 4, a), and not vice versa. We call these lines horizontal, though they can have varying slopes, however, they have a non-trivial projection on the horizontal axis, i.e. are strictly non-vertical. In the original brane theory interpretation these horizontal lines depict the D-branes, while vertical are the N S branes, from this point of view our description applies only to the conformal models (Nf = 2Nc) with de nite Nc = M = # of horizontal lines. Quiver models SU(Ni) with di erent Ni can seem excluded, but in fact they appear after application of the spectral duality: a 90 rotation of the graph, see Fig, 4, b). After this rotation, the in nite horizontal lines get associated with the in nite N S branes, while the vertical segments with D-branes between them. This pattern looks more relevant from the gauge theory point of view, but we emphasize that our free elds live on the in nite horizontal lines, the three-valent vertices (the DIM algebra intertwiners and , also known as topological vertices) act as operators in the Fock spaces horizontally, while the third vertical edge carries a Young-diagram label, not converted into operator language. In result these vertices can look like ? or >, but not like ` or a. All these restrictions can be lifted by switching from Fock to MacMahon modules, which are representations of DIM spanned by 3d partitions, but such a description is only combinatorial so far, no generalization to the full- edged double-loop free eld formalism is available yet. This is what makes tedious the consideration of dotted sections in gure 3, c). We brie y touch this issue at the very end of this text, but detailed presentation is postponed to the future work. Our main purpose here is to describe the powerful free eld formalism for the balanced network as a straightforward generalization of that for the ordinary conformal theories, and explain how the DIM algebra becomes the symmetry of generic Nekrasov functions generalizing the Virasoro/W symmetry of the ordinary conformal blocks and Dotsenko-Fateev matrix models. In the next section 2, we explain how the elementary theory of a harmonic oscillator can be straightforwardly developed and lifted to description of generic networks, i.e. of generic Nekrasov functions. In section 3, in the simplest examples we demonstrate the actual formalism in full detail. It is important that most complications come from sophisticated notation, which are largely no more than a change of variables (normalization { 7 { M L M + 1 M + 2 M + 3 L 1 L 2 L 3 a) Bending of the \horizontal" lines due to tension from the vertical segments is re ected in their slopes marked above them. b) Spectral duality acts by rotating the diagram a). After rotation one can identify the conventional Hanany-Witten (or brane web/geometric engineering) setup with NS5 and D5-branes (). of creation and annihilation operators). The really big change comes in section 4, when one looks at the symmetry : it is indeed essentially deformed. But this deformation actually simpli es things, reducing all the symmetries to the action of the DIM generators, while the Sugawara construction of Virasoro and W-operators and of their sophisticated q-deformations is no more than the simple comultiplication rule. At last, at section 5 we brie y discuss the spectral duality action on symmetry generators. Finally, the appendix contains further details about various DIM algebras and their representations. At present stage of development, di erent parameters are treated as providing di erent algebras, but further studies can promote them to parameters of di erent representations of a single uni ed algebra (like the triple-a ne elliptic Pagoda DIM algebra anticipated in [50]). Notation. Throughout the text we use the notation (1.10) 2 Basic example: theme with variations We assume some familiarity with [50] and do not repeat the general logic, leading to Ward identities like (1.2) in DF and network matrix models. 2.0 The main theme Screening charge Q^, acting on the Fock space F = n Pols( n) o e T0 , is Q^ = I S^(x)dx = resx=0 S^(x); ! r q t { 8 { it gives where the calculation involves 1 2 X m1;m2 m1 m2 I I and i.e. the power of Q^ acts as a character of rectangular Young diagram. This is the old result by [98{104]. The rectangular diagrams arise from the Cauchy formula r Y exp i=1 X n>0 nxin ! n = exp X 1 n>0 n n r X x n i i=1 ! = X f g [~x] with a sum over all Young diagrams (actually, with no more than r lines) after the Vandermonde projection (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) HJEP07(216)3 S^(x) = : ep2 ^(x) : = exp nxn ! n | {z Pn xn nf g } n>0 where nf g are the characters of symmetric representations [n] of sl algebras (the Schur polynomials in this particular case). Applied to a highest-weight state (i.e. the one annihilated by all negative modes a^ n = Residue is non-vanishing, because x2@0 converts jm + 1 > into x m 1. Similarly which is a direct generalization of (2.5). Since the screening charge commutes with the Virasoro generators ^ Ln = X(k + n) k k + X k(n k) (2.10) Yr I i=1 dxi xm+r i (~x)2 [~x] ;[mr] [L^n; Q^] = 0 { 9 { one has In application to (2.6), this gives while the action of gives just the size of the Young diagram: L^nQ^r m + r = Q^rL^n m + r = 0 for n > 0 n > 0 L^0 = X k k k [mr] In the Miwa parametrization n = Pi Xin, this turns into the statement about the Calogero eigenfunctions. Also Qr m + r E are singular vectors in Verma modules and (2.12) can be considered as the simplest version of BPZ equations for correlators with degenerate elds, [12]. Equation (2.12) provides a simple example of the Ward identity for the state Q^rjm + r >, which can be promoted to identity for the matrix element in conformal eld theory, i.e. in the abstract Fock module and corresponding Sugawara energy-momentum tensor (which we denote by Gothic letters), [mr] =< m insertion of the intertwining operator, see below. We are now ready to formulate the main rjC^T (z)Q^rjm + r > by additional theme of the present paper: A trivial symmetry property (2.9) gives rise to a non-trivial equation for the matrix element (2.12), provided one can calculate (2.6). In what follows we extend this simple example to matrix elements of an arbitrary network of intertwining operators, what allows to reveal in a rather explicit form the hidden DIM symmetry of the Seiberg-Witten/Nekrasov theory. We continue in this section with variations on the main theme, developing it at conceptual level. Next sections will describe technical details of the story. 2.1 Variation I: matrix elements in the free- eld theory mode operators a^ n = n=p2, n > 0, i.e. contains Q Actually, in theory of free eld (z), the bra vacuum state is annihilated by all the negative n>0 (a n) in holomorphic represen< m rj Qr jm + r >: this matrix element would not depend on to introduce a special intertwining operator tation. Thus, one can not simply convert (2.6) into a statement that [mr]f g is equal to at all. The way out is which converts the bra vacuum into the coherent state C^fpg = exp X pna^n ! n>0 n hmj ! hmj C^fpg (2.11) (2.12) (2.13) (2.14) (2.15) Symmetry (2.9) actually holds for all n 2 Z. We will also need a \current" T^(z) = X n2Z ^ Ln zn+2 L^ n = k + [L^n; L^m] = (n m)L^n+m + with n J^ n = p ; 2 J^n = p 2n with the property This allows us to rewrite (2.6) as hmj C^fpg a^ n = pn hmj C^fpg Among many complications as compared with (2.6), there is p2, which re ects the fact that the character is extracted here from the screening charge in a single eld (\current") section 3.2 of [72] and section 2.6 below) which involves two scalar elds, and p realization. A more adequate kind of formulas arise within the fermionic realization (see 2 is a result of basis rotation to their symmetric combination. Variation II: generating functions We can make from particular Virasoro generators L^n a single operator (stress tensor) (2.17) (2.18) (2.19) (2.20) (2.21) (2.22) (2.23) (2.24) Positive and zero modes with n 0 are given by (2.10) and (2.13) respectively, negative X n2Z J^0 = p 2 ^ Jn zn+1 [J^n; J^m] = n n+m;0 [L^n; J^m] = mJn+m T^(z) = : 1 J^(z)2 : 2 The two operators are related by the Sugawara relation where normal ordering puts all p-derivatives to the right of all p's (in each term of the formal series). The generating functions satisfy the commutation relations [J^(z); J^(w)] = 0(w=z) (x) = X x n In terms of generating functions, the Ward identity (2.12), i.e. the corollary of symmetry (2.9) becomes or, in other words, a regularity constraint z2 T^(z) z2 T^(z) m r p 2 zJ^(z) m p 2 r zJ^(z) This will be the typical form of Ward identities (regularity condition for qq-characters) for network Nekrasov functions Z generalizing the simple character [mr]. 2.3 Variation III: DF model Expressions (2.6) and (2.1) together imply the integral representation of the matrix element [mr]f g = D m r C^f n= 2g Q^r m + r E = Gf jxg = exp which is the archetypical example of DF or conformal matrix model [70{72, 102{104]. Ward identity (2.27), which is a trivial corollary of commutativity (2.9) looks now like a not-so-obvious set of integral identities: z2 T^(z) = * X k;i z m r p 2 kxik+1 xi zJ^(z) D 1 E + r X DFm;r xixj xi)(z Actually there are two standard ways to derive the l.h.s.: Y(xi i<j xj )2 = D 1 E DFm;r n nf g xj ) (m r) X i xi (z xi) + + DFm;r FFm (2.25) (2.26) (2.27) (2.28) (2.29) (2.30) ( 1 ) by using bosonization, which is the simplest version of free- eld (FF) formalism, i.e. the Wick rule for decomposition of correlators into pair ones, T^(z) 1 D E DFm;r = * C^( n=p2) T (z) Yr I ep2'(xi)dxi where the index m refers to a special way of handling the zero mode of ' and '(z) refers to the scalar eld acting in the abstract Fock module, and (2) by a change of integration variables xi = xin+1 in the multiple integral (2.28), [17, 18, 73{79]: in this case we get the identities in a slightly di erent form: * r X 2 i i<j xn+1 xi xn+1 j xj + X i;k r i kxik+n + X(n + 1 m r)xin = 0; n > 0 + DFm;r (2.31) In this paper we actually need an outdated and tedious third way: ( 3 ) the operator formalism based on an explicit calculation of commutators arising when the stress tensor is carried from the left to the right through the screening operators: this is what we are now doing, starting from section 2.0 and this is what in the simplest case brought us to the Ward identity in the form (2.29). Both the OPE-based and change-of-integration-variables/total-derivative approaches should also work in the network model context, but they still need to be developed. 2.4 Variation IV: multi- eld case The network matrix models can be considered as associated with networks of branes (branewebs [105{110]), which being projected onto the 4 5 plane look like segments with di erent slopes. From the point of view of Yang-Mills theories, interpretation of the di erent slopes is di erent. Surprisingly or not, it is also di erent at the present level of understanding of the DIM symmetry. Throughout the section, we distinguish only between the horizontal and vertical segments, while intermediate slopes appear in this section only in sections 2.8 and 2.9. Our next variations introduce and describe the associated notions. The rst one is horizontal branes. These are associated with di erent free elds. Generalization of the DF model to K- eld case provides WK constraints for models with K horizontal branes. An additional procedure can be applied to separate a \center-ofmass" eld: this explains why in the previous subsection 2.3 the number of elds was one rather than two. The multi- eld conformal model [72] is de ned as * m~ ~r Y ^ a=1 Caf n(a)=p2g Q^raa m~ + ~r = h1iDFm~;~r + (2.32) where the screening charges now carry additional indices labeled by K 1 simple roots ~ a of slK . They are actually associated with segments of the vertical branes ending on two adjacent horizontal branes, gure 3, a), in accordance with the decomposition ~ a = ~ea+1 ~ea. In other words, a better labeling of Q is by pairs of indices ab, each corresponding to a particular horizontal (in fact, any non-vertical, see section 2.5) brane.1 Now the matrix model partition function depends on K sets of times, one of which is associated with the \center of mass" and actually decouples in the DF model (2.28), thus it was actually suppressed in that formula. However, this is not always true: the decoupling will not take 1To avoid possible confusion, note that in [50] an \orthogonal" labeling rule was used, treating horizontal edges of the network as segments between the vertical ones. place already in the Chern-Simons deformation of (2.28) in section 2.5, and all the M sets of times will be relevant in generic DIM considerations. This phenomenon is familiar in the CFT approach to Nekrasov functions, where relevant is the Heis + V irasoro symmetry and its generalizations rather than the V irasoro alone. This is also re ected in appearance of \100 in the popular notation W1+1. Algebraically, the multi- eld generalization is controlled by the comultiplication DIM , which builds all the symmetry generators from a single element of DIM: current algebra Virasoro # # # # W3 : : : WK # : : : Z (rJ )2 + : : : K a=1 X Ja = 0 This comultiplication adds new scalar elds, and non-linearity of the usual 4d Sugawara formulas is mostly due to elimination of the center-of-mass eld; what makes this possible is the exponential form of symmetry generators beyond 4d. Somewhat symbolically, the Sugawara formulas for the stress tensor (at the second level of DIM) arise from the expansion of characters (in fact, q-characters) underlined in the rst two lines are terms appearing due to the center-of-mass reduction Cab is the Cartan matrix for slK , which the K = 1 limit describes a di erence Laplace 2 operator r . Other W-operators made from higher powers of J arise in the same way at K = 2 : K = 3 : : : : K : : : : Tsl2 = Tsl3 = 1 2 1 3 eJ + e J = 1 + 2 1 J 2 + : : : eJ1 + eJ2 J1 + e J2 = 1 + (J12 J1J2 + J22) + : : : 1 3 1 K! X a;b=1 TslK = 1 + CabJaJb + : : : K = 1 : Tsl1 = 1 + const (2.33) (2.34) (2.35) higher levels of DIM, i.e. after several applications of the comultiplication DIM , e.g. at K = 3 the second generator of the W3-algebra is so that the standard W3-generator is a di erence dxiGf jxige (log xi)2 Y(xi xj )2 i<j The parameter controls the brane slope, it vanishes for the horizontal branes, while for the vertical ones it becomes in nite and the story gets a separate twist, see section 2.6 below. From the point of view of DIM symmetry of the network model, the Virasoro/Ward constraints should look similar with and without these logarithmic terms, in the sense that they should be always dictated by the Wick theorem hidden in the algebraic structures of DIM. There is, however, a crucial di erence: in this case, the U( 1 )-mode should not decouple for non-trivial slopes, and two sets of times survive (see section 2.6). This is re ected in the fact that one needs to consider Gf jxg depending on n>0 and n<0 in (2.38), Gf jxg = exp 1 X n2Z nxn ! n in order to construct the Ward identities. Then, a counterpart of (2.31) for (2.38) looks somewhat di erent [122, 123, 125{127]: i i * (n r + 1) Xr xin + Xr xin (log (xijq))0 + 2 X xin+1 i<j xi xn+1 j xj + X k;i kxin+k + = 0 (2.40) where q = exp( 21 ) and (xjq) = P1 Variation VI: correlators with vertex operators The vertical branes are associated with insertions of vertex operators into the DF and CS models. A particular instance of the vertex operator is the screening current. As already mentioned in section 2.4, screening charges are segments of vertical branes between the two neighbour horizontal ones, and they can be considered as contractions of two vertex operators attached to these two branes. However, the relevant operators are special, Variation V: Chern-Simons (CS) model The brane slopes show up in a specially designed 4d limit as additional square-logarithmic terms (log xi)2 in the action of the DF matrix model (2.28), giving rise to what is often called the CS matrix model [111{124]: namely, they are e with = 1: a kind of \fermion vertices" (in fact, intertwining operators) = e . Accordingly, the screening charges should be associated with bilinears a+(x) b (x), \non-local" in the vertical direction: Q^ab = I a+(x) b (x)dx (2.41) exponentials rather than @ -like currents, as well as the emergency of peculiar p This non-locality explains, among other things, why the screening currents are \naturally" 2 in (2.1) coming from the 45 rotation of the basis 1 ; 2 into 1 instead of x and the screening charge is a convolution of these indices (see s.3.2 of [72] for details). Interchanging of + and labels changes the screening charge to the dual one (in algebraic terms, this corresponds to using instead of a positive root the corresponding negative one): as usual in conformal matrix models [70{72], the use of dual charges is unnecessary. In fact, one can connect every screening charge with a simple root: one can associate with each end of leg a a basis vector ~ea, then, the screening charge Qa;a+1 corresponds to a simple root ~ a = ~ea+1 ~ea. In operator formalism the correlator of vertex operators is just a matrix element of an ordinary product of linear operators. A generic vertex operator is constructed from the primary eld V (x) and is labeled by the Young diagram : ^ V = L^ V (x) V^ (x + z) = ezL^ 1 V^ (x)e zL^1 with L^ = Qi L^ i . The conjugation with L^ 1 moves it to an arbitrary point z: However, in CFT the positions of operators does not matter: they can be considered as located at points in the complex z-plane, or, more generally, on a Riemann surface (in the latter case same traces need to be taken in operator formalism). Still, location of the stress-tensor insertion does matter: in the Riemann surface picture, it is associated with a choice of a contour encircling the vertex operator insertions, and correlator depends on the homology class of this contour. Changing the class is equivalent to commutation of T (z) with the vertex operator, which is read o the commutation relations [Ln; V (x)] = xn+1V 0 (x) + 2(n + 1)xnV (x) (2.44) and those of the Virasoro algebra. This is what we did in the derivation of (1.5) placing the stress-tensor to the left, and to the right of vertex operators. Central-charge-preserving comultiplication is provided by the Moore-Seiberg comultiplication MS. The action of Virasoro algebra MS , which is given by the ordinary Leibnitz rule on the negative modes T , but the positive modes act di erently: 1 X zn+1 k k=0 n + 1 k ! MS (Ln)R1 R2 = Lk 1R1 R2 + R1 LnR2 (2.45) (2.42) (2.43) This comultiplication can be read o the conformal Ward identities, [128] and celebrates two important properties: It is parameterized by an arbitrary parameter z, it does not change the central charge, in contrast with the comultiplication in the DIM algebra that we use below. We de ne the Nekrasov function as partition function of the DF/CS network matrix model depending on parameters ~ i, zi and Na, associated respectively with external legs (assumed vertical), horizontal and vertical edges of the graph : schematically, Z = Y V^~ i (zi) exp i=1 K 1 X Q^a;a+1 a=1 !+ DFNa (2.46) and this partition function describes the A1-quiver with obvious modi cations for more sophisticated quivers, [50] (changing the number of vertex operators and adding more screening charges that di er by the choice of the integration contours). The right numbers of screening charges are automatically selected from the series expansion of the exponential by zero mode conditions. On the gauge theory side, this data describes the theory with the gauge group SU(K) and 2K fundamental matter hypermultiplets (i.e. zero -function). Here the numbers Na are the Coulomb moduli, the hypermultiplet masses are parameterized by the vertex operator parameters ~ i and the positions of vertices (rather their double-ratio) control the instanton expansion in the gauge theory. Note that this theory is characterized by zero -function, all other cases are obtained by evident degeneration. The case of adjoint matter hypermultiplets is described by the elliptic DIM algebras2 [140{143] and is out of scope of the present paper. The other quiver theories, say Ak are described, on the physical side, by a product of k gauge groups: Qik SU(ni) with i = 2ni ni 1 ni+1 bifundamental hypermultiplets for each i transforming under the gauge groups SU(ni) and SU(ni+1). There are also 0 and k fundamental hypermultiplets that are transformed under SU(n1) or SU(nk) (we put n0 = nk+1 = 0). These theories have also zero -functions, other cases can be obtained by a degeneration of hypermultiplet masses. Note that the Nekrasov network partition functions typically contain additional singlet elds, which corresponds to U(K) instead of SU(K) group. The contribution of this singlet factorizes out and reduces just to a simple multiplier in the Nekrasov function. While exponentiation of bosonized screenings Q = H e can look somewhat arti cial, the same procedure is very natural in the fermionic version Q = H + : this adds bilinear terms to the free fermion action, i.e. leaves it quadratic. This is the reason for integrability, and in bosonized version this is re ected in integrable properties of Toda like systems with exponential actions. Exponentiation of fermionic screenings makes a new interesting twist after the qdeformation in section 2.10, see eq. (2.50) below. 2By DIM algebras in this paper we mean both DIM and its limits like a ne Yangian [49, 129{139]. n = v1 v2 = 0 Q a) K v2 = 0. The \horizontal" line with spectral parameters u and v is shown in blue. The length of the intermediate edge is determined by the ratio of the spectral parameters on the adjacent edges, Q = uv . b) An example of a 3d Young diagram which contributes to the vertex C[1];[2];[2;1]. The vertex C[1];[2];[2;1] is given by the weighted sum over all 3d Young diagrams with three xed asymptotics shown in blue. 2.8 Variation VIII: network model level. Network as a Feynman diagram Network model is de ned for a planar 3-valent graph with edges parameterized by slopes and lengths. Slopes are given by pairs of numbers (X1; X2), see gure 5, and lengths by parameters Q. The 2-component vectors X~ are conserved at the vertices of : X~ v 0 + X~ v 00 + X~ v 000 = 0 at each vertex v; this is a stability condition for the brane-web. The graph with this structure describes a la [3, 4] the tropical spectral curve of the underlying integrable system, but for our purposes it can be considered just as a Feynman diagram with cubic vertices and momenta QX~ on the edges, associated with some e ective Chern-Simons-type eld theory. Expressions Z for this Feynman diagram (Nekrasov partition function or genagators eralized conformal block) is build by convolution of vertices CIJK (X~ 0; X~ 00; X~ 000jq) and prop IJ (Q), where indices I; J; K are Young diagrams, and CIJK are, in turn, \(re ned) topological vertices" [35{37] given [144{148] by sums over 3d (plane) partitions with three boundary conditions described by three ordinary Young diagrams I; J; K, see gure 5, b). In the generic network matrix model, the exponentials of screening charges no longer turn into exponential of \fermions": it produces an elementary 3-valent vertex (=re ned topological vertex) providing the true DIM intertwiner. Automatic is now not only adjustment of the number of screenings, but also matching between their + and constituents. Screening charges are substituted by vertical lines between pairs of horizontal brains, H exp(~ ij ~), involving two free elds associated with the corresponding branes. Slopes of the horizontal branes enter the matrix model description through (log xi)2 terms in the action, see (3.45) in s.3. The coe cient is made out of the skew product (see gure 5 a)) Xv1 ^ X~ v2 ~ (2.47) where X~ v1 , X~ v2 are associated with the external horizontal lines, one incoming, the other one outgoing. In the case with several horizontal lines, see e.g. (3.42), one has to consider X~ v1 , X~ v2 for di erent horizontal lines, and the answer in this case does not depend on the concrete choice of these lines. We described in this subsection a generic network model. One can consider its particular case: the model that gives rise to the quiver gauge theory (as described in the previous subsection). In this case (for any quiver gauge theory), one can construct a K-theoretic version of the Nekrasov functions, Z , [149{152]. They coincides [36, 37, 153, 154] with the re ned partition functions in the corresponding geometry, which can be constructed via the re ned topological vertex. Another possibility is to consider the quiver theories with zero -functions (so that all other can be obtained via various limiting procedures from these) and all gauge groups coinciding, ni = n 8i. These theories are associated with so called balanced networks and can be immediately described within the representation theory of DIM algebras, and the requirement of all gauge groups having the same rank is implied by a possibility of immediate extension of DIM to the elliptic DIM: this latter describes the quiver gauge theories with adjoint matter, where the condition ni = n is inevitable. We discuss the issue of balanced networks in the next subsection. 2.9 Variation IX: balanced network model As usual, the q; t-deformation leads to overloaded formulas, but in fact it drastically simplies them by providing a very clear and transparent interpretations and unifying seemingly di erent ingredients. Namely, everything gets controlled by the DIM symmetry: the edges of graph carry DIM representations, the topological vertices C become their intertwiners, and symmetries (stress-tensor and its W-counterparts) are just the generators of DIM acting in tensor products of representations and thus de ned by powers of the comultiplication DIM (which is di erent from MS). An exhaustive description of the network models depends on development of representation theory for the double a ne algebra DIM, and it is not yet brought to the generality level of [95{97] for ordinary a ne algebras. In particular, at the moment, it is not immediate to describe within the DIM framework an arbitrary DF or CS matrix model. However, among the DF matrix models there is a subclass that is directly lifted to rather peculiar networks, which we call balanced which are controlled by an analogue of the level one representations of Kac-Moody algebras and allow a drastically simpli ed bosonization and even fermionization. As we already mentioned the balanced networks correspond to special quiver gauge theories with zero -functions. the DIM(gl1) algebra. This is a very powerful method, but it is only at the rst stage of development. There are several restrictions which should be consequently lifted at the next stages. If one considers them as a consequent speci cation of representation types, the list should be read in inverse order. The construction that admits fermionization of intertwiners and at the level one of DIM is much similar to that for the level k = 1 Kac-Moody algebras. Hence, one could expect a straightforward generalization to arbitrary level a la [95{97] involving analogues of the b; c-systems. Note, however, that the requirement on the level does not restrict the value of the Virasoro central charge regulated by : all matrix models -ensembles and, hence, the generic Liouville and WK -conformal blocks are already handled by the existing formalism. Also, at this level the di erence disappears between the vertex operators (in particular, the screening charges) and the stress tensors (including the W-operators): all these are described by exponentials of the free elds, the di erences emerge only in the limit q; t ! 1. The formalism is best developed for the intertwiners, which act as operators between the two \horizontal" Fock modules F is the \vertical" leg associated with F (1;L) and F (1;L 1), while the third representation (0; 1). Such a non-symmetricity is inevitable since the resulting topological vertex of [35] is still asymmetric and remembers about the distinguished vertical direction. Technically this restricts consideration to the balanced networks, what makes many important models, including the quiver ones, treatable only via additional application of the spectral duality. A better treatment should involve in nitely many free elds, giving rise to MacMahon type modules, what should also allow one to de ne skew intertwiners, where all the three legs are non-vertical. An existing description of the MacMahon modules is pure combinatorial, in terms of 3d Young diagrams (plane partitions). A naive free eld formalism would involve elds depending on two coordinates instead of one, and this requires a far-going generalization of holomorphic elds used in the ordinary 2d CFT. Such a formalism is now developing, also with the motivation coming from MHV amplitudes, but its incorporation into the DIM representation theory is a matter of future. Still, it seems important for a full understanding of the spectral dualities and of generic networks, including the sophisticated ones from [165{169]. They can be treated by the existing formalism, but it leaves the underlying symmetries well hidden: they show up only in answers, but not at any of the intermediate stages. A further challenge is further generalization from DIM(gl1) to DIM(gln) and the triple-Pagoda algebras DIM(gcl1) and DIM(gcln). An intriguing problem (see appendix A3) is that already DIM(gln) is built from the a ne Dynkin diagram of gcln, thus, the triple-a ne generalization should involve more sophisticated Dynkin diagrams. We hope that the present text can serve as a good introduction in the DIM-based generalization of conformal theories, where the conformal blocks are the generic Nekrasov functions and the Ward identities are the associated regularity conditions for qq-characters. We hope that it will help to attract more attention to emerging challenging problems, which we have just enumerated. Technical means for this seem to be already at hand. Acknowledgments work on this project. A.M.'s and Y.Z. are grateful for remarkable hospitality at Nagoya University during the Our work is supported in part by Grant-in-Aid for Scienti c Research (# 24540210) (H.A.), (# 15H05738) (H.K.), for JSPS Fellow (# 26-10187) (Y.O.), JSPS Grant-in-Aid for Young Scientists (B) # 16K17567 (T.M.) and JSPS Bilateral Joint Projects (JSPSRFBR collaboration) \Exploration of Quantum Geometry via Symmetry and Duality" from MEXT, Japan. It is also partly supported by grants 15-31-20832-Mol-a-ved (A.Mor.), 15-31-20484-Mol-a-ved (Y.Z.), mol-a-dk 16-32-60047 (And.Mor), by RFBR grants 16-0100291 (A.Mir.) and 16-02-01021 (A.Mor. and Y.Z.), by joint grants 15-51-50034-YaF, 15-51-52031-NSC-a, 16-51-53034-GFEN. A Properties of the DIM algebras and their limits In this appendix, we describe the algebraic structures of DIM algebras and their degenerations. A.1 Constructing DIM(gl1) from W1+1 algebra Let us discuss how one can construct DIM(gl1) starting from the algebra of di erence Algebra W1+1. Consider the algebra W1+1 (as usual, 1 + 1 refers here to adding the Heisenberg algebra to W1 ) given by the generators Wnk = W (znDk), n 2 Z; k 2 Z 0, where D = z@z. One can consider the central extension of this algebra: [W (znDk); W (zmDl)] = W [znDk; zmDl] + c n+m;0 n;kl; n;kl = ( Pjn=1( j)k(n 0 j)l; n > 0 n = 0 or, in the di erent basis of Wnk = W (znDk) with D tD (see (A.22)), [W (znDk); W (zmDl)] = (tmk tnl)W zn+mDk+l c n+m;0 tk+l tmk tnl 1 nct nk n+m;0 k+l;0, see (A.3). c t k 1 Note that, if k + l 6= 0, the second term in the right hand side of (A.2) can be absorbed into the rst term by rede ning the generators W (Dk) with k 6= 0: W (Dk) ! W (Dk) , k 6= 0. However, at k + l = 0 this term can not be absorbed and is equal to (A.1) (A.2) Algebra W1+1. The next step is to consider the algebra W1+1 = n Zo, which is a double of the W1+1 and may have two central extensions: [W (znDk); W (zmDl)] = (tmk tnl)W zn+mDk+l + t nk(nc1 + kc2) m+n;0 k+l;0 (A.3) Automorphisms. The algebra W1+1, (A.3) has the evident automorphisms ; ~ and (Wnk) = t 21 n2Wnk+n ; (c1) = c2 ; ~(c1) = c2 ; (c1) = c1 + c2 ; (c2) = c1 ; ~(c2) = c1 ; (c2) = c2 : x0n; xm = x0n; x0m = n n n qc1n q n q(n jnj)c1=2xn+m q c1n q 1 n+m;0 In particular, and form SL(2; Z) acting on two central charges c1 and c2 . Heisenberg subalgebras. By the commutation relations (A.3), it is easy to see that it contains a Heisenberg subalgebra generated by fWn0; c1gn2Z satisfying [Wn0; W m0] = nc1 n+m;0 : From the viewpoint of the root lattice of W1+1 , this can be seen as the vertical embedding of the Heisenberg algebra. By using the automorphisms and in the above, it is easy to nd the horizontal and the embedding with arbitrary slope 2 Z as follows; [W0n; W0m] = nc2 n+m;0 ; [Wn n; Wmm] = nt n2(c1 + c2) n+m;0 : Chevalley generators and Serre relations. The generators Wn ;0 = W (znD 1;0) form a closed subalgebra: Wn+; Wm Wn0; Wm = (tm t n)W m0+n + (nc1 + c2)t n n+m;0 = (1 t n)Wm+n Wn0; W m0 = nc1 n+m;0 One can generate the whole algebra from this subalgebra provided the Serre relations are added: h Wn ; [Wn+1; Wn 1] = 0 i Quantization: from W1+1 to DIM(gl1). This algebra can be deformed with the deformation parameter q. Let us denote the deformed (properly rescaled) generators through Wn0 ! xn, Wn ! xn . Then, 0 (A.4) (A.5) (A.6) (A.7) (A.8) where and Introducing the series of generators, (z) = (1 X X q1 = t2; q2 = q 2t 2; q3 = q2 (q1q2q3 = 1) (A.10) k z k q c2 exp 1 X x n 0 z n n=1 ! k q c1k=2z k; x (z) = X xn z n n2Z we immediately come to the DIM(gl1) algebra of section 3.1 upon identi cation q1 = q, q2 = t 1. in its terms as Free eld realization. At the values of central charges (c1; c2) = (1; 0), the constructed DIM algebra has the deformed a ne U( 1 ) subalgebra so that the generators are realized x+(z) = exp x (z) = exp +(z) = exp (z) = exp X 1 n>0 X 1 n>0 X(qn n>0 X 1 n>0 t n t n n 1)(1 t n n (1 znpn ! ! nznpn exp ! X(qn n>0 exp ! 2n)!n=2 znpn ! ! ! X(qn n>0 1)! n ! (A.9) (A.11) (A.12) (t 1 N 1) X Y t 1z i i=1 j(6=i) z i zj zj zinq Di = (qt) n 4 n t N x n n;0 (A.15) with n 0. Similarly, at the values of central charges (c1; c2) = (2; 0) this DIM algebra contains a q-deformed subalgebra (Virasoro U[( 1 )) (and is realized by two free elds), at (c1; c2) = (3; 0) it contains a q-deformed subalgebra (W ( 3 ) U[( 1 )) (and is realized by three free elds), etc. After the Miwa transform of variables pn = PiN zin, these expressions reduce to the Macdonald operators (t 1 i=1 j(6=i) zi zj N 1) X Y t 1zi zj zinq Di 2 iz X 1 t n znpn n exp X 1 t n n>0 n z np n exp X(q n n>0 for n > 0 so that Elliptic DIM(gl1) algebra Elliptic version of DIM algebra is generated by the same set of operators as the ordinary DIM: x (z), (z) and the central element . The relations are a copy of eq. (3.1), except for the [x+; x ] relation, which changes to q0 (q; q0) q0 (t 1; q0) (q0; q0)31 q0 (q=t; q0) ( z=w) where p(z) = (p; p)1(z; p)1(p=z; p)1 is the theta-function. Also, most importantly, the structure function G (z) is now not trigonometric, but elliptic: is exactly the same as in the trigonometric case, given by eqs. (3.2). The essential di erence with the trigonometric case appears when one tries to build Fock representation of elliptic DIM: one set of bosons turns out not to be enough. One needs at least two sets of Heisenberg generators a^n and ^bn to reproduce the commutation relations of the elliptic algebra. Concretely, we have for the level one representation: u( +(z)) = '+(z) = exp u( (z)) = ' (z) = exp u( ) = (t=q)1=2 0 n>0 n>0 1 0 X (1 tn)z n n6=0 n(1 q0jnj) ^anA exp @ n6=0 X (1 t n)q0jnjzn ^bnA : n(1 q0jnj) 0 n6=0 X (1 tn)! jnjz n n(1 q0jnj) 1 0 n6=0 X (1 t n)!jnjq0jnjzn ^bnA : n(1 q0jnj) ! X (1 tn)(! n !n)! n=2 n(1 q0n) X (1 t n)(! n !n)! n=2 n(1 q0n) z n^an !nq0nzn^bn zn^a n !nq0nz n^b n 1 ! where the bosons a^n and b^n satisfy the following commutation relations: 1=2w) (A.16) (A.17) 1 (A.18) (A.19) [a^m; a^n] = m [b^m; b^n] = m [a^m; b^n] = 0: (1 (1 q0jmj)(1 1 tjmj q0jmj)(1 qjmj) qjmj) (pq0)jmj(1 tjmj) m+n;0; m+n;0; The dressed current t(z) = (z)x+(z) (z), corresponding to the stress energy tensor is given by exactly the same expression (4.1), as in the ordinary DIM case. Moreover, the dressing operators (z) and (z) are constructed from the generators of the elliptic DIM algebra using the same formulas (4.2) as give above. In the level two representation (2) u1;u2 the element t(z) produces the elliptic Virasoro stress-energy tensor T (z) = : e ^(z)e ^(t 1z) : + t : e ^(tz=q)e^(z=q) : (A.20) where ^ (z) = X n6=0 z n n(1 q0jnj) p a^ n 1 + !jnj X n6=0 n(1 z n q0jnj) (!2q0)jnj=2b^ n Let us also mention that the undressed elliptic DIM charge H x+(z)dz=z also leads to several very interesting objects. In the level one representation it gives elliptic Ruijsenaars Hamiltonian, while in the second level representation it is the di erence version of the intermediate long-wave Hamiltonian [197{202], which itself is a generalization of the Benjamin-Ono system. A.3 Rank > 1: DIM(gln) = quantum toroidal algebra of type gln In complete parallel with the previous consideration, DIM(gln) emerge as a deformation of the universal enveloping algebra of the Lie algebra An = Matn C[z 1; D 1] with i.e. of n n matrices with entries being elements of the algebra of functions on the quantum torus, zD = q1Dz. The deformation of A N introduces another parameter, q2. Providing this deformed algebra with two-dimensional central extension, one arrives at DIM(gln). The set of generators of DIM(gln) is Eik; Fik; Hir; Ki0; q c with k 2 Z, r 2 Z=f0g, 0 i n 1. The generating functions (currents) are: Ei(z) = X Eikz k; Fi(z) = X Fikz k; k2Z k2Z Ki (z) = Ki01 exp q 1) X Hi; rz r 1 r=1 ! The two centers are qc and = Qin=01 Ki0. The commutation relations are dij Gij (z; w) Ki q(1 1)c=2z Ej (w) + Gji(w; z) Ej (w)Ki q(1 c)=2z = 0; dij Gij (z; w) Ei(z)Ej (w) + Gji(w; z) Ej (w)Ei(z) = 0; djiGji(z; w) Fi(z)Fj (w) + Gij (w; z) Fj (w)Fi(z) = 0; djiGji(z; w) Ki q(1 1)c=2z Fj (w) + Gij (w; z) Fj (w)Ki q(1 c)=2z = 0; Gij (q cz; w) Gij (qcz; w) h Ei(z); Fj (w)i = Ki (z)Kj+(w) = h Ki (z); Kj (w)i = 0 q ij q 1 Gji(w; q cz) Gji(w; qcz) qcw z Ki+(z) Ki (z)Kj+(w) qcz w Ki (w) (A.21) (A.22) (A.23) (A.24) HJEP07(216)3 where, in variance with the DIM(gl1)-case, and powers of q are made from entries of the Cartan matrix. The commutation relations can be added with the Serre relations q1 = tq 1; q2 = q2; q3 = t 1q 1 (A.25) for n 3 for n = 2 symz1;z2 Ei(z1); hEi(z2); Ei 1(w)i Ei(z1); Ei(z2); hEi(z3); Ei 1(w)i (A.26) and similarly for F . The q-commutator is [A; B]q = AB qBA. The comultiplication is the same as for DIM(gl1). The structure functions are build from the a ne Dynkin diagrams and for gln-case are de ned as follows: for the simply laced case n 3 Abn Ab1 Gij (z; w) = > 8> (z >< (z > (z > >: (z ( 1 q1w) for i = j q2w) for i = j q3w) for i = j + 1 w) for i 6= j; j 1 1 3 dij = t 1 for i = j 1; n otherwise (A.27) Gg0l02 (z; w) = Gg1l12 (z; w) = (z Gg0l12 (z; w) = Gg1l02 (z; w) = (z q2w) q1w)(z q3w) d00 = d11 = 1; d01 = d10 = 1 (A.28) The a ne Dynkin diagram for n = 2 is not simply laced, and in this case For n = 1 we return to section 3.1, i.e. Gg0l01 (z; w) = (z q1w)(z q2w)(z q3w); d00 = 1 (A.29) One expects in the Pagoda (triple-a ne) case DIM(gcl1) (or Uq;t;et (gbbbl1), hence, the name Pagoda) the Dynkin diagram of the form: A.4 A ne Yangian of gl1 [139] One can consider a \quasiclassical" limit of the DIM(gl1) algebra, q = e~h1 , t 1 = e~h2 , t=q = e~h3 with properly rescaled generators. We also use another parameterizations: 1 = h1 + h2 + h3 = 0; (A.30) 3 = h1h2h3 In the limit of ~ ! 0, one obtains the a ne Yangian, which, on the gauge theory side, describes the 4d theories/Nekrasov functions. It is given by the commutation relations: and two more relations similar to (A.34) with ei substituted by fi and 3 substituted by 3. These commutation relations should be added by the Serre relations symi1;i2;i3 ei1; [ei2; ei3+1] = 0 i and similarly for fi. The commutation relations should be supplemented with the \initial conditions": 0;1 are the central elements, i.e. commute with everything all generators 2 is the grading element, i.e. [ 2; ej] = 2ej; 2fj; (A.36) (A.31) (A.32) (A.33) (A.34) (A.35) (A.37) (A.38) Note that, introducing the generator functions one can rewrite the commutation relations as with (u) = ((uu+hh11))((uu+h2)(u+h3) . h2)(u h3) h 1 i=0 1 i=0 e(u)e(v) f (u)f (v) (u)f (v) e(u)f (v) f (v)e(u) e(u) = X eiu i 1; f (u) = X fiu i 1; (u) (v) (v) (u) 1 i=0 v) e(v)e(u); u) f (v)f (u); v) e(v) (u); u) f (v) (u); v Heisenberg subalgebra The commutation relations of the Virasoro algebra with extended U[( 1 )-algebra, [Jm; Jn] = km m+n nJm+n n)Lm+n + (A.39) can be realized with identi cation: HJEP07(216)3 c 12 1 2 J 1 = e0; J1 = f0; L 1 = e1 + e0; L1 = f1 L 2 = 1 L0 = 2 + 2 1 + From the rst line it follows that k = . The other current mode are constructed by repeated commutators: J 2 = [e1; e0], J2 = [f1; f0] etc. Consistency conditions (e.g. J 3 [L 1; J 2 ] [L 2; J 1]) require 2 = ( 1 ) 3 0 (the dependence on h-parameters comes from relation with [e0; 3], which does not involve e3, because [e3; 0] = 0). Thus, there remains a free parameter . The central charge is c = 2 0 3 03 = 1 (1 1)(1 2)(1 3), where a = 0hbhc with (abc) is a cyclic permutation of (123). Representations: plane partitions The basis of a quasi- nite representation of this a ne Yangian6 can be described by plane partitions (3d Young diagrams). The generators of algebra act on the plane partition as follows: e(u) f (u) (u) adding a box to 3d Young diagram removing a box to 3d Young diagram diagonal action More precisely, the diagonal action is > = (u) = ;(u) Y > 2 u u0 h( ) where h( ) = xh1 + yh2 + zh3 and (x; y; z) are the coordinates of the box within the plane partition; 6Such representations are labeled by a triple of ordinary Young diagrams: \minimal" plane partitions are labeled by boundary conditions, [65, 139]. (A.41) (A.42) the raising (lowering) action is e(u)j f (u)j > = > = X 2 +n X 2 n E( F ( u u0 u0 h( ) h( ) j j + >; A ( 1 ) + A (2) E A ( 1 ) + A (2) + A ( 3 ) = 0 (A.48) (A.43) (A.45) (A.47) h B h A (A.46) n n (u) u h( A) o (u) u h( B) o = where + ( box as compared to . ) denotes arbitrary plane partition with one additional (one subtracted) Here F and E are coe cients which have to be de ned from the commutation relations of the algebra and are some residues of inhomogeneity in the standard spin chain. (u), u0 is a constant shift, a counterpart of and F ( generating functions it looks like Formula (A.42) is derived by acting with the both sides of the commutation relation (u v) e(v) (u) on j >, using (A.43) and then taking the residue at v = h( ) Constraints on the coe cients E and F . Constraints on functions E( ) can be derived from the commutation relations [ei; fj ] = i+j . For the (u) = 1 + 3 X E( h( ) h( ) (A.44) where the second-order pole does not contribute. This relation does not x E and F completely. Imposing an additional requirement of unitarity E( one immediately obtains [139] +) = F ( + ), 3E( 3E( resu !h( ) )2 = resu !h( ) (u) (u) The commutation relation e(u)e(v) v)e(v)e(u) relates adding two boxes in One still has to x the sign (after taking the square root). di erent order: E( E( A)E( + B)E( + A B A + A + B) B) To check that it is satis ed, calculate the square of the l.h.s.: resu !h( A) (u) resu !h( B) + A (u) resu !h( A) (u) resu !h( B) resu !h( B) (u) resu !h( A) + B (u) = resu !h( B) (u) resu !h( A) h( B) h( A) h( A) h( B) h( B) h( A) 2 Similarly one can check the Serre relations by adding three boxes: X hh( A ( 1 ) ) 2S3 E 2h( A (2) ) + h( A ( 3 ) ) E A ( 1 ) i { 54 { B (ei; ej ) < jfiej j >= i+j; < should have only r 1 independent lines, i.e. there is a relation Then, the generating function of eigenvalues r 1 X i=0 i i+k; = 0; k 0 j; u j 1 = 1 j=0 f (u) where f (u) and g(u) are polynomials of degree r and g(u): Consider the case of r = 2, i.e. a single state at the level one and linear functions f (u) (u) = u + 3 0; = 1 + 3 0; A simplest example of the highest-weight representation. tion with the highest weight j >: j j j; j fj j >= 0 Since we consider the quasi- nite representations, there should be linear relations among eij >. Consider vectors in the representation at the rst level with nitely many, r independent vectors. This means that the Shapovalov matrix at the level one, which is (A.49) (A.51) (A.52) (A.53) (A.54) (A.55) (A.56) Then, the commutation relations and the Serre relations implies that there are 3 states at the second level (this is since the function (u) is a ratio of cubic polynomials) and 6 states at the third level. These particular numbers are equal to the number of 3d Young diagrams with a given number of boxes. This means that the highest weight is associated with the trivial plane partition j >= ;, and the single rst level vector is associated with the only one box plane partition j >: eij; >= 0; i > 0; e0j; > ij; >= 0; i > 0 Since 1 is a center and [ 2; e0] = 2e0 one immediately obtains Using these formulas, from the Serre relations that involve j and e0;1;2;3, one gets 2j 1j >= 0; >= 2j (u)j > = u u + 3 0;; '(u)j > > Open Access. 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Hidetoshi Awata, Hiroaki Kanno, Takuya Matsumoto. Explicit examples of DIM constraints for network matrix models, Journal of High Energy Physics, 2016, 103, DOI: 10.1007/JHEP07(2016)103