Refined toric branes, surface operators and factorization of generalized Macdonald polynomials

Journal of High Energy Physics, Sep 2017

We find new universal factorization identities for generalized Macdonald polynomials on the topological locus. We prove the identities (which include all previously known forumlas of this kind) using factorization identities for matrix model averages, which are themselves consequences of Ding-Iohara-Miki constraints. Factorized expressions for generalized Macdonald polynomials are identified with refined topological string amplitudes containing a toric brane on an intermediate preferred leg, surface operators in gauge theory and certain degenerate CFT vertex operators.

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Refined toric branes, surface operators and factorization of generalized Macdonald polynomials

HJE Re ned toric branes, surface operators and factorization of generalized Macdonald polynomials National Research Nuclear University MEPhI 0 1 2 3 4 5 6 7 8 0 Moscow , 115409 , Russia 1 Moscow 117218 , Russia 2 Institute for Theoretical and Experimental Physics , ITEP 3 I-20126 Milano , Italy 4 INFN , sezione di Milano-Bicocca 5 Piazza della Scienza 3 , I-20126 Milano , Italy 6 Dipartimento di Fisica, Universita di Milano-Bicocca 7 Moscow 117312 , Russia 8 Physics Department, Moscow State University We nd new universal factorization identities for generalized Macdonald polynomials on the topological locus. We prove the identities (which include all previously known forumlas of this kind) using factorization identities for matrix model averages, which are themselves consequences of Ding-Iohara-Miki constraints. Factorized expressions for generalized Macdonald polynomials are identi ed with re ned topological string amplitudes containing a toric brane on an intermediate preferred leg, surface operators in gauge theory and certain degenerate CFT vertex operators. Topological Strings; Conformal and W Symmetry; M(atrix) Theories 1 Introduction 1.1 Re ned topological strings and branes 1.2 q-deformed CFT 2 Factorization of generalized Macdonald polynomials 3 4 1 2.1 2.2 2.3 2.4 2.5 3.1 3.2 Schur and Macdonald polynomial factorization. A reminder Generalized Macdonald polynomials factorization on the general topological locus Factorization identities from matrix model averages New formulas for skew generalized Macdonald polynomials Gluing, traces and factorization of instanton sums Toric brane on the intermediate leg and surface operators Re ned topological amplitudes with branes Degenerate elds and surface operators Conclusions and further prospects Introduction The interplay between algebraic structures and geometry has been fundamental to the development of mathematics in the recent decades. In particular, it has led to a cornucopia of new results in mathematical physics. One of examples is the (re ned) topological vertex function [1{9] which on the geometric side describes Gromov-Witten and DonaldsonThomas invariants of toric Calabi-Yau (CY) three-folds, while from the algebraic point of view it is the intertwiner of the Ding-Iohara-Miki (DIM) algebra1 [10, 11]. The second famous example comes from the gauge theory: the equivariant cohomology of the instanton moduli spaces (captured by Nakajima quiver varieties [12{14] and the corresponding Nekrasov partition functions [15{22]) is acted on by a certain vertex operator algebra, which turns out to be the WN -algebra of two dimensional conformal eld theory. This correspondence between the geometric (moduli space) and algebraic (WN -algebra) objects is known as the AGT relation [23{25] and has many known implications and generalizations [26{36]. These two examples are in fact directly related to each other and their relation can be understood on both sides of the algebro-geometric correspondence. On the algebraic side the equivariant cohomology (or more precisely K-theory [37]) of the instanton moduli spaces is a tensor product of Fock representations of the DIM algebra [38], while the q-deformed 1Alternative names include quantum toroidal, elliptic Hall, spherical degenerate DAHA algebras, or simply Uq;t(gbbl1). { 1 { WN -algebra generators are built from the currents of the DIM algebra [39{41] and vertex operators are combinations of topological vertices intertwining the action of the DIM algebra and therefore of the WN -algebra [42]. On the geometric side the 5d gauge theory is obtained by compactifying M-theory on the toric CY three-fold. The parameters of the gauge theory correspond to Kahler moduli of the CY and the cohomology of the moduli space of instantons is identi ed with the Hilbert space of M2 branes stretching between toric xed points. In this paper we explore a particular case of the algebro-geometric correspondence, which is important for topological strings as well as for gauge theories. We consider re ned toric branes wrapping Lagnangian submanifolds inside a toric CY three-fold [43]. As is well-known, this setup corresponds to surface operators in gauge theory and to degenerate elds of the WN -algebra [44{50, 53{55]. We will consider mostly the algebraic side of the problem and relate the stack of re ned branes on the preferred leg of the toric diagram to a particular intertwining operator of DIM algebra, which can be recast into a combination of generalized Macdonald polynomials [56, 59]. The properties of the branes are related to the remarkable factorization identities for generalized polynomials evaluated on a particular submanifold in the brane moduli space called the topological locus. Generalized Macdonald polynomials [56, 59] play the central role in the AGT correspondence. They arise naturally in the study of the DIM algebra representations on tensor producs of Fock modules. In [59, 60] matrix elements between generalized Macdonald polynomials were computed using matrix model techniques. They turned out to reduce to integral factorization identities, which provide a very explicit answer for the q-Selberg averages in terms of Nekrasov functions. In this note we would like to use these integral identities to prove new topological locus factorization identity recently found in [61]. For special values of parameters the integrals disappear and one is left with Macdonald polynomials evaluated at the topological locus. The integral identity implies that those are still given by the factorized formulas. This technique allows us to nd several new identities for generalized Macdonald polynomials on a more general topological locus. In this way we prove and generalize the results of [61]. We then connect the factorization of the polynomials to the re ned topological string picture. To this end we interpret matrix model averages as topological string amplitudes on toric CY threefolds with Lagrangian branes appropriately placed on the legs of the toric diagram (for exact correspondence and explanation see [60]). Factorization of averages relies on the particular properties of the branes residing on the preferred direction of the diagram. The topological locus corresponds to a certain degenerate limit of the CY, which models addition of a stack toric branes on one of the legs in the preferred direction. Factorization of generalized Macdonald polynomials in this picture allows us to understand the amplitudes with toric branes placed on intermediate preferred legs of the toric diagram. One can also interpret factorization formulas for generalized Macdonald polynomials in terms of CFT vertex operators in Dotsenko-Fateev (DF) representation. In this case the radical simpli cation of the formulas occurs due to the particular choice of the dimensions for which the vertex operators do not require any screening currents. In view of the AGT relations this corresponds to a particularly simple surface operator in the corresponding gauge theory. { 2 { In the remaining part of the introduction we discuss the main points of this network of correspondences in more detail. In section 2 we write down and prove the factorization identities, in section 3 we connect this results with topological strings and gauge theories. We present our conclusions in section 4. 1.1 Re ned topological strings and branes Re ned topological string theory [8, 9] is a deformation of the topological string theory living on a toric CY three-fold, which gives additional information on the spin content of the D-brane BPS spectrum of type IIA string theory. Re ned amplitudes are computed using re ned topological vertex, quite similarly to the ordinary topological vertex computations [1{7]. A Young diagram Yi is assigned to each leg i. The vertices, always trivalent, correspond to certain explicit combinations CYiYjYk (q; t) of symmetric functions depending on three Young diagrams Yi;j;k on the adjacent legs i, j, k: There is one crucial point in the computation of re ned amplitudes. Unre ned topological vertex CYiYjYk (q; q) is cyclically symmetric in the three Young diagrams Yi, Yj , Yk, while the re ned vertex CYiYjYk (q; t) for general q 6= t is not. The recipe above, therefore, includes the choice of ordering of Young diagrams in each vertex. This choice is indicated by the double ticks and the labels q and t on the corresponding legs in eq. (1.1). In what follows we will usually omit the indices q and t. It turns out that the choices for the neighbouring vertices should be coordinated, so that the only freedom remaining is the global choice of the preferred direction (horizontal in eq. (1.1)) on the toric diagram. We omit here the concrete expression for CYiYjYk (q; t), which can be easily found in the literature, not to overcomplicate our presentation. To get the nal answer for the amplitude one takes the sum over all the Young diagrams Yi on the intermediate legs, each taken with weight2 ( Qi)jYij, where Qi denotes the exponentiated complexi ed Kahler parameter of the two-cycle associated to the leg i. Let us give the simplest example of two vertices glued together to form the resolved conifold geometry: Yi i t j Yj q Yk k = CYiYjYk (q; t) (1.1) 2In general there are also framing factors which we will not need here. = X( Q1)jY1jCY1Y2Y3 (q; t)CY1TY4TY5T (t; q) (1.2) Y1 { 3 { = X( Q)jY1jCY1?W (q; t)CY1??(t; q) Y1 = Zopen ? W Q1 ? ? ? ? (1.3) (1.4) HJEP09(217) With the external lines one can associate either empty or non-empty diagrams which do not take part in the sums. The former choice gives the closed string amplitude (partition function), while the latter one gives the open string amplitude with stacks of toric branes on the external legs determining the external diagrams, or \boundary conditions" for the theory: W = Q1 Q1 W Zclosed(Q1) = Zopen ? ? Q1 The dashed lines here denote toric branes. The number of branes in the stack sets the maximal possible number of rows in the Young diagram W . The nal answer for the closed string amplitude does not depend on the choice of the preferred direction, though open string amplitudes do.3 In the unre ned case there is also a natural way to put a stack of toric branes on the internal leg (and indeed on any Lagrangian submanifold of the CY). However, in the re ned case the study of branes on the external lines has been so far very limited (see [51], though). In the present paper we will address this problem and propose a recipe to put a stack of branes on the intermediate preferred leg. To do this we will employ the duality between open and closed string amplitudes. Open-closed duality in topological strings allows one to model stacks of toric branes by closed string amplitudes [52{55]. The open string amplitudes should be packed in the Ooguri-Vafa generating function, and the closed strings propagate in the modi ed background containing additional vertical line in the toric diagram. Let us draw the dual pictures in the simplest case of one toric brane. The diagram corresponding to the brane can have at most one column, i.e. it is of the form4 W = [l]. We then have X zl { 4 { In the l.h.s. of eq. (1.5) z plays the role of the holonomy of the (abelian) gauge eld living on the toric brane, while on the r.h.s. it is identi ed with the Kahler parameter of the 3In the algebraic approach of [38] the choice of the preferred direction is associated with the choice of the slope of the coproduct used in the de nition of the DIM algebra. The most relevant choices used e.g. in [62, 63] where the \horizontal" coproduct and the \vertical" (or perpendicular, or Drinfeld) coproduct ?. 4Compared to the notation of [55] we use the transposed diagram W . Another way to obtain our conventions from that of [55] is to exchange the equivariant parameters, q $ t 1. Using the terminology of [43] this amounts to the exchange of q-branes and t-branes. two-cycle obtained by adding an extra vertical line to the geometry. In general, for N toric branes the Kahler parameter in the r.h.s. will change to q 1=2tN+1=2. There are several points requiring clari cation in this approach which are absent for ordinary topological string, i.e. for t = q, and appear only in the re ned case: 1. The additional vertical line necessarily intersects all the parallel legs coming out of the diagram if there happen to be any (see gure 4 b) for an example). One expects that the amplitude should be insensitive to these intersections since they have nothing to do with the toric brane insertion. However, in the re ned case there is no way to make a \trivial crossing" of two lines: no choice of the Kahler parameter gives the desired result. One concludes that for several parallel legs the toric brane attached to one of them also interacts with all the others. 2. Although there is no way to make a \trivial crossing" of lines one can make a crossing, which models the trivial one in some situations. For example, this crossing can be used to set the diagram on one side to vanish if the diagram on the other vanishes (see gure 5 b), c)). However, it works only in one direction: either the left diagram vanishes whenever the right one is empty or vice versa. 3. Because of these features of the re ned theory it is unclear how to put a toric brane on the intermediate preferred leg. The explanation of these puzzles will be the main focus of the present work. We will show that the amplitudes in the presence of the toric brane on the intermediate leg can be identi ed with generalized Macdonald polynomials evaluated on the topological locus. 1.2 q-deformed CFT It was shown in [57{60], that certain re ned topological string amplitudes on toric CY three-folds correspond to conformal blocks of the q-deformed Virasoro or WN -algebras.5 The horizontal legs of the toric diagram represent the Hilbert space of the CFT, on which the conformal algebra acts, and the intersections with vertical legs give vertex operators or intertwiners of the algebra (see gure 1). Naturally, the sums over Young diagrams living on the horizontal lines represent the sums over the complete basis of states in the CFT Hilbert space. There is a natural choice for such a basis | the basis of generalized Macdonald polynomials, which leads to explicit factorized matrix elements for the vertex operators given by Nekrasov formulas. The sums over diagrams on the vertical lines corresponds to the integrals over the positions of the screening currents appearing in the Dotsenko-Fateev representation of the conformal blocks. Therefore, vertical lines correspond not simply to vertex operators, but more concretely to the screened vertex operator insertions [41, 60]. The topological loci, i.e. the submanifold of the moduli space on which generalized Macdonald polynomials factorize into products of monomials, represent the special set of parameters, for which the number of screenings is zero. 5More concretely, to get a conformal block one should consider only balanced toric diagrams, see [40, 41] for details. { 5 { W1 W2 Q1 Q2 QF Y2 = hMY1Y2(QF Q2 1)j V Q1 Q2 jMW1W2(QF Q1)i HJEP09(217) of CFT. Double lines in the r.h.s. denote the CFT Hilbert space on which Virq;t Heis algebra acts. On the l.h.s. it corresponds to the two horizontal lines. The circle in the r.h.s. represents the vertex operator corresponding to the intersection with the vertical line in the toric diagram in the l.h.s. The matrix element on the r.h.s. is computed in the basis of generalized Macdonald polynomials MY1Y2 , which corresponds to the choice of horizontal preferred direction (lines marked by by double ticks) on the l.h.s. Notice the relation between the Kahler parameters Q1;2, QF of the CY on the l.h.s. and the parameters of the vertex operator VQ1=Q2 and the states on the r.h.s. . 2 2.1 Factorization of generalized Macdonald polynomials Schur and Macdonald polynomial factorization. A reminder Let us rst recall the familiar factorization formulas for Schur and Macdonald polynomials. Schur polynomials sY (xi) are symmetric polynomials in the variables xi, i = 1 : : : N labelled by Young diagrams Y . They can be understood as characters of nite-dimensional irreducible representations RY of slN algebra corresponding to the Young diagrams Y : sY (x) = trRY diag(x1; : : : ; xN ) FWoer upsauratlilcyulwarritveaalullessyomfmtehtericvaproialybnleosmliyailnsgasonfunthcteiotnospoolfotghiceaplolwoceurssupmn s=pn 11= AqPnn iN=S1chxuinr. polynomials are given by very simple factorized formulas: sY 1 1 An tn = Y (i;j)2Y ti 1 1 1 Atj i tYi j+YjT i+1 These expressions can be related to \quantum dimensions", or generating functions of the values of the Casimir operators on the corresponding representations. Macdonald polynomials MY(q;t)(pn) provide a natural generalization of Schur polynomials, depending on two parameters q and t. Macdonald polynomials do not have immediate group theory interpretation, but nevertheless have many properties similar to Schur polynomials, to which they reduce for t = q. In particular, they still factorize on the topological locus pn = 11 Atnn : MY(q;t) 1 1 An tn = Y { 6 { Notice that the parameters of the topological locus for Macdonald polynomials are tied with the deformation parameters, so that for given t the locus is one-dimensional. We will see similar e ect in the following sections, where generalized Macdonald polynomials will depend on an additional parameter which will enter the de nition of the topological locus. Generalized Macdonald polynomials factorization on the general topological locus In this section we give general factorization formulas for generalized Macdonald polynomials. Concretely, we have found a generalization of the factorization formula for generalized Macdonald polynomials conjectured in [61] to a wider topological locus. The identity reads:6 0 An tn ; = ( 1)jY2jt t n q 1 q tn The original formula which appeared in [61] is given by 1 qxqAi j tBjT i+1 1 qxq Bi+j 1t AjT+i Y (i;j)2B (1 qArmY (i;j)tLegY (i;j)+1); jY j = l(Y ) X Yi; i=1 jjY jj2 = l(Y ) X Yi2 i=1 = ( 1)jY1jq jjY1jj2 jY1j Q jY2jqjjY2jj2 jY2jt 2 jjY2Tjj22 jY2j G(?qY;t1)(Q 1)G(?qY;t2)(1) CY01 (q; t)CY02 (q; t)G(Yq2;Yt)1 (Q 1) It is obtained from eq. (2.4) in the limit A ! 1 (one should divide both sides by AjY1j+jY2j to get a nite answer). For completeness let us also give the factorization formula where the second argument of the generalized Macdonald polynomial is nontrivial: 0 1 1 t B q tn 6Similar identity for A = Q has already appeared in [59] (see eq. (24) there). There was a minor typo in [59] eq. (24): Kronecker symbol Y2? was missing in the r.h.s. { 7 { (2.4) (2.5) (2.6) (2.7) (2.8) where the integration measure is and the Jackson q-integral is de ned as hf (xi)iu;v;N;q;t = def R01 dqN x (u; v; N; q; tjxi)f (xi) R01 dqN x (u; v; N; q; tjxi) (u; v; N; q; tjxi) = Y Y xxji ; q i6=j k 0 t xxji ; q N 0 Y i=1 u Y (xi; q)1 k 0 (qvxi; q)1 1 A Notice the asymmetry between the two arguments of the generalized Macdonald polynomials: while the rst factorization formula (2.4) has nontrivial dependence on both Young diagrams, the second one (2.8) actually reduces to the formula (2.3) for the ordinary Macdonald polynomials. This peculiar feature can be traced back to the nontrivial choice of coproduct in the DIM algebra [64, 65]. From the form of eqs. (2.4) and (2.8) one could have suspected that there is a general two-parametric factorization formula involving both A and B. However, it turns out that this is not the case, as we explain below. Factorization identities from matrix model averages Before presenting more generalizations of the factorization formulas for generalized polynomials let us give here a short and simple proof for the factorization identities obtained so far. To this end we will employ the integral factorization identities discovered in [59, 60]. These identities give explicit factorized answers for q-Selberg averages of generalized Macdonald polynomials. The Selberg average of a symmetric function f (xi) is given by the following matrix integral: HJEP09(217) Z a 0 dqxg(x) = (1 q)a X qng(qna) One example of the factorized identity for the average considered in [60] is M A(qB;t) q u 1t p n + (t=q)n qnv 1 tn ; p n t n 1 q 1 (t=q)n tn = ( 1)jAjq 2jBj+ujAjtjBj jAjtP(i;j)2B i+2 P(i;j)2A iq P(i;j)2A j u;v;N;q;t = GA? t N q u GA? t N 1qv+1 GB? t N 1 q GB? t N 2qu+v+2 CA0(q; t)CB0(q; t)G(BqA;t) (qu+1t 1) : (2.12) Notice that the parameters of the measure also enter the arguments of the generalized polynomials under the average sign. Let us make a peculiar specialization of eq. (2.12) and take N = 0. What does it mean to have zero number of integrations? There is of course no general answer, but for Selberg averages the de nition we consider seems very natural and can be obtained from analytic continuation in N . The generalized polynomial under { 8 { 1 1 the average is written in terms of power sums pn = PiN=1 xin of integration variables. For N = 0 power sums contain zero terms and therefore should vanish. It is also evident that for N = 0 there are no integrations neither in the numerator, nor in the denominator in the de nition of the average and the integration measure is absent. Thus the l.h.s. of eq. (2.12) reduces to the generalized Macdonald polynomial evaluated at the point pn = 0, while the r.h.s. gives the correct factorized answer, coinciding with eq. (2.4). Notice that the topological locus parametrized by u and v in eq. (2.12) and by Q and A in eq. (2.4) is two-dimensional. This will always be the case in our considerations since the original integral depends on three parameters, u, v, and N and we have to put N to zero. HJEP09(217) More identities can be obtained by using the symmetry of the Selberg measure (u; v; N; q; tjxi) under the change of parameters: Of course, if the function f itself depends on the parameters u, v or N one has to revariables (2.13) in eq. (2.12) and setting Ne = 0 we get the identity (2.8). place them with ue, v or Ne respectively to get the same average. Making the change of e Summarizing, the factorization identities for generalized Macdonald polynomials (2.4), (2.8) follow from the integral identity (2.12) in the limit N = 0. In the next section we will give more factorization identities involving skew generalized Macdonald polynomials. They are proven using a similar argument. hf (x)iu;v;N;q;t = hf (x)iue;ve;Ne;q;t : Since the same, (u; v; N; q; tjxi) = (ue; ve; Ne ; q; tjxi), the average of any function f (x) remains 2.4 New formulas for skew generalized Macdonald polynomials We can also take the specialization N = 0 in more general integral factorization formulas from [60] (see eqs. (93), (94) there). The identities we obtain in this way involve two skew generalized Macdonald polynomials. Skew generalized Macdonald polynomials are de ned similarly to the usual skew Macdonald polynomials: MY(q1;Yt)2=Z1Z2 (Qjpn; pn) = M Z(q1;t) n 1 1 q M Z(q2;t) n 1 1 q MY(q1;Yt)2 (Qjpn; pn) { 9 { (2.13) (2.14) (2.15) where MZ details let us write down the nal results: (q;t) are ordinary Macdonald polynomials. Without giving too much technical 1 1 t q tn n ; 1 1 1 B n tn A = t qB q BQ Y~ ; W~ zbifund (q;t) G(Yq1;Yt)2 (Q)G(Wq;2tW)1 qjjW2jj2 jW2jt 1 Q 2 ; t BQ q 1 2 ; q B t q t BQ t n 1 2 1 2 jjY2jj2 jY2j tjjY2Tjj2 jY2j ( t)jW1j q jjW1jj2 jW1j 2 2 tn n t q 1 1 (2.16) (2.17) (2.18) (2.19) 1 where the conjugate generalized polynomial is de ned as and the norm of Macdonald polynomial is given by an explicit expression jjMY jj2 = CY0 (q; t) CY (q; t) CY (q; t) = (1 qArmY (i;j)+1tLegY (i;j)) The bifundamental Nekrasov function is given by Y~ ; W~ zbifund(Q; P; M ) = G(q;t) Y1W1 Q M P G(q;t) Y1W2 G(q;t) Y2W1 1 M QP G(q;t) Y2W2 1 M QP There is one more identity similar to eq. (2.16): X Z1;Z2 jjMZ1 jj2jjMZ2 jj q tQ Y~ ; W~ zbifund t jW2j (q;t) 2 M jjW2Tjj2 jW2j 2 jY1j jjY1Tjj2 jY1j Qt2 jY2j qjjW2jj2 jW2jt 1 Q 2 ; t AQ q 1 2 ; t A q G(Yq1;Yt)2 (Q)G(Wq;2tW)1 tAQ q A n t n 1 ; 0A jjY2jj2 jY2j tjjY2Tjj2 jY2j ( A)jW1j q jjW1jj2 jW1j 2 2 CY01 (q; t)CY02 (q; t)CW01 (q; t)CW02 (q; t) Identities (2.16), (2.20) are more general than eqs. (2.4), (2.8) and reduce to them in special cases. For Y1;2 = ? eq. (2.16) reduces to eq. (2.8) and for W1;2 = ? it reduces to eq. (2.4). In eq. (2.20) the situation is reversed, i.e. for Y1;2 = ? it reduces to eq. (2.4) and for W1;2 = ? it reduces to eq. (2.8). Gluing, traces and factorization of instanton sums The new identity (2.16) allows one to glue several factorized expressions together and then use Cauchy completeness to obtain a factorized answer for the full sum of factorized terms. As a simplest example we can take the trace over Young diagrams Y~ = W~ in the identity (2.16). In the language of gauge theory this corresponds to making a circular quiver representing a U( 2 ) adjoint theory, while for topological strings this gives the partial compacti cation of the base of the toric bration. In each case, to get a meaningful result we have to set spectral parameters of the generalized Macdonald polynomials equal to each other. For eq. (2.16) this means taking B = qt . Thus, we set B = qt , Y1 = W1 and Y2 = W2 in eq. (2.16) and take the sum over Young diagrams Y1;2 with weight jY1j+jY2j jjMY1 jj2jjMY2 jj 2 . The r.h.s. of eq. (2.16) then takes the form of Nekrasov instanton partitions function for a particular value of the adjoint hypermultiplet mass: X jY1j+jY2j X Y1;Y2 jjMY1 jj2jjMY2 jj2 Z1;Z2 jjMZ1 jj2jjMZ2 jj 2 MY1(qY;2t=)Z1Z2 @Q 1 1 t n q tn ; 1 1 t n 1 q tn A Now we notice that the l.h.s. of eq. (2.21) does not depend on the choice of basis in the space of symmetric polynomials, since it is a trace over this space. This immediately implies that the l.h.s. is in fact independent of Q. Choosing the basis of ordinary Macdonald polynomials we nd that the sum factorizes into a product of two identical sums: X jY1j+jY2j Y1;Y2 jjMY1 jj2jjMY2 jj2 Z1;Z2 jjMZ1 jj2jjMZ2 jj so that the double sum in the r.h.s. turns into a single one: jY j X Y jjMY jj2 X q Z jjMZ jj2 MY(q=;Zt) @ t jZj 0 1 1 t n 1 q tn AMY(q=;Zt) (0) = X Y jY j = Y MY(q1;Yt)2=Z1Z2 (Qj0; 0) = 6X One can immediately notice that jY j Y jjMY jj2 X q Z jjMZ jj2 MY(q=;Zt) @ 0 1 MY(q=;Zt)(0) = Y Z jjMY jj2; MY(q=;Yt)(pn) = jjMY jj 2 t jZj and 0 0 1 1 1 t q t n q tn ; t n 1 q tn 1 1 AMY(q=;Zt) (0)7 (2.22) (2.21) t n 1 A q tn 3 2 5 (2.23) 1 t q k (2.24) Eventually, we get the factorized answer for Nekrasov instanton partition function: X 1 t q This is in fact nothing but the partition function of the corresponding 2d CFT, which contains two bosonic eld, hence power two in the r.h.s. The example we have described is of course a trivial one, since each term in the l.h.s. simpli es due to the identity However, gluing two or more bifundamental contributions from eqs. (2.16), (2.20) together one gets nontrivial factorization identities for linear quiver gauge theories. Also one can take the trace to obtain circular quivers with several nodes. 3 Toric brane on the intermediate leg and surface operators In this section we will demonstrate that factorization identities we have obtained can be thought of as the amplitudes of re ned topological strings in the presence of a stack of toric branes. We will also comment on their relation with surface operators in gauge theory and degenerate vertex operators in 2d CFT. 3.1 Re ned topological amplitudes with branes Selberg averages such as eq. (2.12), which we have used in our proof of factorization in section 2.3, can be identi ed with re ned topological string amplitudes on toric CY depicted in gure 2. Kahler parameters Q1;2 and QF of the CY are related to the matrix integral parameters u, v, N as follows: Q1 = t 21 N q 12 v; Q2 = t 21 N q 21 ; QF = qu+v+ 32 tN 32 (3.1) instead of u, v and N to get the second possible identi cation between the parameters. In eq. (3.1) one can also use ue, v and Ne obtained by the change of variables (2.13) e Factorization happens for a special value of the parameters corresponding to N = 0 | the topological locus. On this locus one of the resolved conifold pieces in the toric diagram of the CY degenerates, i.e. its Kahler parameters becomes q qt or q qt as shown in gure 3. For unre ned amplitudes degenerate resolution factorizes into a product of two pieces as shown in gure 4. Each piece is given by an explicit factorized formula, which coincides with the factorized answer for the polynomial. However, in the unre ned case the answers for the amplitudes from gure 4 b) are not very interesting since generalized Macdonald polynomials in this case reduce to products of Schur functions, and the factorization identities turn into the well-known formulas for quantum dimensions (2.2). They reproduce the known amplitudes in the presence of the stack of toric branes on the intermediate leg in the unre ned theory. W1 W2 Q1 Q2 QF Y2 DP EF MY1Y2=EF MW1W2=EF E u;v;N Macdonald polynomials. The number of integrations N and the parameters of the integral u; v are expressed through the Kahler parameters Q1;2 and QF according to eq. (3.1). HJEP09(217) W1 W2 q t q p q Q 1 t p q t b) Y1 Y2 = p qt in the CY corresponds to setting The lower resolved conifold piece with Kahler parameter p q B 1 can be transformed by geometric transition into geometry containing a stack of M toric branes, where tM = qt B 1 . b) Setting t N = v in the Selberg average. This average is related by the symmetry (2.13) to the integral with N = 0 number of integrations shown in a). For Q1 = q t A and QF = p qt (AQ) 1 one arrives at the factorization formula (2.20). This amplitudes q corresponds to a stack of M 0 toric branes on the upper horizontal leg with M 0 given by tM0 = A. For re ned amplitudes degenerate conifold geometry does not split into two parts. Moreover, there are two di erent degenerations of the resolved conifold with Kahler parameter either q qt or q qt as shown in gure 5 a). The di erence between these two situations is evident from gure 5 b) and c): when some of the legs are empty the amplitudes do factorize and give the same result as in the unre ned case. The amplitudes from gure 3 are still given by the factorized expressions, though they cannot be separated into two noninteracting parts as shown in gure 4 b). We argue that this is the natural de nition of the stack of toric branes placed on the horizontal leg of the diagrams. For a single horizontal leg the corresponding geometry is shown in { 13 { Y1 Y2 into a product of two separate non-interacting lines. Notice that there is no preferred direction in the unre ned case. b) Though the values of the polynomials on the topological locus are factorized into a product of monomials, they do not factorize into a product of independent terms corresponding to two horizontal lines. This happens only in the unre ned limit t ! q. 6 = A B W1 W2 a) b) c) A A ? B Q = 1 C q t q 2 one gets di erent decoupling conditions, b) and c). the diagrams are empty, the amplitude reduces to the unresolved one. For two choices of Kahler W1 W2 D C ? C D ? B = qt A = qt A = B = 1 A Q1 = Q2 = qt the two loci. strip geometry which can also be thought of as Coulomb moduli space of the gauge theory. The moduli space is parametrized by Q, A and B, which are transformed into Q1, Q2 and QF using the formulas Q1 = q qt A, Q2 = p q B 1, QF = p qt (QA) 1. There are two distinct topological t loci, fA = 1g and fB = 1g (shown in red and blue respectively), which intersect on the line A = B = 1. There are also two special lines, A = qt and B = qt , each on its own locus, where and 1. In the unre ned limit t ! q the special lines coalesce with the intersection of eq. (1.5). However, if we have several horizontal legs, the vertical line will intersect several of them. Figure 3 describes precisely this situation: gure 3 a) models the stack of M branes on the lower horizontal line, and gure 3 b) represents the stack of M 0 branes on the upper horizontal line. The intersection of the vertical line with the second horizontal leg is degenerate and in the unre ned limit gives the trivial crossing from gure 4. In the re ned case the crossing trivializes when the corresponding diagram on the left or right of the crossing is empty as depicted in gure 5 b), c). Let us recapitulate our main point. A stack of re ned toric branes sitting on a preferred leg of the diagram interacts with all other parallel legs. The resulting amplitude is given by the factorized value of generalized Macdonald polynomials evaluated on the topological locus. 3.2 Degenerate elds and surface operators Topological loci can be given a natural gauge theory interpretation. The Kahler moduli space of the CY is identi ed with the Coulomb moduli space of the 5d gauge theory. The topological locus corresponds to the root of the Higgs branch inside the Coulomb branch. In other words, degeneration of the resolved conifold pieces of the toric diagram allows one to deform the geometry instead. This deformation corresponds to going on the Higgs branch. The identi cation can also be seen directly by identifying the parameters of the corresponding gauge theory with Kahler parameters of the CY. The geometry in gure 2 corresponds to a single bifundamental eld of mass m charged under two SU( 2 ) gauge groups, HJEP09(217) SU( 2 )L and SU( 2 )R. The parameters of the theory are the Coulomb moduli QL = qaL , QR = qaR and the exponentiated mass Qm = qm. They are given by the following formulas: QL = (Q1QF ) 2 ; 1 QR = QF Q2 1 21 ; Qm = r t q Q1Q2 1 21 (3.2) One can immediately see that on the topological locus where we have either QL = QRQm or QR = QLQm. This indeed corresponds to the origin of Higgs branches. It is well-known that gauge theory at this point is equivalent to a theory on a defect associated with surface operator. Factorization formulas allow us to identify partition function of this theory with the values of the generalized Macdonald polynomials on the One nal interpretation of the factorization formulas is given by the thedegenerate vertex operators in q-deformed 2d CFT. According to the AGT relations, gauge theory we have just described corresponds to vertex operator in the Liouville theory. The Selberg integrals used in section 2.3 are interpreted as integrals in the DF screening charges. Naturally, if N = 0, the screening charges are absent and we return to pure bosonic vertex operator V Q1 . Matrix elements of this operator in the generalized Macdonald basis are given by generalized Macdonald polynomials evaluated on the topolgical locus. Schematically this can be written as follows: hMY1Y2 (QR)jVQm jMW1W2 (QRQm)i = MY1Y2=Z1Z2 (QR)MW1W2=Z1Z2 (QRQm)jtop locus X Z1Z2 As usual degenerate eld obeys a di erence equation. Thus, generalized Macdonald polynomials taken on the topological locus should also obey this equation. We hope to clarify (3.3) this point in the future. 4 Conclusions and further prospects In this paper we have presented new factorization identities for generalized Macdonald polynomials. We proved the identities using the technique of matrix models and related them to re ned topological string amplitudes in the presence of a stack of toric branes. We have also identi ed the corresponding gauge theories and CFT vertex operators. It would be interesting to understand better the meaning of the factorization identities directly in the DIM algebra. Also we would like to investigate the di erence equations satis ed by the polynomials on the topological locus. Nekrasov-Shatashvili limit of our construction might help to understand better the surface operators corresponding to toric branes on the intermediate legs of the toric diagram. Acknowledgments The author thanks Y. Kononov and N. Sopenko for discussions. The work of the author was supported in part by INFN, by the ERC Starting Grant 637844-HBQFTNCER and by RFBR grants 17-01-00585, 15-31-20484-mol-a-ved, 15-51-52031 NSC, 15-51-50034 YaF, 16-51-53034 GFEN, 16-51-45029 Ind. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] A. Iqbal, All genus topological string amplitudes and ve-brane webs as Feynman diagrams, hep-th/0207114 [INSPIRE]. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE]. HJEP09(217) hep-th/0309208 [INSPIRE]. [3] A. Okounkov, N. Reshetikhin and C. Vafa, Quantum Calabi-Yau and Classical Crystals, [4] T. Eguchi and H. Kanno, Topological strings and Nekrasov's formulas, JHEP 12 (2003) 006 [hep-th/0701156] [INSPIRE]. 41 (1997) 181 [q-alg/9608002]. 313 [math/0306198] [INSPIRE]. function, math/0505553 [INSPIRE]. 534 (1998) 549 [hep-th/9711108] [INSPIRE]. [5] A. Iqbal, N. Nekrasov, A. Okounkov and C. Vafa, Quantum foam and topological strings, JHEP 04 (2008) 011 [hep-th/0312022] [INSPIRE]. hep-th/0602087 [INSPIRE]. [6] R. Dijkgraaf, C. Vafa and E. Verlinde, M-theory and a topological string duality, [7] T.J. Hollowood, A. Iqbal and C. Vafa, Matrix models, geometric engineering and elliptic genera, JHEP 03 (2008) 069 [hep-th/0310272] [INSPIRE]. [8] H. Awata and H. Kanno, Instanton counting, Macdonald functions and the moduli space of D-branes, JHEP 05 (2005) 039 [hep-th/0502061] [INSPIRE]. [9] A. Iqbal, C. Kozcaz and C. Vafa, The Re ned topological vertex, JHEP 10 (2009) 069 [10] J. Ding and K. Iohara, Generalization of Drinfeld quantum a ne algebras, Lett. Math. Phys. [11] K. Miki, A (q; ) analog of the W1+1 algebra, J. Math. Phys. 48 (2007) 123520. [12] H. Nakajima and K. Yoshioka, Instanton counting on blowup. 1., Invent. Math. 162 (2005) [13] H. Nakajima and K. Yoshioka, Lectures on instanton counting, math/0311058 [INSPIRE]. [14] H. Nakajima and K. Yoshioka, Instanton counting on blowup. II. K-theoretic partition [15] A. Losev, N. Nekrasov and S.L. Shatashvili, Issues in topological gauge theory, Nucl. Phys. B [16] A. Losev, N. Nekrasov and S.L. Shatashvili, Testing Seiberg-Witten Solution, hep-th/9801061 [INSPIRE]. Math. Phys. 209 (2000) 97 [hep-th/9712241] [INSPIRE]. [17] G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. [18] G.W. Moore, N. Nekrasov and S. Shatashvili, D-particle bound states and generalized instantons, Commun. Math. Phys. 209 (2000) 77 [hep-th/9803265] [INSPIRE]. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE]. [20] R. Flume and R. Poghossian, An Algorithm for the microscopic evaluation of the coe cients SU(N ) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [INSPIRE]. (2010) 1 [arXiv:0908.2569] [INSPIRE]. [26] H. Awata and Y. Yamada, Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra, JHEP 01 (2010) 125 [arXiv:0910.4431] [INSPIRE]. Prog. Theor. Phys. 124 (2010) 227 [arXiv:1004.5122] [INSPIRE]. [27] H. Awata and Y. Yamada, Five-dimensional AGT Relation and the Deformed beta-ensemble, [28] M.-C. Tan, M-Theoretic Derivations of 4d{2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems, JHEP 07 (2013) 171 [arXiv:1301.1977] [INSPIRE]. [29] M.-C. Tan, An M-Theoretic Derivation of a 5d and 6d AGT Correspondence and Relativistic and Elliptized Integrable Systems, JHEP 12 (2013) 031 [arXiv:1309.4775] [INSPIRE]. [30] M.-C. Tan, Higher AGT Correspondences, W-algebras and Higher Quantum Geometric Langlands Duality from M-theory, arXiv:1607.08330 [INSPIRE]. [31] A. Iqbal, C. Kozcaz and S.-T. Yau, Elliptic Virasoro Conformal Blocks, arXiv:1511.00458 [INSPIRE]. [32] F. Nieri, An elliptic Virasoro symmetry in 6d, arXiv:1511.00574 [INSPIRE]. [33] A. Nedelin and M. Zabzine, q-Virasoro constraints in matrix models, JHEP 03 (2017) 098 [arXiv:1511.03471] [INSPIRE]. operator insertion, arXiv:1512.01084 [INSPIRE]. [34] R. Yoshioka, The integral representation of solutions of KZ equation and a modi cation by K [35] A. Mironov, A. Morozov and Y. Zenkevich, On elementary proof of AGT relations from six dimensions, Phys. Lett. B 756 (2016) 208 [arXiv:1512.06701] [INSPIRE]. [36] A. Mironov, A. Morozov and Y. Zenkevich, Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings, JHEP 05 (2016) 121 [arXiv:1603.00304] [INSPIRE]. [37] E. Carlsson, N. Nekrasov and A. Okounkov, Five dimensional gauge theories and vertex operators, Moscow Math. J. 14 (2014) 39 [arXiv:1308.2465] [INSPIRE]. Vertex, JHEP 03 (2012) 041 [arXiv:1112.6074] [INSPIRE]. Ding-Iohara algebra and AGT conjecture, arXiv:1106.4088 [INSPIRE]. (2016) 103 [arXiv:1604.08366] [INSPIRE]. [42] J.-E. Bourgine, Y. Matsuo and H. Zhang, Holomorphic eld realization of SHc and quantum geometry of quiver gauge theories, JHEP 04 (2016) 167 [arXiv:1512.02492] [INSPIRE]. [43] M. Aganagic and S. Shakirov, Re ned Chern-Simons Theory and Topological String, arXiv:1210.2733 [INSPIRE]. [44] L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [INSPIRE]. [arXiv:0911.1316] [INSPIRE]. Operator, Irregular Conformal Blocks and Open Topological String, Adv. Theor. Math. Phys. 16 (2012) 725 [arXiv:1008.0574] [INSPIRE]. [47] A. Marshakov, A. Mironov and A. Morozov, On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles, J. Geom. Phys. 61 (2011) 1203 [arXiv:1011.4491] [INSPIRE]. [48] M. Aganagic, M.C.N. Cheng, R. Dijkgraaf, D. Kre and C. Vafa, Quantum Geometry of Re ned Topological Strings, JHEP 11 (2012) 019 [arXiv:1105.0630] [INSPIRE]. [49] H.-Y. Chen and A. Sinkovics, On Integrable Structure and Geometric Transition in Supersymmetric Gauge Theories, JHEP 05 (2013) 158 [arXiv:1303.4237] [INSPIRE]. [50] H. Mori and Y. Sugimoto, Surface Operators from M-strings, Phys. Rev. D 95 (2017) 026001 [arXiv:1608.02849] [INSPIRE]. [51] A. Iqbal and C. Vafa, BPS Degeneracies and Superconformal Index in Diverse Dimensions, Phys. Rev. D 90 (2014) 105031 [arXiv:1210.3605] [INSPIRE]. [52] J. Gomis and T. Okuda, D-branes as a Bubbling Calabi-Yau, JHEP 07 (2007) 005 [arXiv:0704.3080] [INSPIRE]. [53] C. Kozcaz, S. Pasquetti and N. Wyllard, A & B model approaches to surface operators and Toda theories, JHEP 08 (2010) 042 [arXiv:1004.2025] [INSPIRE]. [54] T. Dimofte, S. Gukov and L. Hollands, Vortex Counting and Lagrangian 3-manifolds, Lett. Math. Phys. 98 (2011) 225 [arXiv:1006.0977] [INSPIRE]. [arXiv:1007.2524] [INSPIRE]. [55] M. Taki, Surface Operator, Bubbling Calabi-Yau and AGT Relation, JHEP 07 (2011) 047 [56] Y. Ohkubo, Existence and Orthogonality of Generalized Jack Polynomials and Its q-Deformation, J. Phys. Conf. Ser. 804 (2017) 012036 [arXiv:1404.5401] [INSPIRE]. arXiv:1309.1687 [INSPIRE]. AGT correspondence in ve dimensions, JHEP 05 (2015) 131 [arXiv:1412.8592] [INSPIRE]. [arXiv:1510.01896] [INSPIRE]. HJEP09(217) quantum toroidal gl(1), arXiv:1603.02765 [INSPIRE]. [65] H. Awata et al., Anomaly in RTT relation for DIM algebra and network matrix models, Nucl. Phys. B 918 (2017) 358 [arXiv:1611.07304] [INSPIRE]. [2] M. Aganagic , A. Klemm , M. Marino and C. Vafa , The Topological Vertex, Commun. Math. [19] N.A. Nekrasov , Seiberg-Witten prepotential from instanton counting, Adv . Theor. Math. of the Seiberg-Witten prepotential , Int. J. Mod. Phys. A 18 ( 2003 ) 2541 [ hep -th/0208176] [21] N. Nekrasov and A. Okounkov , Seiberg-Witten theory and random partitions, Prog . Math. [22] A. Mironov and A. Morozov , The Power of Nekrasov Functions, Phys. Lett. B 680 ( 2009 ) [23] L.F. Alday , D. Gaiotto and Y. Tachikawa , Liouville Correlation Functions from Four-dimensional Gauge Theories , Lett. Math. Phys. 91 ( 2010 ) 167 [arXiv: 0906 .3219] [24] N. Wyllard , AN 1 conformal Toda eld theory correlation functions from conformal N = 2 [25] A. Mironov and A. Morozov , On AGT relation in the case of U(3), Nucl . Phys. B 825 [39] H. Awata , B. Feigin , A. Hoshino , M. Kanai , J. Shiraishi and S. Yanagida , Notes on [40] A. Mironov , A. Morozov and Y. Zenkevich , Ding-Iohara-Miki symmetry of network matrix models , Phys. Lett. B 762 ( 2016 ) 196 [arXiv: 1603 .05467] [INSPIRE]. [41] H. Awata et al., Explicit examples of DIM constraints for network matrix models , JHEP 07 [45] D. Gaiotto , Surface Operators in N = 2 4d Gauge Theories , JHEP 11 ( 2012 ) 090 [46] H. Awata , H. Fuji , H. Kanno , M. Manabe and Y. Yamada , Localization with a Surface [57] M. Aganagic , N. Haouzi , C. Kozcaz and S. Shakirov , Gauge/Liouville Triality, [58] M. Aganagic , N. Haouzi and S. Shakirov , An-Triality, arXiv: 1403 .3657 [INSPIRE]. [60] A. Morozov and Y. Zenkevich , Decomposing Nekrasov Decomposition, JHEP 02 ( 2016 ) 098 [61] Ya. Kononov and A. Morozov , On Factorization of Generalized Macdonald Polynomials, Eur. Phys. J. C 76 ( 2016 ) 424 [arXiv: 1607 .00615] [INSPIRE]. [62] B. Feigin , M. Jimbo , T. Miwa and E. Mukhin , Quantum toroidal gl1 and Bethe ansatz , J. [63] B. Feigin , M. Jimbo , T. Miwa and E. Mukhin , Finite type modules and Bethe Ansatz for [64] H. Awata et al., Toric Calabi-Yau threefolds as quantum integrable systems . R-matrix and


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Yegor Zenkevich. Refined toric branes, surface operators and factorization of generalized Macdonald polynomials, Journal of High Energy Physics, 2017, 70, DOI: 10.1007/JHEP09(2017)070