Toric Calabi-Yau threefolds as quantum integrable systems. \( \mathrm{\mathcal{R}} \) -matrix and \( \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} \) relations

Journal of High Energy Physics, Oct 2016

Abstract \( \mathrm{\mathcal{R}} \)-matrix is explicitly constructed for simplest representations of the Ding-Iohara-Miki algebra. Calculation is straightforward and significantly simpler than the one through the universal \( \mathrm{\mathcal{R}} \)-matrix used for a similar calculation in the Yangian case by A. Smirnov but less general. We investigate the interplay between the \( \mathrm{\mathcal{R}} \)-matrix structure and the structure of DIM algebra intertwiners, i.e. of refined topological vertices and show that the \( \mathrm{\mathcal{R}} \)-matrix is diagonalized by the action of the spectral duality belonging to the SL(2, ℤ) group of DIM algebra automorphisms. We also construct the \( \mathcal{T} \)-operators satisfying the \( \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} \) relations with the \( \mathrm{\mathcal{R}} \)-matrix from refined amplitudes on resolved conifold. We thus show that topological string theories on the toric Calabi-Yau threefolds can be naturally interpreted as lattice integrable models. Integrals of motion for these systems are related to q-deformation of the reflection matrices of the Liouville/Toda theories.

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Toric Calabi-Yau threefolds as quantum integrable systems. \( \mathrm{\mathcal{R}} \) -matrix and \( \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} \) relations

Received: August R-matrix and Hidetoshi Awata 1 2 6 7 8 9 10 11 12 Hiroaki Kanno 1 2 4 6 7 8 9 10 11 12 Andrei Mironov 0 1 2 3 5 7 8 9 10 11 12 Alexei Morozov 0 1 2 3 7 8 9 10 11 12 Andrey Morozov 0 1 2 3 7 8 9 10 11 12 g Yusuke Ohkubo 1 2 6 7 8 9 10 11 12 Yegor Zenkevich 0 1 2 7 8 9 10 11 12 h 1 2 7 8 9 10 11 12 Open Access 1 2 7 8 9 10 11 12 c The Authors. 1 2 7 8 9 10 11 12 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 12 [39] B. Feigin, K. Hashizume , A. Hoshino, J. Shiraishi and S. Yanagida, A commutative algebra R-matrix is explicitly constructed for simplest representations of the DingCalculation is straightforward and signi cantly simpler than the one through the universal R-matrix used for a similar calculation in the Yangian case by A. Smirnov but less general. We investigate the interplay between the R-matrix structure and the structure of DIM algebra intertwiners, i.e. of re ned topological vertices and show that the R-matrix is diagonalized by the action of the spectral duality belonging to the SL(2; Z) group of DIM algebra automorphisms. We also construct the T -operators satisfying the RT T relations with the R-matrix from re ned amplitudes on resolved conifold. We thus show that topological string theories on the toric Calabi-Yau threefolds can be naturally interpreted as lattice integrable models. Integrals of motion for these systems are related to q-deformation of the re ection matrices of the Liouville/Toda theories. relations; Conformal and W Symmetry; Supersymmetric gauge theory; Topological - DIM algebra, generalized Macdonald polynomials and the R-matrix Re ned topological strings and RT T relations R-matrices: from -deformation to (q; t)-deformation A ne Yangian R-matrix from generalized Jack polynomials DIM R-matrix from generalized Macdonald polynomials 1 Introduction 2 3 1.1 RT T relations in the toric diagram Trivial diagrams on vertical legs Arbitrary diagrams on vertical legs 4 Integrals of motion and compacti cation 5 Conclusion A Explicit expressions for DIM R-matrix R-Matrix at level 1 A.1.1 (q; t)-deformed version -deformed version R-Matrix at level 2 A.2.1 (q; t)-deformed version -deformed version B.2 Higher Hamiltonians Action of x+1 on generalized Macdonald polynomials Introduction Integrability plays an exceptional role in modern studies of quantum eld theory and string list of recent examples includes Seiberg-Witten solution of N complex integrable systems [3{6], from integrable spin chains and -models [7], models with new solutions to Yang-Baxter equations [9], topological string calculations and the study of Hurwitz -functions [23{25]. kind of integrability appears. emerging in the description of coproducts of group elements g^ 2 G A(G) [28{31] for quantum groups [32{36], I) = R (g^ diagram corresponding to a DIM intertwiner [40, 41]. of the generalized Macdonald Hamiltonian with known eigenvalues. The notation in this paper follows our paper [41]. + n form the three central rows. n into the zero mode of the raising generator. DIM algebra, generalized Macdonald polynomials and the R-matrix xn+ and xn with n 2 Z together with the \Cartan" generators n, n 2 Z>0 and two central can be, therefore, drawn as an integral point on the plane. The generators, xn+, and their commutators form a lattice, which is sketched in gure 1. The exact de nition There is a nice representation of the DIM algebra on the Fock space Fu izontal direction. There is also the vertical Fock representation Fu (0;1), isomorphic to the Young diagram Y , while x n delete one box, and n act diagonally. The central charges vertical ones and vice versa. The action of S is illustrated in gure 1. Let us construct a natural basis in the tensor product of horizontal modules. which are the eigenfunctions of the Hamiltonian with eigenvalues H1 MfAB = ABMfAB H1 = DIM(x+(z)) AB = u1 In the simplest example, i.e. for the tensor product of two Fock modules Fu1 the spectral parameters uu12 . The Hamiltonian H^1 is the zero mode of the raising generator, x 0+ in the horizonof the rst Cartan generator the Cartan generators n+ acts diagonally on the ordinary Macdonald polynomials. The 1 . As we mentioned above, in the vertical representation given by tensor products of the Macdonald polynomials M A(q;t)(a(1n) )ju1i ju2i, which diagonalize x 0+ = S( 1+), can be thought of Hn = u1 2) are automatically diagonal, see appendix B. IIB S-duality exchanging NS5 and D5 branes, hence, our notation. number of time sets and Young diagrams is correspondingly increased. (x+) = x op(x+) = 1 by an R-matrix: op = R Hence, their eigenfunctions are also related:2 RAB conservation law jAj + jBj = jCj + jDj which makes R block-diagonal with nite-dimensional blocks. The coproducts op di er only by permutation of the two representations on obtained by a simple change of variables, exchanging u1 $ u2, A $ B and pn $ pn: = MfBA polynomials as functions of the variable uu12 , which we call Q, whereas in [86] we denoted uu21 as Q. as was suggested in [88, 89], and this will be the approach we adopt here. Re ned topological strings and RT T relations to the non-self-dual Nekrasov -deformation. In order to reduce it to the ordinary topothreefold into C 3 patches and nd the universal amplitudes, trivalent re ned vertices on topology S1 D2 sitting on the legs of the toric diagram. These boundary conditions are Kahler parameters of the edges. involving descendants to those of the primary elds. In fact, one can show that this following from the regularity of qq-characters. Also, in the Nekrasov-Shatashvili limit systems related to the gauge theory [13{16]. TARBP (Q; u; z) = Calabi-Yau manifold [102{104]. R(12)R(13)R(23) = R(23)R(13)R(12) RT (1)T (2) = T (2)T (1)R trW T (1) trW T (2) = trW T (2) trW T (1) V (vertical leg) and the auxiliary space W (horizontal leg), b) the R-matrix acting on W of motion. rotate back using S . We thus obtain the relation of the form to the spectral dual frame using S, then make the permutation of the spaces and R = S encounter this relation when performing concrete computations of the R-matrices. horizontal directions manifest. S. Shakirov [108]. ing on the direction of compacti cation, it becomes either elliptic or a ne. Eventually, R-matrices: from -deformation to (q; t)-deformation the DIM R-matrix. de nes a trivial R-matrix, which is proportional to the identity matrix:3 RCABD(u)jt=q / A B : C D ne Yangian R-matrix from generalized Jack polynomials of two strands 12. to the Cartan subalgebra of the a ne Yangian: ( ) = 2 n=1 2 n=1 makes dual polynomials di erent. Notice that this term vanishes for = 1, i.e. in the corresponding to the eigenfunctions JABfp; pg are 2 n;m=1 2 n;m=1 (AB) = [2];[1;1]. Thus, one still needs higher Hamiltonians Hn with n 2 to uniquely specify the polynomials, which J[1];; = (1 J;;[1] = (u J[1];; = (1 + u J;;[1] = (u u)p1 + (1 polynomials, one has the following de nition of the opposite ones: NAB(u1 NBA(u2 u2j jp; p) = JBA (u2 the R-matrix is indeed given by u2j jp; p) = RCABD(u1 u2) JCD (u1 or, using the Jack scalar product, RCABD(u1 u2) = u2j jp; p) jJCD (u1 The Jack scalar product is de ned as hf (pn)jg(pn)i = f g(pn)jpn=0 : CD6=AB Notice the conjugate polynomial J oApB in the bra vector in eq. (2.7). we expand JAB in the basis of monomial symmetric functions: CACBD(uj )mC (pn)mD(pn) NAB(uj ) = gAB(uj ) gAB(x) = i) (2.11) The normalization factors satisfy the identity NAB(uj )NBA( uj )jjJAjj2jjJBjj 2 = particular, one has jjJABjj2 = zAveBc (uj ): can be associated to the xed points in the moduli space of SU(2) instantons (or to the then describe stable envelopes of the corresponding xed points. of generalized Jack polynomials: ( ) = B identity block arising from the generalized Jack polynomials at the zeroth level: J?;?(uj jp; p) = J ?;?(uj jp; p) = 1 : The resulting 3 3 matrix 3 block of the standard rational R-matrix 1 A which is the only 3 3 rational solution to the Yang-Baxter equation. This discrepancy is products of the Jack polynomials J the generalized Kostka matrices: A( )(pn)J B( )(pn). The basis is changed with the help of KACBD(uj ) = hJ AB(uj )j(jJ C( )i K CABD(uj ) = (hJ A( )j At the rst level, we have K(uj ) = K (uj ) = The R-matrix in the factorized basis of the ordinary Jack polynomials is given by Rord Jack = K K = ( ) = B K = B whole expression becomes Rord Jack = where NAB CD = NNABBA((uu12 uu21jj )) AC BD = K is lower triangular. (1+ ) 1 . In eq. (2.22) the term in the curly an interpretation in the cohomology of the instanton moduli space [88, 89]. Parts of product over [ 2 ; ]: automorphism S . Also, in [119] it was shown that the generalized Kostka matrices are in decomposition (2.22) is a re ection of the decomposition (1.12), where is accompanied by multiplication with the diagonal matrix N . (z)x+(z) (z). A pair since c = 1 + 6(p but the re ection matrix of the Liouville theory. be described as eigenfunctions of a certain Hamiltonian H1 ( ) sitting inside the a ne Yan2 n;m=1 + nmp(nk+)m @p(nk)@p(mk) 2 n=1 ( ) = H(k1;k2) = H((k)) = di erence is that there are now r 1 R-matrices, which permute the factors in the tensor strands becomes a copy of the two-strand R-matrix: Ri(j ) = where i;j permutes the i-th and j-th strands. Ri(;j) = S H1 = u1 [u1 1(z)+u2 2(z)] + u2exp@ the previous subsection. Two strands. As we described in the Introduction, the generalized Macdonald polynoof Fock representations: H1MfAB = ABMfAB AB = u1 Mf[];[1] = (1 t)(1 Q)p1+(1 t) 1 Mf[];[1] = (1 q) 1 4We again remind the reader of the change of convention Q ! Q 1 as compared to [86]. is the DIM coproduct and u denotes the horizontal Fock representation [41, 63, However, all these Hamiltonians are contained in the expansion of H1. already entrenched. The generalized Macdonald polynomials at the rst level are given by4 Mf[1];[] = (1 t) 1 Mf[1];[] = (1 q)(1 Q)p1 (1 q) 1 X 1 qn zn(q=t)n=2 @ C (q; t) = GAB(ujq; t) = NAB(u1=u2jq; t) NBA(u2=u1jq; t) For the polynomials MAB (without tilde), the de nition of opposite polynomial is polynomials is It is given by R = B q(u1 u2)u2(qu1 tu2) u2(qu1 tu2) u1(qu2 tu1) q(u1 u2)u2 versions of generalized Kostka matrices: K = Rord Mac = K The resulting 2 2 block is the same as the block appearing in the standard trigonometric R-matrix: Rord Mac = B u2(qu1 tu2) u2(qu1 tu2) u2(qu1 tu2) for the q-deformed Virasoro algebra. More strands. Again the discussion here is exactly parallel to the previous section, only is given by H1 = u1 (x+(z)) +u2exp@ +u3exp@ :::+uN exp@ + p(nN)(q=t)(1 N)=2 z n zn(q=t)n=2 @ zn(q=t)n (1 tn=qn)p(n1)+p(n2)(q=t) n=2 z n (1 tn=qn) p(n1)+(q=t) 1=2p(2) +p(n3)(q=t) n z n n (1 tn=qn) p(n1)+(q=t) 1=2p(2)+:::+(q=t)(2 N)n=2p(M 1) n n zn(q=t)(1 N)=2 @ satis ed, as shown in appendix A.1.1. resolved conifold. RT T relations in the toric diagram placed on each external line: TARBP (Q;u;z)= and the generalized Macdonald polynomials are de ned as its eigenfunctions: H1MfA1:::AN = A1:::AN MfA1:::AN A1:::AN = are the intertwiners of DIM algebra [40, 41], jsA; ui denote the basis of polynomials in the vertical Fock space (hence, the sign j) and P denotes the matrix element of for the Macdonald polynomial MP on the vertical leg. Such building blocks in the algebraic form, we have: simpli cation that we use in this subsection and lift in the next one. the automorphism S . This corresponds to the change of basis in the tensor product of moved two R-matrices to the r.h.s. of eq. (3.3)): similar to the general case, which requires one additional observation. relations in any integrable system and can be drawn as follows: : (3.3) nomials is the transposed matrix of B(12): M;;;;[1] M;;[1];; M[1];;; M;;;;[1] M;;[1];; M[1];;; where ki (12) are the proportionality constants between M A(1B2)C and (R12)(MABC ). In the same way, from the formula (x0+) = R23( M;;;;[1] M;;[1];; M[1];;; M;;;;[1] M;;[1];; M[1];;; the representation matrix of u1u2u3 (R23) is B(23) := B@ k(12) = 1; k(23) = k(12) = k(23) = 1. Now we consider the basis change from MABC to power sum symmetric B(ij) := tA(u1; u2; u3) tB(ij) tA 1(u1; u2; u3): e Then Be(ij) have the following form must not appear, one gets the equations b(212) = b(12) = 0. Similarly, b(123) = b(223) = 0. By 3 solving these equations, one can see that In this way, one obtains an explicit expression of the R-Matrix at level 1 ; (A.14) k(12) = k(23) = 1: B(13) = B Indeed, one can check that they satisfy the Yang-Baxter equation B(12)B(13)B(23) = Be(23)B(13)B(12): e e e e e Of course, the same equations for B(ij) also follow. Incidentally, t2u2q qt + t2u2u3q qt + qtu22 + qtu1u2 2 2qtu1u2 2qtu2u3 + t2u1u2 u3(q t)p qt (tu1p qt tu2p qt qu1+tu2)(tu2 qu3) q2t(u1 u2)(u1 u3)(u2 u3) 2qtu1u3 + qtu2u3 2qtu2u3 + t2u2u3 u3)(qu2 x = y = matrix of ( u1 u2 )(R) is the 2 2 matrix block at the lower right corner of tB(12) u2 )(R) = B q(u1 u2)(qu1 tu2) -deformed version Hence, the -deformed version of R-matrix R ( ) is immediately obtained from the results of the last paragraph. For example, for the representation u1 u2 and in the basis of generalized Jack polynomials, ( ) = J;;[1] = p(12) J[1];; = p(11): R(12) = BB R(23) = BB (u1 u3)(u3 u2) (u1 u2)(u1 u3)(u2 u3) ( 1)( +u1 u2 1) 2 2 +u21+u22 2u1u2+1 A +u1 u2+1) 2 2 +u22+u23 2u2u3+1 +u2 u3+1) k2 = lim k(12) = k3 = lim k(12) = and the generalized Jack polynomials are (u1 u2)(u1 u3)(u2 u3) (u1 u2)(u1 u3)( In the basis of power sum symmetric functions, ( ) = B A.2.1 (q; t)-deformed version M;;;;[2] M;;;;[1;1] M;;[1];[1] M[1];;;[1] M;;[2];; M;;[1;1];; M[1];[1];; M[2];;;; M[1;1];;;; = A M;0;;;[2] M;0;;;[1;1] M;0;[1];[1] M[01];;;[1] M;0;[2];; sentation matrix of R. First of all, we choose B(12) at level 2 to be of the form 0 1 0 0 0 0 0 0 0 0 1 B 0 1 0 0 0 0 0 0 0 C B b31 b32 B b61 b62 b63 b64 B b71 b72 b73 b74 b91 b92 b93 b94 0 0 0 0 0 C obtained in this way satis es the Yang-Baxter equation. Examples of the generalized Macdonald polynomials. (q t)u3(tu3q3 tu2q2+tu3q2 u3q2 t2u3q+tu2) qt(qt 1)(u2 u3)(qu3 u2) (q+1)(q t)2(t 1)u32(q2u3 tu2) p qt t2(qt 1)(u2 u3)(u1 qu3)(u2 qu3) (q t)2p qt u23(qu3 t2u2) q2t(tu1 u3)(u2 u3)(tu2 u3) q(q+1)(q t)(t 1)u2 p qt t(qt 1)(u1 qu2) (q 1)(q+1)(q t)(t 1)(t+1)u3 (qt 1)2(qu3 u2) (q t)u3( qu2t3+u3t2+qu2t+q2u3t qu3t qu3) qt(qt 1)(u2 u3)(tu2 u3) (q 1)(q t)(t+1)u3 (q t)u3(qu3 tu2)(qu3 t2u2) q2t(tu1 u3)(tu2 u3)(u3 u2) (q t)u3(qu3 tu2) qt(qu1 u3)(u3 u2) (q t)u2(tu2q3 tu1q2+tu2q2 u2q2 t2u2q+tu1) qt(qt 1)(u1 u2)(qu2 u1) (q 1)(q+1)(q t)(t 1)(t+1)u3(qu3 tu2)(q2u3 tu2) qt(qt 1)2(u2 u3)(qu3 u1)(qu3 u2) C C (q t)u3(qu3 tu2)(qu3 t2u2)( qu1t3+u3t2+qu1t+q2u3t qu3t qu3) C q2t2(qt 1)(u1 u3)(tu1 u3)(tu2 u3)(u3 u2) CC C C a39 C C C (qq(1q)t(q1)t()u(2t+1u)3u)3(t(uqu33 ut1u)2) CCC C (q 1)((qq+t1)1()q2(qt)u(2t 1u)1()t+1)u2 CCC C (q t)u2( qu1t3+u2t2+qu1t+q2u2t qu2t qu2) CC qt(qt 1)(u1 u2)(tu1 u2) C C (q 1)(q t)p qt t(t+1)u2 CC q(qt 1)(u1 tu2) CC C 0 CC C A (q 1)(q+1)(q t)(t 1)(t+1)(q Qt2)( qt3 qt2+qQt2+t2+q2t qQ) q(q Q)(Q 1)t(t Q)(qt 1)2 q2(Q 1)2t2(t Q)(qt 1)(Qt 1) Q(q t)(Qtq3 Qq2+Qtq2 tq2 Qt2q+t) (qQ t)(q2Q t)(qt 1) (q 1)q(q+1)(Q t)(t 1)t(t+1) (q 1)(q+1)Q(q t)2(t 1)(t+1)(Qq2+Qq Qtq+tq t2 t) (Q t)(qQ t)(q2Q t)(qt 1)2 (Q 1)Q(q t)t (qQ t)(qQ t2) Q(q t)( qt3+Qt2+qt+q2Qt qQt qQ) C (qQ t)(qt 1)(qQ t2) Q(q t)2(Qq2+Qtq2 t2q Qtq+tq t2) CC q(qQ 1)(qQ t)t(qQ t2) q(q Q)(qQ 1)t(t Q)(Qt 1) q(q Q)(qQ 1)t(t Q)(Qt 1) q2(Q 1)(qQ 1)2(Q t)t qu23t + qu1u2t u1u2t + qu1u3t u1u3t qu2u3t + u2u3t + qu2u3 1)(q t)2(t + 1)u2u3 u2u3q2 u23q tu1u2q + u1u2q tu1u3q + u1u3q + tu2u3q u2u3q + tu2u3 The representation matrix of R in the basis of generalized Macdonald polynomials v53 = v44 = v34 = v35 = v55 = r11 = r41 = (q 1)(t+1)(q t)p q q3Q+q2 Q3(t 1)+Q2 t2 Qt+t+1 +q5Qt Q3t2 Q2 t2+t+2 +Q t 3t+5 +2t 1 q4Qt Q3t+Q2 3t2 7t+2 +Q 13t2+11t 5 1)(qt 1)(qQ q4Qt(6t2 4t+1)+q3t2 Q3 t2 4t+6 +Q2 5t2+11t 13 +Q 2t2 7t+3 +t +qQt3 Qt2+(Q Qt4+q2t2 Q3(t 2)t+Q2 5t2+3t 1 +Q 2t2+t+1 q(q+1)Q(q t)(t 1) Qq3+Q2tq3 2Q2t2q2+Qt2q2+Q2tq2+2Qtq2 2tq2+Q3t2q 3Q2t2q+t2q 2Q2tq+2Qtq+tq+Q2t3+Q2t2 Qq2 + Qtq2 + Qt2q + Qq + 2Qtq 2t2 + Qt t) Q2q3 + Qq3 + Q2tq3 Qtq3 + Q2q2 Qq2 + Q2tq2 2Qt2q + 2t2q 2Qtq + 2t3 1)(qt 1)(qQ 2(qQ t)(q2Q t)(qQ t2) (Q 1)Q(q t)p qt t(2Qq2 t2q+tq t2 t) 2(qQ t)(q2Q t)(qQ t2) 2(qQ t)(q2Q t)(qQ t2) (Q 1)(q t)p qt t(Qq2+Qtq2+Qq Qtq 2t2) 2(qQ t)(q2Q t)(qQ t2) 1)Q(q t)(t 1)t2 1)Q(q t)(t 1)t 1)t(qt 1)(Q 1)t(qt 1)(Q 1)Q(q t)t2(t+1) (q 1)q(Q 1)Q(q t)t(t+1) Q(q t) Qq3 Q2q2+2Q2t2q2 2Qtq2+2tq2 Q2t3q+Qt3q+t3q Q3t2q+3Q2t2q t2q+2Q2tq q5Q2t+q4Q Q2 2t3+2t2 5t2+t +t2(t+1) +q3t Q4t2+Q3 7t2+t+2 +Q2t t3+13t2+11t+3 Qt t3+3t2+4t+2 +t4 Q3 2t3+4t2+3t+1 +Q2 3t3+11t2+13t+1 +Qt 2t3+t2 qQt4 Q2(t+1)+Q t t2+2t+2 The representation matrix of R in the basis of power sum symmetric functions, 1)Q(q t)(t 1)t q2(q+1)(Q (q 1)q(Q 1)Q(q t)t(t+1) q2(q+1)(Q 1)Q(q t)(t 1) q(Q 1)(q t)(Qq2+Qtq2+Qq Qtq 2t2) q(q+1)(Q 1)Q(q t)(t 1)t q(Q 1)Q(q t)(2Qq2 t2q+tq t2 t) C Q2tq2 + Qtq2 2Qt2q + 2t2q + 2Qtq 2t3 Q2tq2 Q2t3q + 3Qt3q t3q + Qt2q Qt4 Qt3q + t3q Qt2q + t2q 2Qtq + Qt3 + t3 Qt2 + t2 t)(q2Q t)(qQ Qt2q + t2q 2Qtq + Qt3 + t3 + Qt2 t2 t)(qQ t2) The representation matrix of R ( ) in the basis of generalized Jack polynomials ( ) = s33 = s44 = (a 1)2a( +1) 2+ +2a+2) (a 1)a2(a+1)( +1) (a 1)a(a ( +1)( a+ +2)( +1)2 1) (a3+ a2 a2+ 2a 6 a+4a+ 3 3 2+2 ) +2)( +1)(a+ ) ( +1)2( a+ 2a3 + 3 2a2 a + 14 a 4a 3 + 7 3 + 16a 2 + 1)(a + 1)(a a(a + 1)(a 4 + 2a3 + 5a2 + 8a + 4 a(a+1)(a+ ) a(a+1)2(a 1)(a3+ a2 a2+ 2a 6 a+4a+ 3 3 2+2 ) )( +1)(a+ ) 1)(a3+ a2 a2+4 2a 6 a+a 2 2+3 (a 2 +1)(a ( +1)( a+ 1)( a+2 1)(a3+ a2 a2+4 2a 6 a+a 2 2+3 (a 1)(a 2 +1)(a +1)(a+ ) (a 3 +3)( (a+1)(a 2 +1)(a a(a 2 +1)(a +1)( +1) 4 5 8a 4 8 4+5a2 3+20a 3+ 3 2a3 2 5a2 2 16a 2+7 2+a4 The representation matrix of R( ) in the basis of power sum symmetric functions where a = u1 2a(a 2 +2)( (a 2 +1)(a +1)(a a3 2 a2+2a2+ 2a 3 a+a 2 3+7 2 7 +2 (a 2 +1)(a +1)(a +2) ( 1)(2a2 4 a+4a+2 2 5 +2) ( a+ 2)( a+ 1)( a+2 1) (a 2 +1)(a 1) C C C C Macdonald polynomials explicitly realize the spectral dual basis to j~u; ~ i in [142]. Action of x 1 on generalized Macdonald polynomials We use the notation X(1)(z) = X Xn(1)z n = (uN1;):::;uN (x+(z)): mials M~ are de ned to be eigenfunctions of X0(1) with the eigenvalues e(1) = 1)t i= : , in terms of the product of the monomial symmetric m (N) . Their integral forms Mf~ are de ned by G (j); (i) (uj =uijq; t) Y M~ is renormalized as M~ = m~ + functions m~ = m (1) Mf~ = M~ k=1 (i;j)2 (k) DIM algebra, that the action of (uN1;):::;uN where T is the transposed of Young diagram and we use the Nekrasov factor (2.34). It is S(a) on the integral forms Mf~ are the same as where (i; j) 2 R(~ ) respectively. We also use the notation (`) are the coordinates of the box of the Young diagram for the AFLT basis in this original sense. j~j=j~ j 1 y6=x y6=x x y 1(q=t)) y x 1(q=t)) j~j=j~ j+1 c~~ ;~ Mf~ ; i.e. X(11)M~ = P~ c~ ;~ ( ) ? c~( +;~)(q; tju1; : : : ; uN ) = and for the triple (`; i; j), we put (`;i;j) = ( 1)N+`p `+21 t(N `)iq(` N+1)j QkNN=1``u`1+k ; (+) generators f1 and e1 in [142] respectively, i.e., those of x1 and x are the spectral duals of x1+ and x 1 1+ in our notation, which M~ , we can further conjecture that 5 for N = 1, for j~ j 3 for N = 2; 3 and for j~ j with respect to X(11) has been also checked for the same sizes of ~ . Higher Hamiltonians For each integer k Hk = [X(11); [X0(1); According to [27], Hk are spectral dual to k+ and consequently mutually commuting: functions of all Hk, i.e. HkM~ = e(k)M~ . and Hk can be regarded as higher Hamiltoni~ where for the partition we de ne e(k) =? 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Hidetoshi Awata, Hiroaki Kanno, Andrei Mironov, Alexei Morozov, Andrey Morozov, Yusuke Ohkubo, Yegor Zenkevich. Toric Calabi-Yau threefolds as quantum integrable systems. \( \mathrm{\mathcal{R}} \) -matrix and \( \mathrm{\mathcal{R}}\mathcal{T}\mathcal{T} \) relations, Journal of High Energy Physics, 2016, 47, DOI: 10.1007/JHEP10(2016)047