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 b (...truncated)


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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, pp. 47, Volume 2016, Issue 10, DOI: 10.1007/JHEP10(2016)047