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)