Related papers: A mirror symmetric solution to the quantum Toda la…
We investigate the phenomenon known as ``quantum equals affine'' in the setting of $T$-equivariant quantum $K$-theory of the flag variety $G/B$, as established by Kato for any semisimple algebraic group $G$. In particular, we focus on the…
We conjecture that appropriate K-theoretic Gromov-Witten invariants of complex flag manifolds G/B are governed by finite-difference versions of Toda systems constructed in terms of the Langlands-dual quantized universal enveloping algebras…
A relation between semi-direct sums of Lie algebras and integrable couplings of lattice equations is established, and a practicable way to construct integrable couplings is further proposed. An application of the resulting general theory to…
Making use of formulas of J. Moser for a finite-dimensional Toda lattices, we derive the evolution law for moments of the spectral measure of the semi-infinite Jacobi operator associated with the Toda lattice. This allows us to construct…
Let $X$ be a Fano variety, and $D\subset X$ be an snc anticanonical divisor. We study relative mirror symmetry for the log Calabi--Yau pair $(X,D)$. (1) We prove a relative mirror theorem for snc pairs without assuming the divisors are nef.…
We give a detailed account of the N -component Toda lattice hierarchy. This hierarchy is an extended version of the one introduced by Ueno and Takasaki. Our version contains N discrete variables rather than one. We start from the Lax…
We construct functors sending torus-equivariant quasi-coherent sheaves on toric schemes over the sphere spectrum to constructible sheaves of spectra on real vector spaces. This provides a spectral lift of the toric homolgoical mirror…
The integral representations for the eigenfunctions of $N$ particle quantum open and periodic Toda chains are constructed in the framework of Quantum Inverse Scattering Method (QISM). Both periodic and open $N$-particle solutions have…
We introduce a spectral transform for the finite relativistic Toda lattice (RTL) in generalized form. In the nonrelativistic case, Moser constructed a spectral transform from the spectral theory of symmetric Jacobi matrices. Here we use a…
We apply the method of confining phase superpotentials to N = 2 supersymmetric Yang-Mills theory with the exceptional gauge group G_2. Our findings are consistent with the spectral curve of the periodic Toda lattice, but do not agree with…
The problem of construction of the boundary conditions for the Toda lattice compatible with its higher symmetries is considered. It is demonstrated that this problem is reduced to finding of the differential constraints consistent with the…
Representation theory is shown to be incomplete in terms of enumerating all integrable limits of quantum systems. As a consequence, one can find exactly solvable Hamiltonians which have apparently strongly broken symmetry. The number of…
M. Toda in 1967 (\textit{J. Phys. Soc. Japan}, \textbf{22} and \textbf{23}) considered a lattice model with exponential interaction and proved, as suggested by the Fermi-Pasta-Ulam experiments in the 1950s, that it has exact periodic and…
This paper studies solutions to a singular $SU(3)$ Toda system with linear source terms on a compact Riemann surface $\Sigma$ with smooth boundaries $\partial\Sigma$. We establish the existence of solutions when the parameters are not…
We construct a Lagrangian torus fibration on a smooth hypertoric variety and a corresponding SYZ mirror variety using $T$-duality and generating functions of open Gromov-Witten invariants. The variety is singular in general. We construct a…
A class of two-dimensional field theories with exponential interactions is introduced. The interaction depends on two ``coupling'' matrices and is sufficiently general to include all Toda field theories existing in the literature. Lie point…
We study an integrable system related to the relativistic Toda lattice. The bilinear representation of this lattice is given and the B\"ackulund transformation obtained. A fully discrete version is also introduced with its bilinear…
The celebrated Mirror Theorem states that the genus zero part of the A model (quantum cohomology, rational curves counting) of the Fermat quintic threefold is equivalent to the B model (complex deformation, variation of Hodge structure) of…
In this paper, we construct a new even constrained B(C) type Toda hierarchy and derive its B(C) type Block type additional symmetry. Also we generalize the B(C) type Toda hierarchy to the $N$-component B(C) type Toda hierarchy which is…
We discuss an analytic proof of a conjecture (Nakamura) that solutions of Toda molecule equation give those of Ernst equation giving Tomimatsu-Sato solutions of Einstein equation. Using Pfaffian identities it is shown for Weyl solutions…