Related papers: Geometric realizations of the Bethe ansatz equatio…
Let us consider a specialization of an untwisted quantum affine algebra of type $ADE$ at a nonzero complex number, which may or may not be a root of unity. The Grothendieck ring of its finite dimensional representations has two bases,…
The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with non-Lie permutation relations. A method for constructing irreducible…
The graded off-diagonal Bethe ansatz method is proposed to study supersymmetric quantum integrable models (i.e., quantum integrable models associated with superalgebras). As an example, the exact solutions of the $SU(2|2)$ vertex model with…
Practical implementations of quantum computing are always done in the presence of decoherence. Geometric phase is useful in the context of quantum computing as a tool to achieve fault tolerance. Recent experimental progresses on coherent…
We study quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing $\mathfrak{gl}(m|n)$-invariant $R$-matrix. We compute the norm of the Hamiltonian eigenstates. Using the notion of a generalized model we show…
In this paper we formulate a general method for building completely integrable quantum systems. The method is based on the use of the so-called multi-parameter spectral equations, i.e. equations with several spectral parameters. We show…
This is an expository account of Grothendieck's construction of Hilbert and Quot Schemes, following his talk `Techniques de construction et theoremes d'existence en geometrie algebriques IV : les schemas de Hilbert', Seminaire Bourbaki 221…
We propose the notion of q-characters for finite-dimensional representations of quantum affine algebras. It is motivated by our theory of deformed W-algebras. We show that the q-characters give rise to a homomorphism from the Grothendieck…
The quantum geometric tensor, composed of the quantum metric tensor and Berry curvature, fully encodes the parameter space geometry of a physical system. We first provide a formulation of the quantum geometrical tensor in the path integral…
A GR-segment for an artin algebra is a sequence of Gabriel-Roiter measures, which is closed under direct predecessors and successors. The number of the GR-segments indexed by natural numbers $\mathbb{N}$ and integers $\mathbb{Z}$ probably…
Geometric/arithmetic theta correspondences provide correspondences between automorphic forms and cohomology classes/algebraic cycles on Shimura varieties. We give an introduction focusing on the example of unitary groups and highlight…
In this paper we apply two-dimensional supersymmetric gauge theories to directly construct a new Bethe ansatz for the wavefunctions of the q-boson hopping model, and then derive the q-boson algebras from this ansatz.
Recent interest in discrete, classical integrable systems has focused on their connection to quantum integrable systems via the Bethe equations. In this note, solutions to the rational nested Bethe ansatz (RNBA) equations are constructed…
We derive a $K$-theoretic analogue of the Vafa--Intriligator formula, computing the (virtual) Euler characteristics of vector bundles over the Quot scheme that compactifies the space of degree $d$ morphisms from a fixed projective curve to…
We summarize our work on spherically symmetric midi-superspaces in loop quantum gravity. Our approach is based on using inhomogeneous slicings that may penetrate the horizon in case there is one and on a redefinition of the constraints so…
The nested algebraic Bethe ansatz is presented for the anisotropic supersymmetric $U$ model maintaining quantum supersymmetry. The Bethe ansatz equations of the model are obtained on a one-dimensional closed lattice and an expression for…
This is an expanded version of the lecture notes for a minicourse that I gave at a summer school called "Advanced Course on Geometry and Dynamics of Integrable Systems" at CRM Barcelona, 9--14/September/2013. In this text we study the…
This thesis seeks to develop a general method for solving so-called quantum realizability problems, which are questions of the following form: under which conditions does there exist a quantum state exhibiting a given collection of…
We review various aspects of representation theory of affine algebras at the critical level, geometric Langlands correspondence, and Bethe ansatz in the Gaudin models. Geometric Langlands correspondence relates D-modules on the moduli space…
We study the tensor product of the {\it higher spin representations} (see the definition in Sect. 2.2) of the elliptic quantum group $E_{\tau,\eta}(sl_n)$. The transfer matrices associated with the $E_{\tau,\eta}(sl_n)$-module are exactly…