Abstract In this paper, we describe a certain kind of q -connections on a projective line, namely Z -twisted ( G , q ) {(G,q)} -opers with regular singularities using the language of generalized minors. In part one we explored the correspondence between these q -connections and đť‘„đť‘„ \mathit{QQ} -systems/Bethe Ansatz equations. Here we associate to a Z -twisted ( G , q ) {(G,q)} -oper a class of meromorphic sections of a G -bundle, satisfying certain difference equations, which we refer to as ( G , q ) {(G,q)} -Wronskians. Among other things, we show that the đť‘„đť‘„ \mathit{QQ} -systems and their extensions emerge as the relations between generalized minors, thereby putting the Bethe Ansatz equations in the framework of cluster mutations known in the theory of double Bruhat cells. more »« less
We consider the space of solutions of the Bethe ansatz equations of the sl_N XXX quantum integrable model, associated with the trivial representation of sl_N. We construct a family of commuting flows on this space and identify the flows with the flows of coherent rational Ruijesenaars-Schneider systems. For that we develop in full generality the spectral transform for the rational Ruijesenaars-Schneider system.
de Leeuw, Marius; Nepomechie, Rafael I.; Retore, Ana L.
(, Journal of High Energy Physics)
A<sc>bstract</sc> We introduce new classes of integrable models that exhibit a structure similar to that of flag vector spaces. We present their Hamiltonians,R-matrices and Bethe-ansatz solutions. These models have a new type of generalized graded algebra symmetry.
Rupel, Dylan; Stella, Salvatore; Williams, Harold
(, Compositio Mathematica)
We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac–Moody groups. We prove that all cluster monomials with $$\mathbf{g}$$ -vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero–Chapoton description via quiver representations. In type $$A_{1}^{(1)}$$ , we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac–Moody algebras.
A bstract We study solutions of the Thermodynamic Bethe Ansatz equations for relativistic theories defined by the factorizable S -matrix of an integrable QFT deformed by CDD factors. Such S -matrices appear under generalized TTbar deformations of integrable QFT by special irrelevant operators. The TBA equations, of course, determine the ground state energy E ( R ) of the finite-size system, with the spatial coordinate compactified on a circle of circumference R . We limit attention to theories involving just one kind of stable particles, and consider deformations of the trivial (free fermion or boson) S -matrix by CDD factors with two elementary poles and regular high energy asymptotics — the “2CDD model”. We find that for all values of the parameters (positions of the CDD poles) the TBA equations exhibit two real solutions at R greater than a certain parameter-dependent value R * , which we refer to as the primary and secondary branches. The primary branch is identified with the standard iterative solution, while the secondary one is unstable against iterations and needs to be accessed through an alternative numerical method known as pseudo-arc-length continuation. The two branches merge at the “turning point” R * (a square-root branching point). The singularity signals a Hagedorn behavior of the density of high energy states of the deformed theories, a feature incompatible with the Wilsonian notion of a local QFT originating from a UV fixed point, but typical for string theories. This behavior of E ( R ) is qualitatively the same as the one for standard TTbar deformations of local QFT.
SHENG, RICHIE; TRIBONE, TIM
(, Nagoya Mathematical Journal)
Abstract Consider a pair of elementsfandgin a commutative ringQ. Given a matrix factorization offand another ofg, the tensor product of matrix factorizations, which was first introduced by Knörrer and later generalized by Yoshino, produces a matrix factorization of the sum$$f+g$$. We will study the tensor product ofd-fold matrix factorizations, with a particular emphasis on understanding when the construction has a non-trivial direct sum decomposition. As an application of our results, we construct indecomposable maximal Cohen–Macaulay and Ulrich modules over hypersurface domains of a certain form.
Koroteev, Peter, and Zeitlin, Anton M. q -opers, QQ -systems, and Bethe Ansatz II: Generalized minors. Retrieved from https://par.nsf.gov/biblio/10399734. Journal fĂĽr die reine und angewandte Mathematik (Crelles Journal) 0.0 Web. doi:10.1515/crelle-2022-0084.
@article{osti_10399734,
place = {Country unknown/Code not available},
title = {q -opers, QQ -systems, and Bethe Ansatz II: Generalized minors},
url = {https://par.nsf.gov/biblio/10399734},
DOI = {10.1515/crelle-2022-0084},
abstractNote = {Abstract In this paper, we describe a certain kind of q -connections on a projective line, namely Z -twisted ( G , q ) {(G,q)} -opers with regular singularities using the language of generalized minors. In part one we explored the correspondence between these q -connections and đť‘„đť‘„ \mathit{QQ} -systems/Bethe Ansatz equations. Here we associate to a Z -twisted ( G , q ) {(G,q)} -oper a class of meromorphic sections of a G -bundle, satisfying certain difference equations, which we refer to as ( G , q ) {(G,q)} -Wronskians. Among other things, we show that the đť‘„đť‘„ \mathit{QQ} -systems and their extensions emerge as the relations between generalized minors, thereby putting the Bethe Ansatz equations in the framework of cluster mutations known in the theory of double Bruhat cells.},
journal = {Journal fĂĽr die reine und angewandte Mathematik (Crelles Journal)},
volume = {0},
number = {0},
author = {Koroteev, Peter and Zeitlin, Anton M.},
}
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