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Register a new userOn quantum $mathfrak{osp}(1|2ell)$-Toda chain
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The orthosymplectic super Lie algebra $mathfrak{osp}(1|,2ell)$ is the closest analog of standard Lie algebras in the world of super Lie algebras. We demonstrate that the corresponding $mathfrak{osp}(1|,2ell)$-Toda chain turns out to be an instance of a $BC_ell$-Toda chain. The underlying reason for this relation is discussed.
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We classify the finite-dimensional irreducible representations of the Yangians associated with the orthosymplectic Lie superalgebras ${frak{osp}}_{1|2n}$ in terms of the Drinfeld polynomials. The arguments rely on the description of the representations in the particular case $n=1$ obtained in our previous work.
The periodic $OSp(1|2)$ quantum spin chain has both a graded and a non-graded version. Naively, the Bethe ansatz solution for the non-graded version does not account for the complete spectrum of the transfer matrix, and we propose a simple mechanism for achieving completeness. In contrast, for the graded version, this issue does not arise. We also clarify the symmetries of bot
The subject of the present paper is an application of quantum probability to $p$-adic objects. We give a quantum-probabilistic interpretation of the spherical Hecke algebra for ${rm PGL}_2(F)$, where $F$ is a $p$-adic field. As a byproduct, we obtain a new proof of the Fourier inversion formula for ${rm PGL}_2(F)$.
We study algebras and correlation functions of local operators at half-BPS interfaces engineered by the stacks of D5 or NS5 branes in the 4d $mathcal{N}=4$ super Yang-Mills. The operator algebra in this sector is isomorphic to a truncation of the Yangian $mathcal{Y}(mathfrak{gl}_n)$. The correlators, encoded in a trace on the Yangian, are controlled by the inhomogeneous $mathfrak{sl}_n$ spin chain, where $n$ is the number of fivebranes: they are given in terms of matrix elements of transfer matrices associated to Verma modules, or equivalently of products of Baxters Q-operators. This can be viewed as a novel connection between the $mathcal{N}=4$ super Yang-Mills and integrable spin chains. We also remark on analogous constructions involving half-BPS Wilson lines.
We study the $mathfrak{gl}_{1|1}$ supersymmetric XXX spin chains. We give an explicit description of the algebra of Hamiltonians acting on any cyclic tensor products of polynomial evaluation $mathfrak{gl}_{1|1}$ Yangian modules. It follows that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the Drinfeld polynomials. In particular our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. We also give dimensions of the generalized eigenspaces. We show that when the tensor product is irreducible, then all eigenvectors can be constructed using Bethe ansatz. We express the transfer matrices associated to symmetrizers and anti-symmetrizers of vector representations in terms of the first transfer matrix and the center of the Yangian.
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