We consider the overlap of Bethe vectors of the XXX spin chain with a diagonal twist and the modified Bethe vectors with a general twist. We find a determinant representation for this overlap under one additional condition on the twist parameters. Such objects arise in the calculations of nonequilibrium physics.
We consider closed XXX spin chains with broken total spin $U(1)$ symmetry within the framework of the modified algebraic Bethe ansatz. We study multiple actions of the modified monodromy matrix entries on the modified Bethe vectors. The obtained form
ulas of the multiple actions allow us to calculate the scalar products of the modified Bethe vectors. We find an analog of Izergin-Korepin formula for the scalar products. This formula involves modified Izergin determinants and can be expressed as sums over partitions of the Bethe parameters.
The zero modes method is applied in order to get action of the monodromy matrix entries onto off-shell Bethe vectors in quantum integrable models associated with $U_q(mathfrak{gl}_N)$-invariant $R$-matrices. The action formulas allow to get recurrenc
e relations for off-shell Bethe vectors and for highest coefficients of the Bethe vectors scalar product.
We consider $mathfrak{gl}_2$-invariant quantum integrable models solvable by the algebraic Bethe ansatz. We show that the form of on-shell Bethe vectors is preserved under certain twist transformations of the monodromy matrix. We also derive the acti
ons of the twisted monodromy matrix entries onto twisted off-shell Bethe vectors.
We study solutions of the Bethe ansatz equations associated to the orthosymplectic Lie superalgebras $mathfrak{osp}_{2m+1|2n}$ and $mathfrak{osp}_{2m|2n}$. Given a solution, we define a reproduction procedure and use it to construct a family of new s
olutions which we call a population. To each population we associate a symmetric rational pseudo-differential operator $mathcal R$. Under some technical assumptions, we show that the superkernel $W$ of $mathcal R$ is a self-dual superspace of rational functions, and the population is in a canonical bijection with the variety of isotropic full superflags in $W$ and with the set of symmetric complete factorizations of $mathcal R$. In particular, our results apply to the case of even Lie algebras of type D${}_m$ corresponding to $mathfrak{osp}_{2m|0}=mathfrak{so}_{2m}$.
Fix a semisimple Lie algebra g. Gaudin algebras are commutative algebras acting on tensor product multiplicity spaces for g-representations. These algebras depend on a parameter which is a point in the Deligne-Mumford moduli space of marked stable ge
nus 0 curves. When the parameter is real, then the Gaudin algebra acts with simple spectrum on the tensor product multiplicity space and gives us a basis of eigenvectors. In this paper, we study the monodromy of these eigenvectors as the parameter varies within the real locus; this gives an action of the fundamental group of this moduli space, which is called the cactus group. We prove a conjecture of Etingof which states that the monodromy of eigenvectors for Gaudin algebras agrees with the action of the cactus group on tensor products of g-crystals. In fact, we prove that the coboundary category of normal g-crystals can be reconstructed using the coverings of the moduli spaces. Our main tool is the construction of a crystal structure on the set of eigenvectors for shift of argument algebras, another family of commutative algebras which act on any irreducible g-representation. We also prove that the monodromy of such eigenvectors is given by the internal cactus group action on g-crystals.