# Nested Algebraic Bethe Ansatz in integrable models: recent results

### Submission summary

 As Contributors: Eric Ragoucy Arxiv Link: https://arxiv.org/abs/1803.00103v2 Date accepted: 2018-10-23 Date submitted: 2018-07-06 Submitted by: Ragoucy, Eric Submitted to: SciPost Physics Lecture Notes Discipline: Physics Subject area: Mathematical Physics Approach: Theoretical

### Abstract

This short note summarizes the works done in collaboration between S. Belliard (CEA, Saclay), L. Frappat (LAPTh, Annecy), S. Pakuliak (JINR, Dubna), E. Ragoucy (LAPTh, Annecy), N. Slavnov (Steklov Math. Inst., Moscow) and more recently A. Hutsalyuk (Wuppertal / Moscow) and A. Liashyk (Kiev / Moscow). It presents the construction of Bethe vectors, their scalar products and the form factors of local operator for integrable models based on the (super)algebras $gl_n$, $gl_{m|p}$ or their quantum deformations. It corresponds to two talks given by E.R. and N.S. at \textsl{Correlation functions of quantum integrable systems and beyond}, in honor of Jean-Michel Maillet for his 60's (ENS Lyon, October 2017).

Published as SciPost Phys. Lect. Notes 6 (2018)

### List of changes

We thank the referee for his comments. We have modified our manuscript accordingly. More precisely, and considering the different questions raised by the referee:

$\bullet \textit{In the section 2, The transfer matrix t(z)}$
We added a sentence right after eq. (6)

$\bullet \textit{In the section 3, In the case of higher rank n}$
We added references just before eq (12).

$\bullet \textit{In the section 3, Known formulas: the trace formula}$
- We mentioned that the matrices $e_{ij}$ act in $\mathbb{C}^3$.
- We added a reference in the sentence mentioning the Gauss decomposition

$\bullet \textit{In the section 4, Sum formula}$
- We disagree with the referee about the quantum case: the coefficients $W_{part}$ are still rational functions of the Bethe parameters.
We guess the referee had in mind a presentation with additive spectral parameters
(for which indeed the coefficients $W_{part}$ are not rational functions anymore), but we use multiplicative ones, for which we still have rational functions.
To clarify this point we gave the form of the functions $f$ and $g$ in the quantum case, see the new equation at the beginning of section 2.1.
- The common guideline of the article is to have a special paragraph were the references are gathered, so we prefer to keep it this way.
However, we mention after eq (24) the first peoples that got such a formula, and refer to the special paragraph.

$\bullet \textit{In the section 4, Determinant formula}$
In fact the whole paragraph is needed to specify what are the special conditions (we need to introduce the twisted transfer matrix, the twisted BVs, etc..).
Then, we don't see how to fulfill the requirement of the referee. Instead, we
added sentences at the end of the first paragraph to mention that the special requirements will be shown at the end of the subsection.

$\bullet \textit{In the section 5, Bethe vectors and zero modes}$
- We corrected $j$ in $j-1$ in the limit $t^{(j-1)}_k\to\infty$.
- Indeed, the universal FF are the solution, as the referee guessed. We thought it was clear from our section 5.3, but apparently it was not. We modified the end of this section to be clearer.

### Submission & Refereeing History

Resubmission 1803.00103v2 on 6 July 2018
Submission 1803.00103v1 on 2 March 2018