Bethe Vectors of gl(3)-Invariant Integrable Models, Their Scalar Products and Form Factors

Abstract

This short note corresponds to a talk given at Lie Theory and Its Applications in Physics (Varna, Bulgaria, June 2013) and is based on joint works with S. Belliard, S. Pakuliak and N. Slavnov, see arXiv:1206.4931, arXiv:1207.0956,  arXiv:1210.0768, arXiv:1211.3968 and arXiv:1312.1488.

Appendices

Appendix 1: The Matrix $\mathcal{N}$

Diagonal blocks

$$\displaystyle\begin{array}{rcl} \mathcal{N}^{(u)}(u_{ j}^{C },u_{k}^{B })& =& h(\bar{v}^{C },u_{k}^{B })h(u_{k}^{B },\bar{u}^{C })\Big[\kappa t(u_{k}^{B },u_{j}^{C }) {}\\ & & +t(u_{j}^{C },u_{k}^{B })\frac{f(\bar{v}^{B},u_{ k}^{B})} {f(\bar{v}^{C},u_{k}^{B})} \frac{h(\bar{u}^{C},u_{ k}^{B})h(u_{ k}^{B},\bar{u}^{B})} {h(u_{k}^{B},\bar{u}^{C})h(\bar{u}^{B},u_{k}^{B})}\Big]\quad \mbox{ $a \times a$ block}{}\\ \end{array}$$

$$\displaystyle\begin{array}{rcl} \mathcal{N}^{(v)}(v_{ j}^{B },v_{k}^{C })& =& h(v_{k}^{C },\bar{u}^{B })h(\bar{v}^{B },v_{k}^{C })\Big[t(v_{j}^{B },v_{k}^{C }) {}\\ & & +\kappa t(v_{k}^{C },v_{j}^{B })\frac{f(v_{k}^{C},\bar{u}^{C})} {f(v_{k}^{C},\bar{u}^{B})} \frac{h(\bar{v}^{C},v_{ k}^{C})h(v_{ k}^{C},\bar{v}^{B})} {h(v_{k}^{C},\bar{v}^{C})h(\bar{v}^{B},v_{k}^{C})}\Big]\quad \mbox{ $b \times b$ block} {}\\ \end{array}$$

Off-diagonal blocks

$$\displaystyle\begin{array}{rcl} \mathcal{N}^{(u)}(u_{ j}^{C },v_{k}^{C })& =& \kappa t(v_{k}^{C },u_{j}^{C })h(v_{k}^{C },\bar{u}^{C })h(\bar{v}^{C },v_{k}^{C })\quad \mbox{ $a \times b$ block} {}\\ \mathcal{N}^{(v)}(v_{ j}^{B },u_{k}^{B })& =& t(v_{j}^{B },u_{k}^{B })h(\bar{v}^{B },u_{k}^{B })h(u_{k}^{B },\bar{u}^{B })\qquad \mbox{ $b \times a$ block} {}\\ \end{array}$$

Appendix 2: The Matrix Θ ^(s)

First of all we define an (a + b) × (a + b) matrix θ with the entries

$$\displaystyle\begin{array}{rcl} & & \theta _{j,k} = \left. \frac{\partial \varPhi _{j}} {\partial u_{k}^{C}}\right \vert _{{ \bar{u}^{C}=\bar{u} \atop \bar{v}^{C}=\bar{v}} },\qquad k = 1,\ldots,a, \\ & & \theta _{j,k+a} = \left. \frac{\partial \varPhi _{j}} {\partial v_{k}^{C}}\right \vert _{{ \bar{u}^{C}=\bar{u} \atop \bar{v}^{C}=\bar{v}} },\qquad k = 1,\ldots,b,{}\end{array}$$

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where the Φ _j are given by (36) and (37).

Then we extend the matrix θ to an $(a + b + 1) \times (a + b + 1)$ matrix Θ ^(s) with s = 1, 2, 3, by adding one row and one column

$$\displaystyle \begin{array}{llll} &\varTheta _{j,k}^{(s)} =\theta _{ j,k},\quad \qquad \qquad \qquad j,k = 1,\ldots,a + b, \\ &\varTheta _{a+b+1,k}^{(s)} = \frac{\partial \tau (z\vert \bar{u},\bar{v})} {\partial u_{k}},\qquad k = 1,\ldots,a, \\ &\varTheta _{a+b+1,a+k}^{(s)} = \frac{\partial \tau (z\vert \bar{u},\bar{v})} {\partial v_{k}},\qquad k = 1,\ldots,b, \\ &\varTheta _{j,a+b+1}^{(s)} =\delta _{ s1} -\delta _{s2}\qquad \quad j = 1,\ldots,a, \\ &\varTheta _{j+a,a+b+1}^{(s)} =\delta _{ s3} -\delta _{s2}\qquad \quad j = 1,\ldots,b, \\ &\varTheta _{a+b+1,a+b+1}^{(s)} = \left.\frac{\partial \tau _{\bar{\kappa }}(z\vert \bar{u}^{C},\bar{v}^{C})} {\partial \kappa _{s}} \right \vert _{{ \bar{u}^{C}=\bar{u} \atop \bar{v}^{C}=\bar{v}} }. \end{array} $$

Here the $\delta _{sk}$ are Kronecker deltas. Notice that Θ ^(s) depends on s only in its last column.

Appendix 3: The Matrix $\mathcal{N}^{(s,p)}$

For j ≠ p we define the entries $\mathcal{N}_{j,k}^{(s,p)}$ of the (a + b) × (a + b) matrix $\mathcal{N}^{(s,p)}$ as

$$\displaystyle\begin{array}{rcl} \mathcal{N}_{j,k}^{(s)}& =& c\,g^{-1}(w_{ k},\bar{u}^{C })\,g^{-1}(\bar{v}^{C },w_{k})\frac{\partial \tau (w_{k}\vert \bar{u}^{C},\bar{v}^{C})} {\partial u_{j}^{C}}, \\ & & \qquad \quad j = 1,\ldots,a,\quad j\neq p, {}\end{array}$$

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$$\displaystyle\begin{array}{rcl} \mathcal{N}_{a+j,k}^{(s)}& =& -c\,g^{-1}(\bar{v}^{B },w_{k})\,g^{-1}(w_{ k},\bar{u}^{B })\frac{\partial \tau (w_{k}\vert \bar{u}^{B},\bar{v}^{B})} {\partial v_{j}^{B}},{}\end{array}$$

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$$\displaystyle\begin{array}{rcl} & & \qquad j = 1,\ldots,b,\quad j\neq p. {}\\ \end{array}$$

In these formulas one should set $w_{k} = u_{k}^{B}$ for $k = 1,\ldots,a$ and $w_{k+a} = v_{k}^{C}$ for $k = 1,\ldots,b$.

The p-th row has the following elements

$$\displaystyle{ \mathcal{N}_{p,k}^{(s)} = h(\bar{v}^{C },w_{k})h(w_{k},\bar{u}^{B })Y _{k}^{(s)}, }$$

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where again $w_{k} = u_{k}^{B}$ for $k = 1,\ldots,a$ and $w_{k+a} = v_{k}^{C}$ for $k = 1,\ldots,b$, and

$$\displaystyle\begin{array}{rcl} Y _{k}^{(s)}& =& c\,(\delta _{ s1} -\delta _{s2}) + (\delta _{s1} -\delta _{s3})u_{k}^{B }\left (1 -\frac{f(\bar{v}^{B},u_{ k}^{B})} {f(\bar{v}^{C},u_{k}^{B})}\right ),\qquad \qquad k = 1,\ldots,a, \\ Y _{a+k}^{(s)}& =& c\,(\delta _{ s3} -\delta _{s2}) + (\delta _{s1} -\delta _{s3})(v_{k}^{C } + c)\left (1 -\frac{f(v_{k}^{C},\bar{u}^{C})} {f(v_{k}^{C},\bar{u}^{B})}\right ), \\ & & \qquad k = 1,\ldots,b. {}\end{array}$$

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