A-Type Open ${\rm SL}(2,\mathbb{C})$ Spin Chain

TL;DR

The paper develops an algebraic, Yang–Baxter–based framework to solve the noncompact open ${\rm SL}(2,\mathbb{C})$ spin chain, constructing eigenfunctions of a key monodromy-matrix element through local ${\mathcal R}$-operators, a Baxter-like ${Q}$-operator, and raising operators. It establishes a complete, orthogonal eigenbasis via SoV, derives both coordinate and Mellin–Barnes representations, and proves a hidden symmetry under $s\to 1-s$ that manifests through similarity transformations. A complex-field generalization of the Gustafson integral is employed to connect representations and to obtain explicit integral kernels, overlaps, and normalization. The results pave the way for applications to higher-rank noncompact systems and to conformal fishnet diagrams, including BC-type open boundaries.

Abstract

For the noncompact open ${\rm SL}(2,\mathbb{C})$ spin chain, the eigenfunctions of the special matrix element of monodromy matrix are constructed. The key ingredients of the whole construction are local Yang-Baxter $\mathcal{R}$-operators, $Q$-operator and raising operators obtained by reduction from the $Q$-operator. The calculation of various scalar products and the proof of orthogonality are based on the properties of $Q$-operator and demonstrate its hidden role. The symmetry of eigenfunctions with respect to reflection of the spin variable $s \to 1-s$ is established. The Mellin-Barnes representation for eigenfunctions is derived and equivalence with initial coordinate representation is proved. The transformation from one representation to another is grounded on the application of $A$-type Gustafson integral generalized to the complex field.

Figures (7)

• Figure 1: The diagrammatic representation for $Q(\bm z_n,\bm w_n;x)$. The arrow with index $\alpha$ from $z$ to $w$ stands for $[z-w]^{-\alpha}$. The indices are given by the following expressions: $a=1-s+x$, $b=1-s-x$, $c=2s-1$.
• Figure 2: The diagrammatic representation for $\Lambda(\bm z_k,\bm w_{k-1};x)$. The indices are given by the following expressions: $a=1-s+x$, $b=1-s-x$, $c=2s-1$.
• Figure 3: The diagrammatic representation for $\Lambda'(\bm{z}_n,\bm{w}_{n-1};x)$. The indices are given by the following expressions: $a=1-s+x$, $b=1-s-x$, $c=2s-1$.
• Figure 4: Diagrammatic representations for kernels of $\Lambda$- and $\Lambda'$-operators: (1) for $\Lambda(\bm{z}_3, \bm{w}_2;x_3)$, (2) for $\Lambda(\bm{z}_2, w_1;x_2)$, (3) for $\Lambda(z_1;x_1) = [z_{10}]^{-s-x_1}$, (4) for $\Lambda'(\bm{z}_3, \bm{w}_2;x_3)$, (5) for $\Lambda'(\bm{z}_2, w_1;x_2)$, (6) for $\Lambda'(z_1;x_1) = [z_{10}]^{-s-x_1}$. (7) Graphical representation for $\Psi_{\bm{x}_3}(\bm{z}_3) = \Lambda_3(x_3) \Lambda_2(x_2) \Lambda_1(x_1)$. (8) Diagrammatic representation for $\Psi'_{\bm{x}_3}(\bm{z}_3) = \Lambda'_3(x_3) \Lambda_2(x_2) \Lambda'_1(x_1)$. The indices are given by the following expressions: $a_k=1-s+x_k$, $b_k=1-s-x_k$, $c=2s-1$.
• Figure 5: Diagrammatic representation of $[z-w]^{-a}$.