Elliptic hypergeometric function and $6j$-symbols for the $SL(2,{\mathbb C})$ group

Abstract

We show that the complex hypergeometric function describing $6j$-symbols for the $SL(2,\mathbb C)$ group is a special degeneration of the $V$-function—an elliptic analogue of the Euler–Gauss ${}_2F_1$ hypergeometric function. For this function, we derive mixed difference–recurrence relations as limit forms of the elliptic hypergeometric equation and some symmetry transformations. At the intermediate steps of computations, there emerge a function describing the $6j$-symbols for the Faddeev modular double and the corresponding difference equations and symmetry transformations.

Appendix A. Details of the $$b\to i$$ asymptotics computations

In this appendix, we give asymptotic estimates for hyperbolic gamma functions in the limit $\delta\to 0^+$, which in combination reduce the $6j$-symbols for the Faddeev modular double (3.9) to $6j$-symbols for the $SL(2,\mathbb C)$ group (2.11). The constraints on parameters were indicated in the main body of the paper:

$$\begin{aligned} \, &S_{i+\delta}(\mu_a-z)\to e^{(\pi i/2)(T_a+N)^2} (4\pi\delta)^{i(U_a+u)-1}\boldsymbol\Gamma(U_a+u,T_a+N), \\ &S_{i+\delta}(\nu_a+z)\to e^{(\pi i/2)(S_a-N)^2}(4\pi\delta)^{i(R_a-u )-1} \boldsymbol\Gamma(R_a-u,S_a-N), \\ &S_{i+\delta}(\alpha_s+\alpha_2-\alpha_1) \to e^{(\pi i/2)A_1^2}(4\pi\delta)^{i(\sigma_1-\sigma_2+\rho_2)} \boldsymbol\Gamma(\sigma_1-\sigma_2+\rho_2-i,A_1), \\ &S_{i+\delta}(\alpha_1+\alpha_t-\alpha_4)\to e^{(\pi i/2) A_4^2} \frac{(4\pi\delta)^{i(-\sigma_1-\sigma_4-\rho_1)}(-1)^{A_4}} {\boldsymbol\Gamma(\sigma_1+\sigma_4+\rho_1-i,A_4)}, \\ &S_{i+\delta}(\alpha_2+\alpha_t-\alpha_3)\to e^{(\pi i/2)A_2^2} \frac{(4\pi\delta)^{i(-\sigma_2+\sigma_3-\rho_1)}(-1)^{A_2}} {\boldsymbol\Gamma(\sigma_2-\sigma_3+\rho_1-i,A_2)}, \\ &S_{i+\delta}(\alpha_3+\alpha_s-\alpha_4)\to e^{(\pi i/2)A_3^2} (4\pi\delta)^{i(-\sigma_3-\sigma_4+\rho_2)} \boldsymbol\Gamma(-\sigma_3-\sigma_4+\rho_2-i,A_3), \\ &|S_{i+\delta}(2\alpha_t)|^2\to(4\pi\delta)^2\frac{M_1^2+4\rho_1^2}{4}. \end{aligned} $$

(A.1)

Appendix B. A check of Eq. (6.6)

We verify difference equation (6.6) for some particular choices of the parameters. Namely, we remark that integral (4.28) can be explicitly computed if, say, $\mu_4+ \nu_4= Q$ and $\sum_a^3(\mu_a+\nu_a)=Q$. In this case, $\gamma^{(2)}(\mu_4-z;\omega_1,\omega_2)\gamma^{(2)}(\nu_4+z;\omega_1,\omega_2)\!=\! 1$ and

$$\begin{aligned} \, J_h(\underline\mu,\underline\nu) &=\int_{-i\infty}^{i\infty}\prod_{a=1}^3\gamma^{(2)}(\mu_a-z;\omega_1,\omega_2) \gamma^{(2)}(\nu_a+z;\omega_1,\omega_2)\,\frac{dz}{i\sqrt{\omega_1\omega_2}}= \nonumber \\ &=\prod_{a,b=1}^3\gamma^{(2)}(\mu_a+\nu_b;\omega_1,\omega_2). \end{aligned}$$

(B.1)

Similarly, if $n_4+m_4=0$ and $s_4+t_4=-2i$, then $\boldsymbol\Gamma(s_4-y,n_4-N)\boldsymbol\Gamma(t_4+y,m_4+N)=(-1)^{N-n_4}$. As a result, for

$$\sum_{a=1}^3(n_a+m_a)=0,\qquad\sum_{a=1}^3(s_a+t_a)=-2i,$$

we have [19]

$$\begin{aligned} \, \mathcal J_{\mathrm{cr}}(\underline s,\underline n;\underline t,\underline m) &=\frac{1}{4\pi}\sum_{N\in\mathbb Z}(-1)^{N-n_4}\int_{-\infty}^\infty \prod_{a=1}^3\boldsymbol\Gamma(s_a-y,n_a-N)\boldsymbol\Gamma(t_a+y,m_a+N)\,dy= \nonumber \\ &=(-1)^{\sum_{a=1}^4n_a}F(\underline s,\underline n;\underline t,\underline m), \end{aligned}$$

(B.2)

where

$$F(\underline s,\underline n;\underline t,\underline m) =\prod_{a,b=1}^3\boldsymbol\Gamma(s_a+t_b,n_a+m_b). $$

(B.3)

From the equations

$$\boldsymbol\Gamma(x-i,n-1)=\frac{\boldsymbol\Gamma(x,n)(n-ix)}{2},\qquad \boldsymbol\Gamma(x+i,n+1)=\frac{\boldsymbol\Gamma(x,n)}{n/2-ix/2+1} $$

(B.4)

and parameterization (6.7), we find

$$F(s_2-i,n_2-1,s_3+i,n_3+1)=\prod_{a=1}^3 \frac{\beta_2+\gamma_a}{\beta_3+\gamma_a-1} F(\underline s,\underline n;\underline t,\underline m),\qquad\;$$

(B.5)

$$\frac{\boldsymbol\Gamma(s_2-s_4,n_2-n_4)\boldsymbol\Gamma(s_3-s_4,n_3-n_4)} {\boldsymbol\Gamma(s_2-i-s_4,n_2-1-n_4)\boldsymbol\Gamma(s_3+i-s_4,n_3+1-n_4)} =\frac{\beta_3-\beta_4-1}{\beta_2-\beta_4}.$$

(B.6)

As a result, imposing the above constraints on the parameters and relations (B.5), (B.6), we obtain the algebraic equation

$$\begin{aligned} \, &(\beta_2-\beta_4-1)(\beta_3-\beta_4)(\beta_3-\beta_2+1)\biggl[\,\prod_{k=1}^3(\beta_2+\gamma_k)(\beta_3-\beta_4-1) +\prod_{k=1}^3(\beta_3+\gamma_k-1)(\beta_4-\beta_2)\biggr]+{} \nonumber \\ &\quad{}+(\beta_3-\beta_4-1)(\beta_2-\beta_4)(\beta_3-\beta_2-1)\biggl[\,\prod_{k=1}^3(\beta_3+\gamma_k)(\beta_2-\beta_4-1) +\prod_{k=1}^3(\beta_2+\gamma_k-1)(\beta_4-\beta_3)\biggr]-{} \nonumber \\ &\quad{}-(\beta_2-\beta_3)(\beta_3-\beta_2+1)(\beta_3-\beta_2-1) \prod_{k=1}^3(\beta_4+\gamma_k)=0, \end{aligned}$$

(B.7)

which is identically satisfied. Taking the limit $\beta_4\to\infty$ in (B.7), we obtain

$$\begin{aligned} \, &(\beta_3-\beta_2+1)\biggl(\,\prod_{k=1}^3(\beta_2+\gamma_k) -\prod_{k=1}^3(\beta_3+\gamma_k-1)\biggr)+{}\nonumber\\ &\quad{}+(\beta_3-\beta_2-1)\biggl(\,\prod_{k=1}^3(\beta_3+\gamma_k) -\prod_{k=1}^3(\beta_2+\gamma_k-1)\biggr)+{} \nonumber \\ &\quad{}+(\beta_3-\beta_2+1)(\beta_3-\beta_2-1)(\beta_2-\beta_3)=0, \end{aligned}$$

(B.8)

which leads to a difference equation for function (B.3),

$$\begin{aligned} \, &\mathcal V(\underline s,\underline n;\underline t,\underline m) (F(s_2-i,n_2-1,s_3+i,n_3+1)-F(s_2,n_2,s_3,n_3)) +(2\leftrightarrow 3)+F(s_2,n_2,s_3,n_3)=0, \end{aligned}$$

(B.9)

where

$$\mathcal V(\underline s,\underline n;\underline t,\underline m) =\frac{\prod_{k=1}^3(\beta_3+\gamma_k-1)}{(\beta_3-\beta_2-1)(\beta_2-\beta_3)}. $$

(B.10)