Holomorphic factorization for a quantum tetrahedron
This paper studies the space \({\mathcal H}_{\vec j}:=(V_{j_1}\otimes\cdots\otimes V_{j_n})^{SU(n)}\) of \(SU(2)\)-invariant tensors (or intertwiners) where \(SU(2)\) acts on the Hilbert space \(V_{j_1}\otimes\cdots\otimes V_{j_n}\) diagonally, \(\vec j=(j_1,\dots,j_n)\in(\mathbb Z_\geq/2)^n\), and \(V^j\) denotes the \((2j+1)\)-dimensional irreducible representation space of \(SU(2)\). Realizing \(V_j\) as spanned by holomorphic polynomials \(\psi(z)\) and expressing the \(SU(2)\)-action on \(\psi(z)\) in terms of the fractional linear (or Möbius) transformation, it is shown that \(SU(2)\)-invariant states in \({\mathcal H}_{\vec j}\) are automatically \(SL(2,\mathbb C)\)-invariant, where the \(SL(2,\mathbb C)\)-action extends the \(SU(2)\)-action canonically via the Möbius transformation, and hence completely determined by their values on the shape space \(S_{\vec j}:=\{(z_1,\dots,z_n)\mid z_i\neq z_j\}/SL(2,\mathbb C)\). With a change of variables from \((z_1,\dots,z_n)\) to \(Z=(Z_1,\dots,Z_{n-3}) \) of cross-ratio coordinates \(Z_i\), the identity operator on \(\mathcal{H}_{\vec j}\) is expressed explicitly as a multiple of \(\int\widehat{K}_{\vec j}(Z,\overline{Z})|\vec j,Z\rangle\langle\vec j,Z|\,d^2Z\) for coherent states \(|{\vec j} ,Z\rangle\), where the kernel function \(\widehat{K}_{\vec j}(Z,\overline{Z})\) allows an interpretation as an exponential of the Kähler potential on \(S_{\vec j}\) and relates closely to the so-called bulk-to-boundary propagators. In the ``tetrahedral'' case of \(n=4\) involving only one complex cross-ratio variable, it is found that the inner product \(^{ij} C_{\vec j}^k:=^{ij}\langle{\vec j},k|{\vec j},Z\rangle \) between the coherent states \(|{\vec j},Z\rangle \) and the real basis vectors \(|{\vec j},k\rangle^{ij}\) in channel \((ij)\) is a multiple of a shifted Jacobi polynomial, where \(|{\vec j},k\rangle^{ij}\) is the \(k(k+1)\)-eigenvector (with length \(1/\sqrt{d_k}\)) of the second-order differential operator \(J_{ij}:=(J_{(i)}+J_{(j)})^2\) for the canonical \({\mathfrak{su}}(2)\) generators \(J_{(i)}^a\), \(a=1,2,3\), acting on \(V_{j_i}\). This leads to a holomorphically factorized form for \(\langle{\vec j},Z|{\vec j},Z\rangle \).
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