Holomorphic factorization for a quantum tetrahedron
type: Item
instance of: scholarly article
title: Holomorphic factorization for a quantum tetrahedron
zbMATH Open document ID: 1211.22013
author: Laurent Freidel
author: Etera R. Livine
author: Kirill Krasnov
published in: Communications in Mathematical Physics
publication date: 2010-07-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/0905.3627
review text: 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…
Mathematics Subject Classification ID: 22E70
Mathematics Subject Classification ID: 53D50
zbMATH DE Number: 5729797
zbMATH Keywords: irreducible representation
zbMATH Keywords: coherent state
zbMATH Keywords: Kähler potential
zbMATH Keywords: symplectic reduction
zbMATH Keywords: Jacobi polynomial
zbMATH Keywords: bulk-to-boundary propagator
zbMATH Keywords: cross-ratio
zbMATH Keywords: holomorphic quantization
zbMATH Keywords: intertwiner
zbMATH Keywords: Möbius transformation
zbMATH Keywords: Hilbert space
zbMATH Keywords: moduli space
reviewed by: Albert Sheu
MaRDI profile type: Publication
OpenAlex ID: W2146341326
Wikidata QID: Q64007788
arXiv ID: 0905.3627
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DOI: 10.1007/S00220-010-1036-5
description: scientific article