Geometry via Coherent States

It is shown how the coherent states permit to find different geometrical objects as the geodesics, the conjugate locus, the cut locus, the Calabi’s diastasis and its domain of definition, the Euler-Poincaré characteristic, the number of Borel-Morse cells, the Kodaira embedding theorem. 1 Coherent state manifold and coherent vector manifold In this talk the coherent states [] are presented as a powerful tool in global differential geometry and algebraic geometry [], []. For general references and proofs see also [], []. The results are illustrated on the complex Grassmann manifold []. We start with some notation. Let π be a unitary irreducible representation, G a Lie group, H a separable complex Hilbert space. Let also the orbit M̃ = π̃(G)ψ̃0 , where ψ0 ∈ H, ξ : H → P(H) is the projection ξ(ψ0) = ψ̃0 . Then there is a diffeomemorphism M̃ ≈ G/K, where K is the stationary group of ψ̃0. If ι : M̃ →֒ P(L) is a biholomorphic embedding in some projective Hilbert space, then M̃ is called coherent state manifold. If σ : M̃ → S(H) is a local section in the unit sphere in H, let M = σ(M̃) be the holomorphic line bundle associated to the principal holomorphic bundle P → G → G/P by a holomorphic character χ of the parabolic subgroup P of the compexification G of G. M is a quantization bundle over M̃ []. The following assertions are equivalent: there exists the embedding ι; there exists a positive line bundle M over M̃; the line bundle M is ample; there exists m0 such that for m ≥ m0, M = M = ι[1], where [1] is the hyperplane bundle over P(L). M is called coherent vector manifold. The Perelomov’s coherent vectors are eZ,j = exp ∑ φ∈∆ n (ZφF + φ )j, eZ = (eZ , eZ) −1/2 eZ , (1) eB,j = exp ∑ φ∈∆ n (BφF + φ − B̄φF φ )j, eB,j :=eZ,j, (2) where j is an extreme weight vector, ∆n denotes the positive non-compact roots, Z:=(Zφ) ∈ C are local coordinates in the neighbourhood V0 ⊂ M̃ and d = dimCM̃. In eqs. (1), (2) F φ j 6= 0, F φ j = 0, φ ∈ ∆n , and (eZ′, eZ) is the hermitian scalar product of holomorphic sections in the holomorphic line bundle M over M̃.