{
  "id": "2410.11311",
  "version": "v1",
  "published": "2024-10-15T06:15:17.000Z",
  "updated": "2024-10-15T06:15:17.000Z",
  "title": "Symmetry in Deformation quantization and Geometric quantization",
  "authors": [
    "Naichung Conan Leung",
    "Qin Li",
    "Ziming Nikolas Ma"
  ],
  "categories": [
    "math.DG",
    "math.QA",
    "math.RT"
  ],
  "abstract": "In this paper, we explore the quantization of K\\\"ahler manifolds, focusing on the relationship between deformation quantization and geometric quantization. We provide a classification of degree 1 formal quantizable functions in the Berezin-Toeplitz deformation quantization, establishing that these formal functions are of the form $f = f_0 - \\frac{\\hbar}{4\\pi}(\\Delta f_0 + c)$ for a certain smooth (non-formal) function $f_0$. If $f_0$ is real-valued then $f_0$ corresponds to a Hamiltonian Killing vector field. In the presence of Hamiltonian $G$-symmetry, we address the compatibility between the infinitesimal symmetry for deformation quantization via quantum moment map and infinitesimal symmetry on geometric quantization acting on Hilbert spaces of holomorphic sections via Berezin-Toeplitz quantization.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-15T06:15:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "geometric quantization",
      "infinitesimal symmetry",
      "hamiltonian killing vector field",
      "quantum moment map",
      "berezin-toeplitz deformation quantization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}