{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $H_0$ and $H_1$ be complex [[Hilbert spaces|HM2HHE3J51]], and let $Y$ be a topological space.\nLet $y \\mapsto T_y$ be a map from $Y$ to the [[Fredholm operators|8HWGKWD6ZM]] $H_0 \\to H_1$ that is continuous in the operator norm.\nFor a finite-dimensional space $V$ write $\\det V = \\Lambda^{\\dim V} V$, with $\\det 0 = \\C$.\nThe *determinant line* of $T_y$ is\n$$\\det T_y = (\\det \\ker T_y)^* \\otimes \\det \\operatorname{coker} T_y.$$\nThe *determinant line bundle* is the set $\\det T = \\bigsqcup_{y \\in Y} \\det T_y$ with the following topology.\nNear a point $y_0$, choose a finite-dimensional subspace $V \\subseteq H_1$ with $\\operatorname{im} T_y + V = H_1$ for all $y$ near $y_0$.\nThen the spaces $T_y^{-1}(V)$ form a [[vector bundle|P4DZM975BV]] over a neighbourhood of $y_0$.\nThe exact sequence $0 \\to \\ker T_y \\to T_y^{-1}(V) \\to V \\to \\operatorname{coker} T_y \\to 0$ gives a canonical isomorphism $\\det T_y \\cong (\\det T_y^{-1}(V))^* \\otimes \\det V$.\nThe right side is a line bundle near $y_0$.\nThe topology of $\\det T$ is the one that makes these isomorphisms continuous, and it does not depend on the choice of $V$.\n\n## Remarks\n\nSome authors use the dual line $\\det \\ker T_y \\otimes (\\det \\operatorname{coker} T_y)^*$; the two conventions give dual line bundles.\n\nExample: if every $T_y$ is invertible, then $\\ker T_y = 0$ and $\\operatorname{coker} T_y = 0$, so each fibre is $\\C$ and $\\det T$ is the trivial line bundle $Y \\times \\C$.\n\nNon-example: the family $T_y = y \\cdot \\id$ on an infinite-dimensional Hilbert space, for $y \\in [0, 1]$, is not a family of Fredholm operators, since $T_0 = 0$ has infinite-dimensional kernel.\nSo $\\det T_0$ is not defined.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "5DST3VKEMX",
    "kind": "definition",
    "msc": [
      "58J52",
      "47A53"
    ],
    "name": "Determinant line bundle",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Determinant_line_bundle"
      }
    ],
    "tag": "8HDWPN2FXS",
    "uses": [
      "8HWGKWD6ZM",
      "HM2HHE3J51",
      "P4DZM975BV"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/8HDWPN2FXS/",
  "links": [
    {
      "field": "defines",
      "target": "5DST3VKEMX"
    },
    {
      "field": "uses",
      "target": "8HWGKWD6ZM"
    },
    {
      "field": "uses",
      "target": "HM2HHE3J51"
    },
    {
      "field": "uses",
      "target": "P4DZM975BV"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/8HDWPN2FXS.md",
  "msc": [
    "58J52",
    "47A53"
  ],
  "name": "Determinant line bundle",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/8HD/8HDWPN2FXS.md",
  "status": null,
  "tag": "8HDWPN2FXS",
  "uri": "m2://8HDWPN2FXS",
  "url": "https://tamnd.github.io/math2/wiki/determinant-line-bundle-definition/",
  "writing": {
    "tag": "8HDWPN2FXS",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}