{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "1YKCH218MN"
    },
    {
      "field": "about",
      "target": "4WCJ49Z90S"
    },
    {
      "field": "about",
      "target": "CHJ4TAH0QX"
    },
    {
      "field": "about",
      "target": "JR2M58RCW3"
    },
    {
      "field": "uses",
      "target": "1YKCH218MN"
    },
    {
      "field": "uses",
      "target": "4WCJ49Z90S"
    },
    {
      "field": "uses",
      "target": "CHJ4TAH0QX"
    },
    {
      "field": "uses",
      "target": "JR2M58RCW3"
    },
    {
      "field": "defines",
      "target": "A5TDM8F4KB"
    }
  ],
  "body": "\n## Summary\n\nThe Dirac operator of a Riemannian spin manifold, also called the Atiyah–Singer operator, acts on sections of the [[spinor bundle|PYYQQKAZ0C]].\nIt combines the spin connection, induced by the [[Levi-Civita connection|CAZYGE1T4E]], with Clifford multiplication by tangent vectors.\nIt is a first-order elliptic and formally self-adjoint operator whose square is the rough Laplacian plus a quarter of the scalar curvature.\nOn a closed spin manifold of dimension divisible by $4$, the index of its restriction to positive spinors is the $\\hat{A}$-genus.\nThis ties the existence of metrics of positive scalar curvature to topology.\nIt is the main example of a [[Dirac operator|3DXGJTEQZH]].\n\n## Explanations\n\n### L1 human\n\nOn flat $\\R^n$, Dirac found a first-order operator whose square is minus the Laplacian, using matrices that anticommute.\nTo do the same on a curved manifold one needs a bundle on which these matrices can act consistently from point to point.\nA spin structure is exactly the extra data that makes this possible, and the spinor bundle is the resulting bundle.\nThe Dirac operator differentiates a spinor field along each direction of an orthonormal frame and multiplies by the matching Clifford matrix, then adds the results.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "53C27",
      "58J60",
      "15A66"
    ],
    "name": "Dirac operator on a spin manifold",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Spinor_bundle"
      }
    ],
    "specializes": [
      "3DXGJTEQZH"
    ],
    "tag": "QMRAVX14BJ"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/QMRAVX14BJ/",
  "links": [
    {
      "field": "specializes",
      "target": "3DXGJTEQZH"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/QMRAVX14BJ.md",
  "msc": [
    "53C27",
    "58J60",
    "15A66"
  ],
  "name": "Dirac operator on a spin manifold",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QMR/QMRAVX14BJ.md",
  "status": null,
  "tag": "QMRAVX14BJ",
  "uri": "m2://QMRAVX14BJ",
  "url": "https://tamnd.github.io/math2/wiki/dirac-operator-on-a-spin-manifold/",
  "writing": {
    "tag": "QMRAVX14BJ",
    "kind": "concept",
    "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
      }
    ]
  }
}