{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $(M, g)$ be a closed oriented [[Riemannian manifold|7CBT4SM83Y]] of dimension $4k$.\nLet $\\Omega^*(M) = \\bigoplus_p \\Omega^p(M)$ be the space of smooth complex-valued [[differential forms|0E3G0GRPZ6]], with [[exterior derivative|0ZHSTAG8ZY]] $d$, [[Hodge star|1Y4WX4XQWJ]] $\\star$ and codifferential $\\delta$, the formal adjoint of $d$.\nDefine $\\tau \\colon \\Omega^*(M) \\to \\Omega^*(M)$ by\n$$\\tau \\omega = i^{p(p-1) + 2k} \\star \\omega \\quad \\text{for } \\omega \\in \\Omega^p(M).$$\nThen $\\tau^2 = \\id$ and $\\tau (d + \\delta) = -(d + \\delta) \\tau$.\nLet $\\Omega^\\pm(M)$ be the $\\pm 1$ eigenspaces of $\\tau$.\nThe *signature operator* of $M$ is the restriction\n$$D^+ = d + \\delta \\colon \\Omega^+(M) \\to \\Omega^-(M).$$\n\n## Remarks\n\nThe operator $d + \\delta$ is elliptic with square the [[Hodge Laplacian|PWG3GKZG80]], so $D^+$ is an [[elliptic operator|MT9P2X85V8]] of order $1$.\nIts index equals the signature $\\sigma(M)$ of the [[intersection form|WK8GKRKAVD]] on $H^{2k}(M; \\R)$.\nIndeed, the harmonic forms of degrees $p \\ne 2k$ contribute equally to the kernel and the cokernel and cancel.\nOn harmonic $2k$-forms $\\tau = \\star$, whose $\\pm 1$ eigenspaces are the self-dual and anti-self-dual forms.\nThe Atiyah–Singer index theorem applied to $D^+$ gives the [[Hirzebruch signature theorem|TEG1ZX1AGJ]].\n\nExample: on $2k$-forms, $\\tau = i^{2k(2k-1) + 2k} \\star = i^{4k^2} \\star = \\star$.\nFor $M = \\C P^2$ ($k = 1$), the harmonic $2$-forms are the multiples of the Kähler form $\\omega_{FS}$, which is self-dual, so $\\sigma(\\C P^2) = 1$ and the index of $D^+$ is $1$.\n\nNon-example: the operator $d + \\delta \\colon \\Omega^{\\mathrm{even}}(M) \\to \\Omega^{\\mathrm{odd}}(M)$ uses the splitting by degree instead of the splitting by $\\tau$.\nIt is the de Rham or Euler operator, and its index is the Euler characteristic, not the signature; for $\\C P^2$ it is $3$.\n\n## Explanations\n\n### L1 human\n\nThe signature operator is $d + d^*$ on forms, split by a twisted Hodge star, and its index is the signature of the manifold.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "4NCYA7GME1",
    "kind": "definition",
    "msc": [
      "58J20",
      "58A14",
      "57R20"
    ],
    "name": "Signature operator",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Signature_operator"
      }
    ],
    "tag": "A77EHY9DAN",
    "uses": [
      "7CBT4SM83Y",
      "0E3G0GRPZ6",
      "0ZHSTAG8ZY",
      "1Y4WX4XQWJ",
      "PWG3GKZG80",
      "MT9P2X85V8",
      "WK8GKRKAVD",
      "TEG1ZX1AGJ"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/A77EHY9DAN/",
  "links": [
    {
      "field": "defines",
      "target": "4NCYA7GME1"
    },
    {
      "field": "uses",
      "target": "7CBT4SM83Y"
    },
    {
      "field": "uses",
      "target": "0E3G0GRPZ6"
    },
    {
      "field": "uses",
      "target": "0ZHSTAG8ZY"
    },
    {
      "field": "uses",
      "target": "1Y4WX4XQWJ"
    },
    {
      "field": "uses",
      "target": "PWG3GKZG80"
    },
    {
      "field": "uses",
      "target": "MT9P2X85V8"
    },
    {
      "field": "uses",
      "target": "WK8GKRKAVD"
    },
    {
      "field": "uses",
      "target": "TEG1ZX1AGJ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/A77EHY9DAN.md",
  "msc": [
    "58J20",
    "58A14",
    "57R20"
  ],
  "name": "Signature operator",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/A77/A77EHY9DAN.md",
  "status": null,
  "tag": "A77EHY9DAN",
  "uri": "m2://A77EHY9DAN",
  "url": "https://tamnd.github.io/math2/wiki/signature-operator-definition/",
  "writing": {
    "tag": "A77EHY9DAN",
    "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
      }
    ]
  }
}