{
  "id": "1303.5340",
  "version": "v2",
  "published": "2013-03-21T17:36:37.000Z",
  "updated": "2014-11-28T23:52:19.000Z",
  "title": "Duality and universality for stable pair invariants of surfaces",
  "authors": [
    "M. Kool"
  ],
  "comment": "19 pages. Exposition and references to existing literature slightly changed",
  "categories": [
    "math.AG",
    "hep-th",
    "math.SG"
  ],
  "abstract": "Let $\\beta$ be a curve class on a surface $S$. The moduli space of stable pairs on $S$ with class $\\beta$ carries a natural reduced virtual cycle. This cycle is defined when $h^2(L) = 0$ for any effective $L \\in \\mathrm{Pic}^\\beta(S)$ (weak assumption). When $h^2(L) = 0$ for any $L \\in \\mathrm{Pic}^{\\beta}(S)$ (strong assumption), the associated invariants are given by universal functions in $\\beta^2$, $\\beta.c_1(S)$, $c_1(S)^2$, $c_2(S)$, and certain invariants of the ring structure of $H^*(S,\\mathbb{Z})$. In this paper, we show the following. (1) Universality no longer holds when just the weak assumption is satisfied. (2) For any $S,\\beta$ (no conditions), the BPS spectrum of the non-reduced stable pair invariants of $S,\\beta$ with maximal number of point insertions consists of a single number. This number is the Seiberg-Witten invariant of $S, \\beta$. (3) The GW/PT correspondence for $X = K_S$ implies Taubes' GW/SW correspondence in certain cases, e.g. when $\\beta$ is irreducible. (4) When $p_g(S)=0$, the difference between the stable pair invariants in class $\\beta$ and $K_S-\\beta$ is universal. The results of this paper follow from relating certain stable pairs invariants of surfaces to the Poincar\\'e invariants of M. D\\\"urr, A. Kabanov, and Ch. Okonek.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-03-21T17:36:37.000Z",
      "abstract": "Let $\\beta$ be a curve class on a surface $S$. The moduli space of stable pairs on $S$ with class $\\beta$ carries a natural reduced virtual cycle \\cite{KT1, KT2}. This cycle is defined when $h^2(L) = 0$ for any \\emph{effective} $L \\in \\mathrm{Pic}^\\beta(S)$ (weak assumption). When $h^2(L) = 0$ for \\emph{any} $L \\in \\mathrm{Pic}^{\\beta}(S)$ (strong assumption), the associated invariants are given by universal functions in $\\beta^2$, $\\beta.c_1(S)$, $c_1(S)^2$, $c_2(S)$, and certain invariants of the ring structure of $H^*(S,\\Z)$. In this paper, we show the following. (1) Universality \\emph{no longer} holds when just the weak assumption is satisfied. (2) For any $S,\\beta$ (no conditions), the BPS spectrum of the non-reduced stable pair invariants of $S,\\beta$ with maximal number of point insertions consists of a single number. This number is the Seiberg-Witten invariant of $S, \\beta$. (3) The GW/PT correspondence for $X = K_S$ implies Taubes' GW/SW correspondence in certain cases, e.g. when $\\beta$ is irreducible. (4) When $p_g(S)=0$, the difference between the stable pair invariants in class $\\beta$ and $K_S-\\beta$ is universal.",
      "comment": "19 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-11-28T23:52:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14N35"
    ],
    "keywords": [
      "universality",
      "weak assumption",
      "point insertions consists",
      "natural reduced virtual cycle",
      "curve class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1224822,
      "adsabs": "2013arXiv1303.5340K"
    }
  }
}