{
  "id": "1206.2317",
  "version": "v1",
  "published": "2012-06-11T19:02:07.000Z",
  "updated": "2012-06-11T19:02:07.000Z",
  "title": "Wall-Crossing in Supersymmetric Gauge Theories",
  "authors": [
    "Kirill Petunin"
  ],
  "comment": "Ph.D. thesis",
  "categories": [
    "hep-th"
  ],
  "abstract": "We study $\\mathcal{N}=2$ supersymmetric Yang--Mills theory in four dimensions and then compactify it on $\\mathbb{R}^{3}\\times S^{1}$. The gauge symmetry of the theory is broken by a vacuum expectation value of the scalar field, which parametrises the moduli space. The spectrum of BPS states, carrying electric and magnetic charges, is piece-wise constant, changing only when the vacuum expectation value crosses the so-called walls of marginal stability. Kontsevich and Soibelman proposed an algebraic construction relating BPS spectra on both sides of a wall of marginal stability. These formulae are known to correctly relate the strong- and weak-coupling spectra in theories with gauge group SU(2) with and without fundamental flavours; we generalise this result to gauge group SU(n) without flavours in the weak-coupling regime. In addition, we find the walls of marginal stability in the SU(n) theory at the root of the Higgs branch and, employing the wall-crossing formula, determine the BPS spectrum in all regions of the moduli space. Gaiotto, Moore, and Neitzke (GMN) proposed an ansatz expressing the moduli space metric of $\\mathcal{N}=2$ theory on $\\mathbb{R}^{3}\\times S^{1}$ in terms of a set of integral equations. Using the GMN ansatz, we find perturbative and instanton corrections in $\\mathbb{R}^{3}\\times S^{1}$ for gauge group SU(2) with or without flavours and for gauge group SU(n) without flavours. For gauge group SU(n), we also demonstrate that the predicted two-instanton metric is continuous across the walls. Then, we calculate instanton corrections from first principles. We find perfect agreement between the GMN prediction and the first-principles result. We also take the limit of small radius of the compactified dimension finding one- and two-instanton corrections in three dimensions, recovering some of the previously known semiclassical results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-06-11T19:02:07.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "gauge group su",
      "supersymmetric gauge theories",
      "bps spectrum",
      "marginal stability",
      "moduli space"
    ],
    "tags": [
      "dissertation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1117911,
      "adsabs": "2012arXiv1206.2317P"
    }
  }
}