Journal of High Energy Physics

, 2018:106 | Cite as

Large-N \( \mathbb{C}{\mathrm{\mathbb{P}}}^{\mathrm{N}-1} \) sigma model on a finite interval and the renormalized string energy

  • Alessandro Betti
  • Stefano Bolognesi
  • Sven Bjarke GudnasonEmail author
  • Kenichi Konishi
  • Keisuke Ohashi
Open Access
Regular Article - Theoretical Physics


We continue the analysis started in a recent paper of the large-N two-dimensional \( \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} \) sigma model, defined on a finite space interval L with Dirichlet (or Neumann) boundary conditions. Here we focus our attention on the problem of the renormalized energy density \( \mathrm{\mathcal{E}} \) (x, Λ, L) which is found to be a sum of two terms, a constant term coming from the sum over modes, and a term proportional to the mass gap. The approach to \( \mathrm{\mathcal{E}}\left(x,\varLambda,\ L\right)\to \frac{N}{4\pi }{\varLambda}^2 \) at large LΛ is shown, both analytically and numerically, to be exponential: no power corrections are present and in particular no Lüscher term appears. This is consistent with the earlier result which states that the system has a unique massive phase, which interpolates smoothly between the classical weakly-coupled limit for LΛ → 0 and the “confined” phase of the standard \( \mathbb{C}{\mathrm{\mathbb{P}}}^{N-1} \) model in two dimensions for LΛ → ∞.


1/N Expansion Sigma Models Solitons Monopoles and Instantons 


Open Access

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Supplementary material

13130_2018_7472_MOESM1_ESM.mp4 (369 kb)
ESM 1 (MP4 368 kb)


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Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Alessandro Betti
    • 1
  • Stefano Bolognesi
    • 2
    • 3
  • Sven Bjarke Gudnason
    • 4
    Email author
  • Kenichi Konishi
    • 2
    • 3
  • Keisuke Ohashi
    • 5
  1. 1.Dipartimento di Ingegneria dell’Informazione e Scienze MatematicheSienaItaly
  2. 2.Department of Physics “E. Fermi”University of PisaPisaItaly
  3. 3.INFN, Sezione di PisaPisaItaly
  4. 4.Institute of Modern PhysicsChinese Academy of SciencesLanzhouChina
  5. 5.Research and Education Center for Natural SciencesKeio UniversityYokohamaJapan

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