• complex projective model
• monopole-vortex-monopole complex
• quantum field theory

One of the more promising ideas that have been proposed to explain the still openproblem of quark confinement in QCD is that of the dual Meissner effect.
Originally it was proposed by ’t Hooft and Mandelstam [see Nucl. Phys. B 190, 455 (1981); Phys. Lett. 53B, 476 (1975); Phys. Rep. C 23, 245 (1976)], that the ground state of QCD might behave as a dual superconductor with chromoelectric vortices confining the quarks, exactly as the magnetic Abrikosov-Nielsen-Olesen (ANO) vortex confines poles of a magnet in a type ii superconductor. This was basically an Abelian mechanism.
Such an Abelian scenario has never been proven neither by direct physical observations nor by many years of lattice simulation studies. There are also certain difficulties associated with such an Abelian mechanism of confinement. An interesting possible way out is that an essentially nonAbelian mechanism of confinement is at work in QCD. In other words, nonAbelian degrees of freedom may play the central role, within the context of dual superconductor mechanism of confinement, i.e. condensation of certain (nonAbelian) magnetic degrees of freedom. Consequently the “confining string” is a nonAbelian Chromoelectric vortex. This kind of vortices in a Yang-Mills theory with SU(N) gauge group have an internal orientational moduli that is CP(N−1); moreover the low-energy dynamics of the orientational moduli parameters is described by a quantum 2D CP(N−1) theory. This gives us the motivation for a careful study of such a model.
The thesis is organized in three chapters. In the first one, after a general introduction to the problem, we go through some mathematical preliminaries in preparation of the second part of the chapter in which we review the basics of soliton theory. The chapter ends with a section that explains how a theory that posses a nonAbelian vortex confined by nonAbelian monopoles leads to the study of CP(N−1) defined on a spatially bounded spacetime worldstrip.
The second chapter gives an introduction and a review of one of the tools that we will use in the study of the model: The large N expansion. Topological discussions involving the Euler characteristic are formalized using the mathematical concept of CW complex. Such technique is applied to a couple of examples and eventually to the resolution of the CP(N−1) model on the plane (as it has been done by Witten [Nucl. Phys. B 149 (1979)]). Essential properties such as asymptotic freedom and confinement are discussed.
The main part of the work is contained in chapter three in which the complex projective model is studied on a spacetime worldstrip. First of all the effective action and the full gap equations are derived following the recent article by Bolognesi Konishi and Ohashi [ arXiv:1604.05630 [hep-th]]; the problem of boundary conditions is addressed and discussed in the case of Dirichlet, Neumann boundary conditions and in the case of periodic conditions. Then the gap equations are analyzed and it is shown that translationally invariant solutions are compatible only with periodic conditions. The numerical analysis in the case of general boundary conditions is discussed.
In the last section of the chapter we study the energy of the model. We recover the result of Monin, Shifman and Yung [see Phys. Rev. D 92 (2015)] in the case of periodic boundary condition. The energy density for a theory with general boundary conditions is derived and we subsequently prove that it is finite after an appropriate subtraction. We manage to show that from the expression derived for the energy density it is indeed possbile to recover the generalized gap equations, providing us with an important consistency check.
Finally it is proven that the energy density is actually constant and we end with some considerations on the large L behavior of the energy, where L is the spatial width of the worldstrip.