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On the moduli space of semilocal strings and lumps

Minoru Eto, Jarah Evslin, Kenichi Konishi, Giacomo Marmorini, Muneto Nitta, Keisuke Ohashi, Walter Vinci, Naoto Yokoi

Abstract

We study BPS non-abelian semilocal vortices in U(Nc) gauge theory with Nf flavors, Nf > Nc, in the Higgs phase. The moduli space for arbitrary winding number is described using the moduli matrix formalism. We find a relation between the moduli spaces of the semilocal vortices in a Seiberg-like dual pairs of theories, U(Nc) and U(Nf-Nc). They are two alternative regularizations of a "parent" non-Hausdorff space, which tend to the same moduli space of sigma-model lumps in the infinite gauge coupling limits. We examine the normalizability of the zero-modes and find the somewhat surprising phenomenon that the number of normalizable zero-modes, dynamical fields in the effective action, depends on the point of the moduli space we are considering. We find, in the lump limit, an effective action on the vortex worldsheet, which we compare to that found by Shifman and Yung.

On the moduli space of semilocal strings and lumps

Abstract

We study BPS non-abelian semilocal vortices in U(Nc) gauge theory with Nf flavors, Nf > Nc, in the Higgs phase. The moduli space for arbitrary winding number is described using the moduli matrix formalism. We find a relation between the moduli spaces of the semilocal vortices in a Seiberg-like dual pairs of theories, U(Nc) and U(Nf-Nc). They are two alternative regularizations of a "parent" non-Hausdorff space, which tend to the same moduli space of sigma-model lumps in the infinite gauge coupling limits. We examine the normalizability of the zero-modes and find the somewhat surprising phenomenon that the number of normalizable zero-modes, dynamical fields in the effective action, depends on the point of the moduli space we are considering. We find, in the lump limit, an effective action on the vortex worldsheet, which we compare to that found by Shifman and Yung.

Paper Structure

This paper contains 30 sections, 182 equations, 1 figure.

Table of Contents

  1. Introduction and discussion
  2. The $U(N_{\rm C})$ model and the moduli matrix
  3. Moduli spaces and quotients
  4. The lump limit
  5. Some examples
  6. Fundamental semilocal vortices and lumps
  7. Multiple semilocal vortices and lumps
  8. Normalizability of zero-modes and the effective action
  9. Non-normalizable modes
  10. Examples of enhancement of normalizable modes
  11. Abelian case
  12. Non-abelian cases
  13. Dynamics of the effective $k=1$ vortex theory
  14. Duality and symmetry breaking
  15. Relation to the action of Shifman and Yung
  16. ...and 15 more sections

Figures (1)

  • Figure 2: (a) The red dots are numerical computations of $F(|a|^2)$, while the blue line is an interpolation done with the function shown in (\ref{['interpolation']}).(b) The red line is the Kähler metric $\partial_{a, \bar{a}}K_{\rm abelian~semilocal}(z_0,a)$ for an abelian semilocal vortex at finite gauge coupling and is obtained from the interpolation function in (a), while the blue dashed line is the metric in the infinite gauge coupling limit (lump limit). The cut-off has been set to a very big value, $L=10^3$, and $g=\xi=1$.