Semilocal fractional instantons

Minoru Eto,Muneto Nitta

doi:10.1007/jhep03(2016)067

Abstract

We find semi-local fractional instantons of codimension four in Abelian and non-Abelian gauge theories coupled with scalar fields and the corresponding ${\mathbb C}P^{N-1}$ and Grassmann sigma models at strong gauge coupling. They are 1/4 BPS states in supersymmetric theories with eight supercharges, carry fractional (half) instanton charges characterized by the fourth homotopy group $\pi_4 (G/H)$, and have divergent energy in infinite spaces. We construct exact solutions for the sigma models and numerical solutions for the gauge theories. Small instanton singularity in sigma models is resolved at finite gauge coupling (for the Abelian gauge theory). Instantons in Abelian and non-Abelian gauge theories have negative and positive instantons charges, respectively, which are related by the Seiberg-like duality that changes the sign of the instanton charge.

Highlights

In two dimensional sigma models, instantons are lumps characterized by the second homotopy group [14], which is, for the CP N−1 model, π2(CP N−1) = π2
We find semi-local fractional instantons of codimension four in Abelian and non-Abelian gauge theories coupled with scalar fields and the corresponding CP N−1 and Grassmann sigma models at strong gauge coupling
We construct BPS instantons of codimension four that solve a set of 1/4 BPS equations in a U(1) or U(N ) gauge theory coupled with scalar fields in four dimensions that reduces to the CP N−1 model or Grassmann sigma model

Summary

Single spherical solution

We consider U(2) gauge theory with NF = 3 flavors. In the strong gauge coupling limit, the master equation is again exactly solved by. Consider SU(2)C invariant quantities, the determinants of 2 by 2 submatrices taking i and j-th column from H, are given by det H12 =. This is exactly identical to the solution in the Abelian theory given in eq (3.6). From these we can calculate the instanton density. Instanton charge densities g2I are shown with cg2 = 1, ∞ for a = 1

Multiple instantons

Seiberg-like duality

Summary and discussion

A Topological charges in Seiberg-like duality