Instantons in the Higgs Phase

Abstract

When instantons are put into the Higgs phase, vortices are attached to instantons. We construct such composite solitons as 1/4 BPS states in five-dimensional supersymmetric U(Nc) gauge theory with Nf(>=Nc) fundamental hypermultiplets. We solve the hypermultiplet BPS equation and show that all 1/4 BPS solutions are generated by an Nc x Nf matrix which is holomorphic in two complex variables, assuming the vector multiplet BPS equation does not give additional moduli. We determine the total moduli space formed by topological sectors patched together and work out the multi-instanton solution inside a single vortex with complete moduli. Small instanton singularities are interpreted as small sigma-model lump singularities inside the vortex. The relation between monopoles and instantons in the Higgs phase is also clarified as limits of calorons in the Higgs phase. Another type of instantons stuck at an intersection of two vortices and dyonic instantons in the Higgs phase are also discussed.

Figures (3)

• Figure 1: BPS states and their relations. Relations discussed in this paper are denoted by solid lines with arrows and the one not discussed by a dashed line with an arrow. The upper triangle ($\triangle$ ABC) describes the $d=4+1$ gauge theory with massless hypermultiplets, and the lower one ($\triangle$ A$'$B$'$C$'$) the $d=3+1$ gauge theory with massive hypermultiplets. The theories A and A$'$ contain eight supercharges, 1/2 BPS vortices and their effective theories B and B$'$ four supercharges and the 1/4 BPS composite states C and C$'$ two supercharges. The lower theory A$'$ is obtained from the upper one A by the Scherk-Schwarz dimensional reduction (A $\to$ A$'$) preserving SUSY. The effective theory on a single vortex in $d=4+1$ (B) or $d=3+1$ (B$'$) is the SUSY ${\bf C}P^{N-1}$ model without or with a potential, respectively. Here the latter B$'$ coincides with the one obtained from the former B by the Scherk-Schwarz dimensional reduction (B $\to$ B$'$) preserving SUSY. Instantons in C and monopoles C$'$ attached by vortices as $1/4$ BPS states can be interpreted as $1/2$ BPS lumps (B$\to$C) and kinks (B$' \to$C$'$) in the effective theories on the vortex in $d=4+1$ and $d=3+1$, respectively. In $d=4+1$ calorons interpolates between instantons within a vortex and a monopole-string in the Higgs phase (see C). The monopole-string in C in $d=4+1$ can be dimensionally reduced to a monopole C$'$ in the Higgs phase in $d=3+1$.
• Figure 2: Single instanton in the Higgs phase. The size of the vortex is given by $L_{\rm v} \sim 1/g\sqrt c$.
• Figure 3: Energy density of the calorons in terms of the vortex theory.