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 $U({N}_{\mathrm{C}})$ gauge theory with ${N}_{\mathrm{F}}(\ensuremath{\ge}{N}_{\mathrm{C}})$ fundamental hypermultiplets. We solve the hypermultiplet BPS equation and show that all $1/4$ BPS solutions are generated by an ${N}_{\mathrm{C}}\ifmmode\times\else\texttimes\fi{}{N}_{\mathrm{F}}$ 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.