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 UN C gauge theory with N F N C fundamental hypermultiplets. We solve the hypermultiplet BPS equation and show that all 1=4 BPS solutions are generated by an N C N 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.

SO and USp Kähler and hyper-Kähler quotients and lumps

“…[43,47,48,49]. For instance, monopoles (Yang-Mills instantons) become kinks [50,51] (CP N C −1 -lumps [51,52]) in the effective field theory of a vortex-string. Intriguing is also the flux matching between non-Abelian vortices and non-Abelian monopoles and the applications are very interesting in the connection with non-Abelian duality etc.…”

Nonintegrability of self-dual Yang–Mills–Higgs system

“…We obtain Lagrangians in lower dimensions and their solutions by dimensional reduction of this system. Imposing the BPS condition on them, we obtain the instanton-like equation in 4+1 dimensions [7,9], the monopole-like equation in 3+1 [5]- [8] , the wall web equation in 3+1 [11], the vortex equation in 2+1 [3], [22]- [26], and the domain-wall equation in 1+1 [27,28,26].…”

“…This view was extended to a few other cases in susy gauge theories. Another development is the discovery of 1/4 BPS composites of such soliton-like objects [5]- [11] as well as 1/8 BPS states [12,13], see [14,15] as a review.…”

Semilocal fractional instantons

“…When some scalar fields are coupled to gauge fields, the gauge symmetry is spontaneously broken in the Higgs phase, where instantons cannot exist stably in the bulk as a consequence of the Derrick's scaling argument [2]. Instead, they can exist stably inside a non-Abelian vortex; the low-energy effective theory of a non-Abelian vortex in a U(N ) gauge theory coupled with N fundamental scalar fields is the CP N −1 model in two dimensions [3][4][5][6][7], in which instantons in the bulk exist as lumps (or sigma model instantons) [8][9][10]. Such a composite configuration is a 1/4 BPS state preserving a quarter supersymmetry in supersymmetric theories with eight supercharges [9].…”