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and a fermion. Supersymmetry implies that the manifold of the hypermultiplet scalars isa hyper-K¨ahler manifold. When the hypermultiplets are charged under the gauge group,the gauge transformations are isometries of the hyper-K¨ahler manifold, of a special type:they are compatible with the hyper-K¨ahler structure.It will be important for our latter purposes to describe the Higgs effect in this case.When a gauge theory is in the Higgs phase, the gauge bosons become massive by com-bining with some of the massless Higgs modes. The low-energy theory (for energies wellbelow the gauge boson mass) is described by the scalars that have not been devoured bythe gauge bosons. In our case, each (six-dimensional) gauge boson that becomes massive,will eat-up four scalars (a hypermultiplet). The left over low-energy theory of the scalarswill be described by a smaller hyper-K¨ahler manifold (since supersymmetry is not brokenduring the Higgs phase transition). This manifold is constructed by a mathematical pro-cedure known as the hyper-K¨ahler quotient. The procedure “factors out” the isometriesof a hyper-K¨ahler manifold to produce a lower-dimensional manifold which is still hyper-191
K¨ahler.Thus, the hyper-K¨ahler quotient construction is describing the ordinary Higgseffect in six-dimensional N=1 gauge theory.The D5-brane we are about to construct is mapped via heterotic/type-I duality to theNS5-brane of the heterotic theory. The NS5-brane has been constructed  as a solitonof the effective low-energy heterotic action. The non-trivial fields, in the transverse space,are essentially configurations of axion-dilaton instantons, together with four-dimensionalinstantons embedded in the O(32) gauge group.Such instantons have a size that de-termines the “thickness” of the NS5-brane. The massless fluctuations are essentially themoduli of the instantons. There is a mathematical construction of this moduli space, asa hyper-K¨ahler quotient. This leads us to suspect  that the interpretation of this con-struction is a Higgs effect in the six-dimensional world-volume theory. In particular, themathematical construction implies that for N coincident NS5-branes, the hyper-K¨ahlerquotient construction implies that an Sp(N) gauge group is completely Higgsed.For asingle five-brane, the gauge group is Sp(1)∼SU(2). Indeed, if the size of the instantonis not zero, the massless fluctuations of the NS5-brane form hypermultiplets only. Whenthe size becomes zero, the moduli space has a singularity, which can be interpreted as therestoration of the gauge symmetry: at this point the gauge bosons become massless again.All of this indicates that the world-volume theory of a single five-brane should containan SU(2) gauge group, while in the case of N five-branes the gauge group is enhanced toSp(N), .