and a fermion. Supersymmetry implies that the manifold of the hypermultiplet scalars is
a 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 well
below the gauge boson mass) is described by the scalars that have not been devoured by
the 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 scalars
will be described by a smaller hyper-K¨ahler manifold (since supersymmetry is not broken
during 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 isometries
of 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 Higgs
effect in six-dimensional N=1 gauge theory.
The D5-brane we are about to construct is mapped via heterotic/type-I duality to the
NS5-brane of the heterotic theory. The NS5-brane has been constructed [49] as a soliton
of the effective low-energy heterotic action. The non-trivial fields, in the transverse space,
are essentially configurations of axion-dilaton instantons, together with four-dimensional
instantons 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 the
moduli of the instantons. There is a mathematical construction of this moduli space, as
a hyper-K¨
ahler quotient. This leads us to suspect [67] that the interpretation of this con-
struction is a Higgs effect in the six-dimensional world-volume theory. In particular, the
mathematical construction implies that for N coincident NS5-branes, the hyper-K¨ahler
quotient construction implies that an Sp(N) gauge group is completely Higgsed.
For a
single five-brane, the gauge group is Sp(1)
∼
SU(2). Indeed, if the size of the instanton
is not zero, the massless fluctuations of the NS5-brane form hypermultiplets only. When
the size becomes zero, the moduli space has a singularity, which can be interpreted as the
restoration 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 contain
an SU(2) gauge group, while in the case of N five-branes the gauge group is enhanced to
Sp(N), [67].