Supersymmetry implies that the manifold of the hypermultiplet scalars is a

# Supersymmetry implies that the manifold of the

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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
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].