Infra-red fixed points at the boundary

Gauged supergravities (in four and five dimensions) with eight supercharges and with vector supermultiplets have a unique ultra-violet (UV) fixed point on a given physical domain M of the space of the scalar fields. We show that in these models the infra-red (IR) fixed points are located on the boundary of M, where the space-time metric becomes singular. e-mail: [email protected] e-mail: [email protected] Over the past year a lot of attention has been given to elucidate, via AdS/CFT correspondence, the renormalization group (RG) flow of strongly coupled gauge theory in terms of static domain wall solutions in anti-deSitter (AdS) supergravity theories [1, 2, 3, 4]. In particular, the kink solutions of scalar-fields interpolating between the (supersymmetric AdS) extrema of gauged supergravity potentials provide a frame-work to address the aspects of RG flows on the dual gauge theory. (The first examples of supersymmetric domain wall solutions was found in the context of D=4 N=1 supergravity theory [5] and reviewed in [6].) In gauged supergravity theories the scalar potential is of a restricted type, in particular, most of the supersymmetric extrema are maxima of the potential and thus in general it is difficult to obtain scalar kink solutions which at short space-time distances asymptote to the Cauchy horizon of the AdS space-time and which would correspond to infra-red (IR) fixed points. For this latter type of behavior in the IR the potential exhibits the second supersymmetric extremum, which is a saddle point or a minimum. Nevertheless an example of this type has been discussed in [2] for non-Abelian gauged supergravity, associated with the massless supermultiplets of Type IIB supergravity compactified on a five-sphere S (D=5 N=8 gauged supergravity). In [7] another example of this type is due to the scalars of a massive supermultiplet of sphere reductions – those are breathing modes parameterizing the volume of the internal sphere. In general, the maximally supersymmetric extremum of gauged supergravity potentials, which is always a maximum, is responsible for the kink solutions that asymptote at large distances (the ultra-violet (UV) fixed point) to the boundary of the AdS space-time, i.e. the corresponding conformal field theory in the UV is specified at the boundary of AdS space-time. On the other hand, the flows from this UV fixed point generically exhibit a singularity in the IR, i.e. the kink solution approaches at short distances the values of the run-away potential. For such examples within D=5 N=8 gauged supergravity see, e.g., [8, 9]. These latter features of the RG flows seem to be more generic within a set-up of gauged supergravity theories. For N=2 D=5 gauged supergravity with vector supermultiplets only, it has been shown [8, 9] that the potential has only a single extremum for any physical domain M of the moduli space for the scalar fields (φA) and this extremum corresponds to an UV fixed point, where the space-time asymptotes to the AdS boundary. The same holds [9] for N=2 D=4 gauged supergravity with vector supermultiplets, only. On the other hand the kink solution at small distances (in the IR regime) necessarily reaches the singular domain at the boundary of M, where both the scalar potential and the space-time become singular. Flows into such singular regions have been also discussed in [10, 11, 12]. The purpose of this letter is to show that, in spite of the singular nature at the boundary of M, the solutions always exhibit an IR fixed point there, i.e. the β functions vanish there (β ∣∣∣ 0 = 0) and their first derivatives ∂Aβ (φ) ∣∣∣ 0 are positive definite. The ordinary space-time (in D=4,5) also becomes singular at this (small) distance. However, the geodesic distance on M, is infinite. To reiterate, these results hold for D=5 and D=4