φ ( r ) = $_{Ar}$− 1 + $_{Br}$− 3 (2.12) ≡ m r + c r 3 (2.13) Let us consolidate these ideas by looking at the example of the gaugino bilinear ¯ λλ . It is a Lorentz scalar of dimension 3: therefore we need a su- pergravity scalar field with mass dimension 1. It is also a ten dimensional representation of the SU(4) R-symmetry. We find such a field in the reduc- tion of type IIB strings on AdS$_{5}$ [55] if we set m 2 = − 3 ‡ in (2.10). Then the solution becomes where we have just redefined the constants A and B . Supergravity fields do not scale under four-dimensional conformal transformations, so m must have dimension 1 and c must have dimension 3, which corresponds with m being a source (i.e. a mass) for ¯ λλ and c being its VEV (or condensate). The careful reader may ask at this point about existence and uniqueness when it comes to operator-field matching. Fortunately that question has been answered in [59]: for every operator there exists a corresponding field which is unique. However, as is the way with existence-uniqueness theories, there is no easy way to match operators with fields: each one has to be done by hand. And in any gauge theory there are an infinite number of operators. When we come to consider duals to QCD, a lot of the work is in deciding which operators of QCD, from an infinite choice, are to be included in the holographic field theory. $^{‡}$AdS has negative curvature, and so we can have m $^{2}$< 0 without faster-than-light travel 43