We construct a canonical transformation that takes the usual Yang-Mills action into one whose Feynman diagram expansion generates the MHV rules. The off-shell continuation appears as a natural consequence of using light-front quantisation surfaces. The construction extends to include massless fermions.

We demonstrate that the canonical change of variables that yields the MHV lagrangian, also provides contributions to scattering amplitudes that evade the equivalence theorem. This 'ET evasion' in particular provides the tree-level (−++) amplitude, which is non-vanishing off shell, or on shell with complex momenta or in (2, 2) signature, and is missing from the MHV (a.k.a. CSW) rules. At one loop there are ET-evading diagrammatic contributions to the amplitudes with all positive helicities. We supply the necessary regularisation in order to define these contributions (and quantum MHV methods in general) by starting from the light-cone Yang-Mills lagrangian in D dimensions and making a canonical change of variables for all D − 2 transverse degrees of freedom of the gauge field. In this way, we obtain dimensionally regularised three-and four-point MHV amplitudes. Returning to the one-loop (++++) amplitude, we demonstrate that its quadruple cut coincides with the known result, and show how the original light-cone Yang-Mills contributions can in fact be algebraically recovered from the ET-evading contributions. We conclude that the canonical MHV lagrangian, supplemented with the extra terms brought to correlation functions by the non-linear field transformation, provide contributions which are just a rearrangement of those from light-cone Yang-Mills and thus coincide with them both on and off shell.

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We test the Maldacena conjecture for type IIB String Theory/ N = 4 Yang-Mills by calculating the one-loop corrections in the bulk theory to the Weyl anomaly of the boundary CFT when the latter is coupled to a Ricci flat metric. The contributions cancel within each supermultiplet, in agreement with the conjecture.

We give a complete account of the Schrödinger representation approach to the calculation of the Weyl anomaly of N = 4 SYM from the AdS/CFT correspondence. On the AdS side, the 1/N 2 correction to the leading order result receives contributions from all the fields of Type IIB Supergravity, the contribution of each field being given by a universal formula. The correct matching with the CFT result is thus a highly non-trivial test of the correspondence.