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Duality, Dimensions, and Reduction on the Lattice

Montonen and Olive found evidence that a duality could exist in Yang-Mills with adjoint scalars.  In this scheme, the 't Hooft-Polyakov monopole forms a gauge triplet with the photon, leading to a theory equivalent to the Georgi-Glashow model but with magnetic charge replacing electric charge.  The duality is believed to be realized in N=4 super-Yang-Mills.  We are pursuing numerical, nonperturbative evidence for this S-duality using our lattice formulation. Two lines of approach are being taken, which I will discuss.  First, we attempt to show that there is a value of the gauge coupling for which the W boson mass is equal to the monopole mass. Second, we are relating the 't Hooft loop to the Wilson loop at this self-dual coupling. On a somewhat unrelated topic, we also discuss the determination of anomalous dimensions on the lattice.  In the dual gravitational picture these correspond to masses of fields in the bulk, so that some aspects of the gauge-gravity duality could be tested by such determinations.  In particular in N=4 super-Yang-Mills there are predictions for the dimensions of non-protected operators at the self-dual point, based on the superconformal bootstrap.