Recently, Berenstein et al. have proposed a duality between a sector of N = 4 super-Yang-Mills theory with large R-charge J, and string theory in a pp-wave background. In the limit considered, the effective 't Hooft coupling has been argued to be λWe study Yang-Mills theory at small λ ′ (large µ) with a view to reproducing string interactions. We demonstrate that the effective genus counting parameter of the Yang-Mills theory is g4 , the effective two-dimensional Newton constant for strings propagating on the pp-wave background. We identify g 2 √ λ ′ as the effective coupling between a wide class of excited string states on the pp-wave background. We compute the anomalous dimensions of BMN operators at first order in g 2 2 and λ ′ and interpret our result as the genus one mass renormalization of the corresponding string state. We postulate a relation between the three-string vertex function and the gauge theory three-point function and compare our proposal to string field theory. We utilize this proposal, together with quantum mechanical perturbation theory, to recompute the genus one energy shift of string states, and find precise agreement with our gauge theory computation.

We examine noncommutative solutions of the non-Abelian theory on the world-volume of N coincident D-strings. These solutions can be interpreted in terms of noncommutative geometry as funnels describing the non-Abelian D-string expanding out into an orthogonal D3-brane. These configurations are ''dual'' to the bion solutions in the Abelian world-volume theory of the D3-brane. In the latter, a charge N magnetic monopole describes N D-strings attached to the D3-brane with a spike deformation of the world volume. The noncommutative D-string solutions give a reliable account of physics at the core of the monopole, where the bion description is expected to break down. In the large N limit, we find good agreement between the two points of view, including the energy, couplings to background fields, and the shape of the funnel. We also study fluctuations traveling along the D-string, again obtaining agreement in the large N limit. At finite N, our results give a limit on the number of modes that can travel to infinity along the N D-strings attached to the D3-brane.

We study new solutions of the low-energy equations of motion for the non-abelian D-string. We find a "fuzzy funnel" solution consisting of a noncommutative four-sphere geometry which expands along the length of the D-string. We show that this funnel solution has an interpretation as D-strings ending on a set of orthogonal D5-branes. Although not supersymmetric, the system appears to be stable within this framework. We also give a dual description of this configuration as a bion spike in the non-abelian world volume theory of coincident D5-branes.

We study a defect conformal field theory describing D3-branes intersecting over two space-time dimensions. This theory admits an exact Lagrangian description which includes both two-and four-dimensional degrees of freedom, has (4,4) supersymmetry and is invariant under global conformal transformations. Both two-and four-dimensional contributions to the action are conveniently obtained in a two-dimensional (2,2) superspace. In a suitable limit, the theory has a dual description in terms of a probe D3-brane wrapping an AdS 3 ϫS 1 slice of AdS 5 ϫS 5 . We consider the AdS/CFT dictionary for this setup. In particular we find classical probe fluctuations corresponding to the holomorphic curve wyϭc␣Ј. These fluctuations are dual to defect fields containing massless two-dimensional scalars which parametrize the classical Higgs branch, but do not correspond to states in the Hilbert space of the CFT. We also identify probe fluctuations which are dual to BPS superconformal primary operators and to their descendants. A nonrenormalization theorem is conjectured for the correlators of these operators, and verified to order g 2 .