A class of correlation functions of half-BPS composite operators are computed exactly ( at finite JV ) in the zero coupling limit of iV = 4 SYM theory. These have a simple dependence on the fourdimensional spacetime coordinates and are related to correlators in a one-dimensional Matrix Model with complex Matrices obtained by dimensional reduction of iV = 4 SYM on a three-sphere. A key technical tool is Frobenius-Schur duality between symmetric and Unitary groups and the results are expressed simply in terms of U(N) group integrals or equivalently in terms of Littlewood-Richardson coefficients. These correlation functions are used to understand the existence/properties of giant gravitons and related solutions in the string theory dual on AdSs x 5 5 . Some of their properties hint at integrability in N = 4 SYM.

These are expository lectures reviewing (1) recent developments in two-dimensional YangMills theory and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinite-dimensional differential geometry.We emphasize the unifying role of equivariant cohomology both as the underlying principle in the formulation of BRST transformation laws and as a central concept in the geometrical interpretation of topological field theory path integrals.

We construct spherical harmonics for fuzzy spheres of even and odd dimensions, generalizing the correspondence between finite matrix algebras and fuzzy two-spheres. The finite matrix algebras associated with the various fuzzy spheres have a natural basis which falls in correspondence with tensor constructions of irreducible representations of orthogonal groups SO(n). This basis is useful in describing fluctuations around various D-brane constructions of fuzzy spherical objects. The higher fuzzy spheres are non-associative algebras that appear as projections of associative algebras related to Matrices. The non-associativity (as well as the non-commutativity) disappears in the leading large $N$ limit, ensuring the correct classical limit. Some simple aspects of the combinatorics of the fuzzy four-sphere can be accounted by a heuristic picture of giant fractional instantons.Comment: 33 pages (Harvmac big), 1 figure v2: added footnote clarifying radius of fuzzy odd spheres, v3 : fixed minor typo in eq. 6.

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We present a complete basis of multi-trace multi-matrix operators that has a diagonal two point function for the free matrix field theory at finite N. This generalises to multiple matrices the single matrix diagonalisation by Schur polynomials. Crucially, it involves intertwining the gauge group U(N) and the global symmetry group U(M) with Clebsch-Gordan coefficients of symmetric groups S n . When applied to N = 4 super Yang-Mills we consider the U(3) subgroup of the full symmetry group. The diagonalisation allows the description of a dual basis to multi-traces, which permits the characterisation of the metric on operators transforming in short representations at weak coupling. This gives a framework for the comparison of quarter and eighth-BPS giant gravitons of AdS 5 × S 5 spacetime to gauge invariant operators of the dual N = 4 SYM.

We propose gauge theory operators built using a complex Matrix scalar which are dual to brane-anti-brane systems in AdS 5 × S 5 , in the zero coupling limit of the dual Yang-Mills. The branes involved are half-BPS giant gravitons. The proposed operators dual to giant-anti-giant configurations satisfy the appropriate orthogonality properties. Projection operators in Brauer algebras are used to construct the relevant multi-trace Matrix operators. These are related to the "coupled representations" which appear in 2D Yang-Mills theory. We discuss the implications of these results for the quantum mechanics of a complex matrix model, the counting of non-supersymmetric operators and the physics of brane-anti-brane systems. The stringy exclusion principle known from the properties of half-BPS giant gravitons, has a new incarnation in this context. It involves a qualitative change in the map between brane-anti-brane states to gauge theory operators. In the case of a pair of sphere giant and anti-giant this change occurs when the sum of the magnitudes of their angular momenta reaches N.

We describe a topological string theory which reproduces many aspects of the 1/N expansion of SU (N ) Yang-Mills theory in two spacetime dimensions in the zero coupling (A = 0) limit. The string theory is a modified version of topological gravity coupled to a topological sigma model with spacetime as target. The derivation of the string theory relies on a new interpretation of Gross and Taylor's "Ω −1 points." We describe how inclusion of the area, coupling of chiral sectors, and Wilson loop expectation values can be incorporated in the topological string approach.

We formulate simple graphical rules which allow explicit calculation of nonperturbative c = 1 S-matrices. This allows us to investigate the constraint of nonperturbative unitarity, which indeed rules out some theories. Nevertheless, we show that there is an infinite parameter family of nonperturbatively unitary c = 1 S-matrices. We investigate the dependence of the S-matrix on one of these nonperturbative parameters. In particular, we study the analytic structure, background dependence, and high-energy behavior of some nonperturbative c = 1 S-matrices. The scattering amplitudes display interesting resonant behavior both at high energies and in the complex energy plane.