BPS Operators and Brane Geometries.
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In this thesis we explore the finite N spectrum of BPS operators in four-dimensional supersymmetric conformal field theories (CFT), which have dual AdS gravitational descriptions. In the first part we analyze the spectrum of chiral operators in the free limit of quiver gauge theories. We find explicit counting formulas at finite N for arbitrary quivers, construct an orthogonal basis in the free inner product, and derive the chiral ring structure constants. In order to deal with arbitrarily complicated quivers, we develop convenient diagrammatic techniques: the results are expressed by associating Young diagrams and Littlewood-Richardson coefficients to modifications of the original quiver. We develop the notion of a "quiver character", which is a generalization of the symmetric group character, obeying analogous orthogonality properties. In the second part we analyze how the BPS spectrum changes at weak coupling, focusing on the N = 4 supersymmetric Yang-Mills. We find a formal expression for the complete set of eighth-BPS operators at finite N, and use it to derive corrections to a near-BPS operator. In the third part of this thesis we move on to the strong coupling regime, where the dual gravitational description applies. The BPS spectrum on the gravity side includes D3-branes wrapping arbitrary holomorphic surfaces, a generalization of the spherical giant gravitons. Quantizing this moduli space gives a Hilbert space, which, via duality and nonrenormalization theorems, should map to the space of BPS operators derived in the weak coupling regime. We apply techniques from fuzzy geometry to study this correspondence between D3-brane geometries, quantum states, and BPS operators in field theory
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