BPS Operators and Brane Geometries.

PhDIn 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 theoryQueen Mary, University of London studentshi

Invariants of Toric Seiberg Duality

Three-branes at a given toric Calabi-Yau singularity lead to different phases of the conformal field theory related by toric (Seiberg) duality. Using the dimer model/brane tiling description in terms of bipartite graphs on a torus, we find a new invariant under Seiberg duality, namely the Klein j-invariant of the complex structure parameter in the distinguished isoradial embedding of the dimer, determined by the physical R-charges. Additional number theoretic invariants are described in terms of the algebraic number field of the R-charges. We also give a new compact description of the a-maximization procedure by introducing a generalized incidence matrix.Comment: 43 pages, 8 figures, LaTe

On the Classification of Brane Tilings

We present a computationally efficient algorithm that can be used to generate all possible brane tilings. Brane tilings represent the largest class of superconformal theories with known AdS duals in 3+1 and also 2+1 dimensions and have proved useful for describing the physics of both D3 branes and also M2 branes probing Calabi-Yau singularities. This algorithm has been implemented and is used to generate all possible brane tilings with at most 6 superpotential terms, including consistent and inconsistent brane tilings. The collection of inconsistent tilings found in this work form the most comprehensive study of such objects to date.Comment: 33 pages, 12 figures, 15 table

The Beta Ansatz: A Tale of Two Complex Structures

Brane tilings, sometimes called dimer models, are a class of bipartite graphs on a torus which encode the gauge theory data of four-dimensional SCFTs dual to D3-branes probing toric Calabi-Yau threefolds. An efficient way of encoding this information exploits the theory of dessin d’enfants, expressing the structure in terms of a permutation triple, which is in turn related to a Belyi pair, namely a holomorphic map from a torus to a P1 with three marked points. The procedure of a-maximization, in the context of isoradial embeddings of the dimer, also associates a complex structure to the torus, determined by the R-charges in the SCFT, which can be compared with the Belyi complex structure. Algorithms for the explicit construction of the Belyi pairs are described in detail. In the case of orbifolds, these algorithms are related to the construction of covers of elliptic curves, which exploits the properties of Weierstraß elliptic functions. We present a counter example to a previous conjecture identifying the complex structure of the Belyi curve to the complex structure associated with R-charges

From counting to construction of BPS states in N=4 SYM

We describe a universal element in the group algebra of symmetric groups, whose characters provides the counting of quarter and eighth BPS states at weak coupling in N=4 SYM, refined according to representations of the global symmetry group. A related projector acting on the Hilbert space of the free theory is used to construct the matrix of two-point functions of the states annihilated by the one-loop dilatation operator, at finite N or in the large N limit. The matrix is given simply in terms of Clebsch-Gordan coefficients of symmetric groups and dimensions of U(N) representations. It is expected, by non-renormalization theorems, to contain observables at strong coupling. Using the stringy exclusion principle, we interpret a class of its eigenvalues and eigenvectors in terms of giant gravitons. We also give a formula for the action of the one-loop dilatation operator on the orthogonal basis of the free theory, which is manifestly covariant under the global symmetry.Comment: 41 pages + Appendices, 4 figures; v2 - refs and acknowledgments adde

Quivers as calculators: counting, correlators and Riemann surfaces

86 figures (84 pages + Appendices