M-theory is currently our best candidate for a theory of everything, but remains mysterious. We know M-theory has M2- and M5-branes. The low-energy theory on a stack of coincident M2-branes is well-understood: it is maximally supersymmetric Chern-Simons-matter theory. However, the low-energy theory on a stack of coincident M5-branes remains poorly-understood: it is a maximally supersymmetric theory of self-dual strings with zero tension. In this talk I will discuss one type of probe of the M5-brane theory, namely self-dual strings with infinite tension. These play a role analogous to Wilson lines in gauge theories, but are two-dimensional surfaces rather than lines, and hence are called Wilson surfaces. I will describe holographic calculations of entanglement entropy associated with these infinite-tension self-dual strings, from which we extract a key parameter characterizing them, their central charge. This provides a count of the number of massless degrees of freedom living on them, and thus may shed light on some of the fundamental degrees of freedom of M-theory.

I will present a proof for a monotonicity theorem, or c-theorem, for a three-dimensional Conformal Field Theory (CFT) on a space with a boundary, and for a higher-dimensional CFT with a two-dimensional defect. The proof is applicable only to renormalization group flows that preserve locality, reflection positivity, and Euclidean invariance along the boundary or defect, and that are localized at the boundary or defect, such that the bulk theory remains conformal along the flow. The method of proof is a generalization of Komargodski��������s proof of Zamolodchikov��������s c-theorem. The key ingredient is an external �������dilaton������� field introduced to match Weyl anomalies between the ultra-violet (UV) and infra-red (IR) fixed points. Reflection positivity in the dilaton��������s effective action guarantees that a certain coefficient in the boundary/defect Weyl anomaly must take a value in the UV that is larger than (or equal to) the value in the IR. This boundary/defect c-theorem may have important implications for many theoretical and experimental systems, ranging from graphene to branes in string theory and M-theory.

In this talk we consider BPS states in 4D, N=2 gauge theory in the presence of defects. We give a semiclassical description of these `framed BPS states' in terms of kernels of Dirac operators on moduli spaces of singular monopoles. For both framed and ordinary BPS states we present a conjectural map between the data of the semiclassical construction and the data of the low-energy, quantum-exact Seiberg-Witten description. This map incorporates both perturbative and nonperturbative field theory corrections to the supersymmetric quantum mechanics of the monopole collective coordinates. We use it to translate recent developments in the study of N=2 theories, including wall-crossing formulae and the no-exotics theorem, into geometric statements about the Dirac kernels. The no-exotics theorem implies a broad generalization of Sen's conjecture concerning the existence of L^2 harmonic forms on monopole moduli space. This talk is based on work done in collaboration with Greg Moore and Dieter Van den Bleeken.

In this talk we consider BPS states in 4D, N=2 gauge theory in the presence of defects. We give a semiclassical description of these `framed BPS states' in terms of kernels of Dirac operators on moduli spaces of singular monopoles. For both framed and ordinary BPS states we present a conjectural map between the data of the semiclassical construction and the data of the low-energy, quantum-exact Seiberg-Witten description. This map incorporates both perturbative and nonperturbative field theory corrections to the supersymmetric quantum mechanics of the monopole collective coordinates. We use it to translate recent developments in the study of N=2 theories, including wall-crossing formulae and the no-exotics theorem, into geometric statements about the Dirac kernels. The no-exotics theorem implies a broad generalization of Sen's conjecture concerning the existence of L^2 harmonic forms on monopole moduli space. This talk is based on work done in collaboration with Greg Moore and Dieter Van den Bleeken.

The Kondo effect occurs in metals doped with magnetic impurities: in the ground state the electrons form a screening cloud around each impurity, leading to dramatic changes in the thermodynamic and transport properties of the metal. Although the single-impurity Kondo effect is considered a solved problem, many questions remain, especially about the fate of the Kondo effect in the presence of multiple impurities. In particular, for a sufficiently dense concentration of impurities, a competition between the Kondo effect and inter-impurity interactions can lead to quantum criticality and non-Fermi liquid behavior, which remains poorly understood. In this talk I will present a model of the single-impurity Kondo effect based on holography, also known as gauge-gravity duality or the AdS/CFT correspondence, which may serve as a foundation for a new approach to the multiple-impurity system.

Gauge-gravity duality is an extremely useful tool for studying strongly-coupled gauge theories, and has many applications to real-world systems, such as the quark-gluon plasma and quantum critical points. Most gauge-gravity dualities involve a gauge theory with fields only in the adjoint representation of the gauge group. In many strongly-coupled systems, however, such as the quark-gluon plamsa, fields in the fundamental representation of the gauge group, 'flavor fields', are crucially important. For (3+1)-dimensional gauge theories with gravity duals (AdS5/CFT4), the supergravity description of flavor fields is well-understood: flavor fields appear in supergravity as probe D-branes in AdS5. The story for (2+1)-dimensional gauge theories (AdS4/CFT3) is much less developed. Indeed, for supergravity on AdS4 x S7, the dual (2+1)-dimensional field theory (without flavor) was only recently discovered. In this talk I will describe how to add flavor to this theory. In particular, I will present a general recipe to determine the field theory, and in particular the couplings of the flavor fields, given a probe D-brane in AdS4.