We use holographic techniques to study SU (N c ) super Yang-Mills theory coupled to N f ≪ N c flavours of fundamental matter at finite temperature and baryon density. We focus on four dimensions, for which the dual description consists of N f D7-branes in the background of N c black D3-branes, but our results apply in other dimensions as well. A non-zero chemical potential µ b or baryon number density n b is introduced via a nonvanishing worldvolume gauge field on the D7-branes. Ref.[1] identified a first order phase transition at zero density associated with 'melting' of the mesons. This extends to a line of phase transitions for small n b , which terminates at a critical point at finite n b . Investigation of the D7-branes' thermodynamics reveals that (∂µ b /∂n b ) T < 0 in a small region of the phase diagram, indicating an instability. We comment on a possible new phase which may appear in this region.

We construct a new class of entanglement measures by extending the usual definition of Rényi entropy to include a chemical potential. These charged Rényi entropies measure the degree of entanglement in different charge sectors of the theory and are given by Euclidean path integrals with the insertion of a Wilson line encircling the entangling surface. We compute these entropies for a spherical entangling surface in CFT's with holographic duals, where they are related to entropies of charged black holes with hyperbolic horizons. We also compute charged Rényi entropies in free field theories.ArXiv ePrint: 1310.4180 arXiv:1310.4180v2 [hep-th] An interesting exercise to gain better intuition for this background gauge field (2.15) is to expand the coordinates near the spherical entangling surface: t E = ρ sin θ and r = R + ρ cos θ with ρ R. To leading order in ρ/R, one then finds that the potential reduces to B µ E 2π dθ.

We systematically study gapless topological phases of (semi-)metals and nodal superconductors described by Bloch and Bogoliubov-de Gennes Hamiltonians. Using K-theory, a classification of topologically stable Fermi surfaces in (semi-)metals and nodal lines in superconductors is derived. We discuss a generalized bulk-boundary correspondence that relates the topological features of the Fermi surfaces and superconducting nodal lines to the presence of protected zero-energy states at the boundary of the system. Depending on the case, the boundary states are either linearly dispersing (i.e. Dirac or Majorana states) or dispersionless, forming two-dimensional surface flat bands or one-dimensional arc surface states. We study examples of gapless topological phases in symmetry classes AIII and DIII, focusing in particular on nodal superconductors, such as nodal noncentrosymmetric superconductors. For some cases we explicitly compute the surface spectrum and examine the signatures of the topological boundary states in the surface density of states. We also discuss the robustness of the surface states against disorder.

We construct smooth asymptotically AdS 5 ×S 5 solutions of Type IIB supergravity corresponding to all the half-BPS surface operators in N = 4 SYM. All the parameters labeling a half-BPS surface operator are identified in the corresponding bubbling geometry. We use the supergravity description of surface operators to study the action of the SL(2, Z) duality group of N = 4 SYM on the parameters of the surface operator, and find that it coincides with the recent proposal by Gukov and Witten in the framework of the gauge theory approach to the geometrical Langlands with ramification. We also show that whenever a bubbling geometry becomes singular that the path integral description of the corresponding surface operator also becomes singular. 04/20071 jgomis@perimeterinstitute.ca 2 smatsuura@perimeterinstitute.ca