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

In this paper we study surface operators in N = 4 supersymmetric Yang-Mills theory.We compute surface operator observables, such as the expectation value of surface operators, the correlation function of surface operators with local operators, and the correlation function of surface operators with Wilson and 't Hooft loops. The calculations are performed using three different realizations of surface operators, corresponding respectively to the gauge theory path integral definition, the probe brane description in AdS 5 ×S 5 and the "bubbling" supergravity description of surface operators. We find remarkable agreement between the different calculations performed using the three different realizations. 05/2008a drukker@physik.hu-berlin.de b jgomis@perimeterinstitute.ca c smatsuura@perimeterinstitute.ca

Recently holographic techniques have been used to study the thermal properties of N = 2 super-Yang-Mills theory, with gauge group SU (N c ) and coupled to N f ≪ N c flavours of fundamental matter, at large N c and large 't Hooft coupling. Here we consider the phase diagram as a function of temperature and baryon chemical potential µ b . For fixed µ b < N c M q there is a line of first order thermal phase transitions separating a region with vanishing baryon density and one with nonzero density. For fixed µ b > N c M q there is no phase transition as a function of the temperature and the baryon density is always nonzero. We also compare the present results for the grand canonical ensemble with those for canonical ensemble in which the baryon density is held fixed [1].

We develop an approach based on edge theories to calculate the entanglement entropy and related quantities in (2+1)-dimensional topologically ordered phases. Our approach is complementary to, e.g., the existing methods using replica trick and Witten's method of surgery, and applies to a generic spatial manifold of genus g, which can be bipartitioned in an arbitrary way. The effects of fusion and braiding of Wilson lines can be also straightforwardly studied within our framework. By considering a generic superposition of states with different Wilson line configurations, through an interference effect, we can detect, by the entanglement entropy, the topological data of Chern-Simons theories, e.g., the R-symbols, monodromy and topological spins of quasiparticles. Furthermore, by using our method, we calculate other entanglement/correlation measures such as the mutual information and the entanglement negativity. In particular, it is found that the entanglement negativity of two adjacent non-contractible regions on a torus provides a simple way to distinguish Abelian and non-Abelian topological orders. CONTENTS

We consider Rényi entropies S n = 1 1−n log Tr ρ n of conformal field theories in flat space, with the entangling surface being a sphere. The AdS/CFT correspondence relates this Rényi entropy to that of a black hole with hyperbolic horizon; as the Rényi parameter n increases the temperature of the black hole decreases. If the CFT possesses a sufficiently low dimension scalar operator the black hole will be unstable to the development of hair. Thus, as n is varied, the Rényi entropies will exhibit a phase transition at a critical value of n. The location of the phase transition, along with the spectrum of the reduced density matrix ρ, depends on the dimension of the lowest dimension non-trivial scalar operator in the theory.