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.

Three types of thick branes, i.e., Poincaré, de Sitter and Anti-de Sitter brane are considered. They are realized as the non-singular solutions of the Einstein equations with the non-trivial dilatons and the potentials. The scalar perturbations of these systems are also investigated. We find that the effective potentials of the master equations of the scalar perturbations are positive definite and consequently these systems are stable under the small perturbations.

The inflationary scenario for the brane world driven by the bulk inflaton is proposed. The quantum fluctuations of the inflaton is calculated and compared to those of the conventional 4-dimensional inflationary scenario. It is shown that the deviation of the primordial spectrum of this model from that of the conventional one is too small to be observed even if AdS radius is very large. Hence, it turns out that the inflation caused by the bulk inflaton is viable in the context of brane world cosmology.

We investigate cosmological evolutions of the bulk scalar field φ(t) and the radion d(t) in five-dimensional dilatonic two branes model. The bulk potential for the scalar field is taken as the exponential function V bulk ∝ exp(−2 √ 2bφ), where b is the parameter of the theory. This model includes Randall-Sundrum model (with b = 0) and fivedimensional Hořava-Witten theory (with b = 1). We consider matter on both branes and arbitrary potentials on the branes and in the bulk. These matter and potentials induce the cosmological expansion of the brane as well as the time evolution of the bulk scalar field and the radion. Starting with full five-dimensional equations, we derive four-dimensional effective equations which govern the low-energy dynamics of brane worlds. A correspondent five-dimensional geometry is also obtained. The effective fourdimensional theory on a positive tension brane is described by bi-scalar tensor theory.If the radion is stabilized, the effective theory becomes Brans-Dicke (BD) theory with BD parameter 1/2b 2 . On the other hand, if the scalar field is stabilized, the effective theory becomes scalar-tensor theory with BD parameter 3 2(3b 2 +1)where ϕ is the BD field defined by radion d(t). If we do not introduce the stabilization mechanism for these moduli fields, the acceptable late time cosmology can be realized only if the dilaton coupling b is small (b 2 < 1.6 × 10 −4 ) and the negative tension brane is sufficiently away from the positive tension brane. We also construct several models for inflationary brane worlds driven by potentials on the brane and in the bulk.

We study the tachyon condensation of the DD-brane system with a constant tachyon vev in the context of classical solutions of the Type II supergravity. We find that the general solution with the symmetry ISO(1, p) × SO(9 − p) (the three-parameter solution) includes the extremal black p-brane solution as an appropriate limit of the solution by fixing one of the three parameters (c 1 ). Furthermore, we compare the long distance behavior of the solution with the massless modes of the closed strings from the boundary state of the DD-brane system with a constant tachyon vev. We find that we must fix c 1 to zero and only two parameters are needed to express the tachyon condensation of the DD-brane system. This means that the parameter c 1 does not correspond to the tachyon vev of the DD-brane system.

We investigate a three-dimensional gravitational theory on a noncommutative space which has a cosmological constant term only. We found various kinds of nontrivial solutions by applying a similar technique which was used to seek noncommutative solitons in noncommutative scalar field theories. Some of those solutions correspond to bubbles of spacetimes or represent dimensional reduction. The solution which interpolates G µν = 0 and the Minkowski metric is also found. All solutions we obtained are non-perturbative in the noncommutative parameter θ, therefore they are different from solutions found in other contexts of noncommutative theory of gravity and would have a close relation to quantum gravity.

We consider a closed string field theory with an arbitrary matter current as a source of the closed string field. We find that the source must satisfy a constraint equation as a consequence of the BRST invariance of the theory. We see that it corresponds to the covariant conservation law for the matter current, and the equation of motion together with this constraint equation determines the classical behavior of both the closed string field and the matter. We then consider the boundary state (D-brane) as an example of a source. We see that the ordinary boundary state cannot be a source of the closed string field when the string coupling g turns on. By perturbative expansion, we derive a recursion relation which represents the bulk backreaction and the D-brane recoil. We also make a comment on the rolling tachyon boundary state.

We investigated regular black holes with fuzzy sources in three and four dimensions. The density distributions of such fuzzy sources are inspired by noncommutative geometry and given by Gaussian or generalized Gaussian functions. We utilized mass functions to give a physical interpretation of the horizon formation condition for the black holes. In particular, we investigated three-dimensional BTZ-like black holes and four-dimensional Schwarzschild-like black holes in detail, and found that the number of horizons is related to the space-time dimensions, and the existence of a void in the vicinity of the center of the space-time is significant, rather than noncommutativity. As an application, we considered a three-dimensional black hole with the fuzzy disc which is a disc-shaped region known in the context of noncommutative geometry as a source. We also analyzed a four-dimensional black hole with a source whose density distribution is an extension of the fuzzy disc, and investigated the horizon formation condition for it.