規範/重力對應之探討

Abstract

In this work, we consider the various applications of the gauge/gravity correspondence which is a powerful tool to study the strongly-coupled gauge theory. The topics we cover include the applications of the gauge/gravity correspondence on QCD, non-relativistic conformal field theory and the hydrodynamics. At first, we consider the introduction of topological charged membranes in an interesting holographic QCD model, namely Sakai-Sugimoto model. This study is motivated by the lattice simulations of QCD in which people find there are some topological domain wall structures. We find there are some thermodynamic favored and stable phases with the topological charged membrane structures. We find a crossover phase with the limiting baryonic current density and temperature which suggest a Hagedorn-like phase transition of meson dissociation. In addition to the crossover phase, we also find a rich phase structure. Next, we provide an alternative framework of non-relativistic holography. The motivation of studying the non-relativistic holography is to study the strongly-coupled fermionic systems at unitarity. The conventional approach of studying non-relativistic holographic is based on the solution generating technique, the so-called Null Melvin Twist. People obtain the asymptotic Schr¨odinger symmetry by deforming the usual AdS space. So the dual field theory only has the Schr¨odinger symmetry in the UV, but the same relativistic conformal symmetry in the IR. So it is not clear how the Schr¨odinger symmetry can be realized explicitly as the RG flow runs down toward IR. We try to preserve the Schr¨odinger symmetry from the UV to IR by using the so-called Bargmann framework which lift the Newton-Cartan gravity to the one-dimensional higher Einstein gravity. This framework naturally explains why the non-relativistic holography is co-dimension two. We also try to construct the black hole solution dual to the thermal non-relativistic CFT. However, due to a no-go theorem which presents that there is no regular horizon in the bulk if there is a covariant constant null-like Killing vector in the bulk, our black hole solution has a singular horizon. Although the singular horizon of our black hole, we still can get sensible thermodynamics by using the standard approaches. We also evaluate the shear viscosity and find it is zero if we neglect the back reaction of the singular horizon, otherwise, it could be non-zero. The last application of the gauge/gravity correspondence we consider is the shear viscosity/ entropy ratio in the holographic superconductor. It is well-known that there is a universal bound for the shear viscosity/entropy ratio, i.e., η/s ≥ 1/4π for any strongly coupled gauge theory. However, the situations considered in the literature are all the cases without explicit phase transition. Thus, one may wonder if there is any signature on the shear viscosity/entropy ratio when there exists some phase transition. So we numerically check η/s in the s-wave holographic superconductor system and find the universal bound is also satisfied in the superconducting phase. However, we only check for a limited range in the huge parameter space. So we still need some rigorous proof on this universal bound in the superconduting phase.