Perturbative and non-perturbative aspects of the two-dimensional string/Yang-Mills correspondence

It is known that YM_2 with gauge group SU(N) is equivalent to a string theory with coupling g_s=1/N, order by order in the 1/N expansion. We show how this results can be obtained from the bosonization of the fermionic formulation of YM_2, improving on results in the literature, and we examine a number of non-perturbative aspects of this string/YM correspondence. We find contributions to the YM_2 partition function of order exp{-kA/(\pi\alpha' g_s)} with k an integer and A the area of the target space, which would correspond, in the string interpretation, to D1-branes. Effects which could be interpreted as D0-branes are instead stricly absent, suggesting a non-perturbative structure typical of type 0B string theories. We discuss effects from the YM side that are interpreted in terms of the stringy exclusion principle of Maldacena and Strominger. We also find numerically an interesting phase structure, with a region where YM_2 is described by a perturbative string theory separated from a region where it is described by a topological string theory.Comment: 24 pages, 5 figure

Emergent Phase Space Description of Unitary Matrix Model

We show that large $N$ phases of a $0$ dimensional generic unitary matrix model (UMM) can be described in terms of topologies of two dimensional droplets on a plane spanned by eigenvalue and number of boxes in Young diagram. Information about different phases of UMM is encoded in the geometry of droplets. These droplets are similar to phase space distributions of a unitary matrix quantum mechanics (UMQM) ($(0 + 1)$ dimensional) on constant time slices. We find that for a given UMM, it is possible to construct an effective UMQM such that its phase space distributions match with droplets of UMM on different time slices at large $N$. Therefore, large $N$ phase transitions in UMM can be understood in terms of dynamics of an effective UMQM. From the geometry of droplets it is also possible to construct Young diagrams corresponding to $U(N)$ representations and hence different large $N$ states of the theory in momentum space. We explicitly consider two examples : single plaquette model with $\text{Tr} U^2$ terms and Chern-Simons theory on $S^3$. We describe phases of CS theory in terms of eigenvalue distributions of unitary matrices and find dominant Young distributions for them.Comment: 52 pages, 15 figures, v2 Introduction and discussions extended, References adde

Random matrix models for phase diagrams

We describe a random matrix approach that can provide generic and readily soluble mean-field descriptions of the phase diagram for a variety of systems ranging from QCD to high-T_c materials. Instead of working from specific models, phase diagrams are constructed by averaging over the ensemble of theories that possesses the relevant symmetries of the problem. Although approximate in nature, this approach has a number of advantages. First, it can be useful in distinguishing generic features from model-dependent details. Second, it can help in understanding the `minimal' number of symmetry constraints required to reproduce specific phase structures. Third, the robustness of predictions can be checked with respect to variations in the detailed description of the interactions. Finally, near critical points, random matrix models bear strong similarities to Ginsburg-Landau theories with the advantage of additional constraints inherited from the symmetries of the underlying interaction. These constraints can be helpful in ruling out certain topologies in the phase diagram. In this Key Issue, we illustrate the basic structure of random matrix models, discuss their strengths and weaknesses, and consider the kinds of system to which they can be applied.Comment: 29 pages, 2 figures, uses iopart.sty. Author's postprint versio