International Journal of Modern Physics A

Some approaches to 2D gravity which have been developed in the last few years are reviewed. They are physical (Liouville) gravity, topological theories and matrix models. Special attention is paid to matrix models and their interrelations with different approaches. Almost all technical details are omitted, but examples are presented.

We demonstrate how to find modular discrete symmetry groups for Z_N orbifolds. The Z₇ orbifold is treated in detail as a nontrivial example of a (2, 2) orbifold model. We give the generators of the modular group for this case which, surprisingly, does not contain SL(2; Z)³ as had been speculated. The treatment models with discrete Wilson lines are also discussed. We consider examples which demonstrate that discrete Wilson lines affect the modular group in a nontrivial manner. In particular, we show that it is possible for a Wilson line to break SL(2, Z).

We construct bosonized vertex operators (VO's) and conjugate vertex operators (CVO's) of U_q[su(2)_k] for arbitrary level k and representation j ≤ k/2. Both are obtained directly as two solutions of the defining condition of vertex operators — namely that they intertwine U_q[su(2)_k] modules. We construct the screening charge and present a formula for the n-point function. Specializing to j = 1/2 we construct all VO's and CVO's explicitly. The existence of the CVO allows us to place the calculation of the two-point function on the same footing as k = 1; that is, it is obtained without screening currents and involves only a single integral from the CVO. This integral is evaluated and the resulting function is shown to obey the q-KZ equation and to reduce simply to both the expected k = 1 and q = 1 limits.

Vertex operators associated with level 2 modules are explicitly constructed using bosons and fermions. An integral formula is derived for the trace of products of vertex operators. These results are applied to give n-point spin correlation functions of an integrable S = 1 quantum spin chain, extending an earlier work of Jimbo et al. for the case S = 1/2.

The Laplace operator admits infinite self-adjoint extensions when considered on a segment of the real line. They have different domains of essential self-adjointness characterized by a suitable set of boundary conditions on the wave functions. In this paper we show how these extensions can be recovered by studying the continuum limit of certain discretized versions of the Laplace operator on a lattice. Associated to this limiting procedure, there is a renormalization flow in the finite-dimensional parameter space describing the discretized operators. This flow is shown to have infinite fixed points, corresponding to the self-adjoint extensions characterized by scale-invariant boundary conditions. The other extensions are recovered by looking at the other trajectories of the flow.