International Journal of Modern Physics A

A comparison is made between the new solutions of the Yang-Baxter equations, arising from curves of higher genus, and magnetic monopoles of higher charge. It is shown that essentially the same algebraic curves arise in both cases, and this leads to speculations about possibly more general solutions of the Yang-Baxter equations.

The Langevin equations describing the quantization of gauge theories have a geometrical structure. We show that stochastically quantized gauge theories are governed by a single differential operator. The latter combines supersymmetry and ordinary gauge transformations. Quantum field theory can be defined on the basis of a Hamiltonian of the type , where Q has has deep relationship with the conserved BRST charge of a topological gauge theory, and is its adjoint. We display the examples of Yang-Mills theory and of 2D gravity. Interesting applications are for first order actions, in particular for the theories defined by the three dimensional Chern Simons action as well as the “two dimensional” ∫_M2TrϕF.

Current progresses in understanding quantum gravity from the operator viewpoint are reviewed. They are based on the U_q(sl(2))-quantum-group structure recently put forward^1,2, for the chiral components of the metric in the conformal gauge.

Some aspects of strings in critical dimensions are noted. In particular aspects of strings in d=1 are discussed for bosonic and superstrings.

E. Verlinde’s formula for the dimension of the nonabelian θ-functions is discussed from an algebraic geometry point of view and related to certain quotient rings of the representative ring of sums.

In this talk I shall give a fairly geometrical account of the main facts about the KdV equation on the circle, explaining in particular how it is related to the group Diff(S¹), and why it is a completely integrable Hamiltonian system. In §4 I shall describe the theorem of Drinfeld and Sokolov [1] which shows that the KdV system can be regarded as a symplectic quotient of a coadjoint orbit of the loop group of SL₂(R). Finally, in §5 I shall explain how the theory generalizes from second-order equations and SL₂ to n^th-order equations and SL_n: the resulting classical system is the one whose quantization leads to the Zamolodchikov algebra W_n. My aim is purely expository: all the material is well known.

We outline some of the main results derived recently on the representation theory of universal W-algebras, which arise in two dimensional field theory as large N limits of W_N (extended) conformal symmetries. Certain connections with integrable systems of non-linear differential equations, hyper-Kähler geometries in four spacetime dimensions and the infinitely long chain of hermitian matrix models are also discussed.

We present the beginnings of a new consistent gauge theory with spin 2 and spin 3 gauge fields in two dimensions. It is based on a nonlinear Lie algebra, whose “gauging” is discussed in detail. (The present treatment of the gauge sector is expanded as compared to [3]). For the coupling to matter, we introduce the concept of a “nested covariant derivative”, and we obtain an invariant action by solving a functional integrability condition. It reads

An intuitive way of looking at the Adler Bell Jackiw anomaly is reviewed and elaborated. The main point is that the created particles are “in reality” being pumped up from the Dirac sea. This point of view is related to the troubles with regularizing in a chirally invariant way. This problem is stressed as a major paradox on a somewhat analogous footing with the paradox of having simultaneity to the understanding of special relativity. A way out proposed by Eichten and Preskill is discussed, but its ability to really solve the problem remains unclear.