[en] Investigation of 2d conformal field theory in terms of geometric quantization is given. We quantize the so-called model space of the compact Lie group, Virasoro group and Kac-Moody group. In particular, we give a geometrical interpretation of the Virasoro discrete series and explain that this type of geometric quantization reproduces the chiral part of CFT (minimal models, 2d-gravity, WZNW theory). In the appendix we discuss the relation between classical (constant) r-matrices and this geometrical approach. (orig.)

[en] An explanation of the appearance of quantum groups in chiral WZNW models is given. Invariance of the theory under the quantum group action is discussed. (orig.)

[en] Some results from group theory are collected, which are important for free field representation of Kac-Moody algebra and Wess-Zumino-Witten model on the lines of ref. 1. 5 refs.; 3 figs

[en] Geometrical interpretation of the appearance of SL(2,R) symmetry in the 2-d quantum gravity is given in terms of the theory of the coadjoint orbits of the Virasoro group. Equivalence of the 2-d gravity and the special type of gauged SL(2,R) WZNW theory is shown. (orig.)

[en] Kac-Moody algebra representation in free field terms is considered. Bosonization of Wess-Zumino-Witten model for determination of multiloop correlators is used. Kac-Moody algebra representation and bosonization of the model are carried out for arbitrary simple group. 10 refs.; 4 figs

[en] This paper lays groundwork for the detailed study of the non-trivial renormalization group flow connecting supersymmetric fixed points in four dimensions using string theory on AdS spaces. Specifically, we consider D3-branes placed at singularities of Calabi-Yau threefolds which generalize the conifold singularity and have an ADE classification. The N=1 superconformal theories dictating their low-energy dynamics are infrared fixed points arising from deforming the corresponding ADE N=2 superconformal field theories by mass terms for adjoint chiral fields. We probe the geometry with a single D3-brane and discuss the near-horizon supergravity solution for a large number N of coincident D3-branes. (author)

[en] A conjecture relating instanton calculus in four dimensional supersymmetric theories and the deformation theory of Lagrangian submanifolds in C^2r invariant under a (subgroup of) Sp(2r,Z) is formulated. This is a four dimensional counterpart of the mirror symmetry of topological strings (relating Gromov-Witten invariants and generalized variations of Hodge structure). (orig.)

[en] We discuss the relation between the functional integral over open Riemann surfaces and the operator formalism on closed Riemann surfaces. The states in the latter formalism arise in the holomorphic representation of the functional integral. (orig.)

[en] The free field representation of Wess-Zumino-Witten (WZW) model is generalized to the case of arbitrary Riemann surface. The multiloop calculations for free fields on Riemann surfaces are discussed. The special attention is attracted to the bosonic βγ-system, which appears in the bosonization scheme for the Kac-Moody current algebras. We consider the general properties of the multiloop blocks of the WZW and in particular we explain, how the one-loop characters are reproduced by our methods. 21 refs.; 2 figs.; 1 tab

[en] We develop some useful techniques for integrating over Higgs branches in supersymmetric theories with 4 and 8 supercharges. In particular, we define a regularized volume for hyperkaehler quotients. We evaluate this volume for certain ALE and ALF spaces in terms of the hyperkaehler periods. We also reduce these volumes for a large class of hyperkaehler quotients to simpler integrals. These quotients include complex coadjoint orbits, instanton moduli spaces on R⁴ and ALE manifolds, Hitchin spaces, and moduli spaces of (parabolic) Higgs bundles on Riemann surfaces. In the case of Hitchin spaces the evaluation of the volume reduces to a summation over solutions of Bethe ansatz equations for the non-linear Schroedinger system. We discuss some applications of our results. (orig.)