We discuss the modified gravity which includes negative and positive powers of the curvature and which provides the gravitational dark energy. It is shown that in GR plus the term containing negative power of the curvature the cosmic speed-up may be achieved, while the effective phantom phase (with w less than −1) follows when such term contains the fractional positive power of the curvature. The minimal coupling with matter makes the situation more interesting: even 1/R theory coupled with the usual ideal fluid may describe the (effective phantom) dark energy. The account of R 2 term (consistent modified gravity) may help to escape of cosmic doomsday. * Also at TSPU, Tomsk, Russia and at IFT, UNESP, Sao Paulo (temporary)

PrefaceWhile other -more phenomenologically motivated -branches of high energy physics, such as Quantum Chromodynamics, the electroweak model of Glashow, Salam and Weinberg, as well as purely group-theoretical studies have by now been extensively portrayed in a large number of research reports and books, the same has not been the case for two-dimensional quantum field theory. This book is intended to fill this gap. Its aim is to give a fairly detailed survey of the developments in twodimensional quantum field theory since the pioneering work of Thirring, without loosing sight of their relevance to the four-dimensional world. Though many of the properties and techniques to be used are peculiar to two-dimensional space-time, the structural richness of the models portrays the complexity to be expected in realistic quantum field theories. Our emphasis therefore lies on the non-perturbative aspects of the two-dimensional models and our tools shall correspondingly be operator and functional methods, although we shall occasionally make also use of Feynman diagram techniques. The present edition represents an extensively revised version of the first edition, and involves major changes and additions, as well as corrections. In particular, the chapter on Conformal Field Theory has been completely rewritten and split into three separate chapters, 16, 17 and 18. We have found it useful to include in Chapter 17 some recent applications to statistical models, and in particular to the Ising model. Chapter 18, on the other hand, is devoted to the application of Conformal Field Theory to two-dimensional gravity, and includes some discussion on Liouville theory.Chapter 4, on functional determinants and heat-kernel methods, has also been reorganized, and a section on the heat-kernel at finite temperature, as well as the application of heat-kernel methods for obtaining asymptotic mass expansions has been added. Some material originally contained in Chapter 11 on Quantum Chromodynamics concerning the calculation of the QCD2 determinant has also been included.In Chapter 7 we further included a discussion on the algebra obeyed by nonlocal charges (Yangian algebra), and Chapter 8 contains an additional section on S-matrices in the presence of boundaries.Chapter 11 on Quantum Chromodynamics has also been extensively rewritten. We have incorporated recent results concerning the decoupled (coset) formulation of QCD 2 , with much emphasis on the BRST constraints defining the Hilbert space in this formulation, and related questions such as vacuum degeneracy. We have 6also added a section on QCD2 with massive fermions and the related problem of "screening versus confinement" in the non-Abelian case.A new chapter devoted to the finite temperature Schwinger model has also been added in this edition. Chapter 15 (originally 14) on chiral electrodynamics now also includes a section on the Batalin-Fradkin-Tyutin embedding of this theory into a bonafide gauge theory. Furthermore, we ommitted Appendices H, I, N, O of the first edition in favour of ...

We review two-dimensional QCD. We start with the field theory aspects since 't Hooft's 1/N expansion, arriving at the non-Abelian bosonization formula, coset construction and gauge-fixing procedure. Then we consider the string interpretation, phase structure and the collective coordinate approach. Adjoint matter is coupled to the theory, and the LandauGinzburg generalization is analysed. We end with considerations concerning higher algebras, integrability, constraint structure, and the relation of high-energy scattering of hadrons with two-dimensional (integrable) field theories.

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We calculate the quantized nonlocal charge of the GP"-' model in the framework of renormalized l/n perturbation theory and prove that it is not conserved. -- 1800