Topics in low-dimensional field theory [report]

M.J. Crescimanno
1991 unpublished
Conformal field theory is a natural tool for understanding two-dimensional critical systems. This work presents results in the lagrangian approach to conformal field theory. The first sections are chiefly about a particular class of field theories called coset constrctions and the last part is an ex position of the connection between two-dimensioal conformal theory and a three-dimenional gauge theory whose lagrangian is the Chern-Simons density. Since 1984 there has been strong renewed interest
more » ... ng renewed interest in field theory in two and three dimensions. In that year Green and Schwarz discovered an anomaly-free string theory 1 thereby launching hopes that there might be a unique, finite fundamental theory of gauge interactions and gravity. In the intervening years although the prospect of understanding physical interactions in terms of string theory have dimmed there have been vigorous efforts and inroads made in understanding aspects of low-dimensional field theory. String theory is an old subject that was first motivated by an attempt to understand the strong interactions-. In particular it was discovered 3 that, in the large N limit of the gauge theory SU(X). the dominant contributions from perturbation theory come from planar diagrams. This combined with earlier work 4 suggested that one may think of mesons, hadrons and other strongly interacting particles as being essentially quarks tied together by tubes of strongly interacting gauge bosons. This idea was further expanded upon in ref. [5] in which a quantum mechanical theory of such an extended particle was described in terms of a lagrangian that measured the area of the surface swept out by the tubes as they propagate in space. It was thus discovered that because the area of the sheet swept out by the string didn't depend on the co-ordinates used to label the points in the sheet, that the resulting theory had a two-dimensional conformal invariance. The picture is as follows; as the string propagate in some background space, the functions that describe how the string is situated in space at every instant (called the embedding fields) if viewed in terms of the two-dimensional co-ordinates of the sheet, become fields of a conformal field theory. In two dimensions the conformal group is infinitedimensional and its representations have been studied in ref.[6]. Two-dimensional systems that possess conformal symmetries are of interest lor rea sons other than string theory. Indeed, any two-dimensional system undergoing a second order phase transition will be describable at the critical point in terms of a conformal field 11. D. Friedan, Ann. of the Mathematics Department and Professor E. Rabinovici (of the Hebrew University, Jerusalem) for many explanations. It is a pleasure to acknowledge useful incidental conversations with I. Bars (UCLA), D. Bar-Natan (Princeton), M. Bos (Columbia), K. Intrilligator (Harvard), S. Naculich (John Hopkins University), V.P. Nair (Columbia), and A. Polychronakos (Florida State). For innummerable technical discussions on a variety of topics this author wishes to thank A. Landsberg, S. Hsu and N. Obers. Special thanks to A. Landsberg, T. Pipp.S.J. and M.K. Cohan for their warm friendship and support. 10 CHAPTER 2: Monopole Backgrounds on the World Sheet Since the advent of string theory it has been useful! to thing about the string propagat ing in some geometrically fixed, i.e. classical, background. In this first paper we investigate what effect a monopole background will have on a siring. As described in the introduc tion to this work, this is an interesting question because a broad class of models (the so called coset constructions) for the compactification of the theory involve integrals over a non-dynamical gauge field. Thus it is natural to ask what may be learned about the theory by studying it in non-trivial gauge backgrounds. The entire paper is a description of a very particular type of coset i.e. those cosets that may be realized in terms of free fermions coupled to an abelian gauge field (as an example consider SO(2n)/SO(2) cosets). Highlights include a new spectrum for states propagating in this monopole background and also novel modular properties at genus one. integral: 0'(*iW(*>)••-.;'(*.)) = Z _1 [D*D*Dej(! l m'i)---JM £ exp\[<PtC N {z)]. J JV»-oo U J (3.3) In this expression, each j stands for either type of operator given in eq.(3.1) and eq.(3.2), and also, in order to simplify writing, flavor indices are suppressed.
doi:10.2172/5730644 fatcat:c4uywbyaezd5hm3qjx6i7gv2xa