Conformal Covariance and Related Properties of Chiral Qft

Though this be madness, yet there is method in 't. PREFACE This PhD thesis in Mathematical Physics contains both a general overview and some original research results about low dimensional conformal Quantum Field Theory (QFT). The closer area of the work is chiral conformal QFT in the so-called algebraic setting; i.e. the study of Möbius covariant and diffeomorphism (or, as it is sometimes called: conformal) covariant local nets of von Neumann algebras on the circle. From the mathematical point of view, it involves mainly functional analyses, operator algebras, and representation theory (and in particular, the representation theory of infinite dimensional Lie groups). The preliminary chapters give an overview, summarize some of the most important known facts and give some examples for such local nets. This is then followed by the presentation of the new results. The questions considered are all about conformal covariance. First, what algebraic property of the net could ensure the existence of an extension of the Möbius symmetry to full diffeomorphism symmetry? Second, if exists, is this extension of the symmetry unique? Third, what (new, so far unknown) properties of the net are automatically guaranteed by the mere existence of conformal covariance? In relation with the existence, the author pinpoints a certain (but purely algebraic) condition on the " relative position " of three local algebras. (This condition is implied by diffeomorphism covariance and even without assuming diffeomorphism covariance it can be proved to hold, assuming complete rationality, for example.) Then it is shown that the Möbius symmetry of a regular net (i.e. a net which is n-regular for every n ∈ N + ; a property which is weaker than strong additivity and expected to hold always in case of con-formal covariance) can be extended to certain nonsmooth transformations if and only if this condition is satisfied, and that in this case the extension is unique. Although the existence of an even further extension to the full diffeomorphism group is not proved, for the author, as it will be explained, it indicates that the introduced condition together with regularity indeed ensure the existence of conformal covariance. Two different proofs will be given to show that there is at most one way the Möbius symmetry of a net can be extended to Diff + (S 1). One of them is close to the setting of Quantum Fields (and in particular, to Vertex Operator Algebras, as it is …