Ultraviolet limit

Already in the early seventies it was realized that conformal symmetry plays a very distinct role in 1+1-dimensional quantum field theories. In the middle of the eighties the key mathematical tools were developed and it was understood [9 ] that the properties of the Virasoro algebras may be exploited in order to compute correlation functions. By means of the coset construction [ 84 ] one may relate the Virasoro algebra to simple Lie algebras which also hint to a structure on the massive side, in form of ATFTs. There is yet another way to link the underlying Lie algebra of ATFT to conformally invariant theories exploiting certain, sometimes misleadingly referred to as fermionic, formulations of Virasoro characters [ 94 ]. From these characters one may extract a quasi-particle spectrum in a very natural way [8 ]. In [11 ] we demonstrated that the formulation of the Virasoro characters is by no means unique and may lead to different types of spectra associable to several types of statistics. The ambiguity reflects on one hand the different possible relevant perturbations of the conformal field theory and on the other that in 1+1 dimensions the choice of the statistics is ambiguous. In [11 ] we have shown that it is possible to give an anyonic interpretation to the different types of spectra. A saddle point analysis, which may only be carried out once the characters are formulated in a very particular way, yields a system of coupled nonlinear equations which is equivalent to the `` constant'' TBA-equations. Due to this fact it is suggestive to investigate how this structure arises on the massive side. For this reason the TBA was formulated for systems whose statistical interaction is of Haldane type and solved for various ATFTs [ 11 ]. The analysis showed that different statistical interactions may be identified in the scattering matrix from the asymptotic behaviour. The investigations in [14 ] showed that Virasoro characters, normally infinite sums due to their structure of cancelling nullvectors, which can be factorized are very exceptional and play a distinct role. For this reason it was proven [ 14 ] in complete mathematical generality which characters of minimal models and their linear combinations factorize. A new technique was developed which exploits the asymptotic behaviour of so-called quantum dilogarithms in order to identify factorizable combinations. As a by-product one can use these results to derive several new identities between characters in a very economical way and also obtain new Rogers-Ramanujan-Schur identities [ 107 ]. Applications to the construction of quasi-particle spectra, modular invariant partition functions, supersymmetric conformal QFTs and conformal QFTs with boundaries were discussed. The connection between Virasoro characters and ATFTs was investigated further in [13 ], where a particular ADE-structure was identified. Besides the characters already studied in [14 ], it was necessary to include parafermionic models and compactified bosonic theories into the considerations. The structures found, in particular the occurrence of the exponents of the underlying Lie algebra, indicate that the quasi-particle spectra are related to the W-algebras. Once the Virasoro characters are q-deformed [24 ] one may also construct pseudo-particle spectra which include unstable particles. In addition these type of expressions allow for the formulation of a new type of scaling function which contains the same qualitative information as the ones obtained from a TBA or form factor analysis. Even without deformation the constant TBA-equations constitute very interesting systems of equations to be studied in their own right. As described above they arise in various conceptually entirely different ways and describe the system at the ultraviolet fixed point. Remarkably, in many cases they can even be solved completely analytically in terms of some very distinct mathematical objects, that is principally specialized Weyl characters. In [ 24 ] we provided several new solutions to the constant TBA-equations and supplied some proofs missing so far in the literature. From a mathematical point of view a very exceptional situation arises when the resulting Virasoro central charge can be expressed in terms if Rogers dilogarithms and is in addition also rational. In that case the set of equations are then referred to as `` accessible" dilogarithms (for a review see e.g. [ 95 ] and references therein). There are various open issues of a more mathematical nature on the UV-limit. Concerning the constant TBA equations, there are still several models for which solutions in terms of Weyl characters are not known yet.