| dc.description.abstract | The CFT in two dimensions is useful to understand critical phenomena of statistical mechanics in two dimensions [1,2]. The study of CFTs on sphere and higher genus Riemann surfaces is important because of the usefulness in perturbation theory calculations of string theory [4,5]. Also, on their own, CFT on higher genus surfaces describes statistical systems at critical temperature on polygon surfaces with appropriate boundary conditions.
There have been various approaches to the study of CFTs on higher genus Riemann surfaces. One method is the differential equations satisfied by their characters/correlators which has been developed in refs. [21–28]. In this approach, the equation must be derived on a case-by-case basis for each correlator for each individual theory. Moreover, the analysis of relatively simple theories on higher genus surfaces gets involved [27,28]. Characters and correlators can also be calculated by means of a Hamiltonian and a Lagrangian description of constrained fermions [30]. The Hamiltonian approach is not directly amenable to surfaces of genus two or greater. The Lagrangian formulation easily generalizes to arbitrary genus surfaces. Another approach makes use of the fact that the level-one WZ model for any simply laced group G is equivalent to a theory of d = rank G toroidally compactified bosons [30–32]. This equivalence is used to calculate correlation functions on higher genus surfaces. Yet another method is the generalization of the Feigin–Fuchs representation for minimal models [35–37] to a larger class of CFTs. Although one can obtain the correlation functions on arbitrary genus Riemann surfaces explicitly, the calculation is non-trivial as one must project spurious states out of the theory to prevent them from propagating in intermediate channels. Also, the bosonization and multiloop calculations of WZ models using free field representations of Kac–Moody algebra have been performed in ref. [38].
In this thesis, we have developed a simple method to solve conformal field theories on arbitrary genus Riemann surfaces. This approach makes use of factorization property and modular invariance of correlation functions. For the proof of unitarity of string theory, it is essential that the correlators on higher genus surfaces should have proper factorization properties under pinching of zero and non-zero homology cycles of Riemann surfaces.
In the study of these factorization properties of CFTs on higher genus surfaces, we have a technical problem. The holomorphic factorization of characters/correlators of a RCFT does not occur on Riemann surfaces of genus two or higher for arbitrary choice of the metric [17]. The lack of factorization is due to conformal anomaly. However, one can study the ratio of the partition function or any higher point correlators of a CFT with central charge c and the partition function of a single scalar field raised to the power c. This ratio would be free of conformal anomaly, would be metric-independent and modular invariant [20]. The characters and correlators so normalized are then amenable to the study of their factorization properties.
The general idea of the factorization property under ZHP limit of a surface of genus g? + g? is that the correlation function should factorize into two correlation functions, one each on genus g? and genus g? surface and the propagator of the mediating primary field on the cylinder. On the other hand, under NZHP limit of a genus (g + 1) surface, the idea of factorization is that the correlator should factorize into correlator on genus g surface and a propagator on cylinder corresponding to the appropriate primary field going round the loop which is pinched. Also, the grand total of all the normalized characters/correlators (whose number is determined by fusion rules) on a given Riemann surface must be modular invariant. This method allows us to obtain any n-point correlator on a lower-genus surface by suitable set of pinchings performed on appropriate characters on higher genus Riemann surface.
By the above method, we have explicitly obtained the normalized characters and correlators for the following CFT models [39]:
(i) Critical Ising model,
(ii) Level-one SU(2) WZ model,
(iii) Level-two SU(2) WZ model, and
(iv) Level-one SU(3) WZ model.
The critical Ising model is described by three primary fields: identity operator, energy operator ? and spin operator ?. The central charge is ½. We have obtained the expressions for appropriately normalized partition function and one- and two-point correlators on genus-two surface explicitly. Also, we have presented the analysis on arbitrary genus. Fusion rules prohibit ? one-point correlator on higher genus surface. The normalized partition functions are given by all possible even ?-functions, while ? one-point correlators are given by odd ?-functions, ?–? correlators by even ?-functions and ?–? correlators by both even and odd ?-functions. We have made use of factorization property under ZHP limit and modular invariance. We also obtain the 3-point ? correlators ??(z?) ?(z) ?(w)? on torus from one-point ? correlator on genus-two ??(z?)? by NZHP limit.
The level-one SU(2) WZ model is described by two primary fields: an SU(2) singlet and a doublet. The central charge is one. By making use of factorization ZHP limit and modular invariance, we have written down the arbitrary genus expressions for normalized correlators. We have also obtained expressions for 2n-point normalized correlators on arbitrary genus surface by exploiting the factorization property under NZHP limit and modular invariance.
The level-two SU(2) WZ model is characterized by three primary fields: a singlet, a doublet and a triplet of SU(2). The central charge is 3/2. By exploiting factorization property under ZHP limit and modular invariance, we have obtained expressions on genus-two and arbitrary genus surface; and two-point correlators on genus-two surface. Fusion rules forbid ? one-point correlator on higher genus surface. We have also surveyed the NZHP limit of genus-two characters/correlators for specific examples. In particular, the NZHP limit of genus-two correlators leads to the four-point function or two-point function on torus depending on whether representations go round the pinched loop or identity representation goes round the pinched loop.
The level-one SU(3) WZ model is specified by three primary fields, which are respectively a singlet, a triplet and an anti-triplet of SU(3). The central charge is two. We have written down the correlators on a genus-one surface by making use of the monodromy properties of correlators. Genus-two results are constructed next for normalized characters and two-point functions associated with SU(2) by making use of fusion rules, modular invariance and factorization property under ZHP limit.
The results are generalized to arbitrary genus surfaces. We have also obtained the normalized four-point correlators by NZHP limits of the characters with [??] representation going round the two pinched blobs.
To conclude, we have demonstrated by examples that it is enough to know all genus characters to solve a CFT. Any correlator can be obtained from the characters by a suitable set of pinchings along zero-homology and non-zero homology cycles of the Riemann surfaces. In particular, this also provides a simple method of obtaining any point correlators on the sphere from higher genus characters.
We have shown the usefulness of factorization properties and modular invariance only in specific examples where characters are expressed in terms of ?-functions. The factorization and modular invariance are general features of conformal field theories. Therefore, in contrast to some other methods which use structure specific to the models studied, the methods developed here may be applicable to other models not considered here. | |
