Genus two meromorphic conformal field theory (Q2782399)

The object of the paper is the investigation of the meromorphis conformal field theory on the genus two Riemann surface, in particular, the explicit description of the partition function and its modular properties for the meromorphic bosonic conformal field theory. NEWLINENEWLINENEWLINEAt first, the review of the corresponding results for the genus one Riemann surface is given. The partition function in this case is very simple and it is a meromorphic elliptic modular form. The examples of such theories are chiral bosonic string and self-dual meromorphic conformal field theory such as Moonshine Module. For the case of genus two some particular results were earlier obtained, for example, the expression of the partition function in terms of operator determinants (for bosonic string) and the investigation of the \(SU(2)\) WZW model. NEWLINENEWLINENEWLINEThe second section is devoted to consideraion of two modes of parametrization of the moduli space for the genus two Riemann surface. One of them is defined in terms of the period matrix, and the alternative way of parametrizing is provided by a standard sewing procedure for joining together two genus one Riemann surfaces. The relation between these two parametrization is investigated. NEWLINENEWLINENEWLINEIn the third section the review of some results from classical string theory is given. In particular, the genus two central charge \(C=24\) bosonic string partition function is presented as the product of certain universal holomorphic functions of the moduli (with ghost contribution) and the inverse of the Siegel modular form of weight 10. NEWLINENEWLINENEWLINEThe last section is devoted to the investigation of the genus two partition function for general bosonic meromorphic CFT's such as chiral bosonic string, even self-dual lattice theories and the Moonshine Module. The explicit formula for the genus two partition function in terms of appropriate torus one point functions is given. For self-dual theories with central charge 24 the genus two partition function multiplied by the universal holomorphic function is a meromorphic Siegel form of weight 2. NEWLINENEWLINENEWLINEThe conclusion contains a very interesting discussion of the results and statements of several unsolved problems.NEWLINENEWLINEFor the entire collection see [Zbl 0980.00029].