In Rational Conformal Field Theory, the partition function is

$Z(\tau,\overline{\tau})~=~\sum\limits_{p,\overline{q}} F_{p\overline{q}}~\chi_{p}(\tau)\chi_{q}(\overline{\tau})$

where the $\chi_{p}$ are the Virasoro-characters of the primaries, and $F_{pq}$ is the physical invariant. Its element can't be anything, because these are multiplicities, and moreover it can't be arbitrary to get a partiton function which describes a consistent theory.

But I don't know what we mean when we say that the theory is consistent, and what does it mean physically if we know that the partition function gives the vacuum-vacuum transition amplitude. I know it means that the partition function is modular invariant, but I don't know what is the physical background of it, that is, what is the connection between the path integral and the torus geometry.

And generally, when do we choose a higher genus Riemann surface and what is it actually based on?

This post imported from StackExchange Physics at 2015-04-19 11:35 (UTC), posted by SE-user Hajnalka Korka