Conformal Field Theory consistency

Question

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

Comments

What do you mean "choose a higher genus"? CFTs are theories living in two dimensions. When the two-dimensional "spacetime" (or just space) comes in a shape that is a higher genus surface, you have a CFT on a higher genus. You don't "choose" it, it's fixed by the system you want to describe. E.g. CFT on all genera occur in string theory (since the genus of the worldsheet is not restricted).

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

---

Of course I thought of any field theory which can be define on higher genus surface, not in CFT.

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