Modular invariance of CFT I am looking at the Cardy formula for entropy in CFT, and in the article 'Kerr/CFT correspondence and its Extensions' there is a sentence:

In any unitary and modular invariant CFT, the asymptotic growth of states...

I've looked a bit, but couldn't find any simple explanation of the term modular invariant CFT.
All I find is that modular invariance is the invariance under the group $SL(2,\mathbb{Z})$. But what does that mean, what's the physical explanation of it and what's it's effect?
 A: Ref : Polchinski Vol $1$, pages $146-150$
With a torus topology, with identifications $(\sigma_1, \sigma_2) \sim (\sigma_1, \sigma_2) + 2\pi(m,n)$, one may bring the worldsheet metrics to the form $ds^2 = |d\sigma_1 + \tau d\sigma_2|^2 = dw d \bar w$, where $\tau$ is  a complex constant (the moduli). The periodicity is expressed  by $w \sim w + 2\pi(m+n\tau)$, with $m,n$ in $\mathbb{Z}$.
There are additional redundancies. The value $\tau+1$ generates the same kind of indendification for $w$, with $(m,n) \to (m-n,n)$. The value $\frac{1}{\tau}$ also generates the same kind of indendification for $w$, with $w \to \tau w$ and $(m,n) \to (n,m)$. $\tau \to \tau+1$ and $\tau \to \frac{1}{\tau}$ are the two generators of the $SL(2,Z)$ group (some people call it the modular group, while others give this name to $PSL(2,Z)$) , the group of transformations $z \to \frac{az+b}{cz+d}$, with $a,b,c,d$ in $\mathbb{Z}$ and $ad-bc=1$. This represents the diffeomorphism of the torus. 
This allows to take a fundamental region,  for the upper half-plane modulo $PSL(2,Z)$. The fact that we may choose a fundamental region without the origin $z=0$, gives the idea, that string theory may be well-suited to avoid some infinite quantities.
