As far as I understand, there are roughly 2 "common" kinds of 2D conformal field theories:

- Theories that are defined only on the plane, more precisely, on any surface of vanishing genus. Such a theory can be mathematically described by a vertex operator algebra, or by "usual" axiomatic QFT in 2D with the added requirement of conformal invariance
- Theories defined on arbitrary surfaces. Such a theory can be described e.g. by Segal's axioms

In order for a theory of the 1st category to become a theory of the 2nd category, it must pass the test of modular invariance. In my experience, this term usually means that the theory has to be defined in genus 1, i.e. that the would-be torus partition function

$$Tr(q^{L_0-\frac{c}{24}}\bar{q}^{\bar{L_0}-\frac{\bar{c}}{24}}) $$

is invariant under the modular group $SL(2,\mathbb{Z})$

What about higher genus? Is the torus condition sufficient to make the theory well defined there? Or does it impose additional conditions? If so, can they be cast into an anologous elegant algebraic form? Have these conditions been proved for CFTs used in string theory?

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