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Is the value of the partition function of a 2d CFT on a compact conformal manifold well defined? Or is there some kind of "anomaly" that makes it dependent on a bulk or some other kind of further structure (as for some $2+1$-dimensional TQFTs)?

If yes, are there formulas that compute it given some topology-dependent parametrisation (elements of the corresponding conformal moduli space), and some data connected to the CFT? Like, for example, depending on the central charge $c$ and on the ratio $R/r$ or a $R\times r$ torus?

This seems to be the one of the most straight-forward questions about CFT, but somehow I'm having a hard time finding anything, mostly because introductions usually focus on CFT on the plane or cylinder.

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    $\begingroup$ Is this a question about 2d CFT? There's a huge literature on this, which could be caricatured as "the partition function is a section of a line bundle on the moduli stack of curves". In d > 2, the unnormalized partition function is going to be infinite. $\endgroup$
    – user1504
    Oct 23, 2019 at 17:59
  • $\begingroup$ thanks, yes, I mean 2d, will edit. The partition function is a real number for every point in moduli space, so isn't it always trivially a section of the trivial line bundle? Are there references that contain plain numbers for concrete CFTs and conformal manifolds? $\endgroup$
    – Andi Bauer
    Oct 23, 2019 at 19:14
  • $\begingroup$ The partition function is a section of the trivial bundle on the space of Riemannian structures, but if the central charge $c$ is non-zero it descends to a section of a non-trivial line bundle on the space of complex structures. I should probably try to write this up as an answer, but don't have time now. $\endgroup$
    – user1504
    Oct 25, 2019 at 17:26
  • $\begingroup$ Thanks, I can wait ;) I'm a bit confused: Don't all non-trivial CFTs have a non-zero central charge? So are you saying the partition function on a fixed conformal manifold is not a well defined quantity in CFT? $\endgroup$
    – Andi Bauer
    Oct 26, 2019 at 15:51
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    $\begingroup$ Meanwhile I found that CFT does in fact have a "conformal anomaly" (unless $c=0$), meaning that the partition function is only defined up to a phase. If I understand correctly, one could say the partition function actually depends on the full metric (not only the conformal structure), but changes under local rescalings in a fixed way (by the Liouville cocycle). $\endgroup$
    – Andi Bauer
    Oct 30, 2019 at 18:00

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