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The moduli space of CFTs with central charge 26 forms the classical phase space of bosonic string theory, in some sense. Similarily the moduli space of SCFTs with central charge 10 forms the classical phase space of type II superstring theory

We might expect this should make these moduli spaces symplectic, or at least Poisson (super)manifolds. Is there such structure on them?

More generally, the only geometric structure on CFT moduli spaces I encountered is the Zamolodchikov metric. What other structures do they have?

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  • $\begingroup$ The extra structures exist in supersymmetric CFTs and they're consequences of supersymmetry (which makes these moduli spaces complex, hyper Kahler etc.). If you take the most generic (bosonic) CFTs, I think that the Zamoldochikov metric is the only structure of this type. $\endgroup$
    – Luboš Motl
    Feb 15, 2012 at 5:33
  • $\begingroup$ Thx. References? $\endgroup$
    – Squark
    Feb 15, 2012 at 6:41
  • $\begingroup$ It is incorrect/misleading to call CFT moduli space the "classical phase space." Rather, it is the space of classical vacua, i.e. the values of the fields in the effective field theory. But the place where fields live is not the phase space. Phase space is, roughly, the cotangent bundle of the space of maps from spacetime to field space. $\endgroup$
    – Eric Zaslow
    Feb 16, 2012 at 18:53
  • $\begingroup$ @Eric I think you are wrong. The vanishing of the beta function in CFT reduces, to first order in worldsheet perturbation theory, to the equations of motion of classical SUGRA on the target space. Hence the space of CFTs is some kind of "deformation" of the SUGRA phase space. In particular a CFT can be modified by introducing finite perturbations to the bulk of the target space, whereas vacua correspond to different asymptotic conditions $\endgroup$
    – Squark
    Feb 17, 2012 at 7:05
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    $\begingroup$ Something like this story is also hopefully true for 3d CFTs and 4d CFTs, where via AdS/CFT, the space of CFTs (also known as the conformal manifold) is in a correspondence with the set of vacua of quantum gravity in AdS space. On the space of vacua there is a natural metric inherited from the kinetic terms in AdS. $\endgroup$
    – Zohar Ko
    Feb 18, 2012 at 14:09

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Due to the existence of symmetries, one would want to look for a BV-BRST structure (from which symplectic structures would be constructed).

There is the famous old article

where it is pointed out that the space of open 2d CFTs should carry a natural BV-bracket.

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  • $\begingroup$ Thx Urs I read this article. As far as I understand it is a somewhat different approach since Witten constructs an action on the space of all 2D theories (not only CFTs). More precisely the article speaks about open string theory so he considers the space of boundary Lagrangians. The CFT moduli space is the space of solutions to this action (the phase space). As far as I know nothing of the sort has been accomplished for closed strings. However it is possible we can identify structures on the CFT moduli space without going through the ambient history space = field theory space $\endgroup$
    – Squark
    Feb 17, 2012 at 7:13
  • $\begingroup$ Hi Squark: an action and a BV bracket, yes. That's what one needs for a phase space in this description. So it may be a different approach than what you thought of, but I think it goes exactly in the direction of answering your question. Of course this is just an incomplete story, yes. I am not sure if this has been followed up since. $\endgroup$
    – Urs Schreiber
    Feb 17, 2012 at 17:13

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