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Poisson structure on moduli space of CFTs

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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?

This post has been migrated from (A51.SE)
asked Feb 14, 2012 in Theoretical Physics by Squark (1,700 points) [ no revision ]
Sorry! I just don't understand! Classically: gravity is not a symplectic dynamical system unless spacetime is (R x something), so why should its solution space be a symplectic reduction of anything? String-Theoretically: The symplectic structure of BV is fermionic, so wouldn't the solution space only carry an odd structure, too, like the de Rham complex of the solution space? More prosaically, we know that charge (1,1) operators are deformations, i.e. the tangent space of moduli of CFT, but is there any real classical geometry here other than Zamolodchikov's metric (see Lubos, above)?

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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.

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Squark is right, I think. Phase space is precisely the space of classical solutions. (See the references by Witten at http://ncatlab.org/nlab/show/phase+space) Only in nice situations are these given by a global initial value problem that allows identification with a cotangent bundle. Non-nice situations include gauge theories and theories of gravity, and we know that the space of CFTs contains both.

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Thx. References?

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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.

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@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

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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.

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1 Answer

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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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answered Feb 16, 2012 by Urs Schreiber (5,085 points) [ no revision ]
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.

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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

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