Urs Schreiber
Shared publicly10:24 AM
Sombody: "All models of space that I know from physics use real or complex manifolds. I was just wondering if it is still the case at the level of Planck scale. In string theory, physicists still use strings (circles) in a 1 dimensional manifold in order to model particles."
http://www.physicsoverflow.org/27295/space-at-the-planck-scale?show=27351#a27351
No. In perturbative string theory (as well as, in fact, in spectral triples) the input datum is not in general a geometry modeled on manifolds, but is the purely algebraic datum of a 2d SCFT . From this one recovers an "emergent" effective target spacetime at low energies by computing the string perturbation series given by summing up correlators of this abstractly, algebraically defined 2d SCFT and then asking for an ordinary manifold with an ordinary field theory on it which produces the same S-matrix elements at low energy. This may or may not exist! There are plenty of completely non-geometric 2d SCFTs, the famous examples being the Gepner models which model completely non-geometric "phases" of spacetime such as famously the flop transition.
It just so happens that for phenomenological reasons, since one is trying to match to the observed physics well below the Planck scale, there is so much focus on those 2d SCFTs which are constructed geometric as sigma-models from differential geometric input data. But this is human prejudice, not string theory's preference. The moduli space of all 2d SCFTs (the true landscape) has no reason to be dominated by geometric sigma-models. That these are at the focus of attention is because it is easier for us humans to deal with them. One fine day the time will come that mathematical tools have advanced to the point that a genuine analysis of the space of all 2d SCFTs is possible and then we'll be speaking much more about non-geometric backgrounds.
Indeed, for the comparatively simple case of rational 2d CFTs a full mathematical description of the moduli space ("landscape") of all these is already known, by the FRS theorem. And indeed, this classification turns out to be completely algebraic, no differential geometry anywhere. The theorem says that rational 2d CFTs are given essentially by certain Frobenius algebra objects internal to the modular representation categories of rational vertex operator algebras. This does include geometric models such as WZW models, but the general point in this space of rational CFTs has no reason at all to have any relation to smooth manifolds.
http://www.physicsoverflow.org/27295/space-at-the-planck-scale?show=27351#a27351
No. In perturbative string theory (as well as, in fact, in spectral triples) the input datum is not in general a geometry modeled on manifolds, but is the purely algebraic datum of a 2d SCFT . From this one recovers an "emergent" effective target spacetime at low energies by computing the string perturbation series given by summing up correlators of this abstractly, algebraically defined 2d SCFT and then asking for an ordinary manifold with an ordinary field theory on it which produces the same S-matrix elements at low energy. This may or may not exist! There are plenty of completely non-geometric 2d SCFTs, the famous examples being the Gepner models which model completely non-geometric "phases" of spacetime such as famously the flop transition.
It just so happens that for phenomenological reasons, since one is trying to match to the observed physics well below the Planck scale, there is so much focus on those 2d SCFTs which are constructed geometric as sigma-models from differential geometric input data. But this is human prejudice, not string theory's preference. The moduli space of all 2d SCFTs (the true landscape) has no reason to be dominated by geometric sigma-models. That these are at the focus of attention is because it is easier for us humans to deal with them. One fine day the time will come that mathematical tools have advanced to the point that a genuine analysis of the space of all 2d SCFTs is possible and then we'll be speaking much more about non-geometric backgrounds.
Indeed, for the comparatively simple case of rational 2d CFTs a full mathematical description of the moduli space ("landscape") of all these is already known, by the FRS theorem. And indeed, this classification turns out to be completely algebraic, no differential geometry anywhere. The theorem says that rational 2d CFTs are given essentially by certain Frobenius algebra objects internal to the modular representation categories of rational vertex operator algebras. This does include geometric models such as WZW models, but the general point in this space of rational CFTs has no reason at all to have any relation to smooth manifolds.
'via Blog this' The Honourable Schoolboy
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