Can we use the mirror symmetry to define quantum gravity ?

It may be fair to say that so far we don't know how to quantize a Riemannian manifold (or a complex manifold). But a symplectic manifold may be more accessible for quantization because it endows a natural cosymplectic (Poisson) structure, e.g., for deformation quantization. If so, using the mirror symmetry, we may try to quantize a symplectic manifold instead of a complex manifold being mirror to the former. How can this approach for quantum gravity make sense ?


closed as not a real question by Ian Agol, José Figueroa-O'Farrill, David Roberts, S. Carnahan Jul 13 '11 at 1:12

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    $\begingroup$ Dear Hyun S. Yang, would it be possible to give a bit more details (references) on what you mean by "Mirror symmetry", and "quantum gravity"? $\endgroup$ – Dmitri Panov Jul 11 '11 at 16:07
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    $\begingroup$ Unfortunately, I do not think this question is well posed. There are many ways that one could look at quantization, but deformation quantization in particular applies to Poisson algebras and not manifolds. Manifolds are in the picture because algebras come from functions on them. Even if your question is translated along these lines, quantizing gravity does not correspond to deforming the algebra of functions on any complex or riemannian manifold. The "manifold" in question is not the finite dimensional space-time, but the infinite dimensional space of all solutions of Einstein's equations. $\endgroup$ – Igor Khavkine Jul 11 '11 at 19:56
  • $\begingroup$ Ask not 'how can this approach make sense?', but 'does this even make sense?'. You may be confusing two different meanings of the work 'quantize'... $\endgroup$ – David Roberts Jul 11 '11 at 23:39
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    $\begingroup$ Dear Hyun S. Yang: Unfortunately, this website works best with well-posed questions that you may expect to have precise mathematical answers. Right now, I think your question is too speculative to admit a mathematical answer. Please consider asking at physics.stackexchange.com $\endgroup$ – S. Carnahan Jul 13 '11 at 1:12