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In canonical quantum gravity we have Wheeler–DeWitt equation and we can chose homogeneous and isotropic ansatz (it kills many problems) to describe quantum universe.

In string theory, as I understand, we still don't know enough about string field theory and can't write something like WDW, we can only get classical supergravity equations with string corrections, right? We can only apply classical string gravity to cosmology and canonically quantize it.

But maybe we can use homogeneous and isotropic ansatz to find string equation on "wave function" of universe froms string field theory? I am very interesed in it.

What should I read about string quantum cosmology, if it exists? About generalization of gravity in string field theory?

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Very nice interests! extraordinarily important questions.

Unfortunately you're right. So far "string cosmology" means "string inspired cosmology" or "cosmology in supergravity". We are nearly completely ignorant about how a full stringy cosmology really looks like.

The only source that I know that tries to produce a quantum cosmology by writing an analogue of a Wheeler-DeWitt for string field theory is Baby universes, third quantization and the cosmological constant. A really nice (and relatively unknown) paper!

Everything you asked is a very good (and difficult with our current understanding) avenue of research.

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