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Are there any models in string theory which are background independent?

If there are, would this mean that these models could be built in any number of dimensions? (Instead of assuming a fixed number of dimensions as it is usually done in string theory)?

And finally, if there could be string theories which are background independent, would this mean that these models would assume no symmetries, as physicist Lee Smolin says 1:

We have posited that the fundamental theory is background independent, which means there are no symmetries

Would this mean that background independent string theories would not assume any symmetry as fundamental (implying that global, local, gauge or even Lorentz and CPT symmetries are not fundamental)?

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Are there any models in string theory which are background independent?

Non-perturbatively, it is conjectured that there is a unique, underlying framework (M-theory), and all existing calculations in string theory are approximations that in principle should be derivable from this larger framework. So there are not different string theory models, but rather one unified non-perturbative framework. (However, no one knows precisely what this framework is, in the sense that no one can write down a fully non-perturbative definition of M-theory and derive all known string theory results from it).

This non-perturbative framework should be "background independent," in the sense that one should be able to solve its equations of motion to find classical backgrounds, and perturb around these backgrounds.

In practice, since we don't know how to formulate M-theory, many calculations either assume a particular background and work perturbatively, or solve for a background in some limit where the theory is well understood (eg supergravity). This practical limitation is not an "in principle" problem though -- it is not as if M-theory has some assumed, fixed background built into it.

If there are, would this mean that these models could be built in any number of dimensions? (Instead of assuming a fixed number of dimensions as it is usually done in string theory)?

No. The dimension of spacetime is fixed in string theory, in order to have a consistent Lorentz-invariant, unitary quantization of the theory. (although, weirdly, in different limits the apparent number of spacetime dimensions is different)

However the extra dimensions can be compactified, so in practice the low energy physics that we observe is 4-dimensional.

And finally, if there could be string theories which are background independent, would this mean that these models would assume no symmetries, as physicist Lee Smolin says 1:

We have posited that the fundamental theory is background independent, which means there are no symmetries

Would this mean that background independent string theories would not assume any symmetry as fundamental (implying that global, local, gauge or even Lorentz and CPT symmetries are not fundamental)?

To be honest I don't really understand what this quote means. However it is a generic property of quantum gravity that there are no global symmetries. Gauge symmetries are allowed (and indeed necessary if one is going to have a hope of embedding the Standard Model into string theory). They can be implemented in string theory by, for example, having multiple D-branes stacked at the same position. Having said that, gauge symmetries are really redundancies of description, not fundamental symmetries, and it should be possible (albeit probably much more complicated) to formulate the theory without gauge symmetries.

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I recommend the excellent blog post What is background independence and how important is it? to clarify common misconceptions about what is a precise and useful description of the intuition of "background independence" in quantum gravity.

With the later clarification in mind the answer to your actual question is yes.

Background independence in quantum gravity means the possibility to define a quantum theory of gravity at every point in the moduli space of allowed backgrounds, and a unitary mapping between the Hilbert spaces and algebra of observables of any two of them. A famous example within string theory where this intuition is made explicit (and so useful that indeed allow the computation of all the correlators of the theory) is the BCOV Kodaira-Spencer theory of gravity, the closed string field theory of the B-model topological string.

In the above context it was shown possible to explicitly construct a flat and unitary connection on the Hodge bundle over the moduli space of complex structures of a Calabi-Yau threefold (the relevant background dynamical variables). Intuitively the Hodge bundle has as fiber over a point the Hilbert space of the string field theory over the background with complex structure defined by the point; the existence of the connection means that there is no obstruction to stablish a map between the fiber at two points by deforming the path integral measure with exactly marginal operators, exactly what background independence means.

Reference: Quantum background independence in string theory.

For a string field theory example see background independent string field theory.

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