# If a fundamental theory exibits e.g. a mirror symmetry, in what sense it the underlying geometry real?

Are the more recently discovered symmetries in string theory such that the theories based on mirroring geometries are absolutely the same from an observable point of view?

I have mirror symmetry in mind, i.e. by symmetries I mean the dualities encountered in string theory.

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Say you take e.g. take the Schwarzschild metric from general relativity and consider its quivalent in some theory of quantum gravity theory. There, if you don't look close enough, i.e. if you look at averages, the quantum-Schwarzschild space would look classical, namely like the general relativity Schwarzschild space.

Now the target space in string theory is one manifold and it leads to a quantized theory. But if there are now actually two mirroring manifolds where this theory can come from, the I guess the classical version looks like neighter of these geometries. The following question comes from this observation, that there are totally different (mirroring) options with which Calabi-Yau you can start (considered in the Lagrangian) if you go in the quantization direction and the conclusion that neighter of will be the limit, or there are more limits.

If we assume that the world is described by such a theory, do we besically live in two indestingushable geometries at once?

To put it differently, I ask if string theory can be considered a quantum gravity in the sense that there is a classical limit down to classical differential geometry, especially regarding the extra dimensions. And if so, to what geometry in these higher dimensions does the limit lead? I do know that at least some of the geometry must come from the string excitation representing a quantizations of a metric/graviton. But I suspect different metrics in the non-linear sigma model Lagrangian will also affect the classical picture in some essential way, directly or not.

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I think I will withdraw the question then, as I am unable to help you. –  dmckee Jun 7 '12 at 0:20
What do you mean by "indistinguishable"? If you mean mathematically for the mathematical models, than that is incorrect: the two geometries in mirrored Calabi-Yau manifolds are often very different. If you mean physically based on observations, note that two theories can give effectively agree based on present observables yet still be possible to be distinguished by future observations. –  Willie Wong Jun 7 '12 at 10:24
@WillieWong: No, the fact that the two Calabi-Yau manifolds are very different is the exact reason why I ask. If the geometries in which the theories lie are different, while the observable physical quantities are the same, can one speak of reality having eighter of these two geometric forms? I personally have no problem with giving up any geometrical picture of the world, but it would be interesting to know (if neighter of these two different options are our effective classical geometry) which geometry comes about, i.e. the geometry that we eventually see? Seems like it's not one of the two. –  NikolajK Jun 7 '12 at 10:29
Based on your previous comment, it appears that your question is about philosophy, not physics. It is like asking if I know (and only know) that a number's decimal expansion starts with 3.141592653589793, is that (abstract) number a fixed number? Or is that number simultaneously the equivalence class of all numbers with that initial decimal expansion? –  Willie Wong Jun 7 '12 at 10:34
In some sense, your question is similar to the question which drove the formulation (and adoption) of Occam's Razor as a meta-scientific principle. –  Willie Wong Jun 7 '12 at 10:40

For two manifolds related by a mirror symmetry, all the predictions are the same if it is IIA strings on one vs. IIB strings on the other--- the two ideas, compactify IIA on M or compactify IIB on mirror-of-M are identical, and the two manifold/string-theory pairs may not be distinguished in any way--- there is no difference between them, they are the exact same theory in two different languages. It is not meaningful to ask which is right.

But IIA theory on M is not the same as IIA theory on the mirror. The easiest way to understand this is using a circle, where mirror symmetry is T-duality, and the mirror symmetry is just the generalization of this to the general Calabi Yau which is mathematically most interesting, because it tells you that the IIA string spectrum on one manifold is equal to the IIB string spectrum on the other.

The identity of T-duality (known already to Schwarz and collaborators in the 1980s) means that there is no difference really between IIA and IIB theory, they are the same theory with a different geometrical language for the microscopic degrees of freedom. The duality is fascinating, but whenever one geometry becomes large and classical (decompactified), the dual description becomes remote and super-quantum (sub-planckian). So generally, you know which description you should be using.

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Thanks. (But you don't touch the second question here right? Namely if or how one or both of the manifolds are found in the classical limit as the classical geometry.) –  NikolajK Jun 7 '12 at 23:25
@NickKidman: The problem is that neither is found, because the geometry is only defined by probing with string stuff. The geometry is secondary, it is only defined to the extent that the probes you have in the string theory allow you to smell it. If you consider yourself probing with type IIA probes, like 0 branes, you will find a IIA geometry. If you probe with type IIB probes, like D1 branes, you will find the other geometry. Both geometries are only well defined at weak coupling where the strings are perturbative, different probes see different things because they are mutually nonlocal. –  Ron Maimon Jun 8 '12 at 1:52