When studying General Relativity, I learned that we use the Levi-Civita connection, i.e. torsion-less(or just symmetric) and compatible with the metric(the covariant derivative of the metric is equal to zero).
My question is: in other areas of physics, like in high-energy physics in the theory of gauge fields or in condensed matter physics in the study of the Berry phase, do we use the Levi-Civita connection or do we use other connections that might not be symmetric and/or compatible with the metric?
If not, since the Levi-Civita connection uniquely determines a metric, then do we not need a metric in those sub-fields of physics that don't use the Levi-Civita connection?
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asked Dec 27, 2017 at 15:54
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$\begingroup$ Gauge theory use the Ehresmann connection, which doesn't have torsion defined for it since there is no solder form. If you want a general affine connection you can look into affine gravity. $\endgroup$– SlereahCommented Dec 27, 2017 at 15:58
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$\begingroup$ @Slereah Hi and thanks for the reply. What do you mean with your last statement? $\endgroup$– TheQuantumManCommented Dec 27, 2017 at 15:59
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$\begingroup$ Affine gravity is a theory for gravity where the connection is independant from the metric, with both torsion and non-metricity, stemming from the spin tensor of the matter field. $\endgroup$– SlereahCommented Dec 27, 2017 at 16:00
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$\begingroup$ @Slereah Oh OK. It must have something to do with the Einstein-Cartan theory, right? If so, hasn't that theory been experimentally falsified? I thought that General Relativity was incompatible with experiments if we have torsion-full connections $\endgroup$– TheQuantumManCommented Dec 27, 2017 at 16:02
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1$\begingroup$ I want to point out to stop any potential confusion that the connections used in the areas you mentioned in your post are not connections on the tangent bundle, but connections in other bundles that are not "natural" with respect to the tangent bundle. Essentially, "affine" connections act on tensor fields, but connections such as the Yang-Mills connection act on sections of completely unrelated vector bundles, so they are quite different animals. And as Slereah said, you cannot actually define torsion for them. $\endgroup$– Bence RacskóCommented Dec 27, 2017 at 21:10
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1 Answer
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A Levi-Civita connection exists when we also have a metric, since apart from being torsion-less, it's also metric-compatible, in that the covariant derivative of the metric vanishes. We have theories in physics, such as in the field of topological materials where the Berry connection (and related quantities) $A_{mn}=i\langle{m}|\nabla_\vec{k}|{n}\rangle$ (for energy eigenstates $|n\rangle$ and $\vec{k}$ being the crystal momentum) is used, where they do not include any metric, hence the connection cannot be metric-compatible, and hence not Levi-Civita. It must be pointed out that the Berry connection is not a connection we add to our base manifold-spacetime, as it exists in the space of parameters of a given problem, such as the crystal momentum in condensed matter physics.
There are also theories, such as supergravity (the classical approximation of M-theory) that work with a spacetime with non-zero torsion.
answered Mar 30, 2020 at 13:07