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Metric structures in General Relativity
This is a very basic beginners question about Chern-Simons theory. I didn't find this spelled out explicitly anywhere, but I assume one does not sum over different choices of principle bundles when calculating the partition function? So is Chern-Simons theory not only a topological field theory that one can put on any topological manifold, but one that we can additionally put on any topological manifold with a Lie-group principle bundle?
Can one restrict to trivial principle bundles without loosing any of the physical interpretation? If one talks about the "3-manifold invariants" of the Chern-Simons theory, does that refer to the partition function on the trivial principle bundle over those manifolds?
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Or is the partition function independent of the choice of bundle? This is the literal analogue of Chern-Simons theory for a finite gauge group; however, a more interesting analogue is twisted Dijkgraaf-Witten theory, which might be what you were trying to get at. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Importance of the principal bundle in Chern-Simons theory Ask Question.
Asked 3 months ago. Active 3 months ago. Viewed times. We start by recalling some basic facts on the formulation of General Relativity in vacuum. Interestingly enough, Lor M can be empty, as Lorentzian metrics are obstructed if M is closed and its Euler characteristic does not vanish. In fact, in this case, the Euler characteristic of M vanishes if and only if M admits Lorentzian metrics. If non-empty, Lor M may be a very large space: intuitively speaking we can say that it is infinite-dimensional , but making this notion mathematically precise, in those cases where it can be done, requires some work.
The pair M,g is then called a Lorentzian manifold. Every manifold can be endowed with an infinite number of Koszul connections also called covariant derivatives on its tangent bundle. In fact, the space of all Koszul connections on a given manifold is an affine space modelled on the space of one-forms taking values on the endomorphism bundle of the tangent bundle.
This is a contractible infinite-dimensional space. On a Lorentzian manifold M,g there is a preferred choice of Koszul connection, which is given by the unique torsion-free Koszul connection which is compatible with the metric g, namely with the metric structure of M,g.
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This is the so-called Levi-Civita LC connection of M,g , and it is the basic object in which the formulation of General Relativity is based. Note that since the LC connection is metric-compatible, its holonomy is contained in O 3,1.
Quantum groups on fiber bundles - INSPIRE-HEP
Let M be a four-manifold. The vacuum Einstein equations are:. This turns out to be a second-order hyperbolic system of partial differential equations defined over M on the space of Lorentzian metrics of M, that is, the space of Lorentzian metric structures on M. A solution to equation 1 is hence a particular Lorentzian structure on M.
In many practical situations, one starts by finding a local metric g U by solving equation 1 on a contractible open set U of R 4. Notice that this geodesic extension is a highly complicated and non-unique process which has to be performed on a case by case basis. Either way, from the previous remarks it is clear that the formulation of GR crucially depends on equipping M with a metric structure, being particular solutions particular metric structures on M.
The discussion above concerns the standard formulation of GR in the language of vector bundles recall that the tangent bundle of a manifold is a particular type of vector bundle and Koszul connections.
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However, one can consider instead the equivalent but seemingly different set up of principal bundles and Ehresmann connections by using the well-known correspondence between Lorentzian vector bundles and principal O d-1,1 bundles as well as the correspondence between metric-compatible Koszul connections in the former and Ehresmann connections in the latter. The formulation of General Relativity in this set up goes as follows. Let O M denote a reduction of F M to a O 3,1 principal bundle, which, unsurprisingly enough is obstructed by the Euler characteristic of M.
The principal subbundle O M is a particular case of what is known in the literature as a G-structure , which consists of a principal subbundle of F M with structure group G. The concept of G-structure is of fundamental importance in differential geometry. Prominent examples of G-structures in Riemannian geometry are G 2 structures in seven dimensions, Spin 7 structures in eight dimensions or Calabi-Yau structures in even dimensions.
Principal Bundles: The Quantum Case
Ehresmann connections can be defined on arbitrary smooth principal bundles. In the case in which the principal bundle is the bundle of linear frames of a manifold an Ehresmann connection is also called a linear connection. An Ehresmann connection can be understood as an invariant horizontal subbundle of the tangent bundle of F M or, equivalently, as a one form in F M not M! This in particular implies that H is a horizontal vector subbundle of O M , the tangent bundle of O M.