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Register a new userGravitational Chern-Simons and the adiabatic limit
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Authors
Brendan McLellan
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We compute the gravitational Chern-Simons term explicitly for an adiabatic family of metrics using standard methods in general relativity. We use the fact that our base three-manifold is a quasi-regular K-contact manifold heavily in this computation. Our key observation is that this geometric assumption corresponds exactly to a Kaluza-Klein Ansatz for the metric tensor on our three manifold, which allows us to translate our problem into the language of general relativity. Similar computations have been performed in a paper of Guralnik, Iorio, Jackiw and Pi (2003), although not in the adiabatic context.
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In this article we describe the relation between the Chern-Simons gauge theory partition function and the partition function defined using the symplectic action functional as the Lagrangian. We show that the partition functions obtained using these two Lagrangians agree, and we identify the semiclassical formula for the partition function defined using the symplectic action functional.
We compute the semiclassical formulas for the partition functions obtained using two different Lagrangians: the Chern-Simons functional and the symplectic action functional.
By using the Hamilton-Jacobi [HJ] framework the three dimensional Palatini theory plus a Chern-Simons term [PCS] is analyzed. We report the complete set of $HJ$ Hamiltonians and a generalized $HJ$ differential from which all symmetries of the theory are identified. Moreover, we show that in spite of PCS Lagrangian produces Einsteins equations, the generalized $HJ$ brackets depend on a Barbero-Immirzi like parameter. In addition we complete our study by performing a canonical covariant analysis, and we construct a closed and gauge invariant two form that encodes the symplectic geometry of the covariant phase space.
Chern-Simons theory on a closed contact three-manifold is studied when the Lie group for gauge transformations is compact, connected and abelian. A rigorous definition of an abelian Chern-Simons partition function is derived using the Faddeev-Popov gauge fixing method. A symplectic abelian Chern-Simons partition function is also derived using the technique of non-abelian localization. This physically identifies the symplectic abelian partition function with the abelian Chern-Simons partition function as rigorous topological three-manifold invariants. This study leads to a natural identification of the abelian Reidemeister-Ray-Singer torsion as a specific multiple of the natural unit symplectic volume form on the moduli space of flat abelian connections for the class of Sasakian three-manifolds. The torsion part of the abelian Chern-Simons partition function is computed explicitly in terms of Seifert data for a given Sasakian three-manifold.
The role played by Deligne-Beilinson cohomology in establishing the relation between Chern-Simons theory and link invariants in dimensions higher than three is investigated. Deligne-Beilinson cohomology classes provide a natural abelian Chern-Simons action, non trivial only in dimensions $4l+3$, whose parameter $k$ is quantized. The generalized Wilson $(2l+1)$-loops are observables of the theory and their charges are quantized. The Chern-Simons action is then used to compute invariants for links of $(2l+1)$-loops, first on closed $(4l+3)$-manifolds through a novel geometric computation, then on $mathbb{R}^{4l+3}$ through an unconventional field theoretic computation.
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