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Register a new userOn the gauge features of gravity on a Lie algebroid structure
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We present the geometric formulation of gravity based on the mathematical structure of a Lie Algebroid. We show that this framework provides the geometrical setting to describe the gauge propriety of gravity.
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The $(4+4)$-dimensional $kappa$-deformed quantum phase space as well as its $(10+10)$-dimensional covariant extension by the Lorentz sector can be described as Heisenberg doubles: the $(10+10)$-dimensional quantum phase space is the double of $D=4$ $kappa$-deformed Poincare Hopf algebra $mathbb{H}$ and the standard $(4+4)$-dimensional space is its subalgebra generated by $kappa$-Minkowski coordinates $hat{x}_mu$ and corresponding commuting momenta $hat{p}_mu$. Every Heisenberg double appears as the total algebra of a Hopf algebroid over a base algebra which is in our case the coordinate sector. We exhibit the details of this structure, namely the corresponding right bialgebroid and the antipode map. We rely on algebraic methods of calculation in Majid-Ruegg bicrossproduct basis. The target map is derived from a formula by J-H. Lu. The coproduct takes values in the bimodule tensor product over a base, what is expressed as the presence of coproduct gauge freedom.
Higgs fields on gauge-natural prolongations of principal bundles are defined by invariant variational problems and related canonical conservation laws along the kernel of a gauge-natural Jacobi morphism.
Using the quantum construction of the BV-BFV method for perturbative gauge theories, we show that the obstruction for quantizing a codimension 1 theory is given by the second cohomology group with respect to the boundary BRST charge. Moreover, we give an idea for the algebraic construction of codimension $k$ quantizations in terms of $mathbb{E}_k$-algebras and higher shifted Poisson structures by formulating a higher version of the quantum master equation.
We consider the extension problem for Lie algebroids over schemes over a field. Given a locally free Lie algebroid Q over a scheme (X,O), and a sheaf of finitely generated Lie O-algebras L, we determine the obstruction to the existence of extensions 0 --> L --> E --> Q --> 0, and classify the extensions in terms of a suitable Lie algebroid hypercohomology group. In the preliminary sections we study free Lie algebroids and recall some basic facts about Lie algebroid hypercohomology.
Let $K$ be a simply connected compact Lie group and $T^{ast}(K)$ its cotangent bundle. We consider the problem of quantization commutes with reduction for the adjoint action of $K$ on $T^{ast}(K).$ We quantize both $T^{ast}(K)$ and the reduced phase space using geometric quantization with half-forms. We then construct a geometrically natural map from the space of invariant elements in the quantization of $T^{ast}(K)$ to the quantization of the reduced phase space. We show that this map is a constant multiple of a unitary map.
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