Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Observables in terms of connection and curvature variables for
  Einstein's equations with two commuting Killing vectors   Einstein's equations with two commuting Killing vectors and the associated
Lax pair are considered. The equations for the connection $A(\varsigma, \eta,
\gamma)=\Psi_{,\gamma}\Psi^{-1}$, where $\gamma$ the variable spectral
parameter are considered. A transition matrix ${\cal T}= A(\varsigma, \eta,
\gamma)A^{-1}(\xi, \eta, \gamma)$ for $A$ is defined relating $A$ at ingoing
and outgoing light cones. It is shown that it satisfies equations familiar from
integrable pde's theory. A transition matrix on $\varsigma={\mbox constant}$ is
defined in an analogous manner.
  These transition matrices allow us to obtain a hierarchy of integrals of
motion with respect to time, purely in terms of the trace of a function of the
connections $g_{,\varsigma}g^{-1}$ and $g_{,\eta}g^{-1}$. Furthermore a
hierarchy of integrals of motion in terms of the curvature variable
$B=A_{,\gamma}A^{-1}$, involving the commutator $[A(1), A(-1)]$, is obtained.
  We interpret the inhomogeneous wave equation that governs $\sigma=ln N$, $N$
the lapse, as a Klein-Gordon equation, a dispersion relation relating energy
and momentum density, based on the first connection observable and hence this
first observable corresponds to mass. The corresponding quantum operators are
$\frac{\partial}{\partial t}$, $\frac{\partial}{\partial z}$ and this means
that the full Poincare group is at our disposal.

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The categories of this paper are: astro-ph
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The categories of this paper are: atom-ph
25
The categories of this paper are: patt-sol
16
The categories of this paper are: stat
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The categories of this paper are: eess
10
The categories of this paper are: q-bio
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The categories of this paper are: comp-gas
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The categories of this paper are: cmp-lg
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
1
The categories of this paper are: cs
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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
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The categories of this paper are: adap-org
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The categories of this paper are: cond-mat