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Energy-momentum conservation laws in higher-dimensional Chern-Simons models
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Energy-momentum conservation laws in higher-dimensional Chern?C
Simons models
G.Sardanashvily
Department of Theoretical Physics, Moscow State University, 117234 Moscow, Russia
E-mail: sard@grav.phys.msu.su
URL: http://webcenter.ru/??sardan/
Abstract. Though a Chern?CSimons (2k ? 1)-form is not gauge-invariant and it depends on a
background connection, this form seen as a Lagrangian of gauge theory on a (2k?1)-dimensional
manifold leads to the energy-momentum conservation law.
The local Chern?CSimons (henceforth CS) form seen as a Lagrangian of the gauge
theory on a 3-dimensional manifold is well known to lead to the local conservation law of
the canonical energy-momentum tensor. Generalizing this result, we show that a global
higher-dimensional CS gauge theory admits an energy-momentum conservation law in
spite of the fact that its Lagrangian depends on a background gauge potential and that
one can not ignore its gauge non-invariance. The CS gravitation theory here is not
considered (e.g., [1, 3]).
We derive Lagrangian energy-momentum conservation laws from the first variational
formula (e.g., [5, 7, 8, 9, 10]). Let us consider a first order field theory on a fibre bundle
Y ?? X over an n-dimensional base X. Its Lagrangian is defined as a density L = Ldnx
on the first order jet manifold J1Y of sections of Y ?? X. Given bundle coordinates
(x??, yi) on a fibre bundle Y ?? X, its first and second order jet manifolds J1Y and J2Y
are equipped with the adapted coordinates (x??, yi, yi??) and (x
??, yi, yi??, y
i
???), respectively.
We will use the notation ?? = dnx and ??? = ??????.
Given a Lagrangian L on J1Y , the corresponding Euler?CLagrange operator reads
??L = ??iL??
i ?? ?? = (?iL ? d???
??
i )L??
i ?? ??, (1)
where ??i = dyi ? yi??dx
?? are contact forms and d?? = ??? + y
i
???i + y
i
????
??
i are the total
derivatives, which yield the total differential
dH? = dx
?? ?? d??? (2)
acting on the pull-
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