The condensate in commutative

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The condensate in commutative and
noncommutative theories
Dmitri Bykov Moscow State University
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Overview of dimension two condensates in
gluodynamics
and
Contribute to OPEs, for example,
The gluon condensate may be sensitive to various
topological defects such as Dirac strings and
monopoles.
F.V.Gubarev, L.Stodolsky, V.I.Zakharov, 2001
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Curci-Ferrari gauge
K.-I. Kondo, 2001
BRST-invariant on-shell
Landau-type -gauges
D.B., A.A.Slavnov, 2005
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Gauge theory on a noncommutative plane
Weyl ordering
Moyal product of symbols
Product of operators
Review M.Douglas, N.Nekrasov, 2001
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Noncommutative gauge theory as a matrix model
Action
is a field with values in the Lie
algebra
The shift
leads us to conventional gauge theory with action
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Gauge transformations in the matrix model language
(homogeneous!)
Thus, we have a gauge-invariant (non-local)
operator
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In the conventional gauge theory language this
operator is
A.A.Slavnov, 2004
an (infinite) constant
operator with zero v.e.v.
operator with the desired condensate as its v.e.v.
Is the condensate
gauge-invariant?
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Important question arises
?
!
This is not always true
Counterexample the partial isometry operator
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Back to our case
R.N.Baranov, D.B., A.A.Slavnov, 2006
Under a quantum gauge transformation
the variation of the condensate is non-zero
is fixed!
Here
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1. The operator is still gauge-invariant, if one
considers Ward identities.
2. The volume-averaged operator decouples from
Greens functions
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Result

• The condensate is invariant under a limited set
  of gauge transformations.

Thank you for attention.