1. The condensate in commutative and noncommutative theories Dmitri Bykov Moscow State University

2. 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]

3. Curci-Ferrari gauge: BRST-invariant on-shell [K.-I. Kondo, 2001] Landau-type -gauges: [D.B., A.A.Slavnov, 2005]

4. Gauge theory on a noncommutative plane Weyl ordering Product of operators Moyal product of symbols [Review: M.Douglas, N.Nekrasov, 2001]

5. Noncommutative gauge theory as a matrix model is a field with values in the Lie algebra Action: The shift leads us to conventional gauge theory with action

6. Gauge transformations in the matrix model language (homogeneous!) Thus, we have a gauge-invariant (non-local) operator

7. 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?

8. Important question arises: ? This is not always true ! Counterexample – the partial isometry operator

9. Back to our case Under a «quantum» gauge transformation the variation of the condensate is non-zero: Here is fixed! [R.N.Baranov, D.B., A.A.Slavnov, 2006]

10. 1. The operator is still gauge-invariant, if one considers Ward identities. 2. The volume-averaged operator decouples from Green’s functions

11. Result: The condensate is invariant under a limited set of gauge transformations. Thank you for attention.