×

Loop equation in two-dimensional noncommutative Yang-Mills theory. (English) Zbl 1243.81214

Summary: The classical analysis of Kazakov and Kostov of the Makeenko-Migdal loop equation in two-dimensional gauge theory leads to usual partial differential equations with respect to the areas of windows formed by the loop. We extend this treatment to the case of \(U(N)\) Yang-Mills defined on the noncommutative plane. We deal with all the subtleties which arise in their two-dimensional geometric procedure, using where needed results from the perturbative computations of the noncommutative Wilson loop available in the literature. The open Wilson line contribution present in the non-commutative version of the loop equation drops out in the resulting usual differential equations. These equations for all \(N\) have the same form as in the commutative case for \(N\rightarrow \infty \). However, the additional supplementary input from factorization properties allowing to solve the equations in the commutative case is no longer valid.

MSC:

81T75 Noncommutative geometry methods in quantum field theory
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
81T13 Yang-Mills and other gauge theories in quantum field theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] doi:10.1016/0370-2693(79)90747-0 · doi:10.1016/0370-2693(79)90747-0
[2] doi:10.1016/0550-3213(80)90507-6 · doi:10.1016/0550-3213(80)90507-6
[3] doi:10.1016/0370-2693(79)90131-X · doi:10.1016/0370-2693(79)90131-X
[4] doi:10.1016/0550-3213(81)90258-3 · doi:10.1016/0550-3213(81)90258-3
[5] doi:10.1016/S0550-3213(00)00183-8 · Zbl 0984.81102 · doi:10.1016/S0550-3213(00)00183-8
[7] doi:10.1016/0550-3213(80)90072-3 · doi:10.1016/0550-3213(80)90072-3
[10] doi:10.1016/0370-2693(81)91203-X · doi:10.1016/0370-2693(81)91203-X
[12] doi:10.1016/0370-2693(94)90366-2 · doi:10.1016/0370-2693(94)90366-2
[13] doi:10.1016/S0550-3213(99)00474-5 · doi:10.1016/S0550-3213(99)00474-5
[14] doi:10.1016/S0550-3213(99)00708-7 · Zbl 0947.81137 · doi:10.1016/S0550-3213(99)00708-7
[15] doi:10.1016/S0370-2693(01)00270-2 · Zbl 0977.81078 · doi:10.1016/S0370-2693(01)00270-2
[16] doi:10.1016/S0550-3213(01)00437-0 · Zbl 0970.81046 · doi:10.1016/S0550-3213(01)00437-0
[18] doi:10.1016/S0920-5632(03)01726-2 · Zbl 1097.81774 · doi:10.1016/S0920-5632(03)01726-2
[19] doi:10.1016/S0550-3213(01)00477-1 · Zbl 0973.81125 · doi:10.1016/S0550-3213(01)00477-1
[20] doi:10.1103/PhysRevD.66.085012 · doi:10.1103/PhysRevD.66.085012
[22] doi:10.1016/0550-3213(74)90088-1 · doi:10.1016/0550-3213(74)90088-1
[23] doi:10.1103/PhysRevLett.72.3141 · doi:10.1103/PhysRevLett.72.3141
[24] doi:10.1016/0370-2693(77)90762-6 · doi:10.1016/0370-2693(77)90762-6
[25] doi:10.1016/0550-3213(83)90179-7 · doi:10.1016/0550-3213(83)90179-7
[27] doi:10.1016/S0370-2693(98)01319-7 · doi:10.1016/S0370-2693(98)01319-7
[33] doi:10.1016/S0370-2693(00)00391-9 · Zbl 0990.81130 · doi:10.1016/S0370-2693(00)00391-9
[40] doi:10.1016/S0370-1573(03)00059-0 · Zbl 1042.81581 · doi:10.1016/S0370-1573(03)00059-0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.