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SIGMA 5 (2009), 013, 25 pages arXiv:0811.3850
https://doi.org/10.3842/SIGMA.2009.013
Contribution to the Proceedings of the XVIIth International Colloquium on Integrable Systems and Quantum Symmetries
Derivations of the Moyal Algebra and Noncommutative Gauge Theories
Jean-Christophe Wallet
Laboratoire de Physique Théorique, Bât. 210, CNRS, Université Paris-Sud 11,
F-91405 Orsay Cedex, France
Received October 29, 2008, in final form January 17, 2009; Published online January 30, 2009
Abstract
The differential calculus based on the derivations of
an associative algebra underlies most of the noncommutative
field theories considered so far. We review the essential
properties of this framework and the main features of
noncommutative connections in the case of non graded associative
unital algebras with involution. We extend this framework to the
case of Z2-graded unital involutive algebras. We
show, in the case of the Moyal algebra or some related
Z2-graded version of it, that the derivation based
differential calculus is a suitable framework to construct
Yang-Mills-Higgs type models on Moyal (or related) algebras, the
covariant coordinates having in particular a natural
interpretation as Higgs fields. We also exhibit, in one
situation, a link between the renormalisable NC φ4-model
with harmonic term and a gauge theory model. Some possible
consequences of this are briefly discussed.
Key words:
noncommutative geometry; noncommutative gauge theories.
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References
- Douglas M.R., Nekrasov N.A.,
Noncommutative field theory, Rev. Mod. Phys. 73
(2001), 977-1029,
hep-th/0106048.
- Szabo R.J., Quantum field theory on noncommutative spaces,
Phys. Rept. 378 (2003), 207-299,
hep-th/0109162.
- Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994,
available at http://www.alainconnes.org/downloads.html.
- Connes A., Marcolli M., A walk in the noncommutative garden,
2006,
available at
http://www.alainconnes.org/downloads.html.
- Schomerus V., D-branes and deformation quantization, J. High Energy Phys. 1999 (1999), no. 6, 030, 14 pages, hep-th/9903205.
- Seiberg N., Witten E.,
String theory and noncommutative geometry, J. High Energy
Phys. 1999 (1999), no. 9, 032, 93 pages,
hep-th/9908142.
- Gracia-Bondía J.M.,
Várilly J.C., Algebras of distributions suitable for phase space
quantum mechanics. I, J. Math. Phys. 29 (1988), 869-879.
- Várilly J.C., Gracia-Bondía J.M.,
Algebras of distributions suitable for phase-space quantum
mechanics. II. Topologies on the Moyal algebra, J. Math. Phys. 29 (1988), 880-887.
- Minwalla S., Van Raamsdonk M., Seiberg N., Noncommutative
perturbative dynamics, J. High Energy Phys. 2000
(2000), no. 2, 020, 31 pages,
hep-th/9912072.
- Chepelev I., Roiban R.,
Renormalization of quantum field theories on noncommutative
Rd. I. Scalars,
J. High Energy Phys. 2000 (2000), no. 5, 037,
31 pages,
hep-th/9911098.
- Matusis A., Susskind L., Toumbas N., The IR/UV connection in the
non-commutative gauge theories, J. High Energy Phys.
2000 (2000), no. 12, 002, 18 pages,
hep-th/0002075.
- Grosse H., Wulkenhaar R.,
Renormalisation of φ4-theory on noncommutative
R4 in the matrix base,
Comm. Math. Phys. 256 (2005), 305-374,
hep-th/0401128.
- Grosse H., Wulkenhaar R., Power-counting theorem for non-local
matrix models and renormalisation, Comm. Math. Phys.
254 (2005), 91-127,
hep-th/0305066.
- Rivasseau V.,
Non-commutative renormalization,
arXiv:0705.0705.
- Wallet J.C.,
Noncommutative induced gauge theories on Moyal spaces, J.
Phys. Conf. Ser. 103 (2008), 012007, 20 pages,
arXiv:0708.2471.
- Langmann E., Szabo R.J.,
Duality in scalar field theory on noncommutative phase spaces,
Phys. Lett. B 533 (2002), 168-177,
hep-th/0202039.
- Grosse H., Wulkenhaar R., Renormalisation of φ4-theory on
noncommutative
R2 in the matrix base,
J. High Energy Phys. 2003 (2003), no. 12, 019,
hep-th/0307017.
- Langmann E., Szabo R.J., Zarembo K., Exact solution of quantum
field theory on noncommutative phase spaces, J. High Energy
Phys. 2004 (2004), no. 1, 017, 69 pages,
hep-th/0308043.
- Langmann E., Szabo R.J., Zarembo K.,
Exact solution of noncommutative field theory in background
magnetic fields, Phys. Lett. B 569 (2003), 95-101,
hep-th/0303082.
- Vignes-Tourneret F.,
Renormalization of the orientable non-commutative Gross-Neveu
model, Ann. Henri Poincaré 8 (2007), 427-474,
math-ph/0606069.
- Grosse H., Wulkenhaar R., The β-function in
duality-covariant noncommutative φ4 theory, Eur. Phys.
J. C Part. Fields 35 (2004), 277-282,
hep-th/0402093.
- Lakhoua A., Vignes-Tourneret F., Wallet J.C.,
One-loop β-functions for the orientable non-commutative
Gross-Neveu
model, Eur. Phys. J. C Part. Fields 52 (2007), 735-742,
hep-th/0701170.
- Disertori M., Gurau R., Magnen J., Rivasseau V., Vanishing of beta
function of non commutative φ44 theory to all
orders, Phys. Lett. B 649 (2007), 95-102,
hep-th/0612251.
- Gurau R., Magnen J., Rivasseau V., Tanasa A., A translation-invariant renormalizable non-commutative scalar model, arXiv:0802.0791.
- Blaschke D.N., Gieres F., Kronberger E., Schweda M., Wohlgenannt M., Translation-invariant models for non-commutative gauge fields, J. Phys. A: Math. Theor. 41 (2008), 252002, 7 pages, arXiv:0804.1914.
- de Goursac A., Wallet J.C., Wulkenhaar R.,
Noncommutative induced gauge theory, Eur. Phys. J. C Part.
Fields 51 (2007), 977-987,
hep-th/0703075.
- Grosse H., Wohlgenannt M.,
Induced gauge theory on a noncommutative space, Eur. Phys. J.
C Part. Fields 52 (2007), 435-450,
hep-th/0703169.
- de Goursac A., On the effective action of noncommutative
Yang-Mills theory, J. Phys. Conf. Ser. 103 (2008),
012010, 16 pages,
arXiv:0710.1162.
- Grosse H., Wohlgennant M., Noncommutative QFT and renormalization, J. Phys. Conf. Ser. 53 (2006), 764-792, hep-th/0607208.
- de Goursac A., Wallet J.C., Wulkenhaar R., On the vacuum states
for noncommutative gauge theories, Eur. Phys. J. C Part.
Fields 56 (2008), 293-304,
arXiv:0803.3035.
- de Goursac A., Tanasa A., Wallet J.C.,
Vacuum configurations for renormalizable non-commutative scalar
models, Eur. Phys. J. C Part. Fields 53 (2008),
459-466, arXiv:0709.3950.
- Dubois-Violette M., Dérivations et calcul différentiel non
commutatif, C. R. Acad. Sci. Paris Sér. I Math. 307
(1988), 403-408.
- Dubois-Violette M., Michor P.W., Dérivations et calcul
différentiel non commutatif. II, C. R. Acad. Sci. Paris
Sér. I Math. 319 (1994), 927-931.
- Dubois-Violette M., Kerner R., Madore J., Noncommutative
differential geometry of matrix algebras, J. Math. Phys.
31 (1990), 316-322.
- Dubois-Violette M., Michor P.W., Connections on central bimodules
in noncommutative differential geometry, J. Geom. Phys. 20 (1996), 218-232,
q-alg/9503020.
- Dubois-Violette M., Lectures on graded differential algebras and
noncommutative geometry, in Proc. of the Workshop on
Noncommutative Differential Geometry and Its Applications to
Physics (Shonan, Japan, 1999), Editors Y. Maeda et al., Math.
Phys. Stud., Vol. 23, Dordrecht, Kluwer Academic Publishers,
2001, 245-306,
math.QA/9912017.
- Dubois-Violette M., Kerner R., Madore J.,
Noncommutative differential geometry and new models of gauge
theory, J. Math. Phys. 31 (1990), 323-330.
- Dubois-Violette M., Masson T., SU(n)-connections and noncommutative differential geometry, J. Geom. Phys. 25 (1998), 104-118, dg-ga/9612017.
- Masson T., On the noncommutative geometry of the endomorphism
algebra
of a vector bundle, J. Geom. Phys. 31 (1999), 142-152, math.DG/9803088.
Masson T., Submanifolds and quotient manifolds in noncommutative
geometry, J. Math. Phys. 37 (1996), 2484-2497,
q-alg/9507030.
- Masson T., Noncommutative generalization of SU(n)-principal
fiber bundles: a review, J. Phys. Conf. Ser. 103
(2008), 012003, 33 pages,
arXiv:0709.0856.
- Cagnache E., Masson T., Wallet J.C., Noncommutative
Yang-Mills-Higgs actions from derivation-based differential
calculus, arXiv:0804.3061.
- Marmo G., Vitale P., Zampini A., Noncommutative differential calculus for Moyal subalgebras, J. Geom. Phys. 56 (2006), 611-622, hep-th/0411223.
- Scheunert M., Generalized Lie algebras, J. Math. Phys. 20 (1979), 712-720.
- de Goursac A., Masson T., Wallet J.C., Noncommutative
ε-graded connections and application to Moyal space,
arXiv:0811.3567.
- Quillen D., Superconnections and the Chern character, Topology 24 (1985), 89-95.
- de Goursac A., Masson T., Wallet J.C., Work in progress.
- Estrada R., Gracia-Bondía J.M., Várilly J.C., On asymptotic expansions of twisted products, J. Math. Phys. 30 (1989), 2789-2796.
- Bichl A.A., Ertl M., Gerhold A., Grimstrup J.M., Grosse H., Popp L., Putz V., Schweda M., Wulkenhaar R., Non-commutative U(1) super Yang-Mills theory: Perturbative self-energy corrections, Internat. J. Modern Phys. A 19 (2004), 4231-4249, hep-th/0203141.
- Martin C.P., Sánchez-Ruiz D., The one-loop UV divergent structure of U(1) Yang-Mills theory on non-commutative R4, Phys. Rev. Lett. 83 (1999), 476-479, hep-th/9903077.
Grosse H., Krajewski T., Wulkenhaar R., Renormalisation of non-commutative Yang-Mills theories: a simple example, hep-th/0001182.
- Slavnov A.A., Consistent noncommutative quantum gauge theories, Phys. Lett. B. 565 (2003), 246-252,
hep-th/0304141.
- Wallet J.C., Algebraic set-up for the gauge-fixing of BF and SuperBF systems, Phys. Lett. B 235 (1990), 71-78.
- Birmingham D., Blau M., Rakowski M., Thomson G., Topological field theory, Phys. Rep. 209 (1991), 129-340.
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