In string theory, noncommutativity of bosonic spacetime coordinates naturally emerges in a (multiple) D-brane vorldvolume, when a constant NS--NS two-form is turned on. More recently, in the context of the Dijkgraaf-Vafa correspondence, which relates N=1 supersymmetric gauge theories and matrix models, it was suggested that non-anticommutativity of superspace coordinates naturally appears in a D-brane worldvolume when a constant R--R two-form is turned on in ten dimensions. A similar phenomenon was discovered in four dimensions when a constant self-dual graviphoton field strength is taken as a superstring background.
Fermionic non-anticommutativity means that the odd superspace coordinates obey a Clifford algebra instead of being anticommuting. It is then possible to retain the commutativity of the bosonic spacetime coordinates, which renders the field theory in question local and tractable. In particular, half of the supersymmetry and the Lorentz symmetry can be kept in non-anticommutative gauge theories, while the deformation of the Lagrangian is merely polynomial in the deformation parameter. This feature makes non-anticommutative field theories very attractive for researchers simply because they allow for a much easier study as compared to standard (bosonic) noncommutative models which are nonlocal as field theories.
One of the major tasks is to go beyond the formal introduction of noncommutativity at the level of the Lagrangian and to construct noncommutative solutions, both at the perturbative and non-perturbative level. The nonlocality of the standard Moyal star product often gives rise to highly involved equations of motion. The non-anticommutative extension of the Moyal star product in its nilpotent version leads to a finite derivative expansion, which drastically simplifies the equations of motion. Recently, several perturbative studies of non-anticommutative quantum field theories were performed by a number of research groups, resulting in an extension of renormalizability and a derivation of certain non-renormalization theorems. These issues are currently under intense investigation world-wide.
The primary goal of the proposed joint research is to win understanding of the role of noncommutativity in string theory. More specifically, we would like to classify which kind of non(anti)commutativity can descend from string theory to quantum field theory, in order to focus and strengthen the case for non(anti)commutative gauge field theory. To this end we propose to study supersymmetric Born-Infeld actions, the AdS/CFT correspondence in the case of coinciding D-branes, noncommutative deformations of non-linear sigma models, the construction of noncommutative BPS configurations (like instantons and solitons) and the embedding of minimal non-anticommutative N=2 super Yang-Mills into string theory.
André Fischer and Olaf Lechtenfeld:
The noncommutative sine-Gordon breather
arXiv:0811.1048 [hep-th] (J. Math. Phys. 50 (2009) 095201)
Seçkin Kürkçüoglu
Noncommutative Q-lumps
arXiv:0808.3745 [hep-th] ()
Olaf Lechtenfeld, Alexander D. Popov and Richard J. Szabo
SU(3)-equivariant quiver gauge theories and nonabelian vortices
arXiv:0806.2791 [hep-th] (JHEP 0808 (2008) 093)
Seçkin Kürkçüoglu:
Noncommutative nonlinear sigma models and integrability
arXiv:0804.3782 [hep-th] (Phys. Rev. D 78 (2008) 065020)
Sergei V. Ketov and Olaf Lechtenfeld
Non-anticommutative solitons
arXiv:0803.2867 [hep-th] (Phys. Lett. B 663 (2008) 353-359)
Olaf Lechtenfeld:
Noncommutative solitons:
arXiv:0710.2074 [hep-th] (Talk at the "Third Mexican Meeting on Mathematical and Experimental Physics'' at El Colegio Nacional, Mexico City, Mexico, 14-17 September 2007;
AIP Conference Proceedings Vol. 977, pp. 37-51)
Christian Gutschwager, Tatiana A. Ivanova and Olaf Lechtenfeld:
Scattering of noncommutative waves and solitons in a supersymmetric chiral model in 2+1 dimensions
arXiv:0710.0079 [hep-th] (JHEP 0711 (2007) 052)
Seçkin Kürkçüoglu and Olaf Lechtenfeld:
Quantum aspects of the noncommutative sine-Gordon model
arXiv:0708.1310 [hep-th] (JHEP 0709 (2007) 020)
Olaf Lechtenfeld, Alexander D. Popov and Richard J. Szabo:
Quiver gauge theory and noncommutative vortices
arXiv:0706.0979 [hep-th]
(Talk by O.L. at the 21st Nishinomiya-Yukawa Memorial Symposium in Nishinomiya/Kyoto, Japan, 11-15 November 2006;
Proceedings: Prog. Theor. Phys. Suppl. 171 (2007) 258-268)
Olaf Lechtenfeld and Alexander D. Popov:
Noncommutative solitons in a supersymmetric chiral model in 2+1 dimensions
arXiv:0704.0530 [hep-th] (JHEP 0706 (2007) 065)
Johannes Brödel, Tatiana Ivanova and Olaf Lechtenfeld:
Construction of noncommutative instantons in 4k dimensions
hep-th/0703009 (Mod. Phys. Lett. A 23 (2008) 179-189)
Michael Klawunn, Olaf Lechtenfeld and Stefan Petersen:
Moduli-space dynamics of noncommutative abelian sigma-model solitons
hep-th/0604219 (JHEP 0606 (2006) 028)
Olaf Lechtenfeld, Alexander D. Popov and Richard J. Szabo:
Rank two quiver gauge theory, graded connections and noncommutative vortices
hep-th/0603232 (JHEP 0609 (2006) 054)
Tatiana A. Ivanova and Olaf Lechtenfeld:
Noncommutative instantons on CPn
hep-th/0603125 (Phys. Lett. B 639 (2006) 407-410)