Strings, Matrix Models, Self-Dual Field Theories and Integrability

duration: 2000-2008
researchers: Albertsson, Brödel, Ivanova, Jünemann, Kling, Lechtenfeld, Popov, Sämann, Spendig, Uhlmann, Wolf; Siegel
funding: DFG priority program (1 postdoc) (in priority program 00-06, then 07-08), Volkswagen foundation (sabbatical research)
other projects

Abstract:

Since the 1990 work of Ooguri and Vafa it is clear that (non-supersymmetric) self-dual Yang-Mills (SDYM) and self-dual gravity (SDG) in four spacetime dimensions are linked to fermionic string theories with (at least) two world-sheet supersymmetries. The N=2 world-sheet supersymmetry enforces a (2,2) signature in four-dimensional spacetime, so that ultimately some Wick rotation is required to make contact with nature. However, N=2 string theories are radically simpler than superstring theories. In fact, they are exactly solvable and ideally suited as `toy models' with respect to important issues in string theory, such as vacuum structure, global symmetries, relation with quantum field theories (commutative and noncommutative) and nonperturbative aspects (D-branes, solitons, etc.).

In the presence of D-branes, N=2 strings in 2+2 dimensions can be coupled to a Kähler NS-NS two-form B. For a U(n) Chan-Paton group, the Seiberg-Witten limit of open N=2 strings yields noncommutative SDYM reduced to the D-brane world-volume. The choice of D3-, D2- or D1-branes produces the noncommutative SDYM, modified sigma model or standard (principal chiral) sigma model in 2+2, 2+1, 1+1 or 2+0 dimensions, respectively, all of which are integrable for any value of the noncommutativity parameter. In the 2+1 dimensional case, multi-solitons may be interpreted as D0-branes moving inside a stack of D2-branes.

Nonperturbative string dynamics is conveniently studied in the framework of string field theory, where important progress is being made. Surprisingly, open superstring field theory (in their cubic as well as nonpolynomial formulations) is integrable in the sense that its equations of motion can be written as compatibility conditions (zero-curvature conditions) of some linear equations. This allows for the application of powerful solution-generating techniques (the splitting and dressing methods suitably generalized to string field theory) towards the construction of nonperturbative classical superstring configurations. In this way, much-needed solutions to vacuum superstring field theory may be found, most conveniently in the Moyal formulation.

The integrability of open superstring field theory may be related to the well-known integrability of ten-dimensional supersymmetric Yang-Mills theory. The latter's reduction to zero dimensions yields the IKKT matrix model which captures nonperturbative aspects of type IIB superstring theory. A similar reduction should be possible for A- and B-type topological string theories in six dimensions or, equivalently, Chern-Simons and holomorphic Chern-Simons theories. This defines topological matrix models, a special kind of which should contain the N=2 string solitons.

The problems which are supposed to be investigated within the project:

Construction of solitons in N=2 string (field) theory and their D-brane interpretation. The latter will be obtained through a comparison of the spectrum of small fluctuations around the concrete solution with the spectrum of stringy excitations in the appropriate brane configurations. Obtaining the above solitons as solutions of a novel (cubic) matrix model may lead to the same result.
Construction of nonperturbative solutions of open superstring field theory and identification some of them with D-branes via an analysis of fluctuations around these configurations and also by embedding them into the IKKT matrix model. This relates to tachyon condensation and vacuum superstring field theory.
Covariant BV formulation of boundary (super)string field theory.
Investigation of solvability properties of the IKKT matrix model, type IIB superstrings and construction of exact D-brane solutions. Analysis of quantum properties of the IKKT and topological (cubic) matrix models.

Papers:

Tatiana A. Ivanova and Olaf Lechtenfeld:
Yang-Mills instantons and dyons on group manifolds
arXiv:0806.0394 [hep-th] (Phys. Lett. B 670 (2008) 91-94)

Alexander D. Popov:
Bounces/dyons in the plane wave matrix model and SU(n) Yang-Mills theory
arXiv:0804.3845 [hep-th] (Mod. Phys. Lett. A 24 (2009) 349)

Alexander D. Popov:
Explicit non-abelian monopoles and instantons in SU(N) pure Yang-Mills theory
arXiv:0803.3320 [hep-th] (Phys. Rev. D 77 (2008) 125026)

Alexander D. Popov:
Non-abelian vortices on Riemann surfaces: an integrable case
arXiv:0801.0808 [hep-th] (Lett. Math. Phys. 84 (2008) 139-148)

Alexander D. Popov:
Integrability of vortex equations on Riemann surfaces
arXiv:0712.1756 [hep-th] ()

Alexander D. Popov:
Sigma models with N=8 supersymmetries in 2+1 and 1+1 dimensions
hep-th/0702106 (Phys. Lett. B 647 (2007) 509-514)

Alexander D. Popov and Martin Wolf:
Hidden symmetries and integrable hierarchy of the N=4 supersymmetric Yang-Mills equations
hep-th/0608225 (Commun. Math. Phys. 275 (2007) 685-708)

Tatiana A. Ivanova and Olaf Lechtenfeld:
Noncommutative instantons on CPⁿ
hep-th/0603125 (Phys. Lett. B 639 (2006) 407-410)

Christian Sämann:
The mini-superambitwistor space
hep-th/0511251
(Talk at JINR workshop ``Supersymmetries and Quantum Symmetries'', Dubna, Russia, 27-31 July 2005;
Proceedings: SQS '05, pp. )

Martin Wolf:
Twistors and aspects of integrability of self-dual sym theory
hep-th/0511230
(Talk at JINR workshop ``Supersymmetries and Quantum Symmetries'', Dubna, Russia, 27-31 July 2005;
Proceedings: SQS '05, pp. )

Olaf Lechtenfeld and Christian Sämann:
Matrix models and D-branes in twistor string theory
hep-th/0511130 (JHEP 0603 (2006) 002)

Christian Sämann:
On the mini-superambitwistor space and N=8 super Yang-Mills theory
hep-th/0508137 ()

Alexander D. Popov, Christian Sämann and Martin Wolf
The topological B-model on a mini-supertwistor space and supersymmetric Bogomolny monopole equations
hep-th/0505161 (JHEP 0510 (2005) 058)

Martin Wolf:
On hidden symmetries of a super gauge theory and twistor string theory
hep-th/0412163 (JHEP 0502 (2005) 018)

Christian Sämann:
The topological B-model on fattened complex manifolds and subsectors of N=4 self-dual Yang-Mills theory
hep-th/0410292 (JHEP 0501 (2005) 042)

Alexander D. Popov and Martin Wolf:
Topological B-model on weighted projective spaces and self-dual models in four dimensions
hep-th/0406224 (JHEP 0409 (2004) 007)

Olaf Lechtenfeld and Alexander D. Popov:
Supertwistors and cubic string field theory for open N=2 strings
hep-th/0406179 (Phys. Lett. B 598 (2004) 113-120)

Alexander D. Popov and Christian Sämann:
On supertwistors, the Penrose-Ward transform and N=4 super Yang-Mills theory
hep-th/0405123 (Adv. Theor. Math. Phys. 9 (2005) 931-998)

Matthias Ihl, Alexander Kling and Sebastian Uhlmann:
String field theory projectors for fermions of integral weight
hep-th/0312314 (JHEP 0403 (2004) 002)

Alexander Kling and Sebastian Uhlmann:
String field theory vertices for fermions of integral weight
hep-th/0306254 (JHEP 0307 (2003) 061)

Alexander Kling, Olaf Lechtenfeld, Alexander D. Popov and Sebastian Uhlmann:
Solving string field equations: new uses for old tools
hep-th/0212335
(Talk by O.L. at the 35th International Symposium Ahrenshoop on the Theory of Elementary Particles in Berlin, Germany, 26-30 August 2002:
Fortsch. Phys. 51 (2003) 775-780)

Alexander Kling, Olaf Lechtenfeld, Alexander D. Popov and Sebastian Uhlmann:
On nonperturbative solutions of superstring field theory
hep-th/0209186 (Phys. Lett. B 551 (2003) 193-201)

Olaf Lechtenfeld, Alexander D. Popov and Sebastian Uhlmann:
Exact solutions of Berkovits' string field theory
hep-th/0204155 (Nucl. Phys. B 637 (2002) 119-142)

Olaf Lechtenfeld and Warren Siegel:
Light-cone gauge for N=2 strings
hep-th/0204073 (Russ. Phys. J. 45 (2002) 712-718)

Klaus Jünemann and Bernd Spendig:
D-brane scattering of N=2 strings
hep-th/0108069 (Phys. Lett. B 520 (2001) 163-169)

Olaf Lechtenfeld, Alexander D. Popov and Bernd Spendig:
Open N=2 strings in a B-field background and noncommutative self-dual Yang-Mills
hep-th/0012200 (Phys. Lett. B 507 (2001) 317-326)

Olaf Lechtenfeld and Alexander D. Popov:
On the integrability of covariant field theory for the open N=2 string
hep-th/0009144 (Phys. Lett. B 494 (2000) 148-154)