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We discuss noncommutative solitons on a noncommutative torus and their application to tachyon condensation. In the large B limit, they can be exactly described by the Powers-Rieffel projection operators known in the mathematical literature. The resulting soliton spectrum is consistent with T-duality and is surprisingly interesting. It is shown that an instability arises for any D-branes, leading to the decay into many smaller D-branes. This phenomenon is the consequence of the fact that K-homology for type II von Neumann factor is labeled by R.
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Intersecting D-brane models and their T-dual magnetic compactifications provide an attractive framework for particle physics, allowing for chiral fermions and supersymmetry breaking. Generically, magnetic compactifications have tachyons that are usually removed by Wilson lines. However, quantum corrections prevent local minima for Wilson lines. We therefore study tachyon condensation in the simplest case, the magnetic compactification of type I string theory on a torus to eight dimensions. We find that tachyon condensation restores supersymmetry, which is broken by the magnetic flux, and we compute the mass spectrum of vector- and hypermultiplets. The gauge group $text{SO}(32)$ is broken to $text{USp}(16)$. We give arguments that the vacuum reached by tachyon condensation corresponds to the unique 8d superstring theory already known in the literature, with discrete $B_{ab}$ background or, in the T-dual version, the type IIB orientifold with three $text{O}7_-$-planes, one $text{O}7_+$-plane and eight D7-branes coincident with the $text{O}7_+$-plane. The ground state after tachyon condensation is supersymmetric and has no chiral fermions.
In order to assess possible observable effects of noncommutativity in deformations of quantum mechanics, all irreducible representations of the noncommutative Heisenberg algebra and Weyl-Heisenberg group on the two-torus are constructed. This analysis extends the well known situation for the noncommutative torus based on the algebra of the noncommuting position operators only. When considering the dynamics of a free particle for any of the identified representations, no observable effect of noncommutativity is implied.
We examine whether tachyon matter is a viable candidate for the cosmological dark matter. First, we demonstrate that in order for the density of tachyon matter to have an acceptable value today, the magnitude of the tachyon potential energy at the onset of rolling must be finely tuned. For a tachyon potential $V(T)sim M_{Pl}^4exp(-T/tau)$, the tachyon must start rolling at $Tsimeq 60tau$ in order for the density of tachyon matter today to satisfy $Omega_{T,0}sim 1$, provided that standard big bang cosmology begins at the same time as the tachyon begins to roll. In this case, the value of $Omega_{T,0}$ is exponentially sensitive to $T/tau$ at the onset of rolling, so smaller $T/tau$ is unacceptable, and larger $T/tau$ implies a tachyon density that is too small to have interesting cosmological effects. If instead the universe undergoes a second inflationary epoch after the tachyon has already rolled considerably, then the tachyon can begin with $T$ near zero, but the increase of the scale factor during inflation must still be finely tuned in order for $Omega_{T,0} sim 1$. Second, we show that tachyon matter, unlike quintessence, can cluster gravitationally on very small scales. If the starting value of $T/tau$ is tuned finely enough that $Omega_{T,0}sim 1$, then tachyon matter clusters more or less identically to pressureless dust. Thus, if the fine-tuning problem can be explained, tachyon matter is a viable candidate for cosmological dark matter.
We consider noncommutative theory of a compact scalar field. The recently discovered projector solitons are interpreted as classical vacua in the model considered. Localized solutions to the projector equation are pointed out and their brane interpretation is discussed. An example of the noncommutative soliton interpolating between such vacua is given. No strong noncommutativity limit is assumed.
We construct exact solitons on noncommutative tori for the type of actions arising from open string field theory. Given any projector that describes an extremum of the tachyon potential, we interpret the remaining gauge degrees of freedom as a gauge theory on the projective module determined by the tachyon. Whenever this module admits a constant curvature connection, it solves exactly the equations of motion of the effective string field theory. We describe in detail such a construction on the noncommutative tori. Whereas our exact solution relies on the coupling to a gauge theory, we comment on the construction of approximate solutions in the absence of gauge fields.
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