Noncommutative Multi-Instantons on R^{2n} x S^2

Generalizing self-duality on R^2 x S^2 to higher dimensions, we consider the Donaldson-Uhlenbeck-Yau equations on R^{2n} x S^2 and their noncommutative deformation for the gauge group U(2). Imposing SO(3) invariance (up to gauge transformations) reduces these equations to vortex-type equations for an abelian gauge field and a complex scalar on R^{2n}_\theta. For a special S^2-radius R depending on the noncommutativity \theta we find explicit solutions in terms of shift operators. These vortex-like configurations on R^{2n}_\theta determine SO(3)-invariant multi-instantons on R^{2n}_\theta x S^2_R for R=R(\theta). The latter may be interpreted as sub-branes of codimension 2n inside a coincident pair of noncommutative Dp-branes with an S^2 factor of suitable size.Comment: 1+8 pages, v2: reference added, version published in PL

Solutions to Yang-Mills equations on four-dimensional de Sitter space

We consider pure SU(2) Yang-Mills theory on four-dimensional de Sitter space dS$_4$ and construct a smooth and spatially homogeneous magnetic solution to the Yang-Mills equations. Slicing dS$_4$ as ${\mathbb R}\times S^3$, via an SU(2)-equivariant ansatz we reduce the Yang-Mills equations to ordinary matrix differential equations and further to Newtonian dynamics in a double-well potential. Its local maximum yields a Yang-Mills solution whose color-magnetic field at time $\tau\in{\mathbb R}$ is given by $\tilde{B}_a=-\frac12 I_a/(R^2\cosh^2\!\tau)$, where $I_a$ for $a=1,2,3$ are the SU(2) generators and $R$ is the de Sitter radius. At any moment, this spatially homogeneous configuration has finite energy, but its action is also finite and of the value $-\frac12j(j{+}1)(2j{+}1)\pi^3$ in a spin-$j$ representation. Similarly, the double-well bounce produces a family of homogeneous finite-action electric-magnetic solutions with the same energy. There is a continuum of other solutions whose energy and action extend down to zero.Comment: 1+7 pages; v2: introduction extended, gauge group representation dependence added, minor clarifications, 3 more references; v3: title change, published versio

Incoherent quantum feedback control of collective light scattering by Bose-Einstein condensates

It is well known that in the presence of a ring cavity the light scattering from a uniform atomic ensemble can become unstable resulting in the collective atomic recoil lasing. This is the result of a positive feedback due to the cavity. We propose to add an additional electronic feedback loop based on the photodetection of the scattered light. The advantage is a great flexibility in choosing the feedback algorithm, since manipulations with electric signals are very well developed. In this paper we address the application of such a feedback to atoms in the Bose-Einstein condensed state and explore the quantum noise due to the incoherent feedback action. We show that although the feedback based on the photodetection does not change the local stability of the initial uniform distribution with respect to small disturbances, it reduces the region of attraction of the uniform equilibrium. The feedback-induced nonlinearity enables quantum fluctuations to bring the system out of the stability region and cause an exponential growth even if the uniform state is globally stable without the feedback. Using numerical solution of the feedback master equation we show that there is no feedback-induced noise in the quadratures of the excited atomic and light modes. The feedback loop, however, introduces additional noise into the number of quanta of these modes. Importantly, the feedback opens an opportunity to position the modulated BEC inside a cavity as well as tune the phase of scattered light. This can find applications in precision measurements and quantum simulations.Comment: 7 pages, 7 figure

Instantons in six dimensions and twistors

Recently, conformal field theories in six dimensions were discussed from the twistorial point of view. In particular, it was demonstrated that the twistor transform between chiral zero-rest-mass fields and cohomology classes on twistor space can be generalized from four to six dimensions. On the other hand, the possibility of generalizing the correspondence between instanton gauge fields and holomorphic bundles over twistor space is questionable. It was shown by Saemann and Wolf that holomorphic line bundles over the canonical twistor space Tw(X) (defined as a bundle of almost complex structures over the six-dimensional manifold X) correspond to pure-gauge Maxwell potentials, i.e. the twistor transform fails. On the example of X=CP^3 we show that there exists a twistor correspondence between Abelian or non-Abelian Yang-Mills instantons on CP^3 and holomorphic bundles over complex submanifolds of Tw(CP^3), but it is not so efficient as in the four-dimensional case because the twistor transform does not parametrize instantons by unconstrained holomorphic data as it does in four dimensions.Comment: 14 pages; v2: discussion of aims and results extended; v3: published versio

Chern-Simons flows on Aloff-Wallach spaces and Spin(7)-instantons

Due to their explicit construction, Aloff-Wallach spaces are prominent in flux compactifications. They carry G_2-structures and admit the G_2-instanton equations, which are natural BPS equations for Yang-Mills instantons on seven-manifolds and extremize a Chern-Simons-type functional. We consider the Chern-Simons flow between different G_2-instantons on Aloff-Wallach spaces, which is equivalent to Spin(7)-instantons on a cylinder over them. For a general SU(3)-equivariant gauge connection, the generalized instanton equations turn into gradient-flow equations on C^3 x R^2, with a particular cubic superpotential. For the simplest member of the Aloff-Wallach family (with 3-Sasakian structure) we present an explicit instanton solution of tanh-like shape.Comment: 1+17 pages, 1 figur

Instantons on sine-cones over Sasakian manifolds

We investigate instantons on sine-cones over Sasaki-Einstein and 3-Sasakian manifolds. It is shown that these conical Einstein manifolds are K"ahler with torsion (KT) manifolds admitting Hermitian connections with totally antisymmetric torsion. Furthermore, a deformation of the metric on the sine-cone over 3-Sasakian manifolds allows one to introduce a hyper-K"ahler with torsion (HKT) structure. In the large-volume limit these KT and HKT spaces become Calabi-Yau and hyper-K"ahler conifolds, respectively. We construct gauge connections on complex vector bundles over conical KT and HKT manifolds which solve the instanton equations for Yang-Mills fields in higher dimensions.Comment: 1+15 pages, 2 figure

Hidden Symmetries of the Open N=2 String

It is known for ten years that self-dual Yang-Mills theory is the effective field theory of the open N=2 string in 2+2 dimensional spacetime. We uncover an infinite set of abelian rigid string symmetries, corresponding to the symmetries and integrable hierarchy of the self-dual Yang-Mills equations. The twistor description of the latter naturally connects with the BRST approach to string quantization, providing an interpretation of the picture phenomenon in terms of the moduli space of string backgrounds.Comment: 24 pages, no figures; v2: typos correcte