Week 01.03.2009 – 07.03.2009

In some recent work with Kronheimer we have carried over a version of Juhasz' Heegaard Floer Homology for sutured three manifolds to Instanton Floer Homology. This leads to amongst over things a streamlined proofs of property P for all knots and that symplectic manifolds have non-vanishing Donaldson invariants.

In this talk I will describe proposals for quantum simulations in Ion traps and BECs. It will be shown that the radial degree of freedom of strings of trapped ions in the quantum regime may be prepared and controlled accurately through the variation of the external trapping potential while at the same time its properties are measurable with high spatial and temporal resolution. This provides a new test-bed giving access to static and dynamical properties of the physics of quantum-many-body systems and quantum phase transitions that are hard to simulate on classical computers. Furthermore, it allows for the creation of double well potentials with experimentally accessible tunnelling rates and with applications in testing the foundations of quantum physics and precision sensing. A scheme for the study of methods for detecting Unruh-like acceleration radiation effects in a Bose-Einstein condensate in a 1+1 dimensional setup will be descried. In particular, the dispersive effects of the Bogoliubov spectrum on the ideal case of exact thermalization will be explained.

Brane Tilings are known to describe the largest known class of SCFT's in 3+1 dimensions. There is a well established formalism to find AdS_5 x SE_5 duals to these SCFT's and to compare results on both sides. This talk extends this formalism to 2+1 dimensional SCFT's, living on the world volume of M2 branes, which are dual to AdS_4 x SE_7 backgrounds of M theory. The SCFT's are quiver gauge theories with 4 supercharges (N=2 in 2+1 dimensions) and Chern Simons (CS) couplings. They admit a moduli space of vacuum configurations which is a CY4 cone over SE_7. The talk will go over the formalism and look at several examples in detail. The computation of scaling dimensions will be mentioned and relations to regular toric Fano 3-folds if time permits.

We attempt to indirectly deduce the form of quantum geometry on the M5-brane in the presence of a constant C-field 3-form potential. We first show how the noncommutative geometry on the D3-brane in a constant 2-form B-field can be deduced by analysing open strings ending on the D3-brane. We show that the string boundary condition is a modified Nahm equation, and that these modifications can be understood in terms of the noncommutative geometry. Similarly, the boundary condition for M2-branes ending on the M5-brane is given by the Basu-Harvey equation, modified by the C-field. By an analogy, we interpret this modification in terms of a proposed quantum geometry on the M5-brane. This geometry is specified by a 3-bracket, unlike the usual 2-bracket for noncommutative geometry.