Minimal Liouville gravity correlation numbers from Douglas string equation

We continue the study of (q, p) Minimal Liouville Gravity with the help of Douglas string equation. We generalize the results of [1,2], where Lee-Yang series (2, 2s + 1) was studied, to (3, 3s + p 0) Minimal Liouville Gravity, where p 0 = 1, 2. We demonstrate that there exist such coordinates \u3c4 m,n on the space of the perturbed Minimal Liouville Gravity theories, in which the partition function of the theory is determined by the Douglas string equation. The coordinates \u3c4 m,n are related in a non-linear fashion to the natural coupling constants \u3bb m,n of the perturbations of Minimal Lioville Gravity by the physical operators O m,n . We find this relation from the requirement that the correlation numbers in Minimal Liouville Gravity must satisfy the conformal and fusion selection rules. After fixing this relation we compute three- and four-point correlation numbers when they are not zero. The results are in agreement with the direct calculations in Minimal Liouville Gravity available in the literature [3-5]. \ua9 2014 The Author(s)

Multi-scale ordering of self-assembled InAs/GaAs(001) quantum dots

Ordering phenomena related to the self-assembly of InAs quantum dots (QD) grown on GaAs(001) substrates are experimentally investigated on different length scales. On the shortest length-scale studied here, we examine the QD morphology and observe two types of QD shapes, i.e., pyramids and domes. Pyramids are elongated along the [1-10] directions and are bounded by {137} facets, while domes have a multi-facetted shape. By changing the growth rates, we are able to control the size and size homogeneity of freestanding QDs. QDs grown by using low growth rate are characterized by larger sizes and a narrower size distribution. The homogeneity of buried QDs is measured by photoluminescence spectroscopy and can be improved by low temperature overgrowth. The overgrowth induces the formation of nanostructures on the surface. The fabrication of self-assembled nanoholes, which are used as a template to induce short-range positioning of QDs, is also investigated. The growth of closely spaced QDs (QD molecules) containing 2–6 QDs per QD molecule is discussed. Finally, the long-range positioning of self-assembled QDs, which can be achieved by the growth on patterned substrates, is demonstrated. Lateral QD replication observed during growth of three-dimensional QD crystals is reported