# Research Projects of Prof. Lechtenfeld

The research of professor Olaf Lechtenfeld is motivated by or belongs to the area of string theory. His group conducts three main lines of investigations: extended supersymmetric (many-particle) mechanics, solitons in noncommutative field theories, and gauge fields in heterotic flux compactifications. The latter two fit to the profile of the Riemann Center and are briefly presented in the following.

String theory leads to extensions of the very successful theory of local quantum fields, in particular by demanding supersymmetry and by a noncommutative deformation of spacetime. In the simplest settings, open-string dynamics on D-branes in a B-field background produces Yang-Mills fields on a Moyal-deformed brane. Besides fundamental branes, string theory also supports solitonic branes. It is therefore of interest to investigate the properties of solitons on noncommutative spacetimes. This includes not only scalar-field solitons but also vortices, magnetic monopoles and instantons, which are an important part of the gauge-field sector of open and heterotic strings. It turns out that, besides smooth Moyal deformations of commutative solitons (called non-abelian), there exists a rich class of so-called abelian noncommutative solitons, which become singular in the commutative limit. Since 2001, we have been constructing Moyal-deformed sigma-model and sine-Gordon (multi-)solitons, Yang-Mills-Higgs monopoles and Yang-Mills instantons, by employing various methods from the theory of integrable systems (dressing approach, ADHM method, Riemann-Hilbert problem, twistor description). We have investigated their properties, moduli spaces, dynamics and scattering behavior. For further reading, please look for example at

Heterotic string or M-theory compactification from ten respectively eleven to four spacetime dimensions remains a paradigm for making contact with phenomenology. The request for low-energy supersymmetry strongly restricts the compactification manifold, for example by the existence of real Killing spinors, the admission of G-structures and reduced (weak) holonomy. Finding string- or M-theory vacua amounts to solving the supergravity equations of motion (and Bianchi identity) with stringy corrections, on spacetimes containing such manifolds as a factor. These solutions involve the graviton, the Kalb-Ramond two-form, the dilaton and the Yang-Mills field as well as their fermionic partners (gravitino, dilatino, gaugino), which may form condensates. In the graviton and gauge sectors, instanton configurations on the internal space are the main building blocks for constructing such vacua. The first-order instanton equations define a notion of self-duality on manifolds with G-structure beyond the familiar four-dimensional case. Introduced first in 1983, they caught the interest of mathematicians since the late 1990s, but little is known about their moduli spaces. Since 2009, we have constructed Yang-Mills instanton (and non-instanton) configurations on compact six-dimensional nearly-Kähler homogeneous spaces. These solutions were also extended to G_{2}- and Spin(7)-instantons on cylinders, cones and sine-cones over these spaces. We managed to lift them to full solutions of string-corrected heterotic supergravity with fermion condensates, supersymmetric and non-supersymmetric. These represent a new class of heterotic flux vacua on AdS_{4} times a nearly-Kähler coset. By introducing warp factors, NS1+NS5-brane solutions interpolating between AdS_{3} x Y and R^{1,2} x cone(Y) were constructed for geometric Killing-spinor manifolds Y of dimension 3,5,6 and 7. For further reading, please look for example at