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String theory and noncommutative geometry (STRING)
Start date: 01 Jun 2009, End date: 31 May 2013 PROJECT  FINISHED 

"I am interested in various algebraic-geometric aspects of string theory and its implications for gravity, cosmology or gauge theories. I am investigating how different kinds of algebraic structures (e.g. Lie groups, Hopf algebras, supersymmetry, and their deformations appearing in noncommutative geometry) act as symmetries of different physical models.My main research topic is the study of applications of noncommutative geometry to physical systems, in particular to various kinds of gauge field theories, like theories with time-dependent backgrounds as they appear in string cosmology, as well as deformed supersymmetric theories and quantum groups. I have been able to find a cohomological approach which will enable me to solve these systems by using the rigidity of their algebroid structure to construct a suitable homotopy operator.This technique, originally developed in the context of deformation quantization for symplectic and Poisson structures, when combined with other techniques developed in the framework of Hopf algebras, such as the Drinfel’d twist, will allow me to gain me useful insights about these physical models as well as about the differential geometric structure of Lie algebroids and Lie 2-algebras.Another research topic, in which recently I am very interested, is the investigation of the geometry of exceptional Lie groups, such as G2, F4, E6, and their applications, e.g. for the construction of manifolds with G2 holonomy and as gauge groups for field theories which appear as low energy limits of string theories. Theories with G2 gauge group are relevant for quark confinement, while E6 is the most promising candidate as the symmetry for grand unification in particle physics. The same methods can be applied to other exceptional Lie groups, even E8. I have found a technique, which makes use of a suitable fibration of the group to generalize the Euler parametrization for SU(2), allowing me to compute explicitly the metric on the group manifolds."
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