STRING

String theory and noncommutative geometry

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 Obiettivo del progetto (Objective)

'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.'

Introduzione (Teaser)

Evidence that the Universe is made of strings has been elusive for more than 30 years. But the theory offering hopes that we can unify physics had an alluring pull for EU-funded researchers.

Descrizione progetto (Article)

Einstein's mathematical formulation of his general theory of relativity predicted an expanding Universe and the existence of black holes, both of which have since been observed. Ever-increasing sophistication of experimental techniques has also brought an explosion of discoveries related to interactions on very small scales.

However, classical mathematical descriptions often fall short in describing interactions of elementary particles and some behaviours are unexpected when quantum effects are taken into account. One mathematical theory that gained exceptional prominence in an effort to unite classical and quantum descriptions is string theory.

Researchers working on the EU-funded 'String theory and noncommutative geometry' (STRING) project were also attracted by its beautiful mathematical formalism. According to string theory, at the heart of every elementary particle is a tiny string-like filament. The differences between one particle and another arise from how their internal strings vibrate.

The mathematics revealed that one of these notes had properties matching those of the graviton, a hypothetical particle that should carry the force of gravity from one location to another. In other words, gravity and quantum mechanics are playing by the same rules. During the STRING project, researchers explored further implications for physical models related to gravity and quantum field theories.

Specifically, STRING focused on how different kinds of algebraic groups can be defined as symmetries of physical models. The exceptional Lie groups were studied in the context of supergravity and gauge field theories. Knowledge of their geometrical properties was then applied to classify different types of black hole orbits.

The exceptional Lie groups had first been constructed using different algebras to build a so-called 'magic square' that contained simpler Lie groups. Within the STRING project, researchers developed a software programme dedicated to computing the generators of Lie groups entering the magic square. Written in Mathematica, this is freely available http://www.mat.unimi.it/users/cerchiai/MathematicaProgram/ (here).

Project work resulted in the assimilation of important contributions related to mathematical descriptions of physical models of the Universe. The cross-fertilisation between mathematics and physics has been particularly rich with one area shedding light on another. Geometrical features of string theory have been revealed, as well as the theory's value by the influence it has on quantum information theory.