The interplay of Physics and Geometry, one of the oldest stories in our scientific tradition, was firmly established as soon as it was realised that our physical space could be described as a 3-dimensional euclidean space. This interplay has been at the core of most developments in classical physics, including Newton's theory of gravitation. A further impetus in the relation between the two fields was given by Einstein's general relativity, which describes gravity as a purely geometrical force, establishing that at long distances our universe is geometric. As a consequence, areas of mathematics, such as riemannian geometry, which were once remote from the physics mainstream suddenly moved to the center of attention.
Both M- and string theories are gravitational theories which describe the force of gravity in geometrical terms, similar to General Relativity. With the discovery of any new geometric gravitational theory, new branches of geometry are developed to study the theory. Furthermore, the ensuing interaction between the physical and mathematical ways of thinking can sometimes lead to surprising discoveries and conjectures in Physics as well as in Mathematics.
M- and string theories are no exception. These theories contain other fields besides the graviton in their definition. This in turn requires the development of new geometric tools for their description. There are several areas experiencing simultaneous developments in Geometry and M- and string theories. This workshop will concentrate on two related topics.
The first topic is the investigation of classical solutions of M- and string theories. Such solutions have been used in a variety of applications, including the investigation of the basic objects of these theories, such as strings and branes, the study of black holes, and the exploration of various compactifications. Recently, there has been a concerted effort to classify those classical solutions that preserve a fraction of the spacetime supersymmetry. There are several reasons for this, including the aim to understand the basic (solitonic) objects of these theories. These investigations are also related to the understanding of the topology and geometry of higher dimensional black holes and branes. Moreover this circle of ideas also concerns the classification of possible compactifications with fluxes, as well as applications in AdS/CFT. During the last decade, much progress has been done on this project with notable successes, like the understanding of the geometry of all supersymmetric solutions of the heterotic theory. However much more remains to be done especially in relation to the classification of similar solutions for type II string theory and M-theory.
The mathematical tools that have been used in such investigations include techniques from spinorial geometry, G-structures, special and exceptional geometries and generalized geometry. This has given new impetus in the further development of well-known areas of geometry, such as Calabi-Yau and G2 -holonomy manifolds, has led to the development of new areas in Geometry, such as Calabi-Yau and Kähler manifolds with torsion, and has resulted in the exploration of different types of generalized geometries.
Another area in which Geometry and M- and string theories intersect is in the investigation of moduli spaces and special geometric structures. There are well-established areas in mathematics which investigate various geometric structures and the global structure and geometry their moduli spaces, like that of gauge theory instantons, solitons, Calabi-Yau and G2 manifolds. On the other hand, when manifolds with the same structures are used in superstring and M-theory compactifications, the effective lower dimensional theories, which describe the local fluctuations of these backgrounds, contain information about the moduli spaces of these manifolds. In particular, the couplings of the lower-dimensional theories describe the metric and some of the special geometric structures on the moduli spaces.
In this way one obtains new geometric structures on moduli spaces as well as unexpected relations between different geometric structures. Some of the global properties of these structures are currently studied within Mathematics. These parallel investigations have led to many novel developments like mirror symmetry for Calabi-Yau manifolds as well as the understanding of new geometric structures on the moduli spaces, such as the Kähler structure on the moduli space of G2-manifolds, which otherwise would not have been directly accessible.
The more recent emphasis on a new class of compactifications with fluxes in M- and string theories leads to the investigation of new geometric moduli problems. In turn they require the introduction and development of new tools in both Geometry and Physics to explore their properties. An example of such developments is the understanding of the moduli space of Calabi-Yau manifolds with torsion in the context of heterotic theory. This is the beginning of larger programme which will involve the flux compactifications of type II superstring and M-theories.
The goal of the workshop is to bring together experts that work in these two areas both from the Geometry side as well as from the Strings and M-theory side first to gauge the progress that has been made so far in all these research areas, and then to identify the directions in which these areas can develop further.
