Generalized Geometries of Effective Actions

In string theory the quanta of physical fields, like photons and gravitons, are described by oscillation modes of either open or closed tiny strings flying through space. The contributions of a large number of strings sum up to mean values of the physical background fields, like the electromagnetic field or the gravitational field and some others, like the so-called `B-field` or `RR-fields`. The dynamics of the background fields can be described by an `effective theory` considering only macroscopic aspects while disregarding the underlying microscopic string structure, which becomes evident only at very high energies. All the information about a theory can be conveniently collected in an object called `action`. We are therefore dealing with `effective actions`. It is an interesting fact that the mathematical implementation of the string-idea leads to the prediction of a ten- dimensional universe (nine space dimensions and one time). In order not to contradict our observations of four spacetime dimensions, the remaining six dimensions are thought of as being curled up on a very small radius, such that they are not observable under normal conditions. The four-dimensional physics can again be described by an effective theory, which does not care about what exactly happens in the hidden dimensions. However, certain features that are desired in the effective theory (like `supersymmetry` - an exchange symmetry of fermions and bosons) constrain the geometry of the dimensions which are curled up. In addition to strings, there are other extended objects in string theory, called D-branes, which determine, where open strings can end. Some fields (like the electromagnetic field) can live only on such branes. It turns out that in the presence of B-fields (and their corresponding fluxes), the brane-fields effectively feel a `non-commutative space`, i.e. they can be conveniently described within a geometry, where length times width differs from width times length. One therefore says that fluxes `deform` geometry. A similar effect is obtained by the RR-fields, which deform the `superspace` - a concept that helps to implement supersymmetry - in a mathematically elegant manner. In the present project we are going to study several aspects of this flux induced deformation of the brane-effective action and as well the effects of fluxes on the geometry of the hidden dimensions.

In string theory the quanta of physical fields, like photons and gravitons, are described by oscillation modes of either open or closed tiny strings flying through space. The contributions of a large number of strings sum up to mean values of the physical background fields, like the electromagnetic field or the gravitational field and some others, like the so-called `B-field` or `RR-fields`. The dynamics of the background fields can be described by an `effective theory` considering only macroscopic aspects while disregarding the underlying microscopic string structure, which becomes evident only at very high energies. All the information about a theory can be conveniently collected in an object called `action`. We are therefore dealing with `effective actions`. It is an interesting fact that the mathematical implementation of the string-idea leads to the prediction of a ten- dimensional universe (nine space dimensions and one time). In order not to contradict our observations of four spacetime dimensions, the remaining six dimensions are thought of as being curled up on a very small radius, such that they are not observable under normal conditions. The four-dimensional physics can again be described by an effective theory, which does not care about what exactly happens in the hidden dimensions. However, certain features that are desired in the effective theory (like `supersymmetry` - an exchange symmetry of fermions and bosons) constrain the geometry of the dimensions which are curled up. In addition to strings, there are other extended objects in string theory, called D-branes, which determine, where open strings can end. Some fields (like the electromagnetic field) can live only on such branes. It turns out that in the presence of B-fields (and their corresponding fluxes), the brane-fields effectively feel a `non-commutative space`, i.e. they can be conveniently described within a geometry, where length times width differs from width times length. One therefore says that fluxes `deform` geometry. A similar effect is obtained by the RR-fields, which deform the `superspace` - a concept that helps to implement supersymmetry - in a mathematically elegant manner. In the present project we are going to study several aspects of this flux induced deformation of the brane-effective action and as well the effects of fluxes on the geometry of the hidden dimensions.