# Singular spaces in String and M-theory V

Singular spaces play an important role in string theory: The extended nature of the string and the existence of D-branes as higher-dimensional BPS objects render certain types of string compactifications physically well-defined even though the underlying space exhibits a singularity. This is in spectacular contrast to point particle theories, where singularities in spacetime typically lead to unacceptable inconsistencies.

These lectures are intended to give an introduction to the physics of geometric singularities in string compactifications, aimed at an audience who is not necessarily specialized in string theory yet.

We will focus on two prime examples of singular string compactifications: the conifold in Type II compactifications as well compactifications of 11-dimensional M-theory on singular torus-fibrations and their interpretation via F-theory.

The outline is as follows:

Lecture 1:

We will introduce the effective Quantum Field Theory for string theory in ten dimensions and its compactification on Calabi-Yau 3-folds to four dimensions. A special role will be played by the important concept of BPS states from Dp-branes wrapping p-cycles on a Calabi-Yau 3-fold.

Lecture 2:

We will describe the effective dynamics of 4-dimensional string compactifications in the language of N=2 supersymmetric field theory. This includes an introduction to the special Kahler property of the vector multiplet moduli space. Following classic ideas by Strominger and others, the singularities in the physical theory due to collapsing p-cycles is resolved by correctly incorporating massless BPS states from D-branes wrapping the vanishing cycles.

Lecture 3:

The precise match between the physical singularity at a conifold point and the geometric singularities will be worked out, establishing the famous example of the conifold transition as a deformation along the Higgs and Coulomb branch of Type II compactifications.

Lecture 4:

We will introduce F-theory as a particular limit of M-theory compactifications on torus-fibered Calabi-Yau spaces. As the simplest example of a Weierstrass model we will describe elliptically fibered K3 surfaces and the geometric structure of their fibral singularities as classified by Kodaira, as well as the physical interpretation of these singularities as provided by F-theory.

Lecture 5:

The structure of singularities in codimension one, two and three and their physical relevance will be discussed. Time permitting, we will give an outlook on the mathematical description of fluxes and more advanced questions in F/M-theory.