Instructional Program: Titles and Abstracts

Abstract: In this course I will cover some basic results on the emerging theory of generalized complex manifolds and its relation to string theory. The topics covered will be Generalized complex manifolds (basic definitions, local coordinates, submanifolds, examples) Generalized Kaehler manifolds (relation to bihermitian geometry, examples, generalized Hodge theory and Kaehler identities), Reduction of generalized structures, T-duality with NS-flux and generalized complex structures.

Lecture Notes

Abstract: String theory is a rich and complicated subject requiring the knowledge of almost every aspect of quantum field theory, including conformal field theory and supersymmetry. However, some string backgrounds are particularly simple and do not require the above machinery: they exist in two spacetime dimensions with no supersymmetry, and are nonperturbatively well-defined. They are also to a large extent solvable via the use of random matrices and integrability. In this course I will introduce the matrix formulation of these "noncritical" string backgrounds. The solution of these matrix models will be worked out using free fermion techniques and a simple basis for the eigenfunctions. Key physical features of more complicated string backgrounds, including symmetries, dualities and flux compactifications, will be exhibited within this approach.

Lecture Notes | Part 1 | Part 2 |

Abstract: An important aspect of the so--called second string revolution of the mid 1990s was the discovery that all the consistent string theories should be expected to be dual to one another in certain regimes of their respective moduli spaces. The conjectured heterotic -- type IIA duality is a prominent example of this phenomenon, which together with its "decompactification limit", the heterotic -- F-theory duality, leads to a number of surprising mathematical predictions.

Roughly speaking, heterotic string theories are associated to the geometric data that specify a compactification space, namely a Calabi-Yau manifold viewed as a real manifold, and a semistable bundle on it. On the other hand, the geometric data of a type IIA string theory are the complex structure and complexified Kähler parameters of a Calabi-Yau manifold. String-string duality predicts correspondences between the respective moduli spaces. Moreover, the consistency conditions for heterotic string theories pose restrictions on their geometric parameters, which in turn translate into previously unknown properties of their dual Calabi-Yau counterparts on the type IIA or F-theory side. Notably, for certain heterotic theories we have a so--called anomaly cancellation condition which has a beautiful classical mathematical description in terms of the vanishing of the first Chern class associated to an appropriate index bundle. As such, the anomaly cancellation condition is expressed in terms of an Atiyah-Singer family index theorem. Under the string-string duality it implies classically unknown relations between the geometric and topological data of the dual Calabi-Yau manifolds on the type IIA/F-theory side.

The lectures will review aspects of the conjectured heterotic -- type IIA and heterotic -- F-theory dualities from a mathematical point of view. We will discuss anomaly cancellation conditions for heterotic string theories on K3 and in particular the implications of these index calculations for the three-dimensional Calabi-Yau geometry underlying the dual F-theory. It is almost impossible to give credit to all the relevant results in mathematical physics; to mention only some of them, works by Morrison/Vafa, Bershadsky/Intriligator/Kachru/Morrison/Sadov/Vafa, Friedman/Morgan/Witten, Aspinwall/Morrison, and Grassi/Morrison will be central to these lectures, along with some more recent own observations together with Anda Degeratu.