# String Theory and Generalized Geometries (12w5098)

Arriving in Banff, Alberta Sunday, December 2 and departing Friday December 7, 2012

## Organizers

Katrin Becker (Texas A & M University)

Melanie Becker (Texas A & M University)

David Morrison (UCSB)

Daniel Robbins (University of Amsterdam)

Shing-Tung Yau (Harvard University)

## Objectives

The primary goal of the workshop will be to bring together experts from around the world, both mathematicians and physicists, in order to study some of these new classes of string theory solutions and advance both the physical and mathematical understanding of them and the connections between them. In order to narrow the focus of the workshop, we will focus primarily on the introduction of fluxes, both for heterotic and type II string theory, and on the dualities between different solutions. Mathematically these spaces should have descriptions in terms of generalized geometries, perhaps with extra structures (bundles or sheafs).

The objectives of the workshop will relate to two somewhat broad directions.

I) Relating the effective physical description of these string theory solutions to the mathematical structures of the corresponding spaces.

In the case of Calabi-Yau manifolds, there are very beautiful relations between certain topological invariants of the manifold and the massless field content of the effective theory. For example, massless scalar fields typically correspond to unobstructed deformations of some of the mathematical structures (e.g. complex, symplectic, generalizations) on the space. In the case of heterotic flux compactifications, there has been progress made by studying the deformations of a broad class of examples, but there is still much work to be done. In the case of type II string theory with fluxes, similar considerations lead to connections with generalized complex geometry and Hitchin functionals. In both heterotic and type II, our understanding seems to be just in its initial stages. Similarly, the geometry usually dictates what sort of gauge groups appear in the effective theory, and how those gauge symmetries are allowed to break. Very little is known about this issue for generalized geometric spaces.

Obviously, to attack problems like this requires combining a very good grasp of the mathematical structures of the relevant spaces with an understanding of the physical consequences that result. That is why a workshop such as the one being proposed, which brings together mathematicians and physicists who have studied different aspects of the problem, has the potential to make such a large impact. The timing is also very auspicious, since much of the relevant work has started only in the past few years; we are far enough along to know some good questions, but have not yet made much progress on the answers.

II) Identifying the web of physical equivalences between different classes of examples and understanding how those equivalences are realized on the mathematical structures involved.

Mirror symmetry was first noticed as a physical equivalence between string theory on two distinct Calabi-Yau spaces. Working out the consequences of this statement led to a vast amount of compelling mathematics that is still being explored. There seems to be very strong potential for similar stories to come out of the broader classes of examples which would be discussed at the workshop. Indeed, much of the physical intuition for finding examples comes from exploiting these sorts of equivalences, or dualities, in order to generalize known constructions. However, most of the detailed understanding comes merely from equivalence at the local level - for example applying a local version of mirror symmetry to a space that can be described as a Calabi-Yau with added flux (so that it's no longer a Calabi-Yau, but this can be accounted for), one is forced into the world of generalized geometry. Globally, however, the situation is much murkier. There are particualr specific examples in which some more concrete statements can be made, but the general story remains largely unexplored.

Hopefully by bringing together experts with diverse backgrounds, this workshop can foster collaborations that will start to examine these issues more fully. Dualities (relating different mathematical descriptions of the same physics) underly many of the most beautiful stories that can be told about he interface between the physics and mathematics of string theory, and it is almost certain that there are still more stories to be told, especially now as more and more classes of solutions are being considered.

The objectives of the workshop will relate to two somewhat broad directions.

I) Relating the effective physical description of these string theory solutions to the mathematical structures of the corresponding spaces.

In the case of Calabi-Yau manifolds, there are very beautiful relations between certain topological invariants of the manifold and the massless field content of the effective theory. For example, massless scalar fields typically correspond to unobstructed deformations of some of the mathematical structures (e.g. complex, symplectic, generalizations) on the space. In the case of heterotic flux compactifications, there has been progress made by studying the deformations of a broad class of examples, but there is still much work to be done. In the case of type II string theory with fluxes, similar considerations lead to connections with generalized complex geometry and Hitchin functionals. In both heterotic and type II, our understanding seems to be just in its initial stages. Similarly, the geometry usually dictates what sort of gauge groups appear in the effective theory, and how those gauge symmetries are allowed to break. Very little is known about this issue for generalized geometric spaces.

Obviously, to attack problems like this requires combining a very good grasp of the mathematical structures of the relevant spaces with an understanding of the physical consequences that result. That is why a workshop such as the one being proposed, which brings together mathematicians and physicists who have studied different aspects of the problem, has the potential to make such a large impact. The timing is also very auspicious, since much of the relevant work has started only in the past few years; we are far enough along to know some good questions, but have not yet made much progress on the answers.

II) Identifying the web of physical equivalences between different classes of examples and understanding how those equivalences are realized on the mathematical structures involved.

Mirror symmetry was first noticed as a physical equivalence between string theory on two distinct Calabi-Yau spaces. Working out the consequences of this statement led to a vast amount of compelling mathematics that is still being explored. There seems to be very strong potential for similar stories to come out of the broader classes of examples which would be discussed at the workshop. Indeed, much of the physical intuition for finding examples comes from exploiting these sorts of equivalences, or dualities, in order to generalize known constructions. However, most of the detailed understanding comes merely from equivalence at the local level - for example applying a local version of mirror symmetry to a space that can be described as a Calabi-Yau with added flux (so that it's no longer a Calabi-Yau, but this can be accounted for), one is forced into the world of generalized geometry. Globally, however, the situation is much murkier. There are particualr specific examples in which some more concrete statements can be made, but the general story remains largely unexplored.

Hopefully by bringing together experts with diverse backgrounds, this workshop can foster collaborations that will start to examine these issues more fully. Dualities (relating different mathematical descriptions of the same physics) underly many of the most beautiful stories that can be told about he interface between the physics and mathematics of string theory, and it is almost certain that there are still more stories to be told, especially now as more and more classes of solutions are being considered.