Department of Mathematics - UC Santa Barbara

Geometry, Topology, and Physics Seminar, Winter 2008

Organizers: Andreas Malmendier and Dave Morrison.
Meets 4:00 - 5:30 p.m. Fridays in South Hall 6635.

Various topics relating geometry, topology, and physics.

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Jan. 11

Dave Morrison (UCSB)

Normal functions and disk counting

In 1990, Candelas, de la Ossa, Green, and Parkes used the then-new technique of mirror symmetry to predict the number of rational curves of each fixed degree on a quintic threefold. The techniques used in the prediction were subsequently understood in Hodge-theoretic terms: the predictions are encoded in a power-series expansion of a quantity which describes the variation of Hodge structures, and in particular this power-series expansion is calculated from the periods of the holomorphic three-form on the quintic, which satisfy the Picard-Fuchs differential equation.
In 2006, Johannes Walcher made an analogous prediction for the number of holomorphic disks on the complexification of a real quintic threefold whose boundaries lie on the real quintic, in each fixed relative homology class. (The predictions were subsequently verified by Pandharipande, Solomon, and Walcher.) This talk will report on recent joint work of Walcher and the speaker which gives the Hodge-theoretic context for Walcher's predictions. The crucial physical quantity "domain wall tension" is interpreted as a Poincare normal function, that is, a holomorphic section of the bundle of Griffiths intermediate Jacobians. And the periods are generalized to period integrals of the holomorphic three-form over appropriate 3-chains (not necessarily closed), which leads to a generalization of the Picard-Fuchs equations.

Transparencies; Additional lecture notes;
Lecture notes of the entire talk.

Jan. 18

Dan Freed (UT Austin)

Self-duality, K-theory, and Orientifolds

In joint work with Jacques Distler and Gregory Moore we develop the theory of Ramond-Ramond fields on orientifolds in string theory. The mathematics we encounter includes the localization theorem in (twisted) equivariant KO-theory, KO-theoretic Wu classes, the self-duality of KO, etc. We derive the gravitational contribution to Ramond-Ramond charge, generalizing a previous result in Type I (the Green-Schwarz anomaly cancellation). Over the rationals this agrees with the formula derived by Morales-Scrucca-Serone from the worldsheet point of view.

Audio [ mp3, wma ]; Lecture notes [ version 1; version 2].

Jan. 25

Mike Hopkins (Harvard University)

Topological Field Theories

This talk will be an informal follow-up to the talk "Topological Field Theories" given by the speaker in the Mathematics Colloquium on Thursday, 01/24/08.

Audio [ mp3, wma ]; Lecture notes.

Feb. 1

No meeting

Feb. 8

Robert Maier (University of Arizona)

Modular Equations and Special Function Transformations

Many identities in classical function theory involve algebraic changes of the independent variable. This includes transformations of elliptic integrals, of combinatorial generating functions, and in general, transformations of the solutions of Fuchsian equations, such as hypergeometric and Heun equations. We shall explain how many such identities come `from geometry', since they have a modular origin. They are induced by covering relations between (rational) families of elliptic curves, and in fact, are relations between the solutions of the associated Picard-Fuchs equations.

Audio [ mp3, wma ] (first 30 minutes); Transparencies.

Feb. 15

David Morrison (UCSB)

Calabi-Yau Singularities

Calabi-Yau threefolds are compact Riemannian manifolds with holonomy SU(3), and as such, are always projective algebraic varieties. As algebraic varieties, there is a natural generalization to singular Calabi-Yau threefolds (although the existence of generalized metrics on these singular spaces is not known). Moreover, in the application of Calabi-Yau threefolds to the study of string compactification, singular Calabi-Yau threefolds play an important role. We will review what is known about the `Calabi-Yau singularities' on such spaces, with an eye on recent developments and applications.

Audio [ mp3, wma ]; Lecture Notes.

Feb. 21

No Meeting

Feb. 29

Allan Adams (MIT)

SU(3) Manifolds and the Reid Fantasy

In this talk I will discuss recent progress in string theory on the study of the moduli space of complex manifolds with SU(3) structure. In the special case of Kahler manifolds, these are just Calabi-Yaus; more generally, they are non-Kahler (but balanced) manifolds whose SU(3)-holonomy connection is not metric compatible. By studying Heterotic conformal field theories on very special examples of these manifolds (which endow the manifold with several decorations), we will find interesting paths on moduli space along which various 2- and 3-cycles shrink and grow: when the 2-cycle is large, we will find a Calabi-Yau; when the 3-cycle is large, we will find a non-Kahler manifold. Connecting these moduli spaces requires a non-local operation in the conformal field theory - bosonization and fermionization - which can be thought of as a kind of local surgery on the classical geometry.

Audio [ mp3, wma ]; Lecture Notes [ version 1, version 2].

Mar. 7

David Morrison (UCSB)

Calabi-Yau Singularities, II

We will continue our discussion of singularities on Calabi-Yau spaces, focussing on singularities which have a blowup that is a smooth Calabi-Yau space.

Audio [ mp3, wma ]; Lecture Notes [ version 1, version 2].

Mar. 14

Katrin Wendland (University of Augsburg)

Reversing sigma model constructions

To certain geometries string theory associates conformal field theories. We discuss techniques to perform the reverse procedure: To recover geometric data from abstractly defined conformal field theories. This is done by introducing appropriate notions of limits of conformal field theories and their degenerations, and by interpreting the resulting structures by methods known form noncommutative geometry.

Seminar cancelled due to travel problems.