Department of Mathematics - UC Santa Barbara

Geometry, Topology, and Physics Seminar, Fall 2007

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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Sep. 28

Fall Welcome BBQ

Oct. 5

Andreas Malmendier (UCSB)

The topology and geometry of the Seiberg-Witten curve

We show that the Seiberg-Witten family of elliptic curves defines a four-dimensional, Jacobian rational elliptic surface Z over the u-plane with boundary whose signature equals minus the number of massive hypermultiplets. The family of the stable semi-classical BPS states defines a unique flat holomorphic line bundle on Z. We also construct rank-two holomorphic SU(2)/Z_2-bundles and show that the central charges of the corresponding quantum states are half the charges of the BPS states.
We show that the local anomaly of the determinant line bundle of the ∂-operator along the fiber of Z vanishes. We determine the non-trivial global anomaly as the holonomy of the determinant section and the relation to the signature of Z. Moreover, we show that the determinant line bundle extends across the nodal fibers of Z. The extension introduces current contributions to the curvature of the determinant line bundle at the points in the u-plane where the stable BPS states become massless.

Audio [ mp3, wma ]; Lecture notes.

Oct. 12

Anton Kapustin (Caltech)

Holomorphic-topological gauge theories, sigma-models, and duality

Any finite N=2 d=4 gauge theory on a product of two Riemann surfaces C1 and C2 can be twisted into a holomorphic-topological theory which depends only on the complex structure on C_1 and is invariant under arbitrary diffeomorphisms of C2. This leads to an interesting connection between 4d gauge theories and twisted 2d sigma-models.

Audio [ mp3, wma ]; Lecture notes.

Oct. 19

Nora Ganter (Colby College)

Elliptic cohomology, Witten genus, and applications to physics

Elliptic cohomology is a field at the intersection of number theory, algebraic geometry and algebraic topology. Its definition is very technical and highly homotopy theoretic. While its geometric definition is still an open question, elliptic cohomology exhibits striking formal similarities to string theory, and it is strongly expected that a geometric interpretation will come from there.
To illustrate the interaction between the two fields, I will speak about my work on orbifold genera and product formulas: After a very informal introduction to elliptic cohomology, I will discuss string theory on orbifolds and explain how a formula by Dijkgraaf, Moore, Verlinde and Verlinde on the orbifold elliptic genus of symmetric powers of a manifold motivated my work in elliptic cohomology. I will proceed to explain why elliptic cohomology provides a good framework for the study of orbifold genera. Time permitting, I will also sketch conjectural connections to generalized Moonshine and non-linear sigma models.

Audio [ mp3, wma ]; Lecture notes.

Oct. 26

Matthew Ando (UIUC)

Equivariant elliptic cohomology and the Fibered WZW models of Distler and Sharpe

Recently Jacques Distler and Eric Sharpe introduced "Fibered WZW models" as a generalization of the heterotic string. The associated elliptic genera have been studied by Kefeng Liu and myself. I shall explain how these genera fit arise naturally in the context of equivariant ellitpic cohomology.

Audio [ mp3, wma ]; Lecture notes.

Nov. 2

Eric D'Hoker (UCLA)

Exact half-BPS solutions to type IIB supergravity

The complete Type IIB supergravity solutions with 16 supersymmetries are obtained on the manifold AdS4 x S2 x S2 x C with SO(2,3) x SO(3) x SO(3) symmetry in terms of two holomorphic functions on a Riemann surface C, which generally has a boundary. This is achieved by reducing the BPS equations using the above symmetry requirements, proving that all solutions of the BPS equations solve the full Type IIB supergravity field equations, mapping the BPS equations onto a new integrable system akin to the Liouville and Sine-Gordon theories, and mapping this integrable system to a linear equation which can be solved exactly. Amongst the infinite class of solutions, a non-singular Janus solution is identified which provides the AdS/CFT dual of the maximally supersymmetric Yang-Mills interface theory discovered recently. We outline the construction of general classes of globally non-singular solutions, including fully back-reacted AdS5 x S5 and supersymmetric Janus doped with D5 and/or NS5 branes.

Audio [ mp3, wma ]; Lecture notes.

Nov. 9

The Arithmetic and Geometry Seminar will meet at 4 pm.

Paolo Cascini (UCSB)

Rational Points on Algebraic Varieties: Campana's Program II

A classical problem in mathematics is the study of the rational solutions of a Diophantine problem, i.e. a finite system of polynomial equations with integral coefficients. In particular, one might ask when the number of these solutions is infinite. At least conjecturally, the answer to these questions is closely related to the geometry of the varieties defined by such polynomials. Campana's program is a new method to approach these problems. This talk will be accessible to a general audience and no prior knowledge of the subjects involved is required.

Nov. 16

Sergei Gukov (UCSB)

D-branes and representations

Audio [ mp3, wma ]; Lecture notes.

Nov. 23

No meeting: Thanksgiving Holiday

Nov. 30

No meeting.

There will be a Distinguished Lecture by Karen Vogtmann (Cornell) at 3:30 pm.

Dec. 7

Robert Waelder (UCLA)

Singular elliptic genus of complex surfaces

The extension of smooth invariants such as Chern numbers and Hodge numbers to singular varieties has interesting applications in mirror symmetry and algebraic geometry. Two such invariants of particular interest are the stringy E-function of Batyrev, and the singular elliptic genus of Borisov and Libgober. I will discuss these invariants, their limitations, and a promising generalization for the case of normal algebraic surfaces.

Audio [ mp3, wma ]; Lecture notes.