Algebraic Geometry in String theory (Ron Donagi, Penn)
The working group will emphasize concrete applications of algebraic
geometry to string theory. We will start by learning to carry out in
detail some of the algebro-geometric constructions and calculations
which came up in the lecture: construction of Calabi-Yau manifolds
(toric geometry, elliptic fibrations, orbifolds); construction of
interesting vector bundles on a given manifold (Fourier-Mukai
transforms, monads); and explicit calculation of various cohomology
groups and related invariants. Depending on student preferences, we
could also explore some other connections and applications of
algebraic geometry to string theory: moduli spaces in algebraic
geometry and in string theory, superpotentials, quivers, stability
conditions, gerbes, the Strominger-Yau-Zaslow conjecture, integrable
systems, del Pezzo surfaces and string dualities...
On day one, since we won't have the opportunity for the students to
prepare anything in advance, we will survey the available material. I
will present a series of half a dozen or more topics, each capturing
an aspect of algebraic geometry which is an important tool for physics
applications. Each "presentation" will consist of a ten minute
introduction, either followed or preceded by a 5-10 minute exercise
for everyone to try out on the spot, followed by a brief assessment
and review. At the end of the day, the group will decide together
which of these topics would become the themes for each of the four
remaining days.
Examples of possible topics:
* Toric geometry, or the part of it relevant to constructing CYs and
bundles on them, taken e.g. from the Cox paper.
* Elliptic fibrations, Tate's algorithm, cohomology computations, the CY
condition, rational elliptic surfaces and K3s, Schoen's threefolds.
* Orbifolds, very rudimentary quantum cohomology, the Vafa-Witten
orbifold, deformations, fundamental groups.
* Semistable bundles on an elliptic curve, Fourier-Mukai transform,
spectral covers, bundles on elliptic fibrations.
* Computational techniques: using relative duality, base change, Leray and
other spectral sequences, etc to explicitly compute cohomology groups on
fibered varieties and finite group actions on them.
* Abelian varieties, integrable systems, the cubic condition, Hitchin's
system, Calabi-Yau systems, Seiberg-Witten systems.
* Gerbes, torus fibrations, duality, the Strominger-Yau-Zaslow
conjecture.
* Superpotentials, Hilbert schemes, normal functions (paper by Clemens),
maybe a bit about quivers (Aspinwall-Katz).
* Del Pezzo surfaces and string dualities (paper by Iqbal, Neitzke, Vafa.)
* Stability conditions on triangulated categories, D-branes (papers of
Bridgeland, Aspinwall)