Research Proposal in Algebraic Geometry and String Theory

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

Some profound connections have been developed, over the past few years, between algebraic geometry and integrable systems, on the one hand, and quantum field theory and string theory on the other. The aim of this proposal is to suggest some new links, and to explore and expand the existing ones. The first proposed project involves a conjectural extension of the Fourier-Mukai transform, or the spectral construction, to several new situations, including fibrations without sections, noncommutative geometries, and higher dimensional complex or special Lagrangian torus fibers. The special Lagrangian case in particular casts new light on the SYZ approach to mirror symmetry and suggests the existence of an integrable system structure on the stringy moduli space of Calabi-Yaus. The next two projects involve applications of the first to two central issues of quantum field theory and string theory. Specifically, the second project seeks to derive a fully realistic version of the Standard Model of particle physics, starting from particular M-theory vacua. This is based on the construction techniques for vector bundles on genus-1 fibrations discussed in the first project. The third project studies several aspects of the other major conjectural string duality, the one between the heterotic string and F-theory. Proposed work includes the rigorous establishment of the isomorphism on all strata of the geometric boundary in complete generality, and its extension to the interior in one important case where it may also be possible to establish the equality of the superpotentials on both sides. String theory is generally considered to be the leading candidate for a unified physical theory which explains everything we know about the physical world, from the smallest sub-particle scales to the entire universe. Our confidence in the power of string theory to describe the real world relies to a great extent on the recent discovery of string dualities and their understanding via geometric tools. This proposal aims to explore and expand these recent applications of algebraic geometry to string theory and especially to dualities. It also includes a range of educational activities, including curriculum development, the writing of a textbook, and extensive work with undergraduate and graduate students, aimed at the dissemination of new knowledge concerning the interactions of mathematics and high energy physics.