Research Project in Algebraic Geometry and String Theory

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

The recent breakthrough in producing the Standard Model of particle physics within heterotic string theory is a perfect illustration of the power of algebraic geometry at the service of physics. Using techniques for construction of non simply connected Calabi-Yau threefolds and of bundles on them satisfying various constraints on their chern classes and cohomology, the PI and a collaborator have recently constructed the only known example of a heterotic string compactification which produces exactly the Minimal Suppersymmetric Standard Model (MSSM) spectrum of particles and forces, with no unwanted exotic matter. This opens up numerous questions related to investigation of the High Country region of the Landscape of string vacua. These problems are of great interest in both algebraic geometry and string phenomenology. Other physics questions investigated in this proposal using techniques of algebraic geometry include exploration of the duality between the heterotic string and F-theory, and the Large N Duality. The purely geometric issues include several extensions of the spectral construction and Fourier- Mukai transforms to gerbes and non-commutative and related situations where such extensions would have many applications. These yield a proof of Langlands duality for Hitchin systems with arbitrary structure groups, and include several problems about the moduli of Calabi-Yaus, their degenerations, and the integrable systems arising from them. String theory is the leading physical candidate for a unified field theory, incorporating both general relativity and quantum field theory. Algebraic geometry is the mathematical investigation of spaces defined by algebraic equations. It provides an extermely versatile and powerful tool for applications to string theory and to many other areas. The main thrust of this proposal is the use of techniques from algebraic geometry to derive the Standard Model of particle physics as a concrete instance of string theory.