Research Project in Algebraic Geometry and String Theory

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The recent breakthrough in producing the Standard Model of particlephysics within heterotic string theory is a perfect illustration of thepower of algebraic geometry at the service of physics. Using techniquesfor construction of non simply connected Calabi-Yau threefolds and ofbundles on them satisfying various constraints on their chern classes andcohomology, the PI and a collaborator have recently constructed the onlyknown example of a heterotic string compactification which producesexactly the Minimal Suppersymmetric Standard Model (MSSM) spectrum ofparticles and forces, with no unwanted exotic matter. This opens upnumerous questions related to investigation of the High Country region ofthe Landscape of string vacua. These problems are of great interest inboth algebraic geometry and string phenomenology. Other physics questionsinvestigated in this proposal using techniques of algebraic geometryinclude exploration of the duality between the heterotic string andF-theory, and the Large N Duality. The purely geometric issues includeseveral extensions of the spectral construction and Fourier- Mukaitransforms to gerbes and non-commutative and related situations where suchextensions would have many applications. These yield a proof of Langlandsduality for Hitchin systems with arbitrary structure groups, and includeseveral problems about the moduli of Calabi-Yaus, their degenerations, andthe integrable systems arising from them.String theory is the leading physical candidate for a unified fieldtheory, incorporating both general relativity and quantum field theory.Algebraic geometry is the mathematical investigation of spaces defined byalgebraic equations. It provides an extermely versatile and powerful toolfor applications to string theory and to many other areas. The main thrustof this proposal is the use of techniques from algebraic geometry toderive the Standard Model of particle physics as a concrete instance ofstring theory.