Mathematical Sciences: Conformal Field Theory and Its Generalizations

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9627782 Zuckerman The proposed research lies in the general area of mathematical physics. Previously, the investigator worked on two-dimensional quantum field theory, clarifying various aspects of the algebraic structure in quantum cohomology theory. The investigator plans to further pursue this, incorporating also the more geometric aspects such as mirror manifolds and Calabi-Yau theory. Another subproject deals with four-dimensional generalizations of quantum field theory using double loop cocycles. Finally, the investigator intends to consider possible links between conformal field theories and ideal hydrodynamics via the geometry of diffeomorphism groups. Quantum field theory arises in the context of attempting to give a unified formulation of basic forces of nature. Two-dimensional 'conformal' formulation of this theory is relatively well-understood, at least from a purely mathematical standpoint, and the present proposal seeks to further study the two-dimensional model with a view towards applying string theory to the phenomenology of elementary particles.