Cohomological and Homotopical Methods in Mathematical Physics

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9803435Stasheff This research is concerned with application of techniques Stasheffdeveloped earlier in his study of classifying spaces and rational homotopytheory and involves his combining of those techniques with developmentsintroduced ad hoc by physicists. The current project is concernedparticularly with three inter-related classes of problems and aims atelucidating their intrinsic structure as well as computation of significantapplications: (I) homotopy associative differential graded algebras and Lie andcommutative analogs, particularly as they occur in various physicalfield theories, (II) the homological aspects of Lagrangian and more general exteriordifferential systems, both classical and quantum, as embodied in theanti-field formalism of Batalin-Vilkovisky and its generalizations, (III) deformation theory as giving rise to higher homotopy algebra andas applied to physical systems. Specific applications are projectedto the problems of higher spin particles and of mixed open-closedstring field theory. Cohomological physics refers to that part of mathematical physics,primarily gauge and other field theories, in which a variety of cohomologicaltechniques are seeing increasing application. Recently, further developmentof these techniques within the physical context has begun to have an effecton more purely mathematical research, for example, providing new applicationsof the existing theory of ``higher dimensional algebra'' for which1-dimensional diagrams are inadequate. Although defined in greater and moreabstract generality, such structures, as they occur in or are inspired bymathematical physics, are the focus of this research. The results shouldaid in deeper understanding of the mathematical structures essential to thephysics (especially of higher spin particles and of mixed open-closed stringfield theory) and of the inter-relation of physical and mathematicalconcepts. The results should also be of independent mathematical importance.***