Mathematical Sciences: Cohomological and Homotopical Methodsin Mathematical Physics

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Cohomological physics refers to that part of physics, primarily gauge and field theories, in which a variety of cohomological techniques are seeing increasing application. The instant project is concerned with application of techniques developed in the investigator's previous study of classifying spaces and rational homotopy theory as an algebraic topologist. It is particularly directed to four classes of problems: (1) homotopy associative differential graded algebras and Lie analogs, especially as they occur in string field theories and spin n- algebras; (2) quasi-Hopf algebras and deformations of bialgebras; (3) Zamolodcdhikov's tetrahedral equation and generalized classifying spaces; (4) the homological aspects of reduction of constrained Hamiltonian systems, both classical and quantum, as embodied in the BRST formalism and the Batalia-Fradkin-Vilkovisky complex and its generalizations. (1) and (2) bear a very strong resemblance; one major thrust of this research being to understand the underlying reason for this resemblance. (3) involves "higher dimensional algebra," which is the analog of the algebra in (1) and (2) but beginning with structures for which one-dimensional diagrams are inadequate. Although defined in greater and more abstract generality, such structures as occur in mathematical physics are the focus of this work. In another context, a noted mathematical physicist has marveled at the "unreasonable effectiveness of mathematics." This project is devoted to a special case of that observation, namely, the unreasonable effectiveness of algebraic topology. It begins with the discovery that a paper the investigator published almost thirty year ago as an algebraic topologist was slated for a key role in conformal string theory and related matters. The "higher order associativity" of H-spaces that he developed there has gone through a sequence of developments at the hands of MacLane; Brustein, Ne'eman, and Sternberg; Joyal and Street; and finally Drinfeld, who, in 1989, gave deep and ingenious applications to both physics and low dimensional topology. Stasheff has become an enthusiastic participant in the ensuing explosion of interest in this and related areas.