String Topology: Background and Present State - Mathematics > Geometric TopologyReport as inadecuate

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Abstract: The data of a -2D field theory with a closed string compactification- is anequivariant chain level action of a cell decomposition of the union of allmoduli spaces of punctured Riemann surfaces with each component compactified asa pseudomanifold with boundary. The axioms on the data are contained in thefollowing assumptions. It is assumed the punctures are labeled and divided intononempty sets of inputs and outputs. The inputs are marked by a tangentdirection and the outputs are weighted by nonnegative real numbers adding tounity. It is assumed the gluing of inputs to outputs lands on thepseudomanifold boundary of the cell decomposition and the entire pseudomanifoldboundary is decomposed into pieces by all such factorings. It is furtherassumed that the action is equivariant with respect to the toroidal action ofrotating the markings. A main result of compactified string topology is theTheorem closed strings: Each oriented smooth manifold has a 2D field theorywith a closed string compactification on the equivariant chains of its freeloop space mod constant loops. The sum over all surface types of the toppseudomanifold chain yields a chain X satisfying the master equation dX + X*X =0 where * is the sum over all gluings. This structure is well defined up tohomotopy.The genus zero parts yields an infinity Lie bialgebra on the equivariantchains of the free loop space mod constant loops. The higher genus termsprovide further elements of algebraic structure called a -quantum Liebialgebra- partially resolving the involutive identity.There is also a compactified discussion and a Theorem 2 for open strings asthe first step to a more complete theory. We note a second step for knots.

Author: Dennis Sullivan


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