  The data of a "2D field theory with a closed string compactification" is an
equivariant chain level action of a cell decomposition of the union of all
moduli spaces of punctured Riemann surfaces with each component compactified as
a pseudomanifold with boundary. The axioms on the data are contained in the
following assumptions. It is assumed the punctures are labeled and divided into
nonempty sets of inputs and outputs. The inputs are marked by a tangent
direction and the outputs are weighted by nonnegative real numbers adding to
unity. It is assumed the gluing of inputs to outputs lands on the
pseudomanifold boundary of the cell decomposition and the entire pseudomanifold
boundary is decomposed into pieces by all such factorings. It is further
assumed that the action is equivariant with respect to the toroidal action of
rotating the markings. A main result of compactified string topology is the
  Theorem (closed strings): Each oriented smooth manifold has a 2D field theory
with a closed string compactification on the equivariant chains of its free
loop space mod constant loops. The sum over all surface types of the top
pseudomanifold 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 to
homotopy.
  The genus zero parts yields an infinity Lie bialgebra on the equivariant
chains of the free loop space mod constant loops. The higher genus terms
provide further elements of algebraic structure called a "quantum Lie
bialgebra" partially resolving the involutive identity.
  There is also a compactified discussion and a Theorem 2 for open strings as
the first step to a more complete theory. We note a second step for knots.
