Prof. Bong Lian from Math writes:
In the 1950’s, French mathematicians Henri Cartan and Armand Borel defined a new topological invariant that was capable of distinguishing symmetries of certain geometric spaces known as G-manifolds. Cartan and Borel called their invariant the Equivariant Cohomology of a G-manifold. It was new in that it was able to capture essential aspects of geometric operations, called Lie group actions (after Sophus Lie), on manifolds that ordinary cohomology theory was unable to detect. Hence it provides a new conceptual framework for studying symmetries of spaces on the one hand, and offers a powerful tool for computing ordinary cohomology of these spaces, on the other.
In the late 80’s, physicists invented String Theory in their attempt to construct a grand unified field theory. They found that certain solutions to String Theory are essentially governed by an algebraic structure called a Chiral Algebra. This turns out to be a new structure that generalizes many fundamental algebraic constructs in mathematics, including commutative algebras and Lie algebras. A question was then raised as to whether there exists a natural theory that integrates both the Cartan-Borel invariant and Vertex Algebras. This hypothetical theory, which I learned about as a graduate student at Yale University, was dubbed the stringy analogue of the Equivariant Cohomology theory.
In 2004, Andrew Linshaw, a Brandeis PhD student (now Research Fellow, U. Darmstadt), and I constructed such a theory, which we coined the Chiral Equivariant Cohomology (CEC) of a G-manifold. In our latest paper, joint with another Brandeis PhD student, Bailin Song (now Assoc. Prof., Univ. of Science and Technology of China), we showed that not only does the CEC subsumes the Cartan-Borel theory, it goes well beyond that. For example, we have found an infinite family of Lie group actions on spheres that the Cartan-Borel theory is too weak to distinguish, but have non-isomorphic CEC. This proves that the CEC theory is strictly stronger as a topological invariant than the Cartan-Borel invariant. The paper appears in the December 2010 issue of the American Journal of Mathematics (Volume 132, Number 6).
