| Résume | It was demonstrated by Baum—Fulton—MacPherson that the Riemann—Roch theorem may be viewed most naturally as a transition from algebra to topology. They construct, for X a quasi-projective complex variety, a homomorphism from K_0^{alg}(X), the Grothendieck group of coherent sheaves on X to K_0^{top}(X), a K-theoretic analog of Borel—Moore homology. The Grothendieck—Riemann—Roch theorem is then the statement that this homomorphism is covariant for proper maps.
In this talk we will prove an analogous result in the context of Landau—Ginzburg models. Let Y be a smooth quasi-projective complex variety and w: Y —> \CC a regular function. The pair (Y, w) is called a Landau—Ginzburg (LG) model. Associated to an LG model is the category MF(Y, w) of matrix factorizations of w, whose objects are “twisted" complexes of vector bundles, where the square of the differential is equal to multiplication by w. We define a K-theoretic analog of critical cohomology of w, which we call the critical K-theory. We construct a homomorphism from the Grothendieck group of the category MF(Y, w) to the critical K-theory and show that this homomorphism is covariant for proper maps of LG models. |
