The Witten equation, mirror symmetry and quantum singularity theory

For any non-degenerate, quasi-homogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to Gromov-Witten theory and generalizes the theory of r-spin curves, which corresponds to the simple singularity A_{r-1}. We also resolve two outstanding conjectures of Witten. The first conjecture is that ADE-singularities are self-dual; and the second conjecture is that the total potential functions of ADE-singularities satisfy corresponding ADE-integrable hierarchies. Other cases of integrable hierarchies are also discussed.Comment: To appear in Annals of Mathematics. Includes resolution of the Witten ADE integrable hierarchies conjecture and Witten's ADE self-mirror conjecture. Several corrections and clarification

Chern Classes and Compatible Power Operations in Inertial K-theory

Let [X/G] be a smooth Deligne-Mumford quotient stack. In a previous paper the authors constructed a class of exotic products called inertial products on K(I[X/G]), the Grothendieck group of vector bundles on the inertia stack I[X/G]. In this paper we develop a theory of Chern classes and compatible power operations for inertial products. When G is diagonalizable these give rise to an augmented $\lambda$-ring structure on inertial K-theory. One well-known inertial product is the virtual product. Our results show that for toric Deligne-Mumford stacks there is a $\lambda$-ring structure on inertial K-theory. As an example, we compute the $\lambda$-ring structure on the virtual K-theory of the weighted projective lines P(1,2) and P(1,3). We prove that after tensoring with C, the augmentation completion of this $\lambda$-ring is isomorphic as a $\lambda$-ring to the classical K-theory of the crepant resolutions of singularities of the coarse moduli spaces of the cotangent bundles $T^*P(1,2)$ and $T^*P(1,3)$, respectively. We interpret this as a manifestation of mirror symmetry in the spirit of the Hyper-Kaehler Resolution Conjecture.Comment: Many improvements. Special thanks to the referee for helpful suggestions. To appear in Annals of K-Theory. arXiv admin note: text overlap with arXiv:1202.060