Let V be a vertex operator algebra with a category C of (generalized) modules that has vertex tensor category structure, and thus braided tensor category structure, and let A be a vertex operator (super)algebra extension of V. We employ tensor categories to study untwisted (also called local) A-modules in C, using results of Huang-Kirillov-Lepowsky showing that A is a (super)algebra object in C and that generalized A-modules in C correspond exactly to local modules for the corresponding (super)algebra object. Both categories, of local modules for a C-algebra and (under suitable conditions) of generalized A-modules, have natural braided monoidal category structure, given in the first case by Pareigis and Kirillov-Ostrik and in the second case by Huang-Lepowsky-Zhang. Our main result is that the Huang-Kirillov-Lepowsky isomorphism of categories between local (super)algebra modules and extended vertex operator (super)algebra modules is also an isomorphism of braided monoidal (super)categories. Using this result, we show that induction from a suitable subcategory of V-modules to A-modules is a vertex tensor functor. We give two applications. First, we derive Verlinde formulae for regular vertex operator superalgebras and regular (1/2)ℤ-graded vertex operator algebras by realizing them as (super)algebra objects in the vertex tensor categories of their even and ℤ-graded components, respectively. Second, we analyze parafermionic cosets C=Com(V_L,V) where L is a positive definite even lattice and V is regular. If the category of either V-modules or C-modules is understood, then our results classify all inequivalent simple modules for the other algebra and determine their fusion rules and modular character transformations. We illustrate both directions with several examples.

The mirror theorem is generalized to any smooth projective variety X. That is, a fundamental relation between the Gromov-Witten invariants of X and Gromov-Witten invariants of complete intersections Y in X is established.

Let V be a convex vector bundle over a smooth projective manifold X, and let Y be the subset of X which is the zero locus of a regular section of V. This mostly expository paper discusses a conjecture which relates the virtual fundamental classes of X and Y. Using an argument due to Gathmann, we prove a special case of the conjecture. The paper concludes with a discussion of how our conjecture relates to the mirror theorems in the literature.

A reconstruction theorem for genus 0 gravitational quantum cohomology and quantum K-theory is proved. A new linear equivalence in the Picard group of the moduli space of genus 0 stable maps relating the pull-backs of line bundles from the target via different markings is used for the reconstruction result. Examples of calculations in quantum cohomology and quantum K-theory are given.

This work is devoted to the study of the foundations of quantum K-theory, a K-theoretic version of quantum cohomology theory. In particular, it gives a deformation of the ordinary K-ring K(X) of a smooth projective variety X, analogous to the relation between quantum cohomology and ordinary cohomology. This new quantum product also gives a new class of Frobenius manifolds.