Non-birational Calabi-Yau threefolds that are derived equivalent

We argue that the existence of genus one fibrations with multisections of high degree on certain Calabi-Yau threefolds implies the existence of pairs of such varieties that are not birational, but are derived equivalent. It also (likely) implies the existence of non-birational counterexamples to the Torelli problem for Calabi-Yau threefolds.Comment: Final version to appear in IJM. Several references adde

The Mukai pairing, II: the Hochschild-Kostant-Rosenberg isomorphism

We continue the study of the Hochschild structure of a smooth space that we began in our previous paper, examining implications of the Hochschild-Kostant-Rosenberg theorem. The main contributions of the present paper are: -- we introduce a generalization of the usual Mukai pairing on differential forms that applies to arbitrary manifolds; -- we give a proof of the fact that the natural Chern character map $K_0(X) \to HH_0(X)$ becomes, after the HKR isomorphism, the usual one $K_0(X) \to \bigoplus H^i(X, \Omega_X^i)$; and -- we present a conjecture that relates the Hochschild and harmonic structures of a smooth space.Comment: 19 pages, uses diagrams.sty, corrected proofs of Theorems 4.1 and 4.4 following a suggestion of Amnon Yekutiel

D-branes, B fields, and Ext groups

In this paper we extend previous work on calculating massless boundary Ramond sector spectra of open strings to include cases with nonzero flat B fields. In such cases, D-branes are no longer well-modelled precisely by sheaves, but rather they are replaced by `twisted' sheaves, reflecting the fact that gauge transformations of the B field act as affine translations of the Chan-Paton factors. As in previous work, we find that the massless boundary Ramond sector states are counted by Ext groups -- this time, Ext groups of twisted sheaves. As before, the computation of BRST cohomology relies on physically realizing some spectral sequences. Subtleties that cropped up in previous work also appear here.Comment: 23 pages, LaTeX; v2: typos fixed; v3: reference adde

Algebraic deformations arising from orbifolds with discrete torsion

We develop methods for computing Hochschild cohomology groups and deformations of crossed product rings. We use these methods to find deformations of a ring associated to a particular orbifold with discrete torsion, and give a presentation of the center of the resulting deformed ring. This connects with earlier calculations by Vafa and Witten of chiral numbers and deformations of a similar orbifold.Comment: 19 pages, LaTeX2e; final version to appear in JPA