Zeta functions of regular arithmetic schemes at s=0

Lichtenbaum conjectured the existence of a Weil-\'etale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme $\mathcal{X}$ at $s=0$ in terms of Euler-Poincar\'e characteristics. Assuming the (conjectured) finite generation of some \'etale motivic cohomology groups we construct such a cohomology theory for regular schemes proper over $\mathrm{Spec}(\mathbb{Z})$. In particular, we obtain (unconditionally) the right Weil-\'etale cohomology for geometrically cellular schemes over number rings. We state a conjecture expressing the vanishing order and the special value up to sign of the Zeta function $\zeta(\mathcal{X},s)$ at $s=0$ in terms of a perfect complex of abelian groups $R\Gamma_{W,c}(\mathcal{X},\mathbb{Z})$. Then we relate this conjecture to Soul\'e's conjecture and to the Tamagawa number conjecture of Bloch-Kato, and deduce its validity in simple cases.Comment: 53 pages. To appear in Duke Math.

Cluster decomposition, T-duality, and gerby CFT's

In this paper we study CFT's associated to gerbes. These theories suffer from a lack of cluster decomposition, but this problem can be resolved: the CFT's are the same as CFT's for disconnected targets. Such theories also lack cluster decomposition, but in that form, the lack is manifestly not very problematic. In particular, we shall see that this matching of CFT's, this duality between noneffective gaugings and sigma models on disconnected targets, is a worldsheet duality related to T-duality. We perform a wide variety of tests of this claim, ranging from checking partition functions at arbitrary genus to D-branes to mirror symmetry. We also discuss a number of applications of these results, including predictions for quantum cohomology and Gromov-Witten theory and additional physical understanding of the geometric Langlands program.Comment: 61 pages, LaTeX; v2,3: typos fixed; v4: writing improved in several sections; v5: typos fixe