Mathematics – Algebraic Geometry
Scientific paper
2011-10-26
Mathematics
Algebraic Geometry
25 pages, preliminary version
Scientific paper
Let $C$ be a smooth connected projective curve of genus $>1$ over an algebraically closed field $k$ of characteristic $p>0$, and $c \in k \setminus F_p$. Let $\Bun_N$ be the stack of rank $N$ vector bundles on $C$ and $\Ldet$ the line bundle on $\Bun_N$ given by determinant of derived global sections. We construct an equivalence of derived categories of modules for certain localizations of twisted crystalline differential operator algebras $D_{\Bun_N,\Ldet^c}$ and $D_{\Bun_N,\Ldet^{-1/c}}$. The first step of the argument is the same as that of arxiv:math/0602255 for the non-quantum case: based on the Azumaya property of crystalline differential operators, the equivalence is constructed as a twisted version of Fourier--Mukai transform on the Hitchin fibration. However, there are some new ingredients. Along the way we introduce a generalization of $p$-curvature for line bundles with non-flat connections, and construct a Liouville vector field on the space of de Rham local systems on $C$.
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