Derived categories of genus one curves and torsors over abelian varieties

Studying orientifold string theories on elliptic curves or abelian varieties motivates studying the derived category of coherent sheaves on a genus one curve or a torsor over an abelian variety over the reals (as opposed to the complex numbers).

In joint work with Nirnajan Ramachandran (to appear in MRL), we show that a genus one curve over a perfect field determines a class in the relative Brauer group of the Jacobian elliptic curve, and that there is a natural Mukai-type derived equivalence between the original genus one curve and the Jacobian twisted by the Brauer class.  The proof extends to torsors over abelian varieties (of any dimension).