 
Summary: HOMOLOGICAL MIRROR SYMMETRY FOR PUNCTURED SPHERES
MOHAMMED ABOUZAID, DENIS AUROUX, ALEXANDER I. EFIMOV, LUDMIL KATZARKOV,
AND DMITRI ORLOV
Abstract. We prove that the wrapped Fukaya category of a punctured sphere (S2
with
an arbitrary number of points removed) is equivalent to the triangulated category of sin
gularities of a mirror LandauGinzburg model, proving one side of the homological mirror
symmetry conjecture in this case. By investigating fractional gradings on these categories,
we conclude that cyclic covers on the symplectic side are mirror to orbifold quotients of
the LandauGinzburg model.
1. Introduction
1.1. Background. In its original formulation, Kontsevich's celebrated homological mirror
symmetry conjecture [25] concerns mirror pairs of CalabiYau varieties, for which it predicts
an equivalence between the derived category of coherent sheaves of one variety and the
derived Fukaya category of the other. This conjecture has been studied extensively, and
while evidence has been gathered in a number of examples including abelian varieties [15,
27, 22], it has so far only been proved for elliptic curves [33], the quartic K3 surface [36],
and their products [7].
Kontsevich was also the first to suggest that homological mirror symmetry can be ex
tended to a much more general setting [26], by considering LandauGinzburg models. Math
