A gerby deformation of complex tori and the homological mirror symmetry

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Abstract: Let

(X, c h e c k X)

be a mirror pair of a complex torus

X

and its mirror partner

c h e c k X

. This mirror pair is described as the trivial special Lagrangian torus fibrations

X i g h t a r r o w B

and

c h e c k X i g h t a r r o w B

on the same base space

B

by SYZ construction. Then, we can associate a holomorphic line bundle

E (s, m a t h c a l L) i g h t a r r o w X

to a pair

(s, m a t h c a l L)

of a Lagrangian section

s

of

c h e c k X i g h t a r r o w B

and a unitary local system

m a t h c a l L

along it. In this paper, we first construct the deformation

X_{m a t h c a l G}

of

X

by a certain flat gerbe

m a t h c a l G

and its mirror partner

c h e c k X_{m a t h c a l G}

from the mirror pair

(X, c h e c k X)

, and discuss deformations of objects

E (s, m a t h c a l L)

and

(s, m a t h c a l L)

over the deformed mirror pair

(X_{m a t h c a l G}, c h e c k X_{m a t h c a l G})

.

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