For a gentle algebra, indecomposables in the 1-periodic derived category are exactly string objects from curves between marked points and band objects from primitive closed curves with indecomposable k[x,x^-1]-modules.
Morphisms between indecomposable complexes in the bounded derived category of a gentle algebra
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The 1-periodic derived category of a gentle algebra : Part 1 -- Indecomposable objects
For a gentle algebra, indecomposables in the 1-periodic derived category are exactly string objects from curves between marked points and band objects from primitive closed curves with indecomposable k[x,x^-1]-modules.