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.
The triangulated hull of periodic complexes
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abstract
In the terms of an `$n$-periodic derived category', we describe explicitly how the orbit category of the bounded derived category of an algebra with respect to powers of the shift functor embeds in its triangulated hull. We obtain a large class of algebras whose orbit categories are strictly smaller than their triangulated hulls and a realization of the phenomenon that an automorphism need not induce the identity functor on the associated orbit category.
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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.