Hochschild cohomology of graded gentle algebras and intrinsic formality

We describe the (bigraded) Hochschild cohomology of graded gentle algebras along with the Gerstenhaber bracket and cup product. In particular, this yields a description of the Hochschild cohomology of partially wrapped Fukaya categories of surfaces in the sense of Haiden-Katzarkov-Kontsevich which have at least one stop. Our results are an important ingredient in the author's recent description of the derived Picard group of partially wrapped Fukaya categories and graded gentle algebras. As another application we provide a characterisation of intrinsically formal graded gentle algebras under mild assumptions.