The Hochschild cohomology and the Tamarkin-Tsygan calculus of gentle algebras

Abstract

The Tamarkin Tsygan calculus of a finite dimensional algebra is a differential calculus given by the comprehensive data of the Hochschild cohomology, its structure both as a graded commutative algebra under the cup product and as a graded Lie algebra under the Gerstenhaber bracket, together with the Hochschild homology and its module structure over the Hochschild cohomology given by the cap product as well as the Connes differential. In this paper, we calculate the whole of the Tamarkin Tsygan calculus for the class of gentle algebras. Apart from some isolated calculations, this is, to our knowledge, the first complete calculation of this calculus for a family of finite dimensional algebras. Gentle algebras appear in many different areas of mathematics such as the theory of cluster algebras, N=2 gauge theories and homological mirror symmetry of surfaces. For the latter Haiden-Katzarkov-Kontsevich show that the partially wrapped Fukaya category of a graded surface is triangle equivalent to the perfect derived category of a graded gentle algebra. Thus the symplectic cohomology of the surface should be linked to the Hochschild cohomology of the gentle algebra. We give a description of how this connection might be visualised, in that, we show how these structures are encoded in the geometric surface model of the bounded derived category associated to a gentle algebra via its ribbon graph. Finally, we show how to recover structural results for a gentle algebra given its Tamarkin-Tsygan calculus.