Hochschild Cohomology of the Symmetric Square of an Annulus with Stops

We compute the Hochschild cohomology of the partially wrapped Fukaya category of the symmetric square of an annulus with stops. Using an explicit generating set in this category, we give a description of its dg endomorphism algebra ${\tilde{\mathcal{A}}}_{n_1,n_2}$ via a quiver with relations. We show that when one boundary component has a single stop, the dg algebra is formal; however, when both boundaries contain at least two stops, it is not formal, and its minimal $A_{\mathrm{\infty }}$-model carries a nontrivial operation $m_3$. This allows us to compute its Hochschild cohomology via reduction systems and spectral sequences, and to construct a family of dg deformations associated to the resulting Hochschild cocycles.