On the $A_{\mathrm{\infty }}$-Category of a Holomorphic Moment Map

Let $M$ be a hyperKähler manifold equipped with a $U(1)$ hyperKähler isometry, and let $I$ be a complex structure on $M$. In this note, we study the $A_{\mathrm{\infty }}$-category of A-branes for the Landau-Ginzburg model with target space $(M,I)$, and superpotential being the $I$-holomorphic moment map. We show that if $I$ is a generic complex structure, the $A_{\mathrm{\infty }}$-category is semi-simple. For exceptional complex structures, though typically not semi-simple, the category still has no instanton corrections. We illustrate the $A_{\mathrm{\infty }}$-category at both generic and exceptional loci when $M$ is the cotangent bundle of the projective line.