Idempotence of microlocal kernels and $S^1$-equivariant Chiu-Tamarkin invariant

In this article, we present some results and constructions about the Chiu-Tamarkin invariant motivated by the idempotence of microlocal kernels, including: (1) a natural explanation for the definition of the $\mathbb{Z}\mathrm{/}\mathrm{\ell }$-equivariant Chiu-Tamarkin invariant; (2) a graded commutative product on the non-equivariant Chiu-Tamarkin invariant; and (3) a construction of the $S^1$-equivariant Chiu-Tamarkin invariant. As applications, we: (1) construct a sequence of symplectic capacities $({\bar{c}}_k{)}_{k\in \mathbb{N}}$ and prove that it coincides with the symplectic capacities $(c_k{)}_{k\in \mathbb{N}}$ we defined using the $\mathbb{Z}\mathrm{/}\mathrm{\ell }$-equivariant Chiu-Tamarkin invariant under certain conditions; and (2) prove a Viterbo isomorphism. In the Appendix, we provide a proof of admissibility for all open sets in a cotangent bundle under the setup of triangulated categories.