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Idempotence of microlocal kernels and $S^1$-equivariant Chiu-Tamarkin invariant
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abstract
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 $\mathbb{Z}/\ell$-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 $(\overline{c}_k)_{k\in \mathbb{N}}$ and prove that it coincides with the symplectic capacities $({c}_k)_{k\in \mathbb{N}}$ we defined using the $\mathbb{Z}/\ell$-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.
Forward citations
Cited by 2 Pith papers
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Density of fibers for the filtered Fukaya category of $T^*N$
Iterated cones of cotangent fibers are dense in the filtered Fukaya category with respect to the interleaving distance, with a dim N + 1-cone improvement.
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A remark on Continuous K-theory and Fourier-Sato transform
For any conic closed set X in the dual space, the universal localizing invariant of sheaves with microsupport over X equals the compactly supported cohomology of X, and the Tamarkin-category version holds up to suspension.
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