Higher symplectic capacities

Siegel, Kyler

doi:10.2140/agt.2025.25.5205

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Higher symplectic capacities

Kyler Siegel

Algebraic & Geometric Topology 25 (2025) 5205–5278

DOI: 10.2140/agt.2025.25.5205

Abstract

We construct new families of symplectic capacities indexed by certain symmetric polynomials, defined using rational symplectic field theory. In particular, we introduce a sequence of capacities based on an $ℒ_{\infty}$ structure on linearized contact homology and rational curve counts with local tangency constraints. We prove various structural properties of these capacities and give some preliminary computations which show that they give state of the art symplectic embedding obstructions in basic examples.

Keywords

symplectic geometry, symplectic field theory, symplectic capacities, symplectic embeddings, $\mathcal{L}_{\infty}$ algebras, local tangency constraints, Maurer–Cartan elements

Mathematical Subject Classification

Primary: 53D35

References

Publication

Received: 1 September 2020

Revised: 17 October 2023

Accepted: 10 August 2024

Published: 18 December 2025

Authors

Kyler Siegel
Department of Mathematics University of Southern California Los Angeles, CA United States

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Volume 25, issue 9 (2025)