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Higher symplectic capacities

Kyler Siegel

TL;DR

This work develops a novel framework of higher symplectic capacities built from rational SFT, using filtered L_infty algebras on linearized contact homology to count punctured curves with geometric constraints. By encoding point constraints, tangencies, and blowups into RSFT augmentations, the authors construct dimensionally stable capacities g_b and their refinements, proving monotonicity, stabilization, and links to closed-curve GW data. They obtain concrete computations for balls, ellipsoids, and polydisks that yield sharp obstructions in stabilized embedding problems, and they introduce broader families r_b from the full contact homology algebra to capture additional obstructions. Collectively, these capacities provide a sharp, algebraically grounded toolkit for high-dimensional symplectic embedding problems and unify several prior obstruction methods under RSFT, paving the way for a complete obstruction program in stabilization questions. The approach offers a principled way to translate intricate curve counts into quantitative embedding constraints with potential for algorithmic computation in convex toric and related domains.

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-infinity 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.

Higher symplectic capacities

TL;DR

This work develops a novel framework of higher symplectic capacities built from rational SFT, using filtered L_infty algebras on linearized contact homology to count punctured curves with geometric constraints. By encoding point constraints, tangencies, and blowups into RSFT augmentations, the authors construct dimensionally stable capacities g_b and their refinements, proving monotonicity, stabilization, and links to closed-curve GW data. They obtain concrete computations for balls, ellipsoids, and polydisks that yield sharp obstructions in stabilized embedding problems, and they introduce broader families r_b from the full contact homology algebra to capture additional obstructions. Collectively, these capacities provide a sharp, algebraically grounded toolkit for high-dimensional symplectic embedding problems and unify several prior obstruction methods under RSFT, paving the way for a complete obstruction program in stabilization questions. The approach offers a principled way to translate intricate curve counts into quantitative embedding constraints with potential for algorithmic computation in convex toric and related domains.

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-infinity 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.

Paper Structure

This paper contains 43 sections, 21 theorems, 169 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Context
  3. Symplectic embeddings
  4. Symplectic capacities
  5. Stabilized symplectic embeddings
  6. Floer theory perspective
  7. Main results
  8. Stable symplectic capacities
  9. First computations and applications
  10. Further capacities from rational symplectic field theory
  11. Outline
  12. Filtered $\mathcal{L}_\infty$ algebras
  13. $\mathcal{L}_\infty$ algebra basics
  14. Review of $\mathcal{A}_\infty$ algebras
  15. Symmetric tensor coalgebras and $\mathcal{L}_\infty$ algebras
  16. ...and 28 more sections

Key Result

Theorem 1.2

There is a symplectic embedding of $E(a,b)$ into $E(c,d)$ if and only if $\mathfrak{c}_k^{\operatorname{ECH}}(E(a,b)) \leq \mathfrak{c}_k^{\operatorname{ECH}}(E(c,d))$ for all $k \in \mathbb{Z}_{> 0}$.

Figures (3)

  • Figure 1: Top: a collection of curves in $\widehat{X}$ contributing to the coefficient $\langle \Phi^3(w_1,w_2,w_3),x_{\Gamma_1^-}...x_{\Gamma_6^-}\rangle$. Bottom: the corresponding graph $G(I_1,I_2,I_3;J_1,\dots,J_6)$, where $I_1 = \{1,2\}$, $I_2 = \{3,4\}$, $I_3 = \{5,6,7,8\}$, $J_1 = \{1\}$, $J_2 = \{2,3\}$, $J_3 = \{4,5\}$, $J_4 = \{6\}$, $J_5 = \{7\}$, $J_6 = \{8\}$ (note that in general the elements of $J_1,\dots,J_m$ need not appear in consecutive order).
  • Figure 2: A typical configuration contributing to the coefficient $\langle \ell^4_{{{\operatorname{lin}}}}(x_{\gamma_1},x_{\gamma_2},x_{\gamma_3},x_{\gamma_4}),x_{\eta}\rangle$. Here the cylinders labeled by $t$ are trivial cylinders.
  • Figure 3: Left: a typical configuration contributing to the coefficient $\langle \Phi^4_{{{\operatorname{lin}}}}(x_{\gamma_1},x_{\gamma_2},x_{\gamma_3},x_{\gamma_4}),x_{\eta}\rangle$, in the case of a non-exact symplectic cobordism $X$. Right: a typical configuration contributing to the Maurer--Cartan element ${\mathfrak{cl}_{{\operatorname{lin}}}} \in {\operatorname{CH}}_{{\operatorname{lin}}}(X_0)$.

Theorems & Definitions (82)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7: other structures on symplectic cohomology
  • Theorem 1.8
  • Theorem 1.10
  • Remark 1.12
  • Remark 1.13
  • Theorem 1.14
  • ...and 72 more