#### Vol. 8, No. 4, 2015

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On symplectic capacities of toric domains

### Michael Landry, Matthew McMillan and Emmanuel Tsukerman

Vol. 8 (2015), No. 4, 665–676
##### Abstract

A toric domain is a subset of $\left({ℂ}^{n},{\omega }_{std}\right)$ which is invariant under the standard rotation action of ${\mathbb{T}}^{n}$ on ${ℂ}^{n}$. For a toric domain $U$ from a certain large class for which this action is not free, we find a corresponding toric domain $V$ where the standard action is free and for which $c\left(U\right)=c\left(V\right)$ for any symplectic capacity $c$. Michael Hutchings gives a combinatorial formula for calculating his embedded contact homology symplectic capacities for certain toric four-manifolds on which the ${\mathbb{T}}^{2}$-action is free. Our theorem allows one to extend this formula to a class of toric domains where the action is not free. We apply our theorem to compute ECH capacities for certain intersections of ellipsoids and find that these capacities give sharp obstructions to symplectically embedding these ellipsoid intersections into balls.

##### Keywords
symplectic capacities, toric domain, moment space axes
##### Mathematical Subject Classification 2010
Primary: 53D05, 53D20, 53D35
##### Milestones
Received: 20 June 2014
Revised: 30 July 2014
Accepted: 2 August 2014
Published: 23 June 2015

Communicated by Michael Dorff
##### Authors
 Michael Landry Mathematics Department Yale University 10 Hillhouse Avenue New Haven, CT 06511 United States Matthew McMillan Wheaton College 501 College Avenue Wheaton, IL 60187 United States Emmanuel Tsukerman Department of Mathematics University of California, Berkeley 970 Evans Hall Berkeley, CA 94720 United States