Advances in Theoretical and Mathematical Physics

Volume 16 (2012)

Number 4

Toric symmetry of $\mathbb{CP}^3$

Pages: 1291 – 1314

DOI: http://dx.doi.org/10.4310/ATMP.2012.v16.n4.a4

Authors

Dagan Karp (Department of Mathematics, Harvey Mudd College, Claremont, Calif., U.S.A.)

Dhruv Ranganathan (Department of Mathematics, Harvey Mudd College, Claremont, Calif., U.S.A.)

Paul Riggins (Department of Mathematics, Harvey Mudd College, Claremont, Calif., U.S.A.)

Ursula Whitcher (Department of Mathematics, University of Wisconsin, Eau Claire, Wisc., U.S.A.)

Abstract

We exhaustively analyze the toric symmetries of $\mathbb{CP}^3$ and its toric blowups. Our motivation is to study toric symmetry as a computational technique in Gromov–Witten (GW) theory and Donaldson–Thomas (DT) theory. We identify all non-trivial toric symmetries. The induced nontrivial isomorphisms lift and provide new symmetries at the level of GW Theory and DT theory. The polytopes of the toric varieties in question include the permutohedron, the cyclohedron, the associahedron, and in fact all graph associahedra, among others.

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