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The Gamma and Strominger–Yau–Zaslow conjectures: a tropical approach to periods

Mohammed Abouzaid, Sheel Ganatra, Hiroshi Iritani and Nick Sheridan

Geometry & Topology 24 (2020) 2547–2602
Abstract

We propose a new method to compute asymptotics of periods using tropical geometry, in which the Riemann zeta values appear naturally as error terms in tropicalization. Our method suggests how the Gamma class should arise from the Strominger–Yau–Zaslow conjecture. We use it to give a new proof of (a version of) the Gamma conjecture for Batyrev pairs of mirror Calabi–Yau hypersurfaces.

Keywords
mirror symmetry, SYZ conjecture, periods, tropical geometry, Riemann zeta values, Gamma class, Batyrev mirror
Mathematical Subject Classification 2010
Primary: 53D37
Secondary: 11G42, 14J33, 14T05, 32G20
References
Publication
Received: 26 February 2019
Revised: 28 October 2019
Accepted: 26 November 2019
Published: 29 December 2020
Proposed: Richard P Thomas
Seconded: Paul Seidel, Dan Abramovich
Authors
Mohammed Abouzaid
Department of Mathematics
Columbia University
New York, NY
United States
Sheel Ganatra
Department of Mathematics
University of Southern California
Los Angeles, CA
United States
Hiroshi Iritani
Department of Mathematics
Kyoto University
Kyoto
Japan
Nick Sheridan
School of Mathematics
University of Edinburgh
Edinburgh
United Kingdom