Journal
JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES
Volume -, Issue -, Pages -Publisher
WILEY
DOI: 10.1112/jlms.12498
Keywords
14T99 (primary); 13F60; 14N10 (secondary)
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Funding
- Center of Excellence Grant 'Centre for Quantum Geometry of Moduli Spaces' from the Danish National Research Foundation [DNRF95]
- National Science Foundation RTG Grant [DMS-1246989]
- European Research Council [759967]
- European Research Council (ERC) [759967] Funding Source: European Research Council (ERC)
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This paper focuses on the relationships between tropical disks, scattering diagrams, and broken lines in the quantum setting, and how they are used to prove significant mathematical results.
In the Gross-Siebert mirror symmetry program, certain enumerations of tropical disks are encoded in combinatorial objects called scattering diagrams and broken lines. These, in turn, are used to construct a mirror scheme equipped with a canonical basis of regular functions called theta functions. This paper serves to develop the relationships between tropical disks, scattering diagrams, and broken lines in the quantum setting; for example, we express quantum theta functions in terms of refined enumerations of tropical disks.We apply these tropical descriptions to prove a refined version of the Carl-Pumperla-Siebert (Preprint) result on consistency of theta functions, and also to prove the quantum Frobenius Conjecture of Fock and Goncharov (Ann. Sci. ec. Norm. Super. (4) 42 (2009) 865-930).
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