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Refined scattering diagrams and theta functions from asymptotic analysis of Maurer-Cartan equations

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Young,  Matthew B.
Max Planck Institute for Mathematics, Max Planck Society;

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Citation

Leung, N. C., Ma, Z. N., & Young, M. B. (2021). Refined scattering diagrams and theta functions from asymptotic analysis of Maurer-Cartan equations. International Mathematics Research Notices, 2021(5), 3389-3437. doi:10.1093/imrn/rnz220.


Cite as: https://hdl.handle.net/21.11116/0000-0006-E651-B
Abstract
We further develop the asymptotic analytic approach to the study of
scattering diagrams. We do so by analyzing the asymptotic behavior of
Maurer-Cartan elements of a differential graded Lie algebra constructed from a
(not-necessarily tropical) monoid-graded Lie algebra. In this framework, we
give alternative differential geometric proofs of the consistent completion of
scattering diagrams, originally proved by Kontsevich-Soibelman, Gross-Siebert
and Bridgeland. We also give a geometric interpretation of theta functions and
their wall-crossing. In the tropical setting, we interpret Maurer-Cartan
elements, and therefore consistent scattering diagrams, in terms of the refined
counting of tropical disks. We also describe theta functions, in both their
tropical and Hall algebraic settings, in terms of flat sections of the
Maurer-Cartan-deformed differential. In particular, this allows us to give a
combinatorial description of Hall algebra theta functions for acyclic quivers
with non-degenerate skew-symmetrized Euler forms.