Special polynomial rings, quasi modular forms and duality of topological strings
type: Item
instance of: scholarly article
title: Special polynomial rings, quasi modular forms and duality of topological strings
zbMATH Open document ID: 1314.14081
author: Murad Alim
author: Emanuel Scheidegger
author: Jie Zhou
author: Shing-Tung Yau
published in: Advances in Theoretical and Mathematical Physics
publication date: 2015-01-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1306.0002
full work available at URL: https://projecteuclid.org/euclid.atmp/1414414839
review text: The authors define differential polynomial rings using B-model special geometry structure on Calabi-Yau threefolds. Higher genus topological string amplitudes are polynomials in the generators of these rings, and yield generating functions of Gromov-Witten invariants at particular loci. Special choice of generators and of a coordinate \(\tau\) leads to a grading on these rings that has nice properties. Namely, the derivative with respect to \(\tau\) increases the degree by \(2\) and functions corresponding to the amplitudes with \(n\) insertions have degree \(-n\). In a number of non-compact examples, namely \(\mathbb{P}^2\) and local del Pezzo surfaces with singularities of exceptional types, the rings are identified with the rings of the quasi modular forms, and the degree corresponds to the modular weight. The action of the Fricke involution, which exchanges expansions of the quasi modular forms at two different cusps, is shown to correspond to the duality on amplitudes, which exchanges the large complex structure limit with the conifold loci. This is a form of electric-magnetic duality. The construction is then formally extended to the case of the quintic, where it provides a generalization of the ring of quasi modular forms, but without a geometric or physical interpretation of the duality at this time.
reviewed by: Sergiy Koshkin
Mathematics Subject Classification ID: 14J33
Mathematics Subject Classification ID: 81T30
Mathematics Subject Classification ID: 81T45
Mathematics Subject Classification ID: 14J15
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 14J32
zbMATH DE Number: 6386219
zbMATH Keywords: special geometry structure
zbMATH Keywords: electric-magnetic duality
zbMATH Keywords: Fricke involution
MaRDI profile type: Publication
arXiv ID: 1306.0002
DOI: 10.4310/ATMP.2014.V18.N2.A4
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description: scientific article