Computation of Open Gromov–Witten Invariants for Toric Calabi–Yau 3-Folds by Topological Recursion, a Proof of the BKMP Conjecture

By B. Eynard, N. Orantin. 2012

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The BKMP conjecture (2006–2008) proposed a new method to compute closed and open Gromov–Witten invariants for every toric Calabi–Yau 3-folds, through a topological recursion based on mirror symmetry. So far, this conjecture has been verified to low genus for several toric CY3folds, and proved to all genus only for $${\mathbb{C}^3}$$C3. In this article we prove the general case. Our proof is based on the fact that both sides of the conjecture can be naturally written in terms of combinatorial sums of weighted graphs: on the A-model side this is the localization formula, and on the B-model side the graphs encode the recursive algorithm of the topological recursion.One can slightly reorganize the set of graphs obtained in the B-side, so that it coincides with the one obtained by localization in the A-model. Then it suffices to compare the weights of vertices and edges of graphs on each side, which is done in two steps: the weights coincide in the large radius limit, due to the fact that the toric graph is the tropical limit of the mirror curve. Then the derivatives with respect to Kähler radius coincide due to the special geometry property implied by the topological recursion.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/69bb8f283d7212cb40a3e6b8cf9e5b7c5fca91c9

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Random matrices. (2015, methodology): “In particular, in 2008, the Bouchard–Klemm–Mariño–Pasquetti (BKMP) conjecture [18] (since then proved [19]) claimed that topological string amplitudes obey the same recursion relations as matrix models’ correlation functions.”
Identification of the Givental Formula with the Spectral Curve Topological Recursion Procedure (2012, background): “heory is often used to reproduce known partitionfunctions, extractsfromitsomehighergenus correlatorswhich were up to now unreachable and gives new non-trivial relations for the correlators, see e. g. [16]. 1.3. Goals of the paper. As we see, there is a lot of similarity in both theories (which was ﬁ”