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We prove an equivalence between the superpotential defined via tropical geometry and Lagrangian Floer theory for special Lagrangian torus fibres in del Pezzo surfaces constructed by Collins-Jacob-Lin [Duke Math. J. 170 (2021), pp. 1291–1375]. We also include some explicit calculations for the projective plane, which confirm some folklore conjectures in this case.
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A tree formula for the ellipsoidal superpotential of the complex projective plane
The ellipsoidal superpotential of the complex projective plane can be interpreted as a count of rigid rational plane curves of a given degree with one prescribed cusp singularity. In this note we present a closed formula for these counts as a sum over trees with certain explicit weights. This is a step towards understanding the combinatorial underpinnings of the ellipsoidal superpotential and its mysterious nonvanishing and nondecreasing properties.
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- Award ID(s):
- 2105578
- PAR ID:
- 10652294
- Publisher / Repository:
- Gökova Geometry / Topology Proceedings
- Date Published:
- Journal Name:
- Proceedings of Gökova Geometry-Topology Conference 2023
- ISSN:
- 2642-2352
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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