SIGMA 21 (2025), 030, 16 pages      arXiv:2504.20489      https://doi.org/10.3842/SIGMA.2025.030
Contribution to the Special Issue on Integrability, Geometry, Moduli in honor of Motohico Mulase for his 65th birthday

Sign Convention for $A_{\infty}$-Operations in Bott-Morse Case

Kaoru Ono
Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

Received July 14, 2024, in final form March 03, 2025; Published online April 29, 2025

Abstract
We describe the sign and orientation issue appearing the filtered $A_{\infty}$-formulae in Lagrangian Floer theory using de Rham model in Bott-Morse setting. After giving the definition of filtered $A_{\infty}$-operations in a Fukaya category, we verify the filtered $A_{\infty}$-formulae.

Key words: filtered $A_{\infty}$-operation; Kuranishi structure; bordered stable map.

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References

1. Abouzaid M., Fukaya K., Oh Y.G., Ohta H., Ono K., Quantum cohomology and split generation in Lagrangian Floer theory, in preparation.
2. Akaho M., Joyce D., Immersed Lagrangian Floer theory, J. Differential Geom. 86 (2010), 381-500, arXiv:0803.0717.
3. Fukaya K., Unobstructed immersed Lagrangian correspondence and filtered $A_{\infty}$-functor, SIGMA 21 (2025), 031, 284 pages,arXiv:1706.02131.
4. Fukaya K., Oh Y.G., Ohta H., Ono K., Lagrangian intersection Floer theory: Anomaly and obstruction. Part I, AMS/IP Stud. Adv. Math., Vol. 46, American Mathematical Society, Providence, RI, 2009.
5. Fukaya K., Oh Y.G., Ohta H., Ono K., Lagrangian intersection Floer theory: Anomaly and obstruction. Part II, AMS/IP Stud. Adv. Math., Vol. 46, American Mathematical Society, Providence, RI, 2009.
6. Fukaya K., Oh Y.G., Ohta H., Ono K., Kuranishi structures and virtual fundamental chains, Springer Monogr. Math., Springer, Singapore, 2020.
7. Solomon J.P., Tukachinsky S.B., Differential forms, Fukaya $A_\infty$ algebras, and Gromov-Witten axioms, J. Symplectic Geom. 20 (2022), 927-994, arXiv:1608.01304.