Polyfold regularization of constrained moduli spaces (Q2036289)

This paper is concerned with the transverse preimage constructions in the polyfold theory. Polyfold theory was introduced by \textit{H. Hofer} et al. [Polyfold and Fredholm theory. Cham: Springer (2021; Zbl 1479.58002)] as a general framework to deal with analytic constructions of moduli spaces of holomorphic curves in symplectic geometry. The polyfold theory is featured with a new calculus called sc-calculus and retraction-based differential topology, which make both the group action of reparameterization and bubbling/breaking degeneration smooth phenomena in such framework. With such novel notion of smoothness, classical constructions in differential topology, in particular, the transverse preimage construction as in the paper under review, requires a check of the validity. The starting point is that the implicit function theorem, with a level shift, holds for sc-smooth maps if the target is finite-dimensional. This property is based on the triviality of the sc-structure on a finite-dimensional space and a relation between sc-smoothness and classical smoothness. With such observation, the author finds a normal form and slices (Lemma 2.3, 3.3) for any sc-smooth submersion to a finite-dimensional manifold. Using those local slices as the local model, the author gives (Theorem 5.10 (1)) a tame M-polyfold structure on \(f^{-1}(N)\) given that the sc-smooth map \(f:\mathcal{B}\to Y\) is transverse to the submanifold \(N\subset Y\), where \(\mathcal{B}\) is a tame M-polyfold and \(Y\) is a finite dimensional manifold. The analogue for M-polyfold bundles is completely analogous (Theorem 5.10 (2)). In Section 6, the author generalizes the above results to the polyfold setting. There is nothing tricky here but it requires careful checking and bookkeeping. The good news is that the transverse preimage, which is naturally a subset/subcategory, is blessed by a free globalization. The core of the polyfold theory is the theory of sc-Fredholm sections [\textit{H. Hofer} et al., Geom. Funct. Anal. 19, No. 1, 206--293 (2009; Zbl 1217.58005)], which allows one to apply the implicit function theorem with polyfold targets to obtain smooth zero sets. It was observed in [\textit{B. Filippenko} et al., Proc. Natl. Acad. Sci. USA 116, No. 18, 8787--8797 (2019; Zbl 1431.58006)] that the notion of sc-Fredholm sections is not only sufficient, but also rather necessary for the purpose of the implicit function theorem. Now the rigidity forced by the natural globalization by subsets/subcategories poses difficulties to obtain sc-Fredholmness of the restricted section. The author introduces the notion of tame sc-Fredholm sections (Definition 3.7) and compatibility with the smooth map \(f\) (Definition 5.9), then the restricted section is also sc-Fredholm (Theorem 5.10 (3)). It is noticed in Section 5.1 that natural examples from Gromov-Witten invariants satisfy these conditions. The author also discusses several applications of the construction, including realizing Gromov-Witten invariants as counting of zero-dimensional moduli spaces when the constraints can be represented by submanifolds, avoiding sphere bubbles in dimension 0 and 1. He also outlines the joint work with \textit{K. Wehrheim} [``A polyfold proof of the Arnold conjecture'', Preprint, \url{arXiv:1810.06180}] of constructing a polyfold Piunikhin-Salamon-Schwarz morphism to obtain a proof of the weak Arnold conjecture for general symplectic manifolds. The construction is based on fiber product constructions, which are special cases of the transverse preimage construction.