A weak \(\infty\)-functor in Morse theory (Q6652629)

\textit{D. Gaiotto} et al. [``Algebra of the infrared: string field theoretic structures in massive $\mathcal{N}=(2,2)$ field theory in two dimensions'', Preprint, \url{arXiv:1506.04087}] studied a Landau-Ginzburg (LG) model pertaining to a Kähler manifold with a holomorphic Morse function, developing a web-based formalism for such theories. An -category of branes was constructed using only data from the so-called BPS solitons and their interactions, corresponding to the Fukaya-Seidel category in symplectic geometry and singularity theory. They follow and enlarge the vision of Morse theory as supersymmetric quantum mechanics [\textit{E. Witten}, J. Differ. Geom. 17, 661--692 (1982; Zbl 0499.53056)] by going beyond one dimension further. Interestingly enough, when varing the theories in a continuous family, at some instance a kind of topological defect happens and it interpolates the theory left and right. Such defects are called interfaces, and an \(A_{\infty}\)\ 2-category was constructed to comprehend these interfaces. This paper is concerned with such higher structures.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] recalls some basic facts and constructions in Morse theory to fix notations.\N\N\item[\S 3] develops the analysis required for the compactified moduli space \(\overline{\mathcal{M}}_{p,q}^{H\left[ l\right] }\)\ with several parameters involved to arrive at the conclusion that \(\overline{\mathcal{M}} _{p,q}^{H\left[ l\right] }\)\ is a manifold with corners and has expected dimension. The desired property of \(\overline{\mathcal{M}}_{p,q}^{H\left[ l\right] }\)\ is established in Theorem 3.21. It is also established (Theorem 3.31) hat, for any sequence in the moduli space, there exists a subsequence which Floer-Gromov converges to a broken gradient flow line by combining [\textit{U. Frauenfelder} and \textit{R. Nicholls}, ``The moduli space of gradient flow lines and Morse homology'', Preprint, \url{arXiv:2005.10799}; \textit{M. Schwarz}, Morse homology. Basel: Birkhäuser Verlag (1993; Zbl 0806.57020)] if it does not converge to a point belonging to the moduli space.\N\N\item[\S 4] is the algebraic part of the paper, after [\textit{T. Leinster}, Higher operads, higher categories. Cambridge: Cambridge University Press (2004; Zbl 1160.18001)] for the \(\infty\)-categories. After introducing the necessary notions, the authors define an \(n\)-globular set in terms of Morse functions and their higher homotopies with natural composite and identity map. By extending \(n\)\ to \(\infty\), a weak \(\infty\)-category \(\mathcal{A}\)\ is obtained. Another strict \(\infty\)-category \(\mathcal{B}\)\ is also constructed by exploring the Morse complexes and their higher homotopies. Finally the boundary structures of the one-dimensional compactified moduli space of gradient flow line of Morse functions with parameters allow of getting a weak \(\infty\)-functor \(\mathcal{F}:\mathcal{A}\rightarrow\mathcal{B}\)\ (Theorem 4.12) to encode their relations.\N\N\item[\S 5] gives a description concerning the boundaries of two-dimensional compactified moduli space, whose corresponding algebraic structures are left for future study.\N\end{itemize}