A polyfold perspective on APS operator families and topology-changing domains

While proving the sc-Fredholm property of APS-type operators \(D_A=\frac{d}{dt}-A(t)\) on both unweighted and weighted Floer path spaces \(\mathcal W_n=\bigcap_{k+r=n} W^{r,2}(\mathbb R, W_k)\) and \(( \mathcal W_{n+k}^{\delta_n})_{n\geq 0}\), we argue that the latter case requires a bound on the weight sequence \(0=\delta_0 < \delta_1 <...\) whose value \(\delta_\infty\) can be calculated in terms of the operator family \(A(t)\).

Moreover, in an attempt to replace the classical Floer cylinder \(\Sigma =\mathbb R \times S^1\) by a pair-of-pants worldsheet with topology-changing level sets, we prove the sc-smoothness of a retraction \(r_{\Sigma}:(- \epsilon,\epsilon)\oplus W_n^{\oplus 2} \longrightarrow W_n^{\oplus 2}\) that interpolates between topologically distinct fibres \(\textstyle r_{t<0}(W_n^{\oplus 2})\cong W^{n+1,2}(S^1)\) and \(r_{t>0}(W_n^{\oplus 2})\cong W^{n+1,2}(S^1)\oplus W^{n+1,2}(S^1)\), leaving it for future investigation to interpret the Cauchy-Riemann operator \(\partial_{\bar z}\sim \partial_t + i\partial_{\omega}\) as calculating the flow of a sc-smooth vector field \(A(t)\sim i\partial_\omega\) on the M-polyfold \(\text{im}(r_{\Sigma})\).