Regularized moduli spaces

table of contents

WORK IN PROGRESS

Given the moduli spaces of pseudoholomorphic polygons $\overline {\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d})$ for each tuple of Lagrangians $L_{0},\ldots ,L_{d}\subset M$, generators $x_{0}\in {\text{Crit}}(L_{0},L_{d}),x_{1}\in {\text{Crit}}(L_{0},L_{1}),\ldots ,x_{d}\in {\text{Crit}}(L_{{d-1}},L_{d})$, and a fixed compatible almost complex structure $J$,

talk about Fredholm index $\overline {\mathcal {M}}^{k}(\ldots )=\{b\in {\mathcal {M}}(\ldots )\,|\,IND...=k\}$

we need to explain how to obtain regularizations $\overline {\mathcal {M}}^{k}(x_{0};x_{1},\ldots ,x_{d};\nu )$ for expected dimensions $k=0,1$ by a choice of perturbations $\nu$. Moreover, we need to choose these perturbations coherently to ensure that the boundary of each 1-dimensional part is given by Cartesian products of 0-dimensional parts,

$\partial \overline {\mathcal {M}}^{1}(x_{0};x_{1},\ldots ,x_{d};\nu )=\bigsqcup _{{m,n\geq 0}}\bigsqcup _{{y\in {\text{Crit}}(L_{n},L_{{m+n}})}}\overline {\mathcal {M}}^{0}(y;\underline {x}';\nu )\times \overline {\mathcal {M}}^{0}(x_{0};\underline {x},y,\underline {x}'';\nu ).$

Finally, we need to check that for each pair $(b,b')\in \overline {\mathcal {M}}^{0}(x_{0};\underline {x},y,\underline {x}'';\nu )\times \overline {\mathcal {M}}^{0}(y;\underline {x}';\nu )$, when considered as boundary point $(b,b')\in \partial \overline {\mathcal {M}}^{1}(x_{0};x_{1},\ldots ,x_{d};\nu )$, has symplectic area $\omega ((b,b'))=\omega (b)+\omega (b')$ and weight function ${\text{w}}((b,b'))=(-1)^{{\|\underline x\|}}{\text{w}}(b){\text{w}}(b')$.

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Analysis TODO:

when degenerating polygons to create a strip with $L_{i}=L_{j}$ boundary conditions, will need to transfer from Morse-Bott breaking to boundary node