Regularized moduli spaces

table of contents

WORK IN PROGRESS

In order to regularize 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 $J$, any abstract regularization approach (opposed to geometric ones, as contrasted in [section 3, FFGW] and [sections 2.1-2, McDuff-Wehrheim]) proceeds by describing each Gromov-compactified moduli space as

${\text{zero set}}\quad \overline {\mathcal {M}}(x_{0};x_{1},\ldots ,x_{d})=\overline \partial _{{J,Y}}^{{-1}}(0)\quad {\text{of a section}}\quad \overline \partial _{{J,Y}}:{\mathcal {X}}\to {\mathcal {Y}}\quad {\text{of a bundle}}\quad \pi :{\mathcal {Y}}\to {\mathcal {X}}.$

In the abstract regularization approach of [2015-FOOO1], [2017-FOOO2], and many other virtual approaches, a global section $s:{\mathcal {Y}}\to {\mathcal {X}}$ (no longer easily identified with a Cauchy-Riemann operator) is patched together from smooth sections of finite rank bundles over finite dimensional manifolds $s_{i}:Y_{i}\to X_{i}$. While at first glance this resolves most analytic issues (up to the question of how to actually obtain smooth sections near nodal curves from the classical gluing analysis), the notions of transition maps between base manifolds $X_{i}$ of different dimensions raises a number of deep topological issues, as discussed in [McDuff-Wehrheim]. Other undesirable features of this approach are that the Kuranishi charts $s_{i}:Y_{i}\to X_{i}$ are heavily choice-dependent, and that these atlases are generally 'too small' to allow for straight-forward constructions of new moduli spaces (e.g. by restriction to curves having certain intersection properties, or coupling of curves with each other or Morse trajectories) via restrictions or fiber products of the local sections.

we will provide Polyfold Fredholm descriptions for them - as

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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