Contact Homology, Symplectic Field Theory, and Holomorphic-Curve Invariants¶
Job: develop the contact homology / cylindrical (Reeb orbits / Conley–Zehnder), the full SFT (the \(A_\infty\) / \(L_\infty\) / BV / pair-of-pants), the virtual perturbation / polyfold (HWZ), and the string topology (Chas–Sullivan / the SFT) — with the cylindrical contact homology of \(S^{2n+1}\) / Hopf Reeb flow as the worked example.
Core¶
Contact homology and the cylindrical¶
The contact homology \(\mathrm{CH}_*(Y,\xi)\) (the Eliashberg–Givental–Hofer: the homology generated by the closed Reeb orbits, the differential counting the holomorphic curves in the symplectization, the \(\partial\) the count of the cylinders).
| Contact homology / cylindrical | Description |
|---|---|
| Contact homology \(\mathrm{CH}_*\) | generated by the closed Reeb orbits, the differential counts the holomorphic curves in the symplectization |
| Cylindrical | the \(S^1\)-equivariant / the cylindrical, the simplest version (the asymptotic to the closed Reeb orbits) |
| Reeb orbits | the closed Reeb orbits (the generators, the action the period) |
| Conley–Zehnder index | the grading of the Reeb orbits (the Maslov / the Conley–Zehnder) |
Symplectic field theory (SFT) and the full theory¶
The symplectic field theory (the Eliashberg–Givental–Hofer: the invariant of the contact / symplectic cobordisms, the holomorphic curves in the symplectization / the cobordisms, the functor from the contact / symplectic category to the algebra).
| SFT / full / pair-of-pants | Description |
|---|---|
| SFT | the invariant of the contact / symplectic cobordisms (the holomorphic curves, the functor to the algebra) |
| Full SFT | the algebraic structure: the DGA / the \(A_\infty\) / the \(L_\infty\), the full holomorphic-curve counts, the BV / the string topology (Cieliebak–Latschev) |
| Pair-of-pants / rational | the genus-0 SFT (the rational, the pair-of-pants, the moduli of the stable curves) |
The virtual perturbation and the polyfold¶
| Virtual / polyfold / algebraic | Description |
|---|---|
| Virtual perturbation | the transversality / the generic \(J\), the multiple covers / the nodal, the failure of the naive transversality, the virtual |
| Polyfold (HWZ) | the Hofer–Wysocki–Zehnder: the polyfold / the Fredholm / the virtual perturbation, the rigorous SFT |
| Algebraic (\(A_\infty\) / \(L_\infty\)) | the algebraic SFT (the Cieliebak–Latschev / the Ekholm) |
The algebraic structure and the string topology¶
| String topology / SFT / BV | Description |
|---|---|
| String topology (Chas–Sullivan) | the loop space homology the BV / the string product (the relation to the SFT the symplectic / the contact) |
| SFT / string topology | the open / the closed string (the holomorphic curves / the loop space, the Cieliebak–Latschev) |
| BV / bracket | the BV operator \(\Delta\), the bracket \(\{-,-\}\), the string bracket |
See it¶
Cylindrical contact homology of the standard contact sphere \(S^{2n+1}\) and the Hopf Reeb flow. The standard \(S^{2n+1}\) (the contact form \(\alpha_0\), the Reeb flow the Hopf \(S^1\)-action, the closed Reeb orbits the Hopf fibers). The \(\mathrm{CH}_*(S^{2n+1},\xi_{\mathrm{st}})\): the Reeb orbits the Hopf fibers (all the same period, the Conley–Zehnder index the Maslov). The differential: the holomorphic cylinders in the symplectization (\(\mathbb{C}^{n+1}\setminus\{0\}\), the \(J\)-holomorphic curves asymptotic to the Hopf orbits), the count the standard. The result: the \(\mathrm{CH}_*(S^{2n+1})\) the contact homology the \(S^1\)-equivariant homology of the loop space (the string topology). The key point: the contact homology of the standard sphere the Reeb orbits (the Hopf) / the holomorphic cylinders (the \(\mathbb{C}^{n+1}\)), the contact homology the string topology / the loop space — the SFT the holomorphic-curve invariant the contact / symplectic.
Check¶
Show that the cylindrical contact homology of \(S^{2n+1}\) (standard) is the \(S^1\)-equivariant homology of the loop space, and identify the Reeb orbits.
The standard \(S^{2n+1}\): the contact form \(\alpha_0=\frac{1}{2i}\sum(z_j d\bar z_j-\bar z_j dz_j)\), the Reeb flow the Hopf \(S^1\)-action (\(R_\alpha\) the \(S^1\)-action on the Hopf fibers). The closed Reeb orbits: the Hopf fibers (all the orbits are the Hopf fibers, the period the \(2\pi\), the iterates the \(k\cdot 2\pi\)). The Conley–Zehnder index: the Maslov index of the Hopf orbit (the rotation of the contact structure along the orbit, the index the \((n+1)\)-dim the Maslov). The \(\mathrm{CH}_*\): the Reeb orbits the Hopf fibers (the generators), the differential the holomorphic cylinders (the \(J\)-holomorphic curves in \(\mathbb{C}^{n+1}\setminus\{0\}\) asymptotic to the Hopf orbits), the count the standard (the cylinders the lines, the count the multiplicity). The result: the \(\mathrm{CH}_*(S^{2n+1})\) the \(S^1\)-equivariant homology of the loop space \(\mathcal{L}S^{2n+1}\) (the string topology, the Chas–Sullivan / the SFT the correspondence). So the cylindrical contact homology the \(S^1\)-equivariant loop space homology, the Reeb orbits the Hopf fibers.
Explain the virtual perturbation / the polyfold (HWZ), and why the naive transversality fails for the SFT.
The naive transversality: for a generic almost-complex structure \(J\) on the symplectization, the moduli of the holomorphic curves is the cut-out transversely (the Fredholm, the expected dimension). The failure: the multiple covers (the holomorphic curves covering a lower-genus curve, the branch covers, the transversality the failure — the multiple covers the not-transverse for the generic \(J\)) and the nodal curves (the nodal / the broken, the boundary of the moduli, the transversality the issue). The virtual perturbation: the correction / the virtual cycles (the Kuranishi / the polyfold, the virtual perturbation the rigorous count). The polyfold (Hofer–Wysocki–Zehnder): the polyfold / the Fredholm / the virtual perturbation (the general framework for the SFT, the rigorous the holomorphic-curve counts, the sc-smooth / the perturbation). The algebraic: the \(A_\infty\) / the \(L_\infty\) / the BV (the Cieliebak–Latschev / the Ekholm, the algebraic SFT, the \(L_\infty\) the structure). The naive transversality the failure (the multiple covers), the polyfold / the virtual the rigorous (the HWZ the framework).
Pitfalls¶
- The SFT the virtual perturbation (the polyfold the HWZ the rigorous, the naive the failure) — the naive transversality the failure (the multiple covers), the polyfold / the virtual the rigorous.
- The contact homology the cylindrical / the simplest (the full the \(A_\infty\) / the BV, the cylindrical the simplest) — the cylindrical the \(S^1\)-equivariant / the simplest version, the full the \(A_\infty\) / the \(L_\infty\) / the BV.
- The Conley–Zehnder index the grading (the Maslov / the Conley–Zehnder) — the index the grading of the Reeb orbits, the contact homology the graded.
- The string topology / the SFT the Chas–Sullivan (the loop space the BV / the string product), the SFT the holomorphic curves / the loop space (the Cieliebak–Latschev) — the two the related (the SFT the holomorphic, the string topology the loop space), the correspondence the deep.
- The polyfold (HWZ) the rigorous framework (the sc-smooth / the perturbation, the general SFT), the algebraic (the \(A_\infty\) / the \(L_\infty\)) the alternative — the two the approaches to the virtual perturbation.
Refs¶
- Eliashberg–Givental–Hofer, "Introduction to symplectic field theory" (GAFA 2000)
- Hofer–Wysocki–Zehnder, "A general Fredholm theory" / "Polyfolds" (the virtual perturbation)
- Cieliebak–Latschev, "Symplectic field theory and string topology" (2010); Chas–Sullivan, "String topology" (1999)
- Bourgeois–Eliashberg–Hofer–Wysocki–Zehnder, "Compactness results in symplectic field theory" (2003)
Next¶
CG index · Foundations · Legendrian & Transverse · h-Principle & Flexibility · ECH & Weinstein · Symplectic foundations · Mirror symmetry / Fukaya