Topological quantum field theory structure on symplectic cohomology. (English) Zbl 1298.53093
The first aim of this paper is to construct algebraic structures on the twisted symplectic cohomology \(SH^*(M)_\alpha\), introduced in [the author, Geom. Topol. 13, No. 2, 943–978 (2009; Zbl 1175.57019)]. Although in general there exists no topological quantum field theory (TQFT) on \(SH^*(M)_\alpha\), the author proves that the TQFT structure exists and possesses a unit, if the form \(\alpha\in H^1(\mathcal{L}M)\) is the transgression of a class \(\eta\in H^2(M)\).
The paper consists of 15 sections and 4 appendices.
In the introduction the author presents the main results of the paper, concerning the topological quantum field theory (TQFT), TQFT structure on ordinary cohomology, wrapped Floer cohomology, the Arnol’d chord conjecture, obstructions to exact contact embeddings, displaceability of contact hypersurfaces and string topology.
Next, the author recalls the notion of Liouville domain (exact symplectic manifold conical at infinity), Reeb periods, contact type conditions, Hamiltonians, action functional, Floer trajectories and their energy, transversality and compactness results (Section 2).
The symplectic cohomology \(SH^*(M)\) and symplectic homology \(SH_*(M)\) are constructed, and in Theorem 3.4 it is shown that \(SH_*(M)\) is canonically the dual of \(SH^*(M)\), but not vice versa.
Section 4 refers to wrapped Floer cohomology \(HW^*(L)\), the relative analogue of \(SH^*\) for exact Lagrangian submanifolds \(L\) of a Liouville domain \((M,d\theta)\), \(L\) having a transverse intersection with the boundary of \(M\), namely \(\partial L = L \cap\partial M\). There are proved some results concerning Hamiltonian and Reeb chords, wrapped trajectories and their energy, transversality and compactness and wrapped Floer cohomology.
In the next section the author shows that there exist maps \(c^* : H^*(M)\rightarrow SH^*(M)\) and \(c^* : H^*(L)\rightarrow SH^*(L)\) from the ordinary cohomology to the symplectic and wrapped cohomologies. In the wrapped case there is a map \(c^* : H^*(L)\cong HW^*(L;H^\delta)\rightarrow HW^*(L)\), so \(HW^*(L)\neq 0\) and there is an isomorphism \(H^*(L)\otimes\Lambda \rightarrow HW^*(L;H^\delta)\).
The author summarizes the TQFT structure on \(SH^*(M)\) in detail in Appendix A, then he proves some results concerning the graded-commutative and associative product \[ \psi_P:SH^i(M)\otimes SH^j(M)\to SH^{i+j}(M),\;x\dot y=\psi_P(x,y), \] defined by the pair-of-pants surface \(P\). He points out that \(H^*(M)\) and its dual \(H_*(M)\) have a TQFT structure via Morse theory, analogous to the TQFT structures on \(SH^*(M)\) and \(SH_*(M)\) (the non-quantum Morse operations are defined in Subsection 6.8). Theorem 6.6 states that the map \(c^*\) from Section 5, and its dual \(c_*\) are TQFT maps, and hence unital ring maps using the cup product on \(H^*(M)\) and the intersection product on the dual \(H_*(M)\). This result is the analogue of the ring isomorphism \(FH^*(M,H)\rightarrow QH^*(M)\), for weakly monotone closed symplectic manifolds \((M,\omega)\) [S. Piunikhin et al., Publ. Newton Inst. 8, 171–200 (1996; Zbl 0874.53031)] (PSS) .
From Theorems 6.13, 6.17, 7.9 and 9.8 the author concludes that “\(HW^*(L)_\eta\) has a TQFT structure, part of which is a unital ring structure. It is an \(SH^*(M)_{\bar{\eta}}\) module via \(\mathcal{W}_D\). The Viterbo restriction maps \(HW^*(L)_\eta\rightarrow HW^*(L\cap W)_{\eta|W}\) preserve the TQFT and module structure.”
Next the author deals with twisted symplectic (co)homology, twisted TQFT, twisting by closed 1-forms from the base, and twisted wrapped theory in Section 7.
Section 8 refers to the action filtration, and the \(SH^*_+\), \(HW^*_+\) groups.
Concerning the Viterbo functoriality, in Section 9 the author proves that the twisted Viterbo restriction maps from his previous paper [Zbl 1175.57019] are TQFT maps, and in particular, unital ring homomorphisms.
The next three sections are devoted to vanishing criteria, Arnold’s conjecture, and exact contact hypersurfaces, respectively. The applications to exact contact hypersurfaces are based on a vanishing result from the above mentioned work.
In Theorem 11.1 the author shows that if \(HW^*(L)\) vanishes, then the Arnold’s chord conjecture holds for \(\partial L\subset \partial M\). For a generic contact form \(\alpha\) there exist a number of chords equal at least to \(\operatorname{rank}H^*(L)\), and similarly for \(HW^*(L)_\eta\) vanishing for \(\eta\in H^2(M,L;\mathbb{R})\). Theorem 11.3 shows that the chord conjecture holds if \(H^2(T^*N)\rightarrow H^2(L)\) is not injective, and there exist at least \(\operatorname{rank}H^*(L)\) Reeb chords. Next, the author refers to ALE spaces (simply connected hyper-Kähler 4-manifolds which at infinity look like \(\mathbb{C^2}/G\) for a finite subgroup \(G \in \mathrm{SL}(2,\mathbb{C})\)) for which the chord conjecture holds for any \(\partial L\). Theorem 11.4 states that for such a space the number of Reeb chords is at least \(\operatorname{rank}H^*(L)\).
The displaceability of \(\partial M\) and Rabinowitz Floer theory are treated in Section 13, the string topology (the Abbondandolo-Schwarz construction and TQFT structure on \(H_*(\mathcal{L}N)_\eta\) and the Chas-Sullivan loop product) in Section 14.
In the last section of the paper, the author deals with Floer trajectories converging to broken Morse trajectories and he proves that \(c^*\) maps preserve the TQFT structure on \(H^*(M)\) (Theorem 6.14), in a similar way as in the proof of Theorem A.14, but using a PSS-description of the isomorphism \(H^*(L)\otimes\Lambda\rightarrow HW^*(L;H^\delta).\)
Appendix A refers to the TQFT structure on \(SH^*(M)\), dealing with model Riemann surfaces, Floer solutions, Fredholm theory, smoothness and compactness of the moduli spaces, TQFT operations.
Next, the author treats some problems concerning coherent orientations for Floer trajectories, Fredholm operators (on trivial bundles over a cylinder and on bundles \(E\rightarrow \overline{M}\)), orientation signs for \(SH^*(M)\) and for the TQFT, the choice of trivializations over the Hamiltonian orbits, the role of the canonical bundle \(\mathcal{K}\) and dimension counts (Appendix B).
\(SH^*(M)\) and TQFT structures are defined by using nonlinear Hamiltonians in Appendix C.
In the last part of the paper (Appendix D) the author deals with the energy and maximum principle for Floer and wrapped solutions, and he proves Lemma D.3 (No escape Lemma), which states that any solution \(u:S\rightarrow V\) of \((du-X\otimes\beta)^{0,1}=0,\) for which \(u(\partial S)\in \partial V\), must map entirely into \(\partial V\) and must satisfy \(du=X\otimes\beta\). Finally, Lemmas D.5 and D.6 reformulate the No escape Lemma for nonlinear \(H\) and respectively, for an exact Lagrangian.
The paper consists of 15 sections and 4 appendices.
In the introduction the author presents the main results of the paper, concerning the topological quantum field theory (TQFT), TQFT structure on ordinary cohomology, wrapped Floer cohomology, the Arnol’d chord conjecture, obstructions to exact contact embeddings, displaceability of contact hypersurfaces and string topology.
Next, the author recalls the notion of Liouville domain (exact symplectic manifold conical at infinity), Reeb periods, contact type conditions, Hamiltonians, action functional, Floer trajectories and their energy, transversality and compactness results (Section 2).
The symplectic cohomology \(SH^*(M)\) and symplectic homology \(SH_*(M)\) are constructed, and in Theorem 3.4 it is shown that \(SH_*(M)\) is canonically the dual of \(SH^*(M)\), but not vice versa.
Section 4 refers to wrapped Floer cohomology \(HW^*(L)\), the relative analogue of \(SH^*\) for exact Lagrangian submanifolds \(L\) of a Liouville domain \((M,d\theta)\), \(L\) having a transverse intersection with the boundary of \(M\), namely \(\partial L = L \cap\partial M\). There are proved some results concerning Hamiltonian and Reeb chords, wrapped trajectories and their energy, transversality and compactness and wrapped Floer cohomology.
In the next section the author shows that there exist maps \(c^* : H^*(M)\rightarrow SH^*(M)\) and \(c^* : H^*(L)\rightarrow SH^*(L)\) from the ordinary cohomology to the symplectic and wrapped cohomologies. In the wrapped case there is a map \(c^* : H^*(L)\cong HW^*(L;H^\delta)\rightarrow HW^*(L)\), so \(HW^*(L)\neq 0\) and there is an isomorphism \(H^*(L)\otimes\Lambda \rightarrow HW^*(L;H^\delta)\).
The author summarizes the TQFT structure on \(SH^*(M)\) in detail in Appendix A, then he proves some results concerning the graded-commutative and associative product \[ \psi_P:SH^i(M)\otimes SH^j(M)\to SH^{i+j}(M),\;x\dot y=\psi_P(x,y), \] defined by the pair-of-pants surface \(P\). He points out that \(H^*(M)\) and its dual \(H_*(M)\) have a TQFT structure via Morse theory, analogous to the TQFT structures on \(SH^*(M)\) and \(SH_*(M)\) (the non-quantum Morse operations are defined in Subsection 6.8). Theorem 6.6 states that the map \(c^*\) from Section 5, and its dual \(c_*\) are TQFT maps, and hence unital ring maps using the cup product on \(H^*(M)\) and the intersection product on the dual \(H_*(M)\). This result is the analogue of the ring isomorphism \(FH^*(M,H)\rightarrow QH^*(M)\), for weakly monotone closed symplectic manifolds \((M,\omega)\) [S. Piunikhin et al., Publ. Newton Inst. 8, 171–200 (1996; Zbl 0874.53031)] (PSS) .
From Theorems 6.13, 6.17, 7.9 and 9.8 the author concludes that “\(HW^*(L)_\eta\) has a TQFT structure, part of which is a unital ring structure. It is an \(SH^*(M)_{\bar{\eta}}\) module via \(\mathcal{W}_D\). The Viterbo restriction maps \(HW^*(L)_\eta\rightarrow HW^*(L\cap W)_{\eta|W}\) preserve the TQFT and module structure.”
Next the author deals with twisted symplectic (co)homology, twisted TQFT, twisting by closed 1-forms from the base, and twisted wrapped theory in Section 7.
Section 8 refers to the action filtration, and the \(SH^*_+\), \(HW^*_+\) groups.
Concerning the Viterbo functoriality, in Section 9 the author proves that the twisted Viterbo restriction maps from his previous paper [Zbl 1175.57019] are TQFT maps, and in particular, unital ring homomorphisms.
The next three sections are devoted to vanishing criteria, Arnold’s conjecture, and exact contact hypersurfaces, respectively. The applications to exact contact hypersurfaces are based on a vanishing result from the above mentioned work.
In Theorem 11.1 the author shows that if \(HW^*(L)\) vanishes, then the Arnold’s chord conjecture holds for \(\partial L\subset \partial M\). For a generic contact form \(\alpha\) there exist a number of chords equal at least to \(\operatorname{rank}H^*(L)\), and similarly for \(HW^*(L)_\eta\) vanishing for \(\eta\in H^2(M,L;\mathbb{R})\). Theorem 11.3 shows that the chord conjecture holds if \(H^2(T^*N)\rightarrow H^2(L)\) is not injective, and there exist at least \(\operatorname{rank}H^*(L)\) Reeb chords. Next, the author refers to ALE spaces (simply connected hyper-Kähler 4-manifolds which at infinity look like \(\mathbb{C^2}/G\) for a finite subgroup \(G \in \mathrm{SL}(2,\mathbb{C})\)) for which the chord conjecture holds for any \(\partial L\). Theorem 11.4 states that for such a space the number of Reeb chords is at least \(\operatorname{rank}H^*(L)\).
The displaceability of \(\partial M\) and Rabinowitz Floer theory are treated in Section 13, the string topology (the Abbondandolo-Schwarz construction and TQFT structure on \(H_*(\mathcal{L}N)_\eta\) and the Chas-Sullivan loop product) in Section 14.
In the last section of the paper, the author deals with Floer trajectories converging to broken Morse trajectories and he proves that \(c^*\) maps preserve the TQFT structure on \(H^*(M)\) (Theorem 6.14), in a similar way as in the proof of Theorem A.14, but using a PSS-description of the isomorphism \(H^*(L)\otimes\Lambda\rightarrow HW^*(L;H^\delta).\)
Appendix A refers to the TQFT structure on \(SH^*(M)\), dealing with model Riemann surfaces, Floer solutions, Fredholm theory, smoothness and compactness of the moduli spaces, TQFT operations.
Next, the author treats some problems concerning coherent orientations for Floer trajectories, Fredholm operators (on trivial bundles over a cylinder and on bundles \(E\rightarrow \overline{M}\)), orientation signs for \(SH^*(M)\) and for the TQFT, the choice of trivializations over the Hamiltonian orbits, the role of the canonical bundle \(\mathcal{K}\) and dimension counts (Appendix B).
\(SH^*(M)\) and TQFT structures are defined by using nonlinear Hamiltonians in Appendix C.
In the last part of the paper (Appendix D) the author deals with the energy and maximum principle for Floer and wrapped solutions, and he proves Lemma D.3 (No escape Lemma), which states that any solution \(u:S\rightarrow V\) of \((du-X\otimes\beta)^{0,1}=0,\) for which \(u(\partial S)\in \partial V\), must map entirely into \(\partial V\) and must satisfy \(du=X\otimes\beta\). Finally, Lemmas D.5 and D.6 reformulate the No escape Lemma for nonlinear \(H\) and respectively, for an exact Lagrangian.
Reviewer: Simona Druta-Romaniuc (Iaşi)
MSC:
| 53D40 | Symplectic aspects of Floer homology and cohomology |
| 57R56 | Topological quantum field theories (aspects of differential topology) |
| 81T45 | Topological field theories in quantum mechanics |
