Discrete and continuous symmetries in monotone Floer theory
Let \(L\) be a closed, connected, monotone Lagrangian submanifold of a compact symplectic manifold \((X,\omega)\). The author develops two new techniques for computing the Floer cohomology \(H^\ast(L,L)\) when \(L\) has certain symmetries inside \(X\). The techniques are applied to a free \(\operatorname{PSU}(N-1)\) orbit \(L\) in \((\mathbb{CP}^{N-2})^N\), where \(N\geq 3\) and \((\mathbb{CP}^{N-2},\omega_{FS})\) is equipped with the Fubini-Study form. The author's first technique involves the (length-zero) closed-open string map \[ \mathcal{CO}^0:QH^\ast(X;R) \rightarrow HF^\ast(L,L;R) \] from the quantum cohomology to the Lagrangian Floer cohomology, which is a unital \(R\)-algebra homomorphism. The Lagrangians considered are \(K\)-homogeneous, i.e., \(L\) is a Lagrangian orbit of the action of a compact connected Lie group \(K\) on a compact Kähler manifold \(X\) by holomorphic automorphisms. The following theorem is the main result in the paper concerning the closed-open string map, where \(N_L^+\) is the minimal Chern number for rational curves in \(X\) that are holomorphic. Throughout the paper the author assumes that if the characteristic of the coefficient ring \(R\) is not 2, then \(L\) is orientable and equipped with a relative spin structure. \textbf{Theorem 3} (Theorem 3.5.3) Let \(Z \subset X\) be a setwise \(K\)-invariant analytic subvariety of complex codimension \(\leq N_X^+\), disjoint from \(L\) and Poincaré dual to a class \(\alpha \in QH^\ast(X;R)\). Then \[ \mathcal{CO}^0(\alpha) = \lambda \cdot 1_L, \] where \(\lambda\) is a count of axial discs passing through \(Z\). If \(L\) is equipped with the standard spin structure (see Definition 3.1.3) then all of these discs contribute with positive sign. The author's second technique concerns the Oh spectral sequence \(E_\ast\). If \(L\) is a closed Lagrangian submanifold of a compact symplectic manifold \((X,\omega)\), then there is an action of the group Symp\((X,L)\) of symplectomorphisms that preserve \(L\) setwise on the \(E_1\) page of the Oh spectral sequence. The author proves that the action commutes with the differentials in the spectral sequence, and hence there is an action of Symp\((X,L)\) on the spectral sequence \(E_\ast\). More specifically, let \(H^D_2\) be the image of \(\pi_2(X,L)\) in \(H_2(X,L;\mathbb{Z})\) under the Hurewicz homomorphism and let \(R[H_2^D]\) be the group ring. Using the filtration by the Maslov index the author constructs a spectral sequence \[ E_1 \cong H^\ast(L;R) \otimes_R R[H_2^D] \Longrightarrow HF^\ast(L,L;R[H_2^D])\qquad (3) \] and proves the following theorem. \textbf{Theorem 4} (Theorem 4.2.2) There is an action of Symp\((X,L)\) on the \(E_1\) page of (3), which involves its actions on both \(H^D_2\) and \(H^\ast(L;R)\), that induces an action on the whole spectral sequence. A large portion of the paper concerns applications of the author's techniques to a family of Lagrangian submanifolds in products of complex projective spaces. Let \(N \geq 3\). The standard action of \(\operatorname{SU}(N-1)\) on \(\mathbb{C}^{N-1}\) gives a \(\operatorname{PSU}(N-1)\) action on \((\mathbb{C}P^{N-2},\omega_{FS})\), and the diagonal action of the group \(\operatorname{PSU}(N-1)\) on \(X = (\mathbb{CP}^{N-2})^N\) is Hamiltonian. The zero set \(L\subset X\) of the moment map is both a monotone Lagrangian submanifold and a free \(\operatorname{PSU}(N-1)\) orbit. There is also an action of the symmetric group \(S_N\) on \(X\) given by permuting factors, and \(L\) is setwise invariant under the action of \(S_N\). Let \(R\) be a coefficient ring and if \(\operatorname{char} R \neq 2\) assume that \(L\) is orientable and equipped with a choice of a relative spin structure. The author proves the following. \textbf{Theorem 1} (Corollary 5.4.6 and Theorem 5.7.3) For any choice of relative spin structure and local system on \(L\), \(H^\ast(L,L;R)\) vanishes unless \(N = p^r\) or \(2p^r\) for some prime \(p\) and positive integer \(r\). Conversely, if \(R\) is a field of prime characteristic \(p\) and \(N = p^r\) or \(2p^r\) then there exists a relative spin structure on \(L\) such that with the trivial local system we have an isomorphism of vector spaces \(HF^\ast(L,L;R) \cong H^\ast(L;R)\). The non-vanishing of \(HF^\ast(L,L)\) means that \(L\) cannot be displaced from itself by Hamiltonian diffeomorphisms. Surprisingly, the author proves the following for the above family of Lagrangian submanifolds \(L\) in \((\mathbb{CP}^{N-2})^N\). \textbf{Theorem 2} (Theorem 5.8.2) For all values of \(N\) the Lagrangian \(L\) is non-displaceable. The author uses the pearl complex for computing Floer cohomology throughout the paper, as developed by \textit{P. Biran} and \textit{O. Cornea} [``Quantum structures for Lagrangian submanifolds'', Preprint, \url{arXiv:0708.4221}].
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