Equivariant Lagrangian Floer Theory

• Equivariant Lagrangian Floer theory is a framework that integrates group actions into Lagrangian intersection theory to yield refined symplectic invariants.
• It employs Borel and Cartan models to introduce extra equivariant parameters, enhancing computational, algebraic, and categorical aspects of classical Floer theory.
• The theory establishes non-displaceability criteria and connects quantum cohomology with mirror symmetry using precise equivariant differential and Floer–Euler constructions.

Equivariant Lagrangian Floer theory provides a robust framework for studying the interaction of symmetry and Lagrangian intersection theory in symplectic geometry. By incorporating Lie group or finite group actions via the Borel or Cartan models, this theory enriches classical Floer-theoretic invariants and reveals subtle phenomena obscured in the non-equivariant setting, especially regarding non-displaceability under equivariant Hamiltonian isotopy. Recent advances have sharpened the algebraic, categorical, and computational aspects of equivariant Floer theory—especially in the monotone setting with involutions, circle, or torus actions—clarifying formalism relevant for both geometric applications and homological mirror symmetry.

1. Borel Equivariant Setup and Symplectic Involution

Let $(W, \omega)$ be a tame-at-infinity symplectic manifold equipped with a symplectic involution $a: W \to W$, with $a^2 = \mathrm{id}$ and $a^*\omega=\omega$. For a closed monotone Lagrangian $L\subset W$ preserved by $a$ (i.e., $a(L)=L$), with minimal Maslov number $N_L\ge2$, one constructs the Borel-equivariant total space $L_G=EG\times_G L$ for $G=\mathbb{Z}/2$ (the universal $\mathbb{Z}/2$-bundle $EG \to BG \simeq \mathbb{R}P^\infty$). This Borel construction introduces extra equivariant parameters at the chain level, encoded via a degree $+1$ formal variable $e$ corresponding to the generator of $H^*(\mathbb{R}P^\infty)$ (Cant et al., 23 Oct 2025).

2. Construction of the Equivariant Floer-Quantum Complex

The chain complex underlying equivariant Lagrangian quantum cohomology employs data from Morse–Smale function $P$ on $L$, and a generic $\omega$-tame almost complex structure $J$ on $W$ ensuring pearl moduli space transversality. Using $\mathbb{Z}/2$-coefficients, one forms the chain group $$\mathrm{CM}(P) = \bigoplus_{x\in\mathrm{Crit}(P)}\mathbb{Z}/2\langle x \rangle$$, graded by the Morse index.

Introduce a Novikov variable $q$ (with $\deg q = N_L$) and an equivariant variable $e$ ($\deg e=1$), forming the equivariant chain complex

$$\mathrm{CEQ} = \mathrm{CM}(P) \otimes \mathbb{Z}/2[q^{-1},q] \otimes \mathbb{Z}/2[e]$$

over the ring $$\Lambda_e = \mathbb{Z}/2[q^{-1},q]\otimes \mathbb{Z}/2[e]$$. The module structure accommodates Laurent power series in $q$ and polynomials in $e$. This setup precisely models the structure of the Borel equivariant parameter space and is compatible with existing filtration and grading conventions from the underlying Floer theory (Cant et al., 23 Oct 2025).

3. Equivariant Differential and the Floer–Euler Class

The equivariant differential $d_{\mathrm{eq}}$ on $\mathrm{CEQ}$ is defined by

$d_{\mathrm{eq}} = \sum_{i,j\ge0} q^i e^j d_{i,j}$

where

• $d_{0,0}$ is the classical Morse differential,
• $d_{i,0}$ for $i>0$ count ordinary pearl trajectories of total Maslov index $N_L i$, reproducing the Biran–Cornea quantum differential,
• $d_{i,j}$ for $j\ge1$ count Borel-parametrized pearl trajectories, built from families of data $(P_\eta,J_\eta)$ over $\eta\in S^\infty$ satisfying $(P_{-\eta},J_{-\eta}) = (a^*P_\eta, a^*J_\eta)$ and equivariance under the shift on $S^\infty$.

In the so-called “circle-bundle” case (e.g., when $L$ is the preimage of a monotone Lagrangian $\bar L \subset M$ under a free circle-Reeb quotient) the differential simplifies: $$d_{\mathrm{eq}} = \begin{pmatrix} d_{\mathrm{QC}} & e^2 + F \ 0 & d_{\mathrm{QC}} \end{pmatrix}$$ where $F \in \mathrm{QH}^2(\bar L)$ is the Floer–Euler class of the associated circle-bundle (Cant et al., 23 Oct 2025). The $e^2$ term encodes the classical Gysin exact sequence relation $e^2=0$ in $H^*_{S^1}(S^1)\cong \mathbb{Z}/2[e]/(e^2)$, while $F$ captures quantum topological data.

4. Transversality, Compactness, and $d_{\mathrm{eq}}^2=0$ in the Monotone Setting

Monotonicity with $N_L\ge2$ ensures that all non-constant pearls have positive Maslov index, preventing unwanted sphere or disk bubbling in rigid moduli spaces. The theory employs “very regular Borel data” $(P_\eta, J_\eta)$ to guarantee transversality for all relevant moduli spaces, using Sard–Smale transversality arguments with $J_\eta$ perturbed away from $S^\infty$ poles. The boundary strata of one-dimensional parametrized moduli correspond to Morse and pearl breakings, flow-line breakings in $S^\infty$ (which cancel in pairs mod 2), and nodal/desingularizing configurations (which also cancel). Gluing and compactification arguments then yield $d_{\mathrm{eq}}^2=0$, ensuring a well-defined equivariant Floer cohomology (Cant et al., 23 Oct 2025).

5. Equivariant Displacements and Vanishing Theorems

A central theorem is: if a Hamiltonian isotopy $\{\psi_t\}$ commuting with the involution $a$ displaces $L$, i.e., $\psi_1(L)\cap L = \emptyset$, then

$$\mathrm{QH}_{\mathrm{eq}}^*(L) = H^*(\mathrm{CEQ},d_{\mathrm{eq}}) = 0$$

The proof constructs an equivariant continuation map, which becomes chain homotopic to identity but vanishes at the chain level in the case of displacement, forcing the equivariant quantum cohomology to vanish. This provides a powerful non-displaceability criterion: the nonvanishing of $\mathrm{QH}_{\mathrm{eq}}^*(L)$ obstructs equivariant Hamiltonian displaceability (Cant et al., 23 Oct 2025).

6. Explicit Computation: The Circle-Bundle and Floer–Gysin Formalism

If $W\subset\mathbb{C}^n$ is a Liouville domain with free circle-Reeb flow and $L$ is the Hopf preimage of a monotone $\bar L\subset\mathbb{CP}^{n-1}$, the equivariant quantum cohomology is

$$\mathrm{QH}_{\mathrm{eq}}^*(L) \simeq (\mathrm{QH}^*(\bar L) \otimes \mathbb{Z}/2[e]) / (e^2+F)$$

where $F$ is the Floer–Euler class in degree 2. For invertible $F$, a $\mathbb{Z}/2[e]$-basis is given by $$\mathrm{QH}^*(\bar L) \oplus e\,\mathrm{QH}^*(\bar L)$$. Crucially, this ring can be nonzero even if ordinary $\mathrm{QH}^*(L)$ vanishes, since the obstruction $e^2+F=0$ encodes the topological Gysin relation in equivariant cohomology (Cant et al., 23 Oct 2025).

In the standard case $W=\mathbb{C}^n$, $L$ the preimage of a monotone $\bar L\subset\mathbb{CP}^{n-1}$ under the Hopf bundle, and $a(z)=-z$, the theory confirms non-displaceability of $L$ via the nonvanishing of the equivariant cohomology.

---

Summary Table: Key Elements of Equivariant Lagrangian Floer Theory

Aspect	Construction/Formula	Reference
Borel Model	$L_G = EG \times_G L$ over $BG = \mathbb{R}P^\infty$, equivariant parameter $e$	(Cant et al., 23 Oct 2025)
Equivariant Complex	$$\mathrm{CEQ} = \mathrm{CM}(P) \otimes \mathbb{Z}/2[q^{-1},q] \otimes \mathbb{Z}/2[e]$$	(Cant et al., 23 Oct 2025)
Equivariant Differential	$d_{\mathrm{eq}} = \sum q^i e^j d_{i,j}$, with $d_{0,0}$ Morse, $d_{i,0}$ ordinary pearls, $d_{i,j}$ Borel pearls	(Cant et al., 23 Oct 2025)
Floer–Euler Operator & Gysin	$$d_{\mathrm{eq}} = \begin{pmatrix} d_{\mathrm{QC}} & e^2 + F \ 0 & d_{\mathrm{QC}} \end{pmatrix}$$, $e^2+F$ encodes the Gysin relation	(Cant et al., 23 Oct 2025)
Displacement Obstruction	$\mathrm{QH}_{\mathrm{eq}}^*(L) = 0$ if equivariantly displaceable	(Cant et al., 23 Oct 2025)
Explicit Ring Structure	$$\mathrm{QH}_{\mathrm{eq}}^*(L) \simeq (\mathrm{QH}^*(\bar L) \otimes \mathbb{Z}/2[e])/(e^2+F)$$	(Cant et al., 23 Oct 2025)

---

Equivariant Lagrangian Floer theory in the monotone and circle-bundle settings offers a sharp tool for analyzing Hamiltonian displaceability in the presence of symmetries. The theory categorifies classical Gysin relations, exhibits finer detection of non-displaceability than ordinary Floer theory, and directly interfaces with quantum cohomological and functorial structures central to symplectic topology and mirror symmetry (Cant et al., 23 Oct 2025).