The Floer theory introduced by Hohloch, Noetzel and Salamon (2009) associates homology groups to a pair (M,X)(M,X) consisting of a Cartan hypercontact 3-manifold MM and a compact hyperkähler manifold XX, built from the gradient flow of the hypersymplectic action functional on Map(M,X)\mathrm{Map}(M,X), whose critical points solve a nonlinear analogue of the Dirac equation. The construction — and with it the hyperkähler Arnold conjecture, a lower bound on the number of solutions by the Betti numbers of XX — was proved only for flat targets, i.e. finite quotients of hyperkähler tori. For a general compact hyperkähler target such as a K3 surface, the energy identity fails to control the derivatives of solutions, which can blow up along codimension-2 Hopf circles, and compactness is known to fail already for critical points. The problem is to resolve this bubbling and to construct the Floer homology groups HF∗(M,X,σ)\mathrm{HF}^*(M,X,\sigma) for arbitrary compact hyperkähler targets, with perturbation invariance, the expected isomorphism with H∗(X;Z2)H^*(X;\mathbb{Z}_2) in the component of constant maps, and the resulting Arnold-type existence theorem. The literature through 2026 (Walpuski's compactness theorem for Fueter sections, the Fueter 2-category program of Doan–Rezchikov, monopole and polysymplectic variants) shows that the bubbling is now well described but no such theory has been constructed: the problem is open.

Background

Hamiltonian Floer homology studies the action functional on the loop space of a symplectic manifold and proves, via the Arnold conjecture of Hamiltonian mechanics, that a generic Hamiltonian system on a compact symplectic manifold has at least as many periodic orbits as a smooth function must have critical points. Hohloch, Noetzel and Salamon proposed and carried out a quaternionic analogue of this picture in which both the time and the target are upgraded.

On the target side, a hyperkähler manifold is a Riemannian manifold XX carrying three complex structures I,J,KI,J,K satisfying the quaternionic relations I2=J2=K2=IJK=−idI^2=J^2=K^2=IJK=-\mathrm{id}, each of which is Kähler for the given metric, with Kähler forms ω1,ω2,ω3\omega_1,\omega_2,\omega_3. Basic examples are the flat hyperkähler tori Hn/Λ\mathbb{H}^n/\Lambda and their finite quotients (these are exactly the compact flat cases), K3 surfaces, and Hilbert schemes of points on K3 surfaces. On the time side, a hypercontact structure on a closed 3-manifold MM is a triple of contact forms (α1,α2,α3)(\alpha_1,\alpha_2,\alpha_3) with α1∧dα1=α2∧dα2=α3∧dα3\alpha_1\wedge d\alpha_1=\alpha_2\wedge d\alpha_2=\alpha_3\wedge d\alpha_3 and αi∧dαj+αj∧dαi=0\alpha_i\wedge d\alpha_j+\alpha_j\wedge d\alpha_i=0 for i≠ji\neq j; it is called a Cartan structure if the αi\alpha_i are the dual coframe of their Reeb vector fields v1,v2,v3v_1,v_2,v_3 (the unique fields with αi(vj)=δij\alpha_i(v_j)=\delta_{ij} and dαi(vj,⋅)=0d\alpha_i(v_j,\cdot)=0). Cartan hypercontact 3-manifolds are precisely the quotients of S3S^3 by finite subgroups of the unit quaternions; the 3-torus T3T^3 with a constant frame is included as an additional case.

For such a pair (M,X)(M,X) the hypersymplectic action functional on the mapping space Map(M,X)\mathrm{Map}(M,X) is A(f)=∫Mα1∧f∗ω1+α2∧f∗ω2+α3∧f∗ω3\mathcal{A}(f)=\int_M \alpha_1\wedge f^*\omega_1+\alpha_2\wedge f^*\omega_2+\alpha_3\wedge f^*\omega_3. Its critical points solve the perturbed nonlinear Dirac equation ∂φH(f):=I df(v1)+J df(v2)+K df(v3)−∇H(f)=0\partial\varphi_H(f) := I\,df(v_1)+J\,df(v_2)+K\,df(v_3)-\nabla H(f)=0, a nonlinear analogue of the 3-dimensional Dirac equation also known as the (perturbed) Fueter equation, first studied by Taubes; its negative L2L^2-gradient trajectories u:R×M→Xu:\mathbb{R}\times M\to X solve the 4-dimensional analogue ∂su+I du(v1)+J du(v2)+K du(v3)=∇H(u)\partial_s u+I\,du(v_1)+J\,du(v_2)+K\,du(v_3)=\nabla H(u), a special case of the Cauchy–Riemann–Fueter equation on a hyperkähler 4-manifold.

For compact flat XX, the source paper proved: compactness of the moduli spaces of critical points and trajectories; Floer homology groups HF∗(M,X,σ;H)\mathrm{HF}^*(M,X,\sigma;H) in each free homotopy class σ\sigma of maps M→XM\to X, independent of the perturbation HH; the isomorphism HF∗(M,X,σ0)≅H∗(X;Z2)\mathrm{HF}^*(M,X,\sigma_0)\cong H^*(X;\mathbb{Z}_2) on the component σ0\sigma_0 of constant maps; and the hyperkähler Arnold conjecture in this setting: if the contractible solutions are nondegenerate, their number is at least the sum of the Z2\mathbb{Z}_2-Betti numbers of XX. Flatness is essential to the proofs: on a flat target the energy identity E(u)=AH(f−)−AH(f+)E(u)=\mathcal{A}_H(f^-)-\mathcal{A}_H(f^+) together with nonpositive curvature yields uniform derivative bounds for finite-energy trajectories, and bubbling is excluded.

For non-flat XX this mechanism breaks down. If uν:S2→Xu_\nu:S^2\to X is a sequence of JλJ_\lambda-holomorphic curves (where Jλ=λ1I+λ2J+λ3KJ_\lambda=\lambda_1 I+\lambda_2 J+\lambda_3 K for λ∈S2\lambda\in S^2) whose derivatives blow up near a point z0z_0, and h:S3→S2h:S^3\to S^2 is a Hopf fibration (the fibration of S3S^3 over S2S^2 by the circles h−1(z)h^{-1}(z)), then fν=uν∘hf_\nu=u_\nu\circ h is a sequence of solutions of the unperturbed equation whose derivatives blow up along the Hopf circle h−1(z0)h^{-1}(z_0) while the energy stays bounded — codimension-2 bubbling, in contrast to the isolated bubbling points familiar from pseudoholomorphic curves. Compactness in fact already fails for critical points on a K3 surface with H=0H=0 (Remark 3.9 of the source), and it is an open question whether finite energy implies a uniform bound on ∣du∣|du| without flatness (Remark 3.11 of the source). Taming this bubbling is the recognized obstacle to the theory for general compact hyperkähler targets.

Problem Statement

Let MM be either a compact Cartan hypercontact 3-manifold (with Reeb vector fields v1,v2,v3v_1,v_2,v_3) or the 3-torus T3T^3 with a constant frame, and let XX be an arbitrary compact hyperkähler manifold, not assumed flat (for instance a K3 surface). Extend the Floer theory of the hypersymplectic action functional AH(f)=∫Mα1∧f∗ω1+α2∧f∗ω2+α3∧f∗ω3−∫MH(f)\mathcal{A}_H(f)=\int_M \alpha_1\wedge f^*\omega_1+\alpha_2\wedge f^*\omega_2+\alpha_3\wedge f^*\omega_3-\int_M H(f) from flat to general such targets. Concretely:

(i) Establish a compactness theorem for sequences of solutions of the perturbed nonlinear Dirac equation ∂φH(f)=0\partial\varphi_H(f)=0 on MM and of the Floer equation ∂su+I du(v1)+J du(v2)+K du(v3)=∇H(u)\partial_s u+I\,du(v_1)+J\,du(v_2)+K\,du(v_3)=\nabla H(u) on R×M\mathbb{R}\times M with bounded energy — including a bubbling analysis that accounts for codimension-2 bubbling along Hopf circles (the bubbles being holomorphic spheres for the complex structures JλJ_\lambda, λ∈S2\lambda\in S^2).

(ii) Construct, for generic HH and for each free homotopy class σ\sigma of maps M→XM\to X, Floer homology groups HF∗(M,X,σ)\mathrm{HF}^*(M,X,\sigma) generated by the solutions of the perturbed Dirac equation, with boundary operator defined by counting solutions of the Floer equation — excluding or incorporating the bubbling of (i) so that the boundary operator squares to zero — and prove that these groups are independent of the choice of HH.

(iii) Prove the hyperkähler Arnold conjecture for general compact hyperkähler XX: when the contractible solutions are nondegenerate, their number is bounded below by the sum of the Z2\mathbb{Z}_2-Betti numbers of XX, and establish the expected isomorphism HF∗(M,X,σ0)≅H∗(X;Z2)\mathrm{HF}^*(M,X,\sigma_0)\cong H^*(X;\mathbb{Z}_2) on the component σ0\sigma_0 of constant maps.

The source paper additionally conjectures richer structures on this theory — a ring structure on HF∗(S3,X)\mathrm{HF}^*(S^3,X) with module structures over it, Gromov–Witten-type invariants from the moduli spaces of the Cauchy–Riemann–Fueter equation, and a Floer–Donaldson correspondence with adiabatic limits of Donaldson–Thomas theory on M×SM\times S. These are natural extensions of the same program; the completion boundary of the problem stated here is the Floer theory (i)–(iii).

References

  1. Sonja Hohloch, Gregor Noetzel, Dietmar A. Salamon, "Hypercontact structures and Floer homology", Geometry & Topology 13(5) (2009), pp. 2543–2617, DOI 10.2140/gt.2009.13.2543, https://doi.org/10.2140/gt.2009.13.2543 (source paper)
  2. Thomas Walpuski, "A compactness theorem for Fueter sections", Commentarii Mathematici Helvetici 92(4) (2017), pp. 751–776, DOI 10.4171/CMH/423, https://doi.org/10.4171/CMH/423
  3. Viktor L. Ginzburg, Doris Hein, "Hyperkähler Arnold conjecture and its generalizations", International Journal of Mathematics 23(8) (2012), article 1250077, DOI 10.1142/S0129167X12500772, https://doi.org/10.1142/S0129167X12500772
  4. Dietmar A. Salamon, "The three-dimensional Fueter equation and divergence-free frames", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 83(1) (2013), pp. 1–28, DOI 10.1007/s12188-013-0075-1, https://doi.org/10.1007/s12188-013-0075-1
  5. Sonja Hohloch, "Hyperkähler Floer theory as infinite dimensional Hamiltonian system", Proceedings of the American Mathematical Society 143(8) (2015), pp. 3519–3524, DOI 10.1090/S0002-9939-2015-12511-7, https://doi.org/10.1090/S0002-9939-2015-12511-7
  6. Aleksander Doan, Semon Rezchikov, "Holomorphic Floer Theory and the Fueter Equation", preprint (2022), arXiv:2210.12047, https://arxiv.org/abs/2210.12047
  7. Saman Habibi Esfahani, "Towards a monopole Fueter Floer homology I: a compactness theorem", preprint (2023), arXiv:2305.09456, https://arxiv.org/abs/2305.09456
  8. Luca Asselle, Ronen Brilleslijper, "Fueter equations and the search for a higher dimensional Hamiltonian Floer theory I: analytical foundations and compactness", preprint (2026), arXiv:2603.01832, https://arxiv.org/abs/2603.01832
  9. Ronen Brilleslijper, Oliver Fabert, "From Euclidean field theory to hyperkähler Floer theory via regularized polysymplectic geometry", preprint (2023), arXiv:2311.18485, https://arxiv.org/abs/2311.18485