Atiyah–Floer conjecture: equivalence of instanton and Lagrangian Floer homology

Establish an isomorphism between instanton Floer homology HF_inst(Y) of a closed, oriented three-manifold Y and Lagrangian Floer homology HF_Lag(L1, L2) arising from the pair of Lagrangian submanifolds L1 and L2 in the moduli space of flat G-connections on the Heegaard surface Σ determined by a Heegaard splitting Y = H1 ∪_Σ H2, for compact gauge group G (such as SU(2) or SO(3)). Concretely, identify HF_inst(Y) ≅ HF_Lag(L1, L2), where L1 and L2 consist of flat connections on Σ that extend over the handlebodies H1 and H2, respectively.

Background

This conjecture connects two Floer theories constructed in different settings. On the gauge-theoretic side, instanton Floer homology HF_inst(Y) is defined by counting anti-self-dual connections on ℝ×Y and is derived from the Chern–Simons functional on the space of connections. On the symplectic side, a Heegaard splitting Y = H1 ∪_Σ H2 determines Lagrangian submanifolds L1 and L2 in the symplectic moduli space of flat G-connections on the surface Σ, and Lagrangian Floer homology HF_Lag(L1, L2) is defined by counting pseudo-holomorphic strips with boundary on these Lagrangians.

The conjecture asserts that these two constructions yield canonically isomorphic invariants, reflecting a deep equivalence between four-dimensional gauge theory on ℝ×Y, three-dimensional Chern–Simons theory on Y, and two-dimensional symplectic geometry on Σ. Despite substantial progress on analytical frameworks bridging these theories, a complete proof remains open in general.