Eigenvalue spectrum equality for U(3) instanton Floer homology of S^1×Σ_g

Determine whether the simultaneous-eigenvalue spectrum Xi_{g,d}^3 of the operators μ_2(Σ_g), μ_3(Σ_g), μ_2(x), μ_3(x) acting on the U(3) instanton Floer homology V_{g,d}^3 = I_*^3(S^1×Σ_g, γ_d) is exactly the set E_{g,d}^3. Here γ_d = S^1×{x_1,…,x_d} with d coprime to 3; E_{g,d}^3 is defined as { (√3 ζ^k a, √−3 ζ^{2k} b, 3 ζ^{2k}, 0 ) | (a,b)∈C_g, k∈{0,1,2} }, where ζ = e^{2πi/3} and C_g = { (a,b)∈Z^2 : |a|+|b| ≤ 2g−2, a ≡ b (mod 2) }. Proposition 2.4 establishes the inclusion E_{g,d}^3 ⊂ Xi_{g,d}^3; the open problem is to prove the reverse inclusion Xi_{g,d}^3 ⊂ E_{g,d}^3 and thus equality Xi_{g,d}^3 = E_{g,d}^3 for all genera g and all d coprime to 3.

Background

The paper studies the action of universal cohomology operators on U(3) instanton Floer homology for Y = S^1×Σ_g with an admissible bundle determined by γd. Writing V{g,d}^3 = I_*^3(S^1×Σ_g, γd), the authors consider the simultaneous eigenvalues of α_r = μ_r(Σ_g) and β_r = μ_r(x), together with odd-degree operators μ_r(η_i). They define a candidate set of eigenvalues E{g,d}^3 based on lattice constraints coming from the geometry of Σg and root-of-unity symmetries, and prove in Proposition 2.4 that these values do occur in the spectrum (E{g,d}^3 ⊂ Xi_{g,d}^3).

For applications such as sutured decomposition and non‑vanishing results, the authors establish a weaker characterization (Theorem 2.7) of the spectrum on certain slices (where the third coordinate equals a 3rd root of 27 and the fourth coordinate is zero). However, the full identification of the spectrum Xi_{g,d}^3 with E_{g,d}^3 is not proved. Equality would give a complete description of simultaneous eigenvalues and streamline arguments analogous to the N=2 case, where the spectrum is known explicitly.