General validity of the proposed Sp(k) chiral-ring constraints

Establish that, for every five-dimensional N=1 Sp(k) gauge theory with 0 ≤ N_f ≤ 2k+5 fundamental hypermultiplets that admits a UV fixed point, the Higgs-branch chiral ring at infinite coupling is generated by the mesons M (adjoint of SO(2N_f)), the gaugino bilinear S (moment map of the topological U(1) symmetry), and instanton operators I and \tilde{I} (spinors of SO(2N_f)), and that these generators satisfy the relations M^2_{ij} = S^2 δ_{ij}; (Mγ)·I = SI and (Mγ)·\tilde{I} = S\tilde{I}; ε · ∧^p M = S^a (∧^b M) when N_f ≤ 2k+1 with p = a + b, N_f = p + b, and p ≤ k+1; S^c (∧^d M) = (I·\tilde{I})|_{μ_{2d}} with c + d = k + 1; and Sym^2 I = (I·I)|_{μ_{N_f}^2} (and similarly for \tilde{I}).

Background

The paper proposes a structured set of chiral-ring relations at infinite coupling for Sp(k) gauge theories, involving mesons M, the gaugino bilinear S, and instanton operators I, \tilde{I}. These relations include a Casimir-type relation M^2_{ij}=S^2δ_{ij}, eigenvalue-like relations (Mγ)·I = SI and (Mγ)·\tilde{I}=S\tilde{I}, Hodge-dual corrections ε·∧^p M = S^a(∧^b M) below a certain flavour bound, instanton-bilinear corrections S^c∧^d M = (I·\tilde{I})|{μ{2d}}, and symmetric-product constraints on instantons.

The authors test these relations systematically for Sp(2) with N_f = 1,…,9 and for SU(2) (citing prior work), and in sporadic higher-rank cases. They conjecture the relations hold for general Sp(k) with any allowed number of flavours, i.e., up to the UV-complete bound N_f ≤ 2k+5.