One constant to rule them all

In this work, we study the coupling matrix of $\mathcal{N}=2$ Superconformal QCD in vicinity of the special vacuum. We find that although there are $\lfloor\frac{N}{2}\rfloor$ independent coupling constants of the theory, only one of them plays a special role.

💡 Research Summary

In this paper the authors investigate the coupling matrix of four‑dimensional 𝒩=2 super‑conformal QCD with gauge group SU(N) and 2N fundamental hypermultiplets, focusing on the so‑called “special vacuum”. In this vacuum the vacuum expectation values of the adjoint scalar a₁,…,a_N sit at the vertices of a regular N‑gon in the complex plane, a configuration that preserves a residual Weyl‑ℤ_N symmetry. To treat this symmetry efficiently the authors introduce symmetric variables w_k (k=0,…,N‑1) built from elementary symmetric polynomials of the a_u and their Fourier modes v_l (l=1,…,N‑1) defined by a discrete Fourier transform of the small fluctuations δa_u around the special vacuum. The relations between w_k and v_l are worked out explicitly, showing that the linear perturbations are encoded solely in v_l while higher‑order perturbations involve quadratic combinations of the v’s.

The prepotential F, which encodes the low‑energy effective action, must have mass dimension two and be invariant under the residual Weyl‑ℤ_N symmetry. By expanding F up to quadratic order in the fluctuations the authors obtain the most general form (eq. 3.1). This expansion contains a single linear term proportional to a coupling constant τ_IR (later identified as τ₁^IR) and a set of quadratic terms each multiplied by an independent constant τ_l^IR (l=2,…,⌊N/2⌋). The crucial observation is that only τ₁^IR appears in the linear term and therefore survives in the limit where the polygon radius (parameterized by w_{N‑1}) goes to infinity, i.e. when the special vacuum becomes asymptotically large. All other τ_l^IR contribute only to higher‑order corrections. Consequently, although the theory possesses ⌊N/2⌋ independent couplings, a single constant governs the dominant physics in the asymptotic regime.

From the prepotential the authors derive the full coupling matrix τ_{uv}=∂²F/∂a_u∂a_v (eq. 3.11). It can be written as a sum over Fourier modes: \