Non-toric 5d SCFTs from Reid's Pagoda

We construct new families of non-toric 5d SCFTs via abelian orbifolds of the Reid Pagoda, including a surprising infinite family of rank-1 theories, that evade all known classifications. Using the McKay correspondence, we derive their BPS quivers and superpotentials. The hallmark of these theories is a novel sector we dub Pagoda matter, whose vacuum expectation values obstruct the Kaehler moduli. This mechanism freezes the gauge coupling to infinite value, precluding a weak-coupling limit and rendering the theories intrinsically strongly coupled. Finally, we interpret these results as 5d SCFTs deformed by non-constant flavor backgrounds.

💡 Research Summary

The paper challenges the conventional paradigm that five‑dimensional superconformal field theories (5d SCFTs) must be either toric or admit a weakly‑coupled gauge theory description. Starting from the non‑toric Reid Pagoda singularity defined by uv = z² − w^{2k} (k∈ℕ), the authors generate an infinite family of new 5d SCFTs by taking abelian orbifolds of the form (Y_Pagoda)/H, where H is a finite abelian group such as ℤ_N or ℤ_m × ℤ_N.

Using the McKay correspondence, they translate the group action into a BPS quiver: each irreducible representation of H gives a node, and the transformation properties of the ambient coordinates dictate the arrows. The construction is illustrated in detail for the familiar case C²/ℤ_2 × C, where the resulting quiver reproduces the known A₁ affine quiver with the expected superpotential. For the Pagoda orbifolds, the superpotential acquires additional higher‑order terms proportional to w^{k+1}, reflecting the intrinsic non‑toric nature of the geometry. Solving the F‑term equations reproduces the original Pagoda equation, confirming that the quiver data encodes the singularity precisely.

A central new ingredient is the “Pagoda matter” sector. When the scalar components of this sector acquire non‑zero vacuum expectation values, the F‑terms force w^{k} ≠ 0, which in turn forces the Kähler modulus controlling the volume of the exceptional curve to shrink to zero. In field‑theoretic language, the inverse gauge coupling m₀ ∝ Vol(C) → 0, i.e. the gauge coupling freezes at infinity. Consequently, there is no point on the Coulomb branch where a perturbative gauge theory description exists; the theory is intrinsically strongly coupled.

The authors reinterpret this phenomenon as a deformation by a non‑constant flavor background. By promoting the global flavor parameters to position‑dependent Higgs fields Φ(w), the geometry is deformed from a toric orbifold (e.g. local F₂) to an isolated point‑like singularity that supports the Pagoda matter. A D2‑brane probe analysis, using 3d mirror symmetry, shows that the Pagoda matter is “trapped” by the geometry, providing a concrete string‑theoretic picture of the coupling freeze.

Finally, the paper extends the construction to a broad class of abelian orbifolds, producing infinite families of rank‑N theories with arbitrarily high flavor rank. In particular, ℤ_N × ℤ_2 orbifolds yield new BPS spectra, while more general ℤ_m × ℤ_N orbifolds give rise to conformal matter sectors and suggest extensions to higher‑length flops such as the Laufer length‑two flop.

In summary, the work delivers three major advances: (1) a systematic McKay‑based method to derive BPS quivers and superpotentials for non‑toric 5d SCFTs; (2) the identification of a novel Pagoda‑matter sector that freezes the gauge coupling, eliminating any weak‑coupling limit; and (3) a physical interpretation in terms of non‑constant flavor backgrounds, supported by D‑brane probe calculations. These results open a new landscape of intrinsically strongly‑coupled, non‑toric 5d SCFTs that lie outside all previously known classifications.