An explicit four-corner dictionary for (2,p) minimal Liouville gravity

Abstract

(2,p) minimal Liouville gravity admits four algebraic descriptions interrelated by dualities. On the Frobenius manifold (FM) side the theory is built either on the $A_{1}$ manifold of $Q(y)=y^{2}+u_{1}$ ($y$-side) or on the $A_{p-1}$ manifold in the variable $x$ ($x$-side). On the spectral curve / topological recursion (SC) side the analogous choice is the Chebyshev curve and its $x\leftrightarrow y$ swap. FM-$y$, SC-$y$ require a resonance transformation between the KdV times and the Liouville couplings, while the other two do not. We work out the explicit correspondence between the four approaches at the level of the dispersionless tau-function. The normalisation-independent three-point ratios agree with conformal field theory in all four formulations, and matching the full amplitudes fixes a single per-insertion factor reproducing the signed Verlinde matrices. We use the Kharchev--Marshakov integral transform of the basis functions, whose determinant gives an explicit relation between the corresponding tau-functions and realises the $x\!\leftrightarrow\! y$ swap at genus zero. Once the independently fixed linear SC-$x$ normalisation is imposed, the Laurent deformation in the second coordinate gives an alternative genus-zero derivation of the known compact resonance transformation for the whole (2,p) series, including all mixed coefficients.