Obstruction of symmetric saddle-point contributions

Determine why the symmetric Young-tableau saddle-point solutions do not contribute to the instanton path integral, by characterizing the obstruction between their Lefschetz thimbles and the defining integration contour, and clarify how their contributions arise in semiclassical conformal-field-theory calculations such as the monodromy method.

Background

The large-instanton-number saddle-point equations admit symmetric solutions whose apparent contributions are more dominant than the asymmetric saddle that reproduces the forbidden-singularity radius of convergence. This conflicts with the independently verified CFT result that the radius is z*=1−exp(−2π/α_H).

The paper argues that the symmetric saddles therefore must not intersect the original integration contour, possibly because their associated singularities lie on nonprincipal branches of the multivalued free energy. Establishing this contour and Lefschetz-thimble mechanism would explain why apparently dominant complex saddles do not control the physical expansion and would connect the instanton description more directly to monodromy calculations.

References

It is a fascinating future problem to understand this obstruction in more details, and how the symmetric solution contribution arises in the semi-classical CFT computations, e.g. the monodromy method.

— Phase transition from eigenstate thermalization: forbidden singularity and instanton proliferation via AGT correspondence  (2608.13246 - Xu et al., 13 Aug 2026) in Section 4.1, “Additional solutions”

One possibility is that the run-away direction of the two-dimensional Young tableaux towards larger λ might be stabilized at finite c, and the two-dimensional Young tableaux may eventually dominate at large ν. This is certainly a reasonable conjecture at c∼O(1), since there is no parametric distinctions between the dynamics along the horizontal and vertical directions. If this is true, it might point to a distinct nature of z→1 singularity than that of the heavy-light result (\ref{eq:HL_UV_sing}), possibly suggesting some subtleties in the order of limits between z→1 and c→∞.

— Phase transition from eigenstate thermalization: forbidden singularity and instanton proliferation via AGT correspondence  (2608.13246 - Xu et al., 13 Aug 2026) in Section 5, “High fugacity phase”