Confirm d-type instanton contributions via Lefschetz thimble analysis

Determine, using a Lefschetz–thimble analysis of the large-N superconformal index matrix model for N=4 SYM at higher truncation levels (p ≥ 2), whether eigenvalue-tunnelling configurations to the additional d-type pinching points (the roots d_i and d_i^{-1} of the spectral curve y(z) in the single-cut ansatz) contribute to the original holonomy integral and generate multi-cut saddle contributions to the partition function.

Background

In higher truncations of the superconformal index matrix model, new pairs of pinching points d_i and d_i{-1} appear in the spectral curve, beyond the π-type pinching point at z = −1. These give rise to prospective d-type instantons corresponding to eigenvalue tunnelling to these points.

While π-type instantons are argued to contribute based on continuity with the p = 1 Gross–Witten–Wadia model and numerical evidence, the authors note that the contribution of d-type instantons to the original integration contour is not established. A rigorous determination requires a Lefschetz thimble analysis to see if the associated multi-cut configurations lie on relevant thimbles for the original contour.

References

It is not obvious whether configurations that correspond to eigenvalue tunnelling to the new pinching points actually contribute to the original partition function. However, a full Lefschetz thimble analysis would be required to confirm that the new d-type instantons also contribute and lead to new multi-cut solutions that contribute to the original contour.

— Instanton condensation and a new phase of BPS black holes  (2604.00095 - Holden, 31 Mar 2026) in Section 4.2, Higher truncations and classifying pinching points