Nonperturbative structure of lower-degree discrete-volume components

Determine the nonperturbative structure of the lower-degree components of the discrete volumes N_{g,n} governing ETH matrix-model amplitudes, in order to obtain subleading-in-λ corrections beyond those captured by the q-deformed Weil–Petersson volumes V^q_{g,n}.

Background

The leading nonperturbative correction is protected because only the top-degree part of the discrete volume contributes at leading order in the small-λ expansion; this top-degree part is represented by the q-deformed Weil–Petersson volume.

Subleading corrections require the nonperturbative behavior of the lower-degree remainder of N_{g,n}, which the paper does not determine in general. This unresolved input prevents the authors from extending their general method to subleading powers of λ.

References

Section~\ref{sec:N11-N12} already shows what this leaves out: subleading-in-$$ corrections would need the lower-degree remainder's own non-perturbative structure, which is not currently known in general.

— The ETH matrix model for DSSYK: non-perturbative corrections and intersection theory  (2608.23298 - Alfinito et al., 24 Aug 2026) in Section 5, From DSSYK discrete volumes to V^q_{g,n}(L)

An open question which we explored in Section \ref{N=1DiscreteWPVolSec} is whether the $\mathcal{N}=1$ q-deformed volumes have some type of interpretation in terms of the highest degree terms of some discrete analogue of $\mathcal{N}=1$ super Weil--Petersson volumes. To answer this question we employed an approach similar to Okuyama in . After finding the associated spectral curve to $\mathcal{N}=1$ DSSYK we computed $\omega_{1,1}(z)$ and demonstrated that it will contain terms with an ill defined degree. We do not know whether this is a fundamental property of $\mathcal{N}=1$ theories or an artifact of our naive generalization of the bosonic case\footnote{In a companion work by a subset of the authors, we have studied $\mathcal{N}=2$ DSSYK and the corresponding $q$-deformed spectral curves. Interestingly, the full discrete curve gives well-behaved quasi-polynomials, and the local edge expansions give the bosonic and $\mathcal{N}=2$ top-degree polynomial volumes, respectively. In particular, the lower edge, which retains the pole associated with the BPS states, gives $q$-deformed $\mathcal{N}=2$ super Weil--Petersson volumes that reduce to those of Turiaci and Witten in the continuum limit.}. In the latter case, Okuyama's approach may need to be modified in the presence of a hard edge to properly arrive at some notion of $\mathcal{N}=1$ analogues of discrete super Weil--Petersson volumes which also nicely embeds the $\mathcal{N}=1$ q-deformed volumes of Do and Norbury.

— $q$-deformed Weil-Petersson Volumes from the Spectral Edge of DSSYK and Supersymmetric Extensions  (2610.13899 - Ahmed et al., 9 Oct 2026) in Section 4, Discussion and Conclusion, paragraph beginning “An open question which we explored”