All-order continuation of the twisted-superpotential exponential sectors

Establish whether the reconstructed divisor-sum formula for the exponentially small sectors of the ABJM on-shell twisted superpotential at Chern–Simons levels k=1, 2, and 4 continues to all instanton orders m, thereby validating the proposed all-order expression with coefficients c_m^(k).

Background

The paper reconstructs finitely many exponentially suppressed sectors of the selected ABJM Bethe-vacuum twisted superpotential: 51 at k=1, 67 at k=2, and 93 at k=4. These coefficients agree with a divisor-sum formula involving sigma_3(m) and its level-dependent rescalings.

The proposed continuation would organize the exponential tail through an Eichler-integral series associated with weight-four Eisenstein forms. However, only the finite reconstructed ranges are directly supported by the Bethe-vacuum data, so the validity of the formula beyond those orders is unresolved.

References

The reconstructed coefficients of the twisted superpotential obey a divisor-sum formula whose conjectural all-order continuation is an Eichler-integral series associated with a weight-four Eisenstein form.

— Constant Maps and Exponentially Small Sectors in ABJM Bethe Observables  (2609.09273 - Hosseini, 8 Sep 2026) in Section 1, Results and their scope; Eq. (intro-W-continuation); Section 3.4, Eq. (Wnp)

The exact agreement over the resolved coefficients motivates the conjecture that R_k(q) admits the all-order continuation

— Constant Maps and Exponentially Small Sectors in ABJM Bethe Observables  (2609.09273 - Hosseini, 8 Sep 2026) in Section 6.4, A modular-product conjecture for the first generator; Eq. (Rmodular)

The present deep reconstruction is restricted to k=1,2,4, so whether comparable divisor-sum closures occur at other levels remains open.

— Constant Maps and Exponentially Small Sectors in ABJM Bethe Observables  (2609.09273 - Hosseini, 8 Sep 2026) in Section 3.4, The exponentially small tail

This sharper comparison leads to a concrete open question. It remains to determine whether the divisor sequence and its universal $t_{\mathcal W}+1/m$ dependence can be obtained from a distinguished projection or reorganization of the ordinary and Nekrasov--Shatashvili topological-string data appropriate to the selected Bethe observable.

— Constant Maps and Exponentially Small Sectors in ABJM Bethe Observables  (2609.09273 - Hosseini, 8 Sep 2026) in Section 8.2, Relation to the sphere transseries and spectral modularity