Derive a non-asymptotic modular S-transform for chiral deformations

Develop an exact, non-perturbative derivation of the modular S-transformation for the generalized partition function ⟨exp(λ W_0)⟩ of a two-dimensional conformal field theory deformed by a holomorphic local field W(z), avoiding an order-by-order expansion in the fugacities and establishing the transformation without relying on asymptotic series.

Background

The paper proves a universal recursive description of the modular S-transform for generalized partition functions of chiral deformations in an asymptotic expansion in the fugacities. This establishes the modular behavior order by order and generalizes previous conjectures.

However, an exact (non-asymptotic) S-transform is known only in special, essentially free, models (e.g., Ising and symplectic fermions). Extending the result beyond asymptotics would provide a complete modular characterization of chiral deformations in general conformal field theories.

References

It is also important to remember that our results are obtained in an asymptotic expansion in the fugacities. A broader understanding of how to derive the modular transformation of the deformed conformal field theory, without working term by term in the asymptotic expansion, remains an outstanding problem.

— Universal Modular Properties of Generalized Gibbs Ensembles and Chiral Deformations  (2603.28244 - Ashok et al., 30 Mar 2026) in Section 6: Discussion, Outlook

An unresolved question is the non-perturbative completion of the hierarchy. The present construction determines formal expansions around vanishing chemical potential, whereas the full modular transform can contain additional contributions that are invisible to these expansions. The non-perturbative TBA solutions discussed in offer a starting point, and there is ongoing work looking to understand the full structure of these non-perturbative solutions in more detail . Based on the analysis in these works, and the free cases where they were worked out in detail in general we expect that the spectrum of the hierarchy actually becomes complex. Understanding how they contribute to the higher charges, and whether the resulting completed charges retain the commutativity established perturbatively here, would clarify the relation between the formal hierarchy and the full modular-transformed GGE.

— Modular Properties of Generalised Gibbs Ensembles: The Deformed KdV Hierarchy  (2610.13875 - Karimi, 9 Oct 2026) in Section “Conclusion and Outlook”

A further question is whether this defect and its higher-spin charges coincide with those studied in .

— Modular Properties of Generalised Gibbs Ensembles: The Deformed KdV Hierarchy  (2610.13875 - Karimi, 9 Oct 2026) in Section “Conclusion and Outlook”