SL(3,Z) modular bootstrap

Determine the appropriate analogue, in the $(\tau,\sigma)$ variables, of the modular bootstrap problem based on differential functionals at the self-dual point for SL(3,Z) modularity.

Background

For two-dimensional conformal theories, modular crossing equations can be acted on by differential functionals evaluated at the self-dual point, producing bootstrap constraints. The authors state that it is unresolved how to formulate an analogous bootstrap problem for SL(3,Z) modularity, where the modular variables are the pair (τ,σ)(\tau,\sigma). The Mellin-space reflection identities are proposed as a possible starting point.

References

For SL$(3,\mathbb Z)$ modularity, it is not yet clear what the appropriate analogue of such a bootstrap problem should be in $(\tau,\sigma)$ variables.

— Mellin space reflections, modularity of elliptic Gamma functions and beyond  (2608.24083 - Lei et al., 25 Aug 2026) in Section 4, Discussion and outlook