Holographic thermal propagator at finite energy from modularity and triality

Abstract

We extend the modularity-based approach of arXiv:2509.02226 for the holographic thermal propagator to the physically relevant case of nonzero energy $ω$. Via the AGT correspondence, the problem is mapped to $\mathcal{N}=2$ $SU(2)$ gauge theory with $N_f=4$ hypermultiplets. Turning on $ω$ activates all four hypermultiplet masses, so that the full $SO(8)$ flavor symmetry constrains the low-temperature expansion, rather than the subgroup left unbroken at $ω=0$. As a result, the building blocks of the expansion are no longer just Eisenstein series but also Jacobi theta functions. We use this fact, together with the Molien series of the resulting invariant ring and the modular anomaly equation, to fix the low-temperature expansion coefficients order by order, matching only a finite number of terms in the Zamolodchikov's $q$-recursion at each order. We obtain the low-temperature expansion of the propagator and find agreement with the result of a completely different method, the near-boundary expansion of the bulk wave equation.