Semiclassical reconstruction of the Toda pole lattice via analytic continuation of the Probabilistic Approach
Abstract
We develop a probabilistic construction of Toda field theory correlators from the Coulomb gas representation. Reinterpreting n-point functions of primaries as w-moments of random variables with respect to a Gaussian measure, we derive constraints on the Toda momenta and coupling constant. L2 and Lp convergence conditions for the associated random variables, triviality constraints on the correlators, and local integrability conditions for the cross-moments are derived. These constraints define an extended region of convergence for the Toda correlators and provide a systematic, probabilistic interpretation of the Seiberg bounds. We proceed to analytically continue the moment indices by employing a Mellin-Barnes representation of the random variables. The local convergence boundaries become meromorphic local integrability divisors, and, in the semiclassical b→0 limit, these divisors reconstruct the Liouville charge pole lattice. For An Toda theory, we show that natural Wn-closure conditions in this semiclassical limit select rank-one, Virasoro-like submodules inside the full Toda theory. On these slices, the scalar integrability conditions reduce to Liouville-like divisors and recover the corresponding Toda charge pole lattices.
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BibTeX
@misc{distler2026semiclassical,
title = {{Semiclassical reconstruction of the Toda pole lattice via analytic continuation of the Probabilistic Approach}},
author = {Jacques Distler and Bahaa Elshimy},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22717},
eprint = {2609.22717},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22717},
}
RIS
TY - UNPB
TI - Semiclassical reconstruction of the Toda pole lattice via analytic continuation of the Probabilistic Approach
AU - Jacques Distler
AU - Bahaa Elshimy
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22717
UR - https://arxiv.org/abs/2609.22717
AB - We develop a probabilistic construction of Toda field theory correlators from the Coulomb gas representation. Reinterpreting n-point functions of primaries as w-moments of random variables with respect to a Gaussian measure, we derive constraints on the Toda momenta and coupling constant. L^2 and L^p convergence conditions for the associated random variables, triviality constraints on the correlators, and local integrability conditions for the cross-moments are derived. These constraints define an extended region of convergence for the Toda correlators and provide a systematic, probabilistic interpretation of the Seiberg bounds. We proceed to analytically continue the moment indices by employing a Mellin-Barnes representation of the random variables. The local convergence boundaries become meromorphic local integrability divisors, and, in the semiclassical b→0 limit, these divisors reconstruct the Liouville charge pole lattice. For A_n Toda theory, we show that natural W_n-closure conditions in this semiclassical limit select rank-one, Virasoro-like submodules inside the full Toda theory. On these slices, the scalar integrability conditions reduce to Liouville-like divisors and recover the corresponding Toda charge pole lattices.
ER -