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. $L^2$ 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 $b\to 0$ 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.