  We develop a novel nonperturbative approach to a class of three-point
functions in planar $\mathcal{N}=4$ SYM based on Thermodynamic Bethe Ansatz
(TBA). More specifically, we study three-point functions of a non-BPS
single-trace operator and two determinant operators dual to maximal Giant
Graviton D-branes in AdS$_5\times$S$^{5}$. They correspond to disk one-point
functions on the worldsheet and admit a simpler and more powerful integrability
description than the standard single-trace three-point functions. We first
introduce two new methods to efficiently compute such correlators at weak
coupling; one based on large $N$ collective fields, which provides an example
of open-closed-open duality discussed by Gopakumar, and the other based on
combinatorics. The results so obtained exhibit a simple determinant structure
and indicate that the correlator can be interpreted as a generalization of
$g$-functions in 2d QFT; an overlap between an integrable boundary state and a
state corresponding to the single-trace operator. We then determine the
boundary state at finite coupling using the symmetry, the crossing equation and
the boundary Yang-Baxter equation. With the resulting boundary state, we derive
the ground-state $g$-function based on TBA and conjecture its generalization to
other states. This is the first fully nonperturbative proposal for the
structure constants of operators of finite length. The results are tested
extensively at weak and strong couplings. Finally, we point out that
determinant operators can provide better probes of sub-AdS locality than
single-trace operators and discuss possible applications.
