  The 4503 rovibronic term values belonging to the mutually perturbed
$A^1\Sigma^+_u$ and $b^3\Pi_u$ states of Cs$_2$ were extracted from laser
induced fluorescence (LIF) $A\sim b\rightarrow X^1\Sigma^+_g$ Fourier transform
spectra with the 0.01 cm$^{-1}$ uncertainty. The experimental term values of
the $A^1\Sigma^+_u\sim b^3\Pi_u$ complex covering the rotational levels $J\in
[4,395]$ in the excitation energy range $[9655,13630]$ cm$^{-1}$ were involved
into coupled-channel (CC) deperturbation analysis. The deperturbation model
takes explicitly into account spin-orbit coupling of the
$A^1\Sigma^+_u(A0^+_u)$ and $b^3\Pi^+_{0_u}(b0^+_u)$ states as well as
spin-rotational interaction between the $\Omega=0$, $1$ and $2$ components of
the $b^3\Pi^+_{\Omega_u}$ state. The \emph{ab initio} relativistic calculations
on the low-lying electronic states of Cs$_2$ were accomplished in the framework
of Fock space relativistic coupled cluster (FSRCC) approach to provide the
interatomic potentials of the interacting $A0^+_u$ and $b0^+_u$ states as well
as the relevant $A\sim b$ spin-orbit coupling function. To validate the present
CC deperturbation analysis solely obtained by energy-based data, the $A\sim b
\to X(v^{\prime\prime}_X)$ LIF intensity distributions were measured and
compared with their theoretical counterparts obtained by means of the
non-adiabatic vibrational wave functions of the $A\sim b$ complex and the FSRCC
$A\sim b \to X$ transition dipole moments calculated by the finite-field
method.
