  We study thermal conductivity for one-dimensional electronic fluid. The
many-body Hilbert space is partitioned into bosonic and fermionic sectors that
carry the thermal current in parallel. For times shorter than bosonic Umklapp
time, the momentum of Bose and Fermi components are separately conserved,
giving rise to the ballistic heat propagation and imaginary heat conductivity
proportional to $T / i\omega$. The real part of thermal conductivity is
controlled by decay processes of fermionic and bosonic excitations, leading to
several regimes in frequency dependence. At lowest frequencies or longest
length scales, the thermal transport is dominated by L{\'e}vy flights of
low-momentum bosons that lead to a fractional scaling, $\omega^{-\frac{1}{3}}$
and $L^{1/3}$, of heat conductivity with the frequency $\omega$ and system size
$L$ respectively.
