  In-medium masses of the light vector $\omega$, $\rho$ and axial-vector $A_1$
mesons are studied in the magnetized hot nuclear matter, accounting for the
effects of (inverse) magnetic catalysis. The in-medium partial decay widths for
the $A_1\rightarrow \rho \pi$ channels are studied from the in-medium masses of
the initial and the final state particles, by applying a phenomenological
Lagrangian to account for the $A_1\rho\pi$ interaction vertices. The masses are
calculated within the QCD sum rule framework, with the medium effects coming
through the light quark ($\sim \langle \bar{q}q \rangle$) and the scalar gluon
condensates ($\sim \langle G^2 \rangle$), as well as the light four-quark
condensate ($\sim \langle \bar{q}q\rangle^2 $). The condensates are calculated
within the chiral $SU(3)$ model in terms of the medium modified scalar fields:
isoscalar $\sigma$, $\zeta$, isovector $\delta$ and the dilaton field $\chi$.
The effects of magnetic fields are incorporated through the magnetized Dirac
sea contribution as well as the Landau energy levels of protons. The
incorporation of the magnetic field through the Dirac sea of nucleons lead to
an enhancement (reduction) of the light quark condensates with magnetic field,
give rise to the phenomenon of magnetic (inverse) catalysis. The effects of
(inverse) magnetic catalysis at finite temperature nuclear matter are studied
on the spectral functions and production cross-sections of the neutral $\rho$
and $A_1$ mesons. This may affect the production of the light vector and
axial-vector mesons in the peripheral heavy-ion collision experiments, where
estimated magnetic field is very large at the early stages of collisions with
very high temperature.
