  The real part of optical conductivity $\text{Re}\sigma(\omega)$ of the Mott
insulators has a large amount of information on how spin and charge degrees of
freedom interact with each other. By using the time-dependent density-matrix
renormalization group, we study $\text{Re}\sigma(\omega)$ of the
two-dimensional Hubbard model on a square lattice at half filling. We find an
excitonic peak at the Mott-gap edge of $\text{Re}\sigma(\omega)$ not only for
the two-dimensional square lattice but also for two- and four-leg ladders. For
the square lattice, however, we do not clearly find a gap between an excitonic
peak and continuum band, which indicates that a bound state is not well
defined. The emergence of an excitonic peak in $\text{Re}\sigma(\omega)$
implies the formation of a spin polaron. Examining the dependence of
$\text{Re}\sigma(\omega)$ on the on-site Coulomb interaction and next-nearest
neighbor hoppings, we confirm that an excitonic peak is generated from a
magnetic effect. Electron scattering due to an electron-phonon interaction is
expected to easily suppress an excitonic peak since spectral width of an
excitonic peak is very narrow. Introducing a large broadening in
$\text{Re}\sigma(\omega)$ by modeling the electron-phonon coupling present in
La$_{2}$CuO$_{4}$ and Nd$_{2}$CuO$_{4}$, we obtain $\text{Re}\sigma(\omega)$
comparable with experiments.
