We present an exact-diagonalization study of the frequency- and wave-vector-dependent conductivity $\sigma (\mathrm{q}, \omega )$ in small clusters in the two-dimensional $t-J$ model. Unlike the related dynamical density correlation function, $\sigma (\mathrm{q}=0, \omega )$ in the underdoped regime has the exchange constant $J$ as its characteristic energy scale and is dominated by a resonancelike excitation with frequency $\sim 1.7J$. We interpret this as transition to a $p$-like excited state of a spin-bag-type quasiparticle (or, alternatively, a tightly bound spinon-holon pair) and show that a simple calculation based on the string picture explains the numerical results semiquantitatively. For doping levels $>~25\%t$ remains the only energy scale of $\sigma (\mathrm{q}=0, \omega )$.

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