  The Heisenberg model for S=1/2 describes the interacting spins of electrons
localized on lattice sites due to strong repulsion. It is the simplest
strong-coupling model in condensed matter physics with wide-spread
applications. Its relevance has been boosted further by the discovery of curate
high-temperature superconductors. In leading order, their undoped parent
compounds realize the Heisenberg model on square-lattices. Much is known about
the model, but mostly at small wave vectors, i.e., for long-range processes,
where the physics is governed by spin waves (magnons), the Goldstone bosons of
the long-range ordered antiferromagnetic phase. Much less, however, is known
for short-range processes, i.e., at large wave vectors. Yet these processes are
decisive for understanding high-temperature superconductivity. Recent reports
suggest that one has to resort to qualitatively different fractional
excitations, spinons. By contrast, we present a comprehensive picture in terms
of dressed magnons with strong mutual attraction on short length scales. The
resulting spectral signatures agree strikingly with experimental data
