  The quantum chromodynamics (QCD) axion may solve the strong CP problem and
explain the dark matter (DM) abundance of our Universe. The axion was
originally proposed to arise as the pseudo-Nambu Goldstone boson of global
$\mathrm{U}(1)_{\rm PQ}$ Peccei-Quinn (PQ) symmetry breaking, but axions also
arise generically in string theory as zero modes of higher-dimensional gauge
fields. In this work we show that string theory axions behave fundamentally
differently from field theory axions in the early Universe. Field theory axions
may form axion strings if the PQ phase transition takes place after inflation.
In contrast, we show that string theory axions do not generically form axion
strings. In special inflationary paradigms, such as D-brane inflation, string
theory axion strings may form; however, their tension is parametrically larger
than that of field theory axion strings. We then show that such QCD axion
strings overproduce the DM abundance for all allowed QCD axion masses and are
thus ruled out, except in scenarios with large warping. A loop-hole to this
conclusion arises in the axiverse, where an axion string could be composed of
multiple different axion mass eigenstates; a heavier eigenstate could collapse
the network earlier, allowing for the QCD axion to produce the correct DM
abundance and also generating observable gravitational wave signals.
