  We calculate the energy spectrum of a confining flux tube that is closed
around a spatial torus, as a function of its length l. We do so for various
SU(N) gauge theories in 3+1 dimensions, and for various values of spin, parity
and longitudinal momentum. We are able to present usefully accurate results for
about 20 of the lightest such states, for a range of l that begins close to the
(finite volume) deconfining phase transition, and extends up to l.sqrt(K)~6
(where K is the string tension). We find that most of these low-lying states
are well described by the spectrum of the Nambu-Goto free string theory in flat
space-time. Remarkably, this is so not only at the larger values of l, where
the gap between the ground state energy and the low-lying excitations becomes
small compared to the mass gap, but also down to much shorter lengths where
these excitation energies become large compared to sqrt(K), the flux-tube no
longer `looks' anything like a thin string, and an expansion of the effective
string action in powers of 1/l no longer converges. All this is for flux in the
fundamental representation. We also calculate the k=2 (anti)symmetric ground
states and these show larger corrections at small l. So far all this closely
resembles our earlier findings in 2+1 dimensions. However, and in contrast to
the situation in D=2+1, we also find that there are some states, with J,P=0,-
quantum numbers, that show large deviations from the Nambu-Goto spectrum. We
investigate the possibility that (some of) these states may encode the massive
modes associated with the internal structure of the flux tube, and we discuss
how the precocious free string behaviour of most states constrains the
effective string action, on which much interesting theoretical progress has
recently been made.
