  Hybrid inflation can be realised in low-energy effective string theory, as
described using supergravity. We find that the coupling of moduli to F-term
hybrid inflation in supergravity leads to a slope and a curvature for the
inflaton potential. The epsilon and eta parameters receive contributions at
tree and one loop level which are not compatible with slow roll inflation.
Furthermore the coupling to the moduli sector can even prevent inflation from
ending at all. We show that introducing shift symmetries in the inflationary
sector and taking the moduli sector to be no-scale removes most of these
problems. If the moduli fields are fixed during inflation, as is usually
assumed, it appears that viable slow-roll inflation can then be obtained with
just one fine-tuning of the moduli sector parameters. However, we show this is
not a reasonable assumption, and that the small variation of the moduli fields
during inflation gives a significant contribution to the effective inflaton
potential. This typically implies that eta is approximately -6, although it may
be possible to obtain smaller values with heavy fine-tuning.
