A note on zeroes of superpotentials in F-theory

Dimers and orientifolds

“…Focusing in the latter case, the D-brane instanton typically has fermion zero modes α a , β b transforming in the a , b under the 4d gauge factors a, b. If there is a coupling α a O ab β b where O ab is a 4d chiral operator in the ( a , b ), integration over the fermion zero modes leads to insertions of O in the superpotential [107].…”

Stringy instantons at orbifold singularities

“…Thus, the only massless modes present in the neutral sector are four bosons x µ , denoting the upper-left component of a µ , two fermions θ α denoting the upper-left component of M α4 and two more fermions λα denoting the upper-left component of λα 4 . We have identified the non zero entries of a µ and M α4 with the super-coordinates x µ and θ α since they precisely correspond to the Goldstone modes of the super-translation symmetries broken by the instanton and do not appear in S 1 + S 2 (cfr.…”

Update of D3/D7-brane inflation on

“…57 As already mentioned in section 7, this uplifting potential would generically modify the inflaton mass because of the following two effects:…”

“….) entering the non-perturbative superpotential (2.11) vanishes if the D3-brane sits on top of the D7-branes responsible for the gaugino condensation [57]. If the gaugino condensation D7-branes and those with worldvolume flux were the same, this would lead to volume destabilization at the end of inflation, when the D3-brane dissolves as an instanton on the D7-branes.…”

“…For the K3 volume this stabilization might proceed via gaugino condensation on a stack of D7-branes wrapped around the K3. In order to avoid a destabilization of the volume after inflation, this stack of D7-branes should be placed away from the FI D7-brane, as the non-perturbative superpotential becomes zero when the D3-brane hits the gaugino condensate D7-branes [57]. In the simplest setup depicted in figure 2, there is a single stack of volume stabilizing D7-branes, placed at fixed point #2 (i.e.…”