Axionic D3-D7 Inflation
  We study the motion of a D3 brane moving within a Type IIB string vacuum compactified to 4D on K3 x T_2/Z_2 in the presence of D7 and O7 planes. We work within the effective 4D supergravity describing how the mobile D3 interacts with the lightest bulk moduli of the compactification, including the effects of modulus-stabilizing fluxes. We seek inflationary solutions to the resulting equations, performing our search numerically in order to avoid resorting to approximate parameterizations of the low-energy potential. We consider uplifting from D-terms and from the supersymmetry-breaking effects of anti-D3 branes. We find examples of slow-roll inflation (with anti-brane uplifting) with the mobile D3 moving along the toroidal directions, falling towards a D7-O7 stack starting from the antipodal point. The inflaton turns out to be a linear combination of the brane position and the axionic partner of the K3 volume modulus, and the similarity of the potential along the inflaton direction with that of racetrack inflation leads to the prediction n_s \le 0.95 for the spectral index. The slow roll is insensitive to most of the features of the effective superpotential, and requires a one-in-10^4 tuning to ensure that the torus is close to square in shape. We also consider D-term inflation with the D3 close to the attractive D7, but find that for a broad (but not exhaustive) class of parameters the conditions for slow roll tend to destabilize the bulk moduli. In contrast to the axionic case, the best inflationary example of this kind requires the delicate adjustment of potential parameters (much more than the part-per-mille level), and gives inflation only at an inflection point of the potential (and so suffers from additional fine-tuning of initial conditions to avoid an overshoot problem). 
C.P. Burgess, J.M. Cline, M. Postma