An index for flux vacua
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One of the main challenges for string theory phenomenology is to reliably identify consistent four-dimensional vacua of string theory with low or broken supersymmetry. This task is difficult as one expects no such vacua in the asymptotic limits where one sends all coupling constants to zero or (equivalently) all moduli fields to infinity. On the other hand, once these fields take finite values there exist various quantum and string corrections to the low-energy effective action that are difficult to compute and often unknown.
In this talk I will propose an index formulation based on contour integration techniques that relates the existence of critical points of effective potentials to their behaviour at the asymptotic boundaries of the field space. I will demonstrate this technique and discuss possible challenges for the specific example of IIB / M-theory flux compactifications with a one-dimensional complex structure moduli space and the resulting N=1 supersymmetric F-term potential.
