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Jafferis, Daniel

We give an elementary physical derivation of the Kontsevich-Soibelman wall crossing formula, valid for any theory with a 4d N = 2 supergravity description. Our argument leads to a slight generalization of the formula, which relates monodromy to the BPS spectrum.

A scalar field in four-dimensional deSitter spacetime (dS4) has quasinormal modes which are singular on the past horizon of the south pole and decay exponentially towards the future. These are found to lie in two complex highest-weight representations of the dS4 isometry group SO(4, 1). The Klein-Gordon norm cannot be used for quantization of these modes because it diverges. However a modified ‘R-norm’, which involves reflection across the equator of a spatial S 3 slice, is nonsingular. The quasinormal modes are shown to provide a complete orthogonal basis with respect to the R-norm. Adopting the associated R-adjoint effectively transforms SO(4, 1) to the symmetry group SO(3, 2) of a 2+1-dimensional CFT. It is further shown that the conventional Euclidean vacuum may be defined as the state annihilated by half of the quasinormal modes, and the Euclidean Green function obtained from a simple mode sum. Quasinormal quantization contrasts with some conventional approaches in that it maintains manifest dS-invariance throughout. The results are expected to generalize to other dimensions and spins.