Analytic continuation and supersymmetry

Real-valued parameters of quantum field theory, such as Planck constant, coupling constants, temperature and spacetime metric, chemical potentials or background gauge fields can be made complex. In perturbative string theory, the worldsheet metric is sometime Lorentzian, somewhere Euclidean, and likely complex in between. Less often, integral, or quantized, parameters, such as the number of flavors or species (as in replica method), the number of colors (as in matrix models, master field), spacetime dimension (dimensional regularization), or angular momentum (Regge poles) are continued to complex values. We observe that many, if not all of these analytic continuations can be realized with conventional supersymmetric quantum field (or M-) theory. The supersymmetry is softly broken near a defect supporting the theory being analytically continued in some of its parameters.