Analytic continuation and supersymmetry

Real-valued parameters of quantum field theory, such as Planck constant h, the coupling constants g⁻²,θ,…, temperature β⁻¹ and spacetime metric g_μν, chemical potentials or background gauge fields A^aμ^,canbe 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 n of flavors or species (as in replica method), the number of colors N (as in matrix models, master field), spacetime dimension D (dimensional regularization), or angular momentum l (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. Somewhat abridged talk at Strings-Math’2022 in Warsaw.