Exact-WKB analysis for SUSY and quantum deformed potentials: Quantum mechanics with Grassmann fields and Wess-Zumino terms

Quantum deformed potentials arise naturally in quantum mechanical systems of one bosonic coordinate coupled to $N_f$ Grassmann valued fermionic coordinates, or to a topological Wess-Zumino term. These systems decompose into sectors with a classical potential plus a quantum deformation. Using exact WKB, we derive exact quantization condition and its median resummation. The solution of median resummed form gives physical Borel-Ecalle resummed results, as we show explicitly in quantum deformed double- and triple- well potentials. Despite the fact that instantons are finite action, for generic quantum deformation, they do not contribute to the energy spectrum at leading order in semi-classics. For certain quantized quantum deformations, where the alignment of levels to all order in perturbation theory occurs, instantons contribute to the spectrum. If deformation parameter is not properly quantized, their effect disappears, but higher order effects in semi-classics survive. In this sense, we classify saddle contributions as fading and robust. Finally, for quantum deformed triple-well potential, we demonstrate the P-NP relation, by computing period integrals and Mellin transform.