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In the context of supersymmetric quantum mechanics we formulate newsupersymmetric localization principle, with application to trace formulas for afull thermal partition function. Unlike the standard localization principle,this new principle allows to compute the supertrace of non-supersymmetricobservables, and is based on the existence of fermionic zero modes. We describecorresponding new invariant supersymmetric deformations of the path integral;they differ from the standard deformations arising from the circle action andrequire higher derivatives terms. Consequently, we prove that the path integrallocalizes to periodic orbits and not not only on constant ones. We illustratethe principle by deriving bosonic trace formulas on compact Lie groups,including classical Jacobi inversion formula.

Supersymmetry and trace formulas I. Compact Lie groups