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