The Witten index is a concept in theoretical physics, specifically in the contexts of supersymmetry and quantum field theory. It is named after the physicist Edward Witten, who introduced it in the context of supersymmetric quantum mechanics. The Witten index is defined as a particular counting of the number of ground states (or lowest energy states) of a supersymmetric quantum system.
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With a well-defined regulated supertrace, positive-energy states of supersymmetric quantum mechanics are paired by an odd invertible charge. Their opposite fermion parities cancel, leaving , independent of . A nonzero index proves the existence of a supersymmetric vacuum; zero does not prove its absence. Discrete-spectrum or suitable infrared conditions matter, because continuum thresholds and states escaping to infinity can invalidate naive trace manipulations or deformation invariance.