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In [[quantum field theory]] and [[statistical mechanics]], the '''Witten index''' at the [[inverse temperature]] β is defined as a modification of the standard partition function:

:<math>Tr[(-1)^F e^{-\beta H}]</math>

Note the [[(-1)^F|(-1)<sup>F</sup>]] operator, where F is the [[fermion]] [[number operator]]. This is what makes it different from the ordinary [[partition function (mathematics)|partition function]].

In a [[supersymmetry|supersymmetric]] theory, each nonzero energy [[eigenvalue]] contains an equal number of bosonic and fermionic states. Because of this, the Witten index is independent of the temperature and gives the number of zero energy bosonic [[vacuum state]]s minus the number of zero energy fermionic vacuum states.  In particular, if [[supersymmetry breaking|supersymmetry is spontaneously broken]] then there are no zero energy ground states and so the Witten index is equal to zero.

The Witten index of a supersymmetric field theory is an example of a quasi-topological quantity, which is a quantity that depends only on [[F-term]]s and not on [[D-term]]s in the [[Lagrangian]].  A more refined invariant in 2-dimensional theories, constructing using only the right-moving part of the fermion number operator together with a 2-parameter family of variations, is the [[Genus of a multiplicative sequence|elliptic genus]].

==References==
* Edward Witten ''Constraints on Supersymmetry Breaking'',  Nucl. Phys. '''B202''' (1982) 253-316

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[[Category:Statistical mechanics]]
[[Category:Supersymmetry]]


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