quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...

and

statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...

, the Witten index at the inverse temperature β is defined as a modification of the standard partition function: :

\textrm -1)^F e^/math>

Note the (-1)

^F operator, where F is the

fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...

number operator. This is what makes it different from the ordinary partition function. It is sometimes referred to as the

spectral asymmetry In mathematics and physics, the spectral asymmetry is the asymmetry in the distribution of the spectrum of eigenvalues of an operator. In mathematics, the spectral asymmetry arises in the study of elliptic operators on compact manifolds, and is ...

. In a

supersymmetric In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theorie ...

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 states minus the number of zero energy fermionic vacuum states. In particular, if 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 the supersymmetric sigma model on a manifold is given by the manifold's Euler characteristic.* p191 (10.124) :

\sum_(-1)^pb_p=\chi(M) \ .

It is an example of a quasi-topological quantity, which is a quantity that depends only on F-terms and not on D-terms in the

Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...

. A more refined invariant in 2-dimensional theories, constructed using only the right-moving part of the fermion number operator together with a 2-parameter family of variations, is the elliptic genus.

References

* Edward Witten ''Constraints on Supersymmetry Breaking'', Nucl. Phys. B202 (1982) 253-316 Supersymmetric quantum field theory Quantum field theory Statistical mechanics {{Quantum-stub

See also

References