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:
Note the (-1)F operator, where F is the fermion number operator. This is what makes it different from the ordinary partition function. It is sometimes referred to as the spectral asymmetry.
In a 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 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.[1]
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. 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.
^* Hori, Kentaro; Sheldon Katz; Albrecht Klemm; Rahul Pandharipande; Richard Thomas; Cumrun Vafa; Ravi Vakil; Eric Zaslow (2003). Mirror Symmetry. CIMM 1. ISBN 978-0-8218-2955-4. p191 (10.124)
In quantum field theory and statistical mechanics, the Wittenindex at the inverse temperature β is defined as a modification of the standard partition...
Edward Witten (born August 26, 1951) is an American theoretical physicist known for his contributions to string theory, topological quantum field theory...
the Wittenindex can be recognized as the sharp trace of the generalized transfer operator. It also links the Wittenindex to the Lefschetz index, I L...
model Non-linear sigma model Wess–Zumino–Witten model Wess–Zumino model Supersymmetric Yang–Mills Seiberg–Witten theory Super QCD (sQCD) N = 4 supersymmetric...
so-emerging supersymmetric theory of stochastic dynamics can be recognized as the Witten-type topological field theory. The meaning of the topological supersymmetry...
spectral asymmetry of the fermion operator (−1)F, where it is known as the Wittenindex. Alternatively, a 2005 paper by Robert Jaffe of MIT states that "Casimir...
and Edward Witten. She is married to software engineer Ari Steinberg. Daniela Witten publications indexed by Google Scholar Daniela Witten at the Mathematics...
fundamental matter have been solved by Nathan Seiberg and Edward Witten, see Seiberg–Witten gauge theory. However, in this subsection we are considering the...
Witten (1991), and generalized in Witten (1993). Witten's original conjecture was proved by Maxim Kontsevich in the paper Kontsevich (1992). Witten's...
Ian Hugh Witten (4 March 1947 – 5 May 2023) was a computer scientist at the University of Waikato, New Zealand. He was a Chartered Engineer with the Institute...
{\displaystyle \Phi =(\phi ,\psi )} . Such a multiplet can be used to define Seiberg–Witten theory concisely. The N = 2 {\displaystyle {\mathcal {N}}=2} hypermultiplet...
Supersymmetry breaking Konishi anomaly Seiberg duality Seiberg–Witten theory Wittenindex Wess–Zumino gauge Localization Mu problem Little hierarchy problem...
dimensions. Itzhak Bars[citation needed] gives this emphasis. In 1981 Ed Witten showed 11 as the smallest number of dimensions big enough to contain the...
solve and so this has confounded the top theoretical physicists. Edward Witten has popularised the concept of a theory in 11 dimensions, called M-theory...
depending on whether arguments are even or odd. These fall into four classes, indexed by the number of odd elements: No odd elements. The statement is just that...
Supersymmetry breaking Konishi anomaly Seiberg duality Seiberg–Witten theory Wittenindex Wess–Zumino gauge Localization Mu problem Little hierarchy problem...
the cancellation of infinities in quantum field theory, Witten's work on the Atiyah-Singer index theorem, and more recent applications to mirror symmetry...
Theory. Delphi, Greece 1997. Ian H Witten, Alistair Moffat, and Timothy C. Bell. Managing Gigabytes: Compressing and Indexing Documents and Images. New York:...
Supersymmetry breaking Konishi anomaly Seiberg duality Seiberg–Witten theory Wittenindex Wess–Zumino gauge Localization Mu problem Little hierarchy problem...
Global Information![{\displaystyle {\textrm {Tr}}[(-1)^{F}e^{-\beta H}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6787e085e9a4eab5949fb14c8dec9a724f68102e)
![{\displaystyle {\textrm {Tr}}[(-1)^{F}e^{-\beta H}]=\sum _{p\in \mathbb {Z} }(-1)^{p}b_{p}=\chi (M)\ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49d9e5bc8a4ee8b43617df4c06b9f0b25849ee08)