Understanding the Witten Index for SU(N)

• BenTheMan
In summary, the Witten index is a useful tool for determining whether supersymmetry remains unbroken beyond perturbation theory. It is represented as Tr(-1)^F where F is the fermion number and can be found in most textbooks on supersymmetry or in the original paper by Witten. Some recommended sources include the Wikipedia entry, Weinberg's book on supersymmetry, and Witten's paper.
BenTheMan
Does anyone know where I can find a brief and useful description of the Witten index? The Wikipedia entry isn't bad, but didn't give me a real good understanding of it.

(I have ascertained that for SU(N), the Witten index is N. Does this mean that SU(N) has N supersymmetric vacua, or that SU(N) has more than zero supersymmetric vacua?)

Theres probably a decent discussion in most textbooks on supersymmetry. Weinberg has a few pages devoted to it (pg 250-). For your purposes, its just Tr(-1)^F where F is the fermion number.

Its primary use is to figure out whether or not supersymmetry remains unbroken beyond perturbation theory.

If you hate reading Weinberg, maybe track down the original paper. I'd imagine Witten probably has a more than readable gist

If you hate reading Weinberg, maybe track down the original paper. I'd imagine Witten probably has a more than readable gist

I'm affraid you're right. I was trying to avoid walking to the library (we don't have a subscription to Science Direct), but I fear it is unavoidable.

Thanks!

1. What is the Witten Index for SU(N)?

The Witten Index for SU(N) is a mathematical concept developed by physicist Edward Witten in the field of quantum field theory. It is a topological invariant that characterizes the supersymmetric quantum field theories associated with the special unitary group SU(N).

2. How is the Witten Index for SU(N) calculated?

The Witten Index for SU(N) is calculated by summing over the eigenvalues of the Hamiltonian operator in the supersymmetric quantum field theory. This sum is then multiplied by a factor of (-1)^F, where F is the fermion number operator. The result is a numerical value that is independent of the choice of basis.

3. What does the Witten Index for SU(N) tell us about a supersymmetric quantum field theory?

The Witten Index for SU(N) is a topological invariant that provides information about the ground state of a supersymmetric quantum field theory. It can tell us about the number of supersymmetric ground states and the existence of any topological obstructions in the theory.

4. What is the significance of the special unitary group SU(N) in the calculation of the Witten Index?

The special unitary group SU(N) plays a crucial role in the calculation of the Witten Index because it is the symmetry group associated with supersymmetric quantum field theories. The group structure of SU(N) allows for the simplification of the calculation and provides deeper insights into the underlying physics of the theory.

5. How is the Witten Index for SU(N) used in theoretical physics?

The Witten Index for SU(N) is used in theoretical physics to study supersymmetric quantum field theories and their properties. It has applications in various areas such as string theory, condensed matter physics, and mathematical physics. It is also a useful tool for classifying and distinguishing between different supersymmetric theories.

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