Question about the expression of Witten Index

  1. The idea behind the notation is that the operator $F$ is supposed to count the number of fermions in an expression, i.e. $$[F,A_n]= n A_n$$ if the operator $A_n$ contains $n$ fermions, what that means.

  2. Then $$[f(F),A_n]= f(n) A_n$$ for a sufficiently well-behaved function $f:\mathbb{C}\to \mathbb{C}$.

  3. In particular, for $f(x)=(-1)^x $, one has $$[(-1)^F,A_n]= (-1)^n A_n.$$

  4. The operator $(-1)^F$ has eigenvalue $+1$ ($-1$) for Grassmann-even (Grassmann-odd) operators, respectively.

  5. The notation $(-1)^F$ is used even if the operator $F$ itself is not well-defined.