quantum field theory - The index of a Dirac operator and its physical meaning

I recently read Witten's paper from the 1980s and he often uses the notion of the index of a Dirac operator in K-theory.

What is the meaning of the index of a Dirac operator? What exactly is the Dirac operator here?

Is it to be understood in the usual sense in quantum field theory?

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I guess index here means a sort of Atiyah-Singer index or maybe a Witten index. As far as I know, the Witten index counts the number of ground states.

What is the "index", mathematically and physically? Can the Atiyah-Singer index and the index of a Dirac operator be interpreted similarly?

I heard that the index of a Dirac operator counts the number of zero-modes of the diarc operator. But I am not sure about it.

Answer

The Dirac operator, as we know it, is $D_\mu\gamma^\mu$ with $D$ as the gauge covariant derivative. Using the Fujikawa method of deriving the Adler-Bell-Jackiw or chiral anomaly, one finds that the anomaly of the chiral current is given by

$$ \partial_\mu \langle (j^5)^\mu\rangle = 2 \mathrm{i}A(x)$$

where

$$ A(x) = \int \sum_n \psi_n^\dagger \gamma^5 \psi_n \mathrm{d}^4 x$$

and the $\psi_n$ are Dirac eigenstates as per

$$ D_\mu\gamma^\mu\psi_n = \lambda_n \psi_n \; \Rightarrow \; D_\mu\gamma^\mu\gamma^5\psi_n = -\lambda_n\gamma^5\psi_n$$

since $\{\gamma^\mu,\gamma^5\} = 0$. For $\lambda_n \neq 0$, $\psi_n$ and $\gamma^5\psi_n$ are therefore orthogonal as per

$$ \int \psi^\dagger_n \gamma^5 \psi_n\mathrm{d}^4x = 0 \; \text{if} \; \lambda_n \neq 0$$

since $\psi_n$ and $\gamma^5\psi_n$ are states with different eigenvalues. Therefore, only the zero modes contribute. Of these, there are two kinds:

$$ \gamma^5 \psi_0^{+,i} = \psi_0^{+,i} \; \text{and} \; \gamma^5 \psi_0^{-,i} = - \psi_0^{-,i}$$

where $i$ labels the linearly independent zero modes with the respective property. Let $n_+$ and $n_-$ denote their respective number. For normalized eigenstates, we thus obtain

$$ A(x) = n_+ - n_- $$

This is precisely the analytical index of $D_\mu \gamma^\mu$ appearing in the Atiyah-Singer index theorem. Mathematically, it counts the difference between the kernels of $D_\mu \gamma^\mu$ and $(D_\mu \gamma^\mu)^\dagger$. Physically, it counts the difference between chiral and anti-chiral zero modes (since this is what the $\psi_0^+$ and $\psi_0^-$ are).

By the Atiyah-Singer index theorem, one relates this to the topological index $-\frac{1}{8\pi}\int F \wedge F$, i.e. the second chern class of the underlying principal bundle of our gauge theory, but one can also get this by "brute force" physical calculation through regularising $A(x)$ by suppressing the Dirac eigenmodes of high eigenvalues exponentially and carrying out the integral.