If you haven't found something strange: Atiyah Singer strikes back

We will introduce another parameter t to help us out as a regulator.

$Tr(\gamma^5e^{tD^2})=N_+e^{t0}-N_-e^{t0}+\sum_{n=1}^\infty e^{-t\Sigma_n^2} (1-1)=N_+-N_-$

This holds only for nonzero t, but we want the t=0 case so we will calculate at small t then take the limit as t goes to 0.

$-D^2\equiv H_0+V=-\partial^2 + i\Sigma_{\mu\nu}F^{\mu\nu}$

Time for some perturbation theory. Define V(t) to be V conjugated by the free Hamiltonian exponentiated.

$\small Tr[\gamma^{5}e^{-tH}]\approx Tr[\gamma^{5}e^{-tH_0}]+Tr[\gamma^{5}e^{-tH_0}\int V(t')dt']+Tr[\gamma^{5}e^{-tH_0}\int V(t')V(t'')dt'dt'']$

Which of these terms will contribute? It will be the one with enough gamma matrices so that the trace does not vanish. Remember V has a $\small \Sigma_{\mu\nu}$ so each power of V gives two gamma's. To get the 4 we need for nonvanishing trace we need two powers of V. That is the third term above. We need to take the trace of this:

$\small \inline \dpi{100} \gamma^5\int <x|e^{-(t-t')H_0}|x'>V(x')<x'|e^{-(t'-t'')H_0}|x''>V(x'')<x''|e^{-t''H_0}|x>$

What do about those propagators from x to x' etc.? Well we know that unperturbed hamiltonian pretty well. It's the free particle which we know.

$\small <y k| e^{-tH_0} |xi> = \frac{1}{(2\sqrt{\pi t})^4}e^{-(x-y)^2/4t}$

Notice that when x and y are even a little bit separated the exponential becomes tiny. So we might as well take x' and x'' to be basically x. The corrections to those propagators will be exponentially suppressed.

$\small \inline Tr[\int \gamma^5<x|e^{-tH_0}|x>V(x)V(x)]=Tr[\gamma^5\Sigma^{\mu\nu}\Sigma^{\rho\tau}]\int \frac{1}{16\pi^2t^2}F_{\mu\nu}F_{\rho\tau} d^4x dt' dt''$

$\small \inline Tr[\gamma^5\Sigma^{\mu\nu}\Sigma^{\rho\tau}]\int \frac{1}{16\pi^2t^2}F_{\mu\nu}F_{\rho\tau}d^4xdt'dt''=\frac{2\epsilon^{\mu\nu\rho\tau}}{16\pi^2}\int F_{\mu\nu}F_{\rho\tau} \frac{1}{t^2}\int dt'dt''$

$\small \frac{1} {16\pi^2}\int F \wedge F$

Success. We have gotten from the index of that elliptic operator to integrating the second Chern class. This is just a chiral anomaly. Indexes could screw our theory in other ways too. A gauge or gravitational anomaly would be more evil.

ch is the Chern character and Td is the Todd, but you can get them both from chern classes. This is a big gun., but no kill like overkill.

If you haven't found something strange

Thursday, August 11, 2011

Atiyah Singer strikes back

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