MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
show/hide this revision's text 4 added 38 characters in body

In my opinion, the most "physical" recasting of the index theorem is the Witten index for supersymmetric quantum theories. It is superficially similar to the heat kernel, but much more general. The Witten index of a supersymmetric quantum mechanical system is the regularised supertrace $$\mathrm{Tr} (-1)^F \exp(-\beta H)$$ where $(-1)^F$ is the grading operator which is $-1$ on fermionic states and $+1$ on bosonic states and $H$ is the hamiltonian. The trace is taken over the Hilbert space of states.

One can show that this does not depend on $\beta$ and hence can be evaluated both at small $\beta$ ("large temperature expansion") or large $\beta$ ("small temperature expansion"). In one expansion one sees that it computes the trace of $(-1)^F$ on zero modes of the hamiltonian, since for a supersymmetric system the dimensions of the bosonic and fermionic positive-energy eigenstates are equal. In the other expansion one writes the Witten index in terms of a functional integral, which (when formally manipulated) becomes a geometric integral. The formal manipulations can be justified as in Getzler's proof of the local Atiyah-Singer index theorem.

The relation with the heat kernel comes from taking a particular supersymmetric model in which the hamiltonian is the laplacian. The relation with the Gauss-Bonnet theorem comes from considering a supersymmetric sigma model in which the hamiltonian is the Hodge laplacian acting on ($L^2$) differential forms. The Witten index then is computing the index of the elliptic operator $d + \delta$ from the odd to the even rank differential forms, which is the Euler characteristic of the manifold.

There are supersymmetric models for which the Witten index computes the index of a generalised Dirac operator as in the original work of Atiyah and Singer.

Witten introduced "his" index in order to study supersymmetry breaking. A nonzero value of the index shows that there is an imbalance of fermionic and bosonic vacua exists a vacuum state (=states =a state of minimal energy), energy) which is invariant under supersymmetry and hence supersymmetry is not (spontaneously) broken.

There is yet another relation between the heat kernel and the index theorem and it comes from a certain regularisation of the functional integral measure as in Fujikawa's celebrated derivation of the chiral anomaly.
show/hide this revision's text 3 added 9 characters in body

In my opinion, the most "physical" recasting of the index theorem is the Witten index for supersymmetric quantum theories. It is superficially similar to the heat kernel, but much more general. The Witten index of a supersymmetric quantum mechanical system is the regularised supertrace $$\mathrm{Tr} (-1)^F \exp(-\beta H)$$ where $(-1)^F$ is the grading operator which is $-1$ on fermionic states and $+1$ on bosonic states and $H$ is the hamiltonian. The trace is taken over the Hilbert space of states.

One can show that this does not depend on $\beta$ and hence can be evaluated both at small $\beta$ ("large temperature expansion") or large $\beta$ ("small temperature expansion"). In one expansion one sees that it computes the trace of $(-1)^F$ on zero modes of the hamiltonian, since for a supersymmetric system the dimensions of the bosonic and fermionic energy positive-energy eigenstates are equal. In the other expansion one writes the Witten index in terms of a functional integral, which (when formally manipulated) becomes a geometric integral. The formal manipulations can be justified as in Getzler's proof of the local Atiyah-Singer index theorem.

The relation with the heat kernel comes from taking a particular supersymmetric model in which the hamiltonian is the laplacian. The relation with the Gauss-Bonnet theorem comes from considering a supersymmetric sigma model in which the hamiltonian is the Hodge laplacian acting on ($L^2$) differential forms. The Witten index then is computing the index of the elliptic operator $d + \delta$ from the odd to the even rank differential forms, which is the Euler characteristic of the manifold.

There are supersymmetric models for which the Witten index computes the index of a generalised Dirac operator as in the original work of Atiyah and Singer.

Witten introduced "his" index in order to study supersymmetry breaking. A nonzero value of the index shows that there is an imbalance of fermionic and bosonic vacua (=states of minimal energy), hence supersymmetry is broken.

There is yet another relation between the heat kernel and the index theorem and it comes from a certain regularisation of the functional integral measure as in Fujikawa's celebrated derivation of the chiral anomaly.
show/hide this revision's text 2 added 5 characters in body

In my opinion, the most "physical" recasting of the index theorem is the Witten index for supersymmetric quantum theories. It is superficially similar to the heat kernel, but much more general. The Witten index of a supersymmetric quantum mechanical system is the regularised supertrace $$\mathrm{Tr} (-1)^F \exp(-\beta H)$$ where $(-1)^F$ is the grading operator which is $-1$ on fermionic states and $+1$ on bosonic states and $H$ is the hamiltonian. The trace is taken over the Hilbert space of states.

One can show that this does not depend on $\beta$ and hence can be evaluated both at small $\beta$ ("large temperature expansion") or large $\beta$ ("small temperature expansion"). In one expansion one sees that it computes the trace of $(-1)^F$ on zero modes of the hamiltonian, since for a supersymmetric system the dimensions of the bosonic and fermionic energy eigenstates are equal. In the other expansion one writes the Witten index in terms of a functional integral, which (when formally manipulated) becomes a geometric integral. The formal manipulations can be justified as in Getzler's proof of the local Atiyah-Singer index theorem.

The relation with the heat kernel comes from taking a particular supersymmetric model in which the hamiltonian is the laplacian. The relation with the Gauss-Bonnet theorem comes from considering a supersymmetric sigma model in which the hamiltonian is the Hodge laplacian acting on ($L^2$) differential forms. The Witten index then is computing the index of the elliptic operator $d + \delta$ from the odd to the even rank differential forms, which is the Euler characteristic of the manifold.

There are supersymmetric models for which the Witten index computes the index of a generalised Dirac operator as in the original work of Atiyah and Singer.

Witten introduced "his" index in order to study supersymmetry breaking. A nonvalue nonzero value of the index shows that there is an imbalance of fermionic and bosonic vacua (=states of minimal energy), hence supersymmetry is broken.

There is yet another relation between the heat kernel and the index theorem and it comes from a certain regularisation of the functional integral measure as in Fujikawa's celebrated derivation of the chiral anomaly.
show/hide this revision's text 1