Please consider the following integrality theorem for differentiable manifolds due to K H Mayer:

I am trying to prove this theorem using Heterotic Super-symmetric Quantum Mechanics described by a Lagrangian density with the form

$$L={\phi}^{T}Q\phi+{\theta}^{T}P\theta$$

where $\phi$ describes bosonic degrees of freedom with an effective propagator denoted $Q$ and $\theta$ describes fermionic degrees of freedom with an effective propagator denoted $P$. The Witten index for this heterotic Susy QM is given by:

$${\it index}=\int \!\!\!\int \!{{\rm e}^{-{\phi}^{T}Q\phi-{\theta}^{T}P

\theta}}{d\theta}\,{d\phi}={\it integer}

$$

Computing the path integrals we obtain:

\[\int \!{{\rm e}^{-{\theta}^{T}P\theta}}{d\theta}=\sqrt {{\it Det} \left( P \right) }=\sqrt {\prod _{i=1}^{s} \left( 4\,\prod _{n=0}^{ \infty } \left( 1+{\frac {{y_{{i}}}^{2}}{ \left( 2\,n+1 \right) ^{2}{ \pi }^{2}}} \right) ^{2} \right) }\\={2}^{s}\prod _{i=1}^{s}\cosh \left( \frac{y_{{i}}}{2} \right)\]

\[\int \!{{\rm e}^{-{\phi}^{T}Q\phi}}{d\phi}={\frac {1}{\sqrt {{\it Det} \left( Q \right) }}}={\frac {1}{\sqrt {\prod _ j \left( \prod _{n=1}^{\infty }(1+{\frac {{x_{{j}}}^{2}}{4{\pi }^{2}{n}^{2}} } )\right) ^{2} }}}\\=\prod _ j{\frac {\frac{x_{{j}}}{2}}{\sinh \left( \frac{x_{{j}}}{2} \right) }} = \hat{A}(M)\]

Then we have:

\[\mathrm{index}=\int \!{{\rm e}^{-{\phi}^{T}Q\phi}}{d\phi}\int \!{{\rm e}^{-{\theta}^{ T}P\theta}}{d\theta}=\int \!{\frac {\sqrt {{\it Det} \left( P \right) }}{\sqrt {{\it Det} \left( Q \right) }}}{dM}\\=\int \hat{A} \left( M \right) {2}^{s}\prod _{i=1}^{s}\cosh \left( \frac{y_{{i}}}{2} \right) {dM} =\\ \mathrm{integer}\]

Then my questions are:

1. Is this heterotic susy proof correct?.

2. This Mayer theorem has applications to the problem of anomaly for the fivebrane in 11-dimensional M-Theory?

3. This Mayer Theorem has applications to the problem of anomaly for the sevenbrane in 12-dimensional F-Theory?