# Heterotic Supersymmetric derivation of an integrality theorem for differentiable manifolds [closed]

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:

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

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?

## closed as off-topic by ACuriousMind♦, Brandon Enright, Kyle Kanos, JamalS, JimNov 13 '14 at 15:13

• This question does not appear to be about physics within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.

• This question appears to be off-topic because it is a check-my-work question. – ACuriousMind Nov 13 '14 at 0:40
• An update was made and new questions were formulated – Juan Ospina Nov 13 '14 at 13:48