Heterotic Supersymmetric derivation of an integrality theorem for differentiable manifolds

Question

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?

Comments

Hi juancho, the longer equations of your nice question look a bit truncated. This can be fixed by using the TEX button (and no dollar signs) of the editor and inserting the equation as centered block equation (I tried but messed up). Maybe @UrsSchreiber can give you an answer if he has time ...?

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Fixed the equations by adding \\ line breaks; if you need to edit the equations for some reason, just double-click them in the editor.

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Hi Dilaton, thanks for your comment, you are very kind. All the bestl

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Hi Dimension10, many thanks for your latex editions. The equations look better now. All the best.

Answer 1

The answer to the second question :  "This Mayer theorem has applications to the problem of anomaly for the fivebrane in 11-dimensional M-Theory?", is yes.

The five-brane world-volume is a six-manifold W in an eleven-manifold Q .  

Perturbative anomalies in 2n dimensions are always related to characteristic classes in 2n + 2 dimensions, so in the case of the five-brane the world-volume  W is six-dimensional  and then the anomalies will involve eight-dimensional characteristic classes.   It is obvious that \(TQ|_ W = TW ⊕ N\) ,  where TW is the tangent bundle to the world-volume W and N is the normal bundle to W in Q with structure group SO(5).  

The fermions that are living on the  world-volume of the five-brane of the M-Theory are (four-component) chiral spinors on W with values in a bundle denoted S(N) constructed from N by using the spinor representation of SO(5).  The contribution of these fermion fields to the anomaly is given by  \[I_D ={\frac {1}{2}} 2^2 Mayer(S(N)). \hat{A}(W) \]

where

\[Mayer(S(N)) = \prod _{i=1}^{2}\cosh \left( \frac{y_{{i}}}{2} \right)\]

and

\[p(N) =\prod _{i=1}^{2}(1+{y_{i}}^2)\]

Reference:

FIVE-BRANE EFFECTIVE ACTION IN M-THEORY, Edward Witten, hep-th/9610234, IASSNS-HEP-96-101, page 31, equations (5.1) and (5.3).

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