• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,075 questions , 2,226 unanswered
5,348 answers , 22,757 comments
1,470 users with positive rep
818 active unimported users
More ...

  Heterotic Supersymmetric derivation of an integrality theorem for differentiable manifolds

+ 4 like - 0 dislike

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


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?

asked Nov 12, 2014 in Theoretical Physics by juancho (1,130 points) [ revision history ]
edited Nov 16, 2014 by juancho

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 ...?

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

Hi Dilaton, thanks for your comment, you are very kind. All the bestl
Hi Dimension10, many thanks for your latex editions. The equations look better now. All the best.

1 Answer

+ 3 like - 0 dislike

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  \[I_D ={\frac {1}{2}} 2^2 Mayer(S(N)). \hat{A}(W) \]


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


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


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


answered Nov 15, 2014 by juancho (1,130 points) [ no revision ]

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights