# How to Derive Yang-Mills Theories from a Kaluza-Klein Perspective?

• BVM
In summary, the author is trying to solve a problem involving the divergence of Killing vectors, but gets stuck and needs help.
BVM

## Homework Statement

I am going through the book "Symmetries in Fundamental Physics" by Kurt Sundermeyer and in his part on deriving Yang-Mills theories from a Kaluza-Klein perspective I seem to be stuck on a small step in the derivation.

## Homework Equations

He expands the metric in a typical Kaluza-Klein-like style
## \begin{cases}
&\hat{g}^{0}_{\mu \nu} (x) = W(\phi) g_{\mu \nu} + f(\phi)h_{\iota \kappa} \mathcal{B}^{\iota}_{\mu}\mathcal{B}^{\kappa}_{\nu}\\
&\hat{g}^{0}_{\mu \kappa} (x) = f(\phi) h_{\iota \kappa}\mathcal{B}^{\iota}_{\mu}\\
&\hat{g}^{0}_{\iota \kappa} (x) = f(\phi)h_{\iota \kappa}
\end{cases}##
and derives from it that under diffeomorphism, the B-field should transform as
##\mathcal{B}'^{\iota}_{\mu} = \frac{\partial x^{\nu}}{\partial x'^{\mu}}\left( \frac{\partial \theta'^{\iota}}{\partial \theta^{\kappa}} \mathcal{B}^{\kappa}_{\nu} - \frac{\partial \theta'^{\iota}}{\partial x^{\nu}} \right).##
Now, by considering the isometries infinitesimally
##\theta'^{\iota} = \theta^{\iota} + \epsilon^a(x)K^{\iota}_a(\theta)##
and expanding the B-fields in terms of the Killing vectors
##\mathcal{B}^{\iota}_{\mu} = g K^{\iota}_{a} \mathcal{A}^{a}_{\mu}##
he gets
##g K^{\iota}_{d} \mathcal{A}^{d}_{\mu} = (\delta^{\iota}_{\kappa} + \epsilon^a \partial_{\kappa}K^{\iota}_a) g K^{\kappa}_{b} \mathcal{A}^{b}_{\mu} - K^{\iota}_{a}\partial_{\mu}\epsilon^a.##

This i can follow, but now he claims that via
##K^{\kappa}_{a} \partial_{\kappa} K^{\iota}_{b} - K^{\kappa}_{b} \partial_{\kappa} K^{\iota}_{a} = f_{abc} K^{\iota}_{c}##
he can get
##\mathcal{A}'^{a}_{\mu} = \mathcal{A}^{a}_{\mu} + \frac{1}{g}f^{bca}\mathcal{A}^b_{\mu}\epsilon^c - \frac{1}{g} \partial_{\mu}\epsilon^a.##

3. The attempt at a solution
By just plugging in the relation I can get as far as
##\mathcal{A}'^{a}_{\mu} = \mathcal{A}^{a}_{\mu} + f^{bca}\mathcal{A}^b_{\mu}\epsilon^c + \epsilon^c(K^{\iota}_a)^{-1}K^{\kappa}_{c}(\partial_{\kappa} K^{\iota}_b)\mathcal{A}^b_{\mu} - \frac{1}{g} \partial_{\mu}\epsilon^a##
but I don't see how the extra term cancels or how the ##\frac{1}{g}## appears in the second term.

BVM said:
By just plugging in the relation I can get as far as
##\mathcal{A}'^{a}_{\mu} = \mathcal{A}^{a}_{\mu} + f^{bca}\mathcal{A}^b_{\mu}\epsilon^c + \epsilon^c(K^{\iota}_a)^{-1}K^{\kappa}_{c}(\partial_{\kappa} K^{\iota}_b)\mathcal{A}^b_{\mu} - \frac{1}{g} \partial_{\mu}\epsilon^a##
but I don't see how the extra term cancels or how the ##\frac{1}{g}## appears in the second term.

Performing the contraction, we find

$$\epsilon^c(K^{\iota}_a)^{-1}K^{\kappa}_{c}(\partial_{\kappa} K^{\iota}_b)\mathcal{A}^b_{\mu} = \epsilon^a(\partial_{\kappa} K^{\kappa}_b)\mathcal{A}^b_{\mu},$$

but the divergence of a Killing vector vanishes, so this is zero.

Oh my god how did I miss this.

Thanks a lot!

## What are Yang Mills fields?

Yang Mills fields are a type of quantum field theory that describes the interactions between elementary particles, specifically those that have a spin of 1 or higher. They are named after physicists Chen Ning Yang and Robert Mills, who first proposed this theory in the 1950s.

## What are isometries in relation to Yang Mills fields?

Isometries are mathematical transformations that preserve the distance between points. In the context of Yang Mills fields, isometries refer to the symmetries or invariances of the theory, which are essential for understanding its behavior and predicting the interactions between particles.

## What is the significance of Yang Mills fields from isometries?

The connection between Yang Mills fields and isometries is important because it allows for the formulation of a gauge theory, which is a type of theory that describes the fundamental forces of nature. In this case, isometries are used to construct the gauge symmetry group, which is a fundamental concept in Yang Mills theory.

## What are some real-world applications of Yang Mills fields from isometries?

Yang Mills fields have been used to successfully describe the strong nuclear force, which is responsible for holding the nucleus of atoms together. This theory has also been used in the development of the Standard Model of particle physics, which is a widely accepted model for understanding the fundamental particles and their interactions.

## What are some current research topics related to Yang Mills fields from isometries?

Some current research topics in this field include the application of Yang Mills theory to quantum gravity, the study of topological aspects of Yang Mills fields, and the search for new particles and interactions that can be described by this theory. There is also ongoing research in developing more efficient techniques for calculating and predicting the behavior of Yang Mills fields.

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