N=4 super yang mills on the light cone.

[bartadam]

I'm currently trying to read a paper and it's not making much sense. Don't feel I expect anyone to read it in detail but it might give you an idea of the lack of understanding I am having. In all honesty I don't think it's terribly well written, coupled with the fact that I'm thick and only half know what I'm doing. The paper is...

Lars Brink et al 'N=4 yang mills theory on the light cone' nucl phys B212(1983) 401-412.

They write a scalar superfield as follows

$$\Phi(x,\theta)=\frac{1}{\partial^+}A(y)+i{\partial_+}\theta^m\bar{\chi}_{m}(y)+\sqrt{1/2}i\theta^{m}\theta^{n}\bar{C}_{mn}(y)+\frac{1}{6}\sqrt{2}\theta^m\theta^n\theta^p \epsilon_{mnpq}\chi^{q}+\frac{1}{12}\theta^{m}\theta^{n}\theta^{p}\theta^{q}\epsilon_{mnpq}\partial^{+}\bar{A}(y)$$

Where $$\partial^+$$ is just the derivative in light cone coords, that's not a problem. A(y) is the vector component (but it's in the light cone gauge so A and its conjugate are just scalars) $$\chi$$ is the spinor component and $$\bar{C}_{mn}$$ is a matrix of the 6 real scalar fields and is anti symmetric in m and n. m and n run from 1 to 4 and $$\theta$$ are grassman numbers.

y is given by $$y=(x,\bar{x},x^+,x^{-}-\sqrt{1/2}i \theta^m \bar{\theta_{m}})$$

It does not say if the $$\theta$$s are majorana spinors or not, and contracting the m indices like that in the expression for y doesn't make sense to me anyway. Isn't there supposed to be a gamma matrix in between?

Basically I want to taylor expand all the components fields so they are in terms of x, not y so I can repeat their calculations for myself. Any hints or help is appreciated. This is the first time I have studied SUSY in detail. I know the basics.

 

[AndyW]

Thank you for sharing your difficulties with understanding the paper "N=4 yang mills theory on the light cone" by Lars Brink et al. I understand that it can be frustrating when a paper is not well written or when you are not familiar with the subject matter. However, I would like to encourage you to keep trying and not feel discouraged by your current level of understanding.

Firstly, I would like to address your concern about the notation used in the paper. It is common in physics papers to use notation and conventions that may not be familiar to everyone. In this case, the authors are using light cone coordinates and Majorana spinors. It may be helpful for you to review these concepts before attempting to read the paper.

Next, I would like to address your question about the contraction of indices in the expression for y. The authors are using the Einstein summation convention, which means that repeated indices are automatically summed over. This is a common notation in physics and can be a bit confusing at first, but once you get used to it, it can make calculations much simpler.

In terms of taylor expanding the components fields, I would suggest looking for any additional information or notation that the authors may have provided in the paper. If not, you may need to refer to other sources or consult with a colleague who is familiar with the subject matter. It is always helpful to have someone to discuss and clarify difficult concepts with.

Overall, I want to encourage you to continue learning and studying SUSY. It may seem challenging now, but with persistence and effort, you will be able to understand and apply the concepts in this paper and beyond. Good luck!

 

[Entropia]

First of all, don't feel discouraged if you are having trouble understanding this paper. N=4 super Yang-Mills theory is a highly advanced and complex topic, so it's completely normal to have difficulty grasping it at first. Additionally, as you mentioned, the paper may not be well-written, which can make it even more challenging to understand.

To address your specific questions, let's break down the expression for the scalar superfield \Phi(x,\theta) and try to make sense of it.

The first term, \frac{1}{\partial^+}A(y), represents the vector component A(y) in the light cone gauge. The notation \partial^+ is just a shorthand for the derivative in light cone coordinates, so that shouldn't be a problem. The vector component A(y) is a scalar field, which means it has no spin or internal structure, and it is in the light cone gauge, which simplifies the calculations.

The next term, i{\partial_+}\theta^m\bar{\chi}_{m}(y), represents the spinor component \chi_m(y) of the superfield. This term involves the derivative \partial_+ acting on the spinor \theta^m, and the spinor \chi_m(y) is the conjugate of the spinor \theta^m. Again, this notation may seem confusing, but it is a common way to represent spinors in supersymmetric theories.

The third term, \sqrt{1/2}i\theta^{m}\theta^{n}\bar{C}_{mn}(y), represents the 6 real scalar fields \bar{C}_{mn}(y) that are anti-symmetric in m and n. This term also involves the spinors \theta^m and \theta^n, and the factor of \sqrt{1/2} is just a normalization constant.

The fourth term, \frac{1}{6}\sqrt{2}\theta^m\theta^n\theta^p \epsilon_{mnpq}\chi^{q}, represents the spinor \chi^q multiplied by the spinor \theta^m, \theta^n, and \theta^p. The factor of \epsilon_{mnpq} is the Levi-Civita symbol and it ensures that this term is invariant under rotations.

Finally, the last term, \frac{1}{12}\theta^{m}\theta^{n}\theta^{p