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Invariance of supersymmetric Yang-Mills theory under supersymmetry

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I was following Brink, Scherk and Schwartz, "Supersymmetric Yang-Mills theories". The variation of the Lagrangian w.r.t a supersymmetry transformation can be reduced to $$ \delta L = -igf_{a b c} \bar{\lambda}^a \gamma_\mu \lambda^b \delta A^{\mu c} = gf_{a b c}( \bar{\lambda}^a \gamma_\mu \lambda^b \bar{\alpha}\gamma^\mu\lambda^c - \bar{\lambda}^a \gamma_\mu \lambda^b \bar{\lambda^c}\gamma^\mu \alpha). \tag{2.7} $$ It says that this term vanishes for $D = 4$ with the Majorana (or Weyl) condition, for $D=6$ with the Weyl condition and for $D=10$ with the Majorana-Weyl condition

Please can someone explain this or point me to further references. Fierz identities are used to derive the results, but I really did not understand how they are obtained.

This question is cross posted here.

This post imported from StackExchange Physics at 2016-01-29 15:22 (UTC), posted by SE-user Prathyush
asked Jun 12, 2015 in Theoretical Physics by Prathyush (695 points) [ no revision ]
Hi Prathyush. If you haven't already done so, please take a minute to read the definition of when to use the homework-and-exercises tag, and the Phys.SE policy for homework-like problems.

This post imported from StackExchange Physics at 2016-01-29 15:22 (UTC), posted by SE-user Qmechanic
@Qmechanic If you or others think the tag is appropriate go aheard. I don't like the idea because, the implication seems to be that the question will not be answered fully, but only hints will be provided the general direction. I would much rather prefer a full answer with all the details involved. It would also be useful as resource material for anyone interested in this question.

This post imported from StackExchange Physics at 2016-01-29 15:22 (UTC), posted by SE-user Prathyush

1 Answer

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The ten-dimensional case is explained in some detail in appendix 4.A of volume one of Green-Schwarz-Witten. Let me therefore consider the 4d case here. The calculation works essentially the same in 3d, 6d and 10d, though you can sometimes make use of Majorana and/or Weyl conditions of the spinors to simplify things.

We want to check that the expression \begin{equation*} f_{abc} \bigl( \bar{\lambda}^a \gamma_{\mu} \lambda^b \, \bar{\epsilon} \, \gamma^{\mu} \lambda^c - \bar{\lambda}^a \gamma_{\mu} \lambda^b \, \bar{\lambda}^c \gamma^{\mu} \epsilon \bigr) \tag{1} \end{equation*} vanishes, where $\lambda^a$ and $\epsilon$ are four-dimensional Majorana spinors. To show this the following identities involving four-dimensional gamma matrices and arbitrary spinors will be useful \begin{align*} \bar{\varphi} \, \psi &= + \bar{\psi} \, \varphi , \tag{2a} \\ \bar{\varphi} \, \gamma^{\mu} \, \psi &= - \bar{\psi} \, \gamma^{\mu} \, \varphi , \tag{2b} \\ \bar{\varphi} \, \gamma^{\mu\nu} \, \psi &= - \bar{\psi} \, \gamma^{\mu\nu} \, \varphi , \tag{2c} \\ \bar{\varphi} \, \gamma^{\mu\nu\rho} \, \psi &= + \bar{\psi} \, \gamma^{\mu\nu\rho} \, \varphi , \tag{2d} \\ \bar{\varphi} \, \gamma^{0123} \, \psi &= + \bar{\psi} \, \gamma^{0123} \, \varphi , \tag{2e} \end{align*} where, eg., $\gamma^{\mu\nu} = \frac{1}{2} ( \gamma^\mu\gamma^\nu - \gamma^\nu\gamma^\mu)$. Using the second identity we see that the two terms in (1) are equal.

Using the antisymmetry of $f_{abc}$ we can rewrite (1) as \begin{equation*} ( \bar{\lambda}^a \gamma_\mu \lambda^b ) ( \bar{\epsilon} \, \gamma^\mu \lambda^c ) + ( \bar{\lambda}^c \gamma_\mu \lambda^a ) ( \bar{\epsilon} \, \gamma^\mu \lambda^b ) + ( \bar{\lambda}^b \gamma_\mu \lambda^c ) ( \bar{\epsilon} \, \gamma^\mu \lambda^a ) , \end{equation*} or, if we remove the spinors, \begin{equation*} (\gamma^0 \gamma_\mu)_{\alpha\beta} (\gamma^0 \gamma^\mu)_{\gamma\delta} + (\gamma^0 \gamma_\mu)_{\delta\alpha} (\gamma^0 \gamma^\mu)_{\gamma\beta} - (\gamma^0 \gamma_\mu)_{\beta\delta} (\gamma^0 \gamma^\mu)_{\gamma\alpha} . \end{equation*} Finally we contract this expression with two arbitrary spinors $\psi^\alpha$ and $\varphi^\beta$ to obtain (here I have also removed an overall $\gamma^0$ and reordered the second term slightly) \begin{equation*} ( \bar{\psi} \, \gamma_\mu \varphi ) ( \gamma^\mu )^\gamma{}_\delta + ( \bar{\psi} \, \gamma_\mu )_\delta ( \gamma^\mu \varphi )^\gamma - ( \bar{\varphi} \, \gamma_\mu )_\delta ( \gamma^\mu \psi )^\gamma . \tag{3} \end{equation*} The idea is to now think of this as a $4 \times 4$ matrix. A basis of such matrices is given by the 16 matrices \begin{equation*} 1 , \quad \gamma^\mu , \quad \gamma^{\mu\nu} , \quad \gamma^{\mu\nu\rho} , \quad \gamma^{0123} . \tag{4} \end{equation*} This basis is orthogonal in the sense that if we call the sixteen matrices $\gamma^{(I)}$, with $I=1,\dots,16$, then \begin{equation*} \operatorname{Tr}\, \bigl( \gamma^{(I)} \gamma^{(J)} \bigr) = \operatorname{Tr} \, \bigl( \gamma^{(I)} \gamma^{(I)} \bigr) \, \delta^{IJ} . \end{equation*} Hence, to prove that (3) vanishes we can check that we get zero when contracting with each of the matrices in (4). It is easy to see that the first term in (3) is zero unless we contract with a single gamma matrix, so let us first contract with $\gamma^\nu$. We then get \begin{equation*} ( \bar{\psi} \, \gamma_\mu \varphi ) \, \operatorname{Tr}\, \bigl( \gamma^\mu \gamma^\nu \bigr) + \bar{\psi} \, \gamma_\mu \gamma^\nu \gamma^\mu \varphi - \bar{\varphi} \, \gamma_\mu \gamma^\nu \gamma^\mu \psi = 4 \eta^{\mu\nu} \bar{\psi} \, \gamma_\mu \varphi - 2 \, \bar{\psi} \, \gamma^\nu \varphi + 2 \, \bar{\varphi} \, \gamma^\nu \psi = 0 , \end{equation*} where I have used equation (2b) as well the second of the following useful relations for four-dimensional gamma matrices \begin{align*} \gamma_\mu \gamma^\mu &= 4 , \tag{5a} \\ \gamma_\mu \gamma^{\nu} \gamma^\mu &= -2\gamma^\nu , \tag{5b} \\ \gamma_\mu \gamma^{\nu\rho} \gamma^\mu &= 0 , \tag{5c} \\ \gamma_\mu \gamma^{\nu\rho\sigma} \gamma^\mu &= +2\gamma^{\nu\rho\sigma} , \tag{5d} \\ \gamma_\mu \gamma^{0123} \gamma^\mu &= -4\gamma^{0123} . \tag{5e} \end{align*} If we instead contract (3) with $\gamma^{\nu\rho}$ each term vanishes separately because of equation (5c). In the remaining three cases (contracting with $1$, $\gamma^{\nu\rho\sigma}$ or $\gamma^{0123}$), the first term in (3) is zero and the second two terms cancel each other because of the positive sign in (2a), (2d) and (2e).

We have now shown that all components of the $4 \times 4$ matrix in (3) vanish. Since the spinors $\psi$ and $\varphi$ were arbitrary this means that (1) is zero as well.

The above procedure of expanding in a basis of (products of) gamma matrices is equivalent to, but more transparent than, using Fierz identities.

This post imported from StackExchange Physics at 2016-01-29 15:22 (UTC), posted by SE-user Olof
answered Jul 5, 2015 by Olof (210 points) [ no revision ]

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