3
$\begingroup$

Following Green, Schwarz and Witten's book on Superstrings, the BRST charge is given by $$Q_B = c^i K_i-\frac{1}{2}f_{ij}^{~~~k}c^ic^jb_k\tag{3.2.4}$$ with $$[K_i, K_j] = f_{ij}^{~~~k}K_k,\tag{3.2.1}$$ $$\{c^i, b_j\}=\delta^{i}_j,\tag{3.2.2}$$ where the $K_i$ are the generator of the gauge group, $c, b$ are ghosts and anti-ghosts respectively.

Now, the action of the BRST charge on a Yang-Mills gauge field $A^i$ goes like $$Q_B A^i = D c^i$$ with $D$ being the covariant derivative. I wish to know, how to derive the last expression starting from the definition of the BRST charge.

$\endgroup$
1
$\begingroup$

Hint: The Dirac conjecture states that the generators $K_i$ of the gauge group are given by the first-class constraints, i.e. Gauss law $(D\cdot E)_i$. OP's last equation then follows by recalling the CCR for the conjugate pair $A^i$ and $E_j$. (Here deWitt's condensed notation is assumed everywhere.)

References:

  1. M. Henneaux & C. Teitelboim, Quantization of Gauge Systems, 1994.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.