% File: brst.tex % Section: ? % Title: BRST % Last modified: 03.04.2001 % \documentclass[a4paper,12pt]{article} \usepackage{amsmath,amssymb} \textwidth 16cm \textheight 25cm \oddsidemargin 0cm \topmargin -1.5cm \pagestyle{empty} \begin{document} \begin{center} \Large\textbf{BRST} \end{center} \vspace{0cm} \begin{center} \large\textbf{BRST operator} \end{center} \vspace{.1cm} To indicate an algebraic ancestor of the BRST operator, let us consider a representation $\rho$ of a Lie algebra $\mathfrak{g}$ in a vector space $V$\,. Let $\{e_\alpha\}$ be a basis of $V$ and $\{e_i\}$ of $\mathfrak{g}$\,, $f_{ij}^{k}$ the structure constants of the latter, $[e_i,e_j]=f_{ij}^{k}e_k$\,, and $t_i$ the generators of the representation $\rho$\,: $t_i=\rho(e_i)$\,. Then a $V$-valued $n$-\textit{cochain} is an antisymmetric linear map $u:\mathfrak{g}^n\rightarrow V$ specified by \begin{equation} \label{cochain} u(e_{i_1},\,\ldots\,,e_{i_n})=u_{i_1\ldots i_n}^{\alpha}e_\alpha\,, \qquad\qquad (n\geqslant0) \end{equation} coefficients $u_{i\ldots j}^{\alpha}$ being totally antisymmetric in lower indices. The \textit{coboundary operator} $\delta$ increases the valence of arbitrary $n$-cochain $u$ by one, sending it to $(n\!+\!1)$-cochain $\delta u$\,. Explicitly ($x_k\in\mathfrak{g}$)\,, \begin{multline} \label{cobound} \delta u\,(x_1,\,\ldots\,,x_{n+1}) \doteq \sum_{k=1}^{n+1}(-)^{k+1}\rho(x_k)u(x_1,\,\ldots\,,{\not\!x}_k,\, \ldots\,,x_{n+1}) \\ + \sum_{j