Baulieu, L.; Singer, I. M. Topological Yang-Mills symmetry. (English) Zbl 0958.58500 Nucl. Phys., B, Proc. Suppl. 5B, 12-19 (1988). From the text: The classical action: (1) \(I_t=\int_{M^4}\text{Tr}F\wedge F\) is determined by the gauge symmetry \(\delta A_\mu=\epsilon_\mu+D_\mu\epsilon\). It can be fully gauge fixed into an action which is: (2) \(I_{\text{GF}}=I_t+\int_{M^4}s(\cdots)=\int_{M^4}d^4x[\text{Tr}(F_{\alpha\beta})^2+\cdots]\). This quantization is a standard BRST quantization. The independence of physical quantities with respect to variations of the background \(g_{\alpha\beta}\) follows from the form of (2). The relevant ghost spectrum is displayed. The gauge functions are: (3) \(\partial_\alpha A^\alpha\), \(F^{\alpha\beta}\pm\epsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}\), \(D_\alpha\psi^\alpha\). In Witten’s work, the role of \(\overline\phi\) and \(\overline\eta\) is played, respectively, by \(\lambda\) and \(\eta\). The Faddeev-Popov ghosts \(c\) and \(\overline c\) are missing and consequently a gauge-fixing term for implementing the gauge function \(\partial_\alpha A^\alpha\cdot\overline\eta\) should be viewed as a Lagrange multiplier while \(\overline\chi_{\alpha\beta}\) and \(\overline\phi\) are antighosts. \(\phi\) is a ghost of ghost due to the degeneracy of the gauge transformation \(\epsilon_\mu\sim \epsilon_\mu+D_\mu\lambda\). The breaking of the nilpotency of \(s\) on \(\overline\chi{}^{\alpha\beta}\) is due to the elimination of the Lagrange multiplier for the gauge function \(F^{\alpha\beta}\pm\epsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}\). Finally, we have interpreted the BRST cohomology in terms of curvature and characteristic forms on \({\mathcal A}/{\mathcal G}\times M\). The functional integral representation for the Donaldson polynomials arises, from this viewpoint, because the Green function \((D_A^*D_A)^{-1}\) occurs in the formula of the curvature. Most of the features which are presented here can be generalized for ‘topological’ classical actions, i.e. classical actions which are topological invariants. In all of these cases, our gauge fixing procedure based on a direct construction of a BRST symmetry operator permits us to perturbatively determine a partition function out of such actions although there are pure derivatives. We shall present elsewhere the cases of the \(\sigma\)-model and of gravity [the authors, Commun. Math. Phys. 125, 227-237 (1989; Zbl 0684.57016)]. Cited in 1 ReviewCited in 47 Documents MSC: 58A12 de Rham theory in global analysis 57R25 Vector fields, frame fields in differential topology 58J70 Invariance and symmetry properties for PDEs on manifolds Citations:Zbl 0684.57016 PDFBibTeX XMLCite \textit{L. Baulieu} and \textit{I. M. Singer}, Nucl. Phys., B, Proc. Suppl. 5B, 12--19 (1988; Zbl 0958.58500) Full Text: DOI References: [2] Baulieu, L.; Bellon, M., Nucl. Phys., B266, 75 (1986) [3] Atiyah, M. F.; Singer, I. M., (Proc. Nat. Acad. Sci. USA, 81 (1984)), 2597 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.