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Topological Yang-Mills symmetry. (English) Zbl 0958.58500

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)].

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
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References:

[1] E. Witten, “Topological Quantum Field Theory”, Princeton Preprint. · Zbl 0656.53078
[2] Baulieu, L.; Baulieu, L.; Baulieu, L.; Bellon, M., (), Phys. rep., Nucl. phys., B266, 75, (1986)
[3] Atiyah, M.F.; Singer, I.M., (), 2597
[4] L. Baulieu and I.M. Singer, in preparation.
[5] S. Donaldson, “Polynomials Invariants for Smooth Four Manifolds”, Oxford Preprint. · Zbl 0715.57007
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