BRS cohomology and topological anomalies
G denotes the (finite-dimensional, usually compact) gauge group, A the Hilbert space of all external gauge potentials (i.e. connections on the principal G-bundle) on \(S^ 4\), G the group of gauge transformations and g its Lie algebra, \(O:=A/G\), (\({\mathbb{C}}*(A)){\mathbb{C}}(A)\) are the spaces of (nowhere vanishing) complex functionals on A, and \((C^*)C\) and Z are the sheaves of germs of (nowhere vanishing) complex and integer-valued functions on O. A multiplicative cohomology with coefficients in \({\mathbb{C}}^*(A)\) is defined on G. It is compared with the BRS cohomology with coefficients in \({\mathbb{C}}(A)\) on g by showing \(H^ 1(G,{\mathbb{C}}*(A))\cong H^ 1(g,{\mathbb{C}}(A)).\) It is then shown that the elements f in any cohomology class in \(H^ 1(G,{\mathbb{C}}^*(A))\) define isomorphic line bundles on O and that there exists \(W\in {\mathbb{C}}(A)\) such that \(f(g,A)=W(A)/W(Ag)\) where \(A\in {\mathcal A}\), \(g\in G\). The nonabelian anomalies \(a(\tau,A):=\lim_{t\to 0}[(f(\exp t \tau,A)-1)/t]\) are then in 1-1 correspondence with these line bundles (for \(\tau\in G)\). Therefore, it can be shown that the cohomology groups \(H^ 1(g,{\mathbb{C}}(A))\), \(H^ 1(G,{\mathbb{C}}^*(A))\), \(H^ 1(O,C^*)\), \(H^ 1(G,{\mathbb{Z}})\) and \(H^ 2(O,{\mathbb{Z}})\) are all isomorphic. If G is compact and \(\pi_ 4(G)\) is trivial, then each of these groups is isomorphic to \(\pi_ 5(G)\). The last of the cohomology groups above is the first Chern class on the line bundle on O, which is given by a closed 2-form G which in turn, by pullback, leads to an exact 2-form \(d(Tc_ 1)\) on A. \(Tc_ 1\) is the ''transgression'' of the Chern 2-form \(c_ 1\). Thus we have: the topological anomaly is obtained by restricting the transgression of the first Chern form of the line bundle on O to the orbits of G in A.
cohomology of multiplicative abelian groups
scientific article; zbMATH DE number 3992391