On the gauge orbit types for theories with classical compact gauge group
Let \(X\) be a CW-complex of dimension 4 and without 2-torsion in \(H^4(X, \mathbb{Z})\). In this paper the authors give a classification of principal bundles with structure groups \(O(3), \;O(5)\) and \(SO(4)\) over \(X\) in terms of cohomology characteristic classes. Let \(Sq^1\) and \(\mathfrak{P}\) denote the Steenrod square and the Pontrjagin square. Set \[ \mathfrak{R}(u_1, u_2, u_4, v_4)=\rho_4v_4-\mathfrak{P}u_2+i_*(u_1Sq^1u_2+u_4) \] in \(H^4(X, \mathbb{Z}_4)\) for \(u_k \in H^k(X, \mathbb{Z}_2)\), \(v_4 \in H^4(X, \mathbb{Z})\) where \(i_* : H^4(X, \mathbb{Z}_2) \to H^4(X, \mathbb{Z}_4)\) and \(\rho_4 : H^4(X, \mathbb{Z}) \to H^4(X, \mathbb{Z}_4)\) are induced from the inclusion \(\mathbb{Z}_2 \to \mathbb{Z}_4\) and reduction mod 4, respectively. Then for example, in the case of \(O(5)\), the result is that the map \(\xi \mapsto (w_1(\xi), w_2(\xi), w_4(\xi), p_1(\xi))\) becomes a bijection between \([X, BO(5)]\) and the set of the 4-tuples in \(H^1(X, \mathbb{Z}_2) \oplus H^2(X, \mathbb{Z}_2)\oplus H^4(X, \mathbb{Z}_2)\oplus H^4(X, \mathbb{Z})\) satisfying the relation \(\mathfrak{R}(u_1, u_2, u_4, v_4)=0\). The authors explain in Sections 1 and 2 how doing this relates to a pure classical gauge theory. Let \(P\) be a principal \(G\)-bundle over a closed 4-dimensional manifold \(M\) with connection on \(G\) where \(G\) is a classical compact Lie group. Let \(\mathcal{A}\) and \(\mathcal{G}\) denote the affine space of connection forms and the group of authomorphisms of \(P\) covering the identity on \(M\), respectively. According to the authors, since for \(O(n)\)-bundles the case \(O(5)\) is the stable case, in order to classify bundles over \(M\) with a Howe subgroup of \(G\), it suffices to classify \(O(k)\)-bundles for \(k=3, 4, 5\). This means that this work contributes to determining the orbit types of the natural action of \(\mathcal{G}\) on \(\mathcal{A}\) via pull-back.
scientific article; zbMATH DE number 5904012
scientific article