String connections and Chern-Simons theory (Q2847041)

Let \(P \to M\) be a principal \(\mathrm{Spin}(n)\)-bundle over a smooth manifold \(M\) (where \(n=3\) or \(n >4\)). We know that the homomorphism \(H^4(B\mathrm{SO}(n), \mathbb{Z}) \to H^4(B\mathrm{Spin}(n), \mathbb{Z})\) induced by the natural map \(B\mathrm{Spin}(n) \to B\mathrm{SO}(n)\) sends the Pontryagin class \(p_1 \in H^4(B\mathrm{SO}(n), \mathbb{Z})\cong\mathbb{Z} \) to twice a generator of \(H^4(B\mathrm{Spin}(n), \mathbb{Z})\cong\mathbb{Z}\), which is denoted by \(\frac{1}{2}p_1\). We say that \(P\) is string if \(\frac{1}{2}p_1(P)=0\) and call a lifting of the classifying map \(M \to B\mathrm{Spin}(n)\) of \(P\) to \(B\mathrm{String}(n)\) a string structure on \(P\), where \(B\mathrm{String}(n)\) is the homotopy fiber of \(\frac{1}{2}p_1 : B\mathrm{Spin}(n) \to K(4, \mathbb{Z})\). However, here the following equivalent definition is used for the reason that the group \(\mathrm{String}(n)\) is not a compact Lie group: A string structure on \(P\) is a cohomology class \(\xi \in H^3(P, \mathbb{Z})\) that restricts fiberwise to the standard generator of \(H^3(\mathrm{Spin}(n), \mathbb{Z})\).NEWLINENEWLINEThe aim of this paper is to propose a new formulation of a string structure in terms of bundle gerbes for the study of a string connection. Put \(G=\mathrm{Spin}(n)\) and consider a bundle 2-gerbe \(\mathbb{CS}_P\), called the Chern-Simons 2-gerbe, associated to \(P\) and a certain multiplicative bundle gerbe over \(G\). The author shows first that its characteristic class \(\text{CC}(\mathbb{CS}_P)\) is equal to \(\frac{1}{2}p_1(P)\) and that the nullness of \(\text{CC}(\mathbb{CS}_P)\) can be detected using the notion of a trivialization of \(\mathbb{CS}_P\). More precisely, it is shown that such a trivialization \(\mathbb{T}\) defines a string structure \(\xi_\mathbb{T} \in H^3(P, \mathbb{Z})\) and that, moreover, both of them determine each other bijectively via the assignment \(\mathbb{T} \mapsto \xi_\mathbb{T}\). This means that a trivialization can be regarded as a string structure. That is, we are led to the following definition: A string structure on \(P\) is a trivialization of \(\mathbb{CS}_P\).NEWLINENEWLINEThis motivates a new definition of a string connection on \(\mathbb{CS}_P\). It is first observed that any connection \(A\) on \(P\) determines a connection \(\nabla_A\) on \(\mathbb{CS}_P\). Based on this, a string connection for \((\mathbb{T}, A)\) is defined as a connection \(\blacktriangledown\) on \(\mathbb{T}\) that is compatible with \(\nabla_A\) and we call the pair \((\mathbb{T}, \blacktriangledown)\) a geometric string structure on \((P, A)\). We then have the result that if \(\frac{1}{2}p_1(P)=0\) then \((P, A)\) admits a geometric string structure, which provides a starting point of subsequent arguments. At the beginning of Section 2.2, it is emphasized that the main concern here is to present the advantages exhibited therein of using the definition of a string structure proposed here.