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Characteristic classes of loop group bundles and generalized string classes. (English) Zbl 0806.57014
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 33-66 (1992).
Let \(M\) be a smooth manifold, \(G\) a Lie group with the Lie algebra \({\mathfrak g}\), \(\xi= \{g_{UV}\}\) a \(G\)-bundle over \(M\). Then \(\xi^ \#= \{g_{UV}^ \#\}\), \((g_{UV}^ \# (\gamma)) (t)= g_{UV} (\gamma(t))\), \(\gamma\in \Omega M\), is an \(\Omega G\)-bundle over \(\Omega M\). Here \(\Omega M\) and \(\Omega G\) are loop space and loop group over \(M\) and \(G\), respectively. If \(\xi\) is the tangent bundle \(\tau\) of \(M\), then \(\tau^ \#\) is the tangent bundle of \(\Omega M\). So the study of loop group bundles has meanings. Let \(\widetilde{\Omega} G\) be the basic central extension of \(\Omega G\). Then the obstruction \(\widetilde {c}^ 1(\eta)\in H^ 2(X, S^ 1)\cong H^ 3(X,\mathbb{Z})\) for the lifting of the structure group of an \(\Omega G\)-bundle \(\eta= \{h_{UV}\}\), over a space \(X\), to \(\widetilde{\Omega} G\) has been defined and is named ‘string class’. If \(X\) is \(\Omega M\) and \(\eta\) is \(\tau^ \#\), then the transgression image of this class is the first Pontryagin class of \(M\) [T. P. Killingback, World-sheet anomalies and loop geometry, Nucl. Phys., B – Part. Phys. 288, 578-588 (1987)]. Precisely saying, the torsion part needs more delicate discussions [cf. K. Pilch and N. P. Warner, Commun. Math. Phys. 115, 191-212 (1988; Zbl 0661.58038)].
In this paper, assuming \(G= U(n)\), characteristic classes \(\widetilde{c}^ p (\eta)\in H^{2p+1} (X,\mathbb{R})\), \(p=1,2,\dots\), of \(\eta\) are defined as the generalization of the (real) string class. Their definitions are as follows: Let \(\{\theta_ U\}\) be a connection form of \(\eta\) with the curvature form \(\{\Theta_ U\}\). Then there is a 0-cochain of \(2p\)-forms \(\{\Psi_{p,U}\}\) such that \(\int^ 1_ 0 \text{tr} (\Theta^ p_ U g_{UV}^ \prime g_{UV}^{-1}) dt= \Psi_{p,V}- \Psi_{p,U}\) for any \(p\). Here \('\) means derivation with respect to the loop variable \(t\). Then \(\Phi_{p,U}= \int^ 1_ 0 \text{tr} (\Theta^ p_ U \wedge \Theta_ U')dt- d\Psi_{p,U}\) defines a global closed \((2p+1)\)-form on \(X\) and its de Rham class is determined by \(\eta\). \(\widetilde{c}^ p (\eta)\) is defined to be the de Rham class of \(\{\Phi_{p,U}\}\) (Lemma 6 and 7). It is shown that \(\widetilde{c}^ 1(\eta)\) coincides with the above string class (as a real class, Theorem 3) and the following is shown:
(i) Define a \(G\)-bundle \(\eta^ \natural\) over \(X\times S^ 1\) by \(\eta^ \natural= \{h_{UV}^ \natural\}\), \(h_{UV}^ \natural (x,t)= (h_{UV}(x)) (t)\); then \(\widetilde{c}^ p (\eta)=- (2\pi \sqrt{-1})^{p+1} p! \int_ S 1\text{Ch}^{p+1} (\eta^ \natural)dt\), holds. Here \(\text{Ch}^{p+1} (\eta^ \natural)\) is the \((p+1)\)-th Chern character of \(\eta^ \natural\) (Theorem 2).
(ii) Let \(\tau^{-1}: H^{q+1} (M,\mathbb{R})\to H^ q (\Omega M,\mathbb{R})\) be the inverse of the transgression map. Then \(\widetilde{c}^ p (\xi^ \#)=- (2\pi \sqrt{-1})^{p+1} p! \tau^{-1} (\text{Ch}^{p+1} (\xi))\) holds (Theorem 4).
(iii) \(\eta\) has the characteristic map \(g: X\to G\) (Theorem 1) and \[ \widetilde{c}^ p (\eta)= -(2\pi \sqrt{-1})^{p+1} (2p+1)!/ (p+1)! g^* (e_{p+1}) \] holds. Here \(e_{p+1}\) is the \((p+1)\)th generator of \(H^* (U(n),\mathbb{Z})\) (Theorem 6).
To extend (iii) to the relation between string classes and Chern-Simons classes, the notion of 2-dimensional non-abelian de Rham cocycle with respect to \(\Omega G\) is introduced. Then the above results are extended to 2-dimensional non-abelian de Rham cocycles with respect to \(\Omega G\). In this case, string classes may become fractional classes and the characteristic map may become a many-valued map. Detailed definitions of this cocycle and its connection and curvature are given in Section 1. Precise properties of \(\widetilde{\Omega} {\mathfrak g}\)-valued connections are also given. Here \(\widetilde{\Omega} {\mathfrak g}\) means the basic central extension of \(\Omega {\mathfrak g}\). Properties of the central extension part of a \(\widetilde{\Omega} {\mathfrak g}\)-valued connection give the prototype of the definition of \(\widetilde{c}^ p (\eta)\). The rest of the paper is devoted to the proofs of (i), (ii) and (iii).
For the entire collection see [Zbl 0764.00002].

57R20 Characteristic classes and numbers in differential topology
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
53C99 Global differential geometry