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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].

##### MSC:
 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