Flat vector bundles and analytic torsion on orbifolds

Abstract

This article is devoted to a study of flat orbifold vector bundles. We construct a bijection between the isomorphic classes of proper flat orbifold vector bundles and the equivalence classes of representations of the orbifold fundamental groups of base orbifolds.We establish a Bismut-Zhang like anomaly formula for the Ray-Singer metric on the determine line of the cohomology of a compact orbifold with coefficients in an orbifold flat vector bundle.We show that the analytic torsion of an acyclic unitary flat orbifold vector bundle is equal to the value at zero of a dynamical zeta function when the underlying orbifold is a compact locally symmetric space of the reductive type, which extends one of the results obtained by the first author for compact locally symmetric manifolds. CONTENTS 5.4. Semisimple orbital integral 41 5.5. Locally symmetric spaces 42 5.6. Ruelle dynamical zeta functions 44 5.7. Reductive group with δ(G) = 1 and with compact center 45 5.8. Auxiliary virtual representations of K 47 5.9. Evaluation of Tr s [γ] [exp(−tC g,X, η /2)] 47 5.10. Selberg zeta functions 50 5.11. The proof of Theorem 5.8 when G has compact center and δ(G) = 1 51 References 51

On full asymptotics of analytic torsions for compact locally symmetric orbifolds

“…We also refer to [37,38] and [1,Chapter 1] for more details. Then we recall the definition of Ray-Singer analytic torsion for compact orbifolds, where we refer to [27], [41] for more details.…”

“…If (F, ∇ F ) is an orbifold vector bundle over Z with a connection ∇ F , we call (F, ∇ F ) a flat vector bundle if the curvature R F = ∇ F,2 vanishes identically on Z. A detailed discussion for the flat vector bundles on Z is given in [41,Section 2.5]. Let (Z, g T Z ) be a compact Riemannian orbifold of dimension m. Let (F, ∇ F ) be a flat complex orbifold vector bundle of rank r on Z with Hermitian metric h F .…”

“…If F is proper, recall that the numbers ρ i , i = 0, • • • , l are defined in previous subsection as the extension of the rank of F . Then by [41,Theorem 4.3], we have…”

“…) is a representation of Γ ′ , which can be viewed as a representation of Γ via the projection Γ → Γ ′ = Γ/S, then F = X Γ ′ × C k is a proper flat orbifold vector bundle on Z with the flat connection ∇ F,f induced from the exterior differential d X on C k -valued functions. By [41,Theorem 2.31], all the proper orbifold vector bundle on Z of rank k comes from this way. Now let ρ : Γ → GL(C k ) be a representation of Γ, we do not assume that it comes from a representation of Γ ′ .…”

“…In particular, if F is acyclic, and if m is odd, then T (Z, F ) is independent of the metric data (cf. [41,Corollary 4.9]). We refer to [27], [41], etc for more details.…”