Refined analytic torsion as analytic function on the representation variety and applications
aa r X i v : . [ m a t h . SP ] M a r REFINED ANALYTIC TORSION AS ANALYTICFUNCTION ON THE REPRESENTATION VARIETYAND APPLICATIONS
MAXIM BRAVERMAN AND BORIS VERTMAN
Abstract.
We prove that refined analytic torsion on a manifoldwith boundary is a weakly holomorphic section of the determi-nant line bundle over the representation variety. As a fundamentalapplication we establish a gluing formula for refined analytic tor-sion on connected components of the complex representation spacewhich contain a unitary point. Finally we provide a new proof ofBr¨uning-Ma gluing formula for the Ray-Singer torsion associatedto a non-Hermitian connection. Our proof is quite different fromthe one given by Br¨uning and Ma and uses a temporal gauge trans-formation.
Contents
1. Introduction and statement of the main results 12. Refined analytic torsion on manifolds with boundary 33. Holomorphic structure on the determinant line bundle 94. The graded determinant as a holomorphic function 125. Refined analytic torsion as a holomorphic section 186. Gluing formula for refined analytic torsion 257. Gluing formula for Ray-Singer analytic torsion 308. Appendix: Temporal Gauge Transformation 32References 351.
Introduction and statement of the main results
The Ray-Singer conjecture has been formulated in the seminal pa-per of Ray and Singer [38] and proved independently by Cheeger [17]and M¨uller [35] for unitary representations. Its importance stems fromthe fact that as in the Atiyah-Singer index theorem, it equates ana-lytic with combinatorial quantities, the analytic Ray-Singer and thecombinatorial Reidemeister torsions.
Date : This document was compiled on: September 18, 2018.
By construction, both the analytic Ray-Singer and the combinato-rial Reidemeister torsions provide canonical norms on the determinantline of cohomology. There have been various approaches to obtaina canonical construction of analytic and Reidemeister torsions as el-ements instead of norms of the determinant line of the cohomology.These constructions seek to refine the notion of analytic and Reide-meister torsion norms on that determinant line, which basically corre-sponds to fixing a complex phase in the family of complex vectors oflength one.In case of the Reidemeister torsion this has been done by Farber andTuraev [18] and [19]. Refinement of analytic torsion has been studiedby the first author jointly with Kappeler in [7] and [5], as well as byBurghelea and Haller in [15] and [16]. Both notions have subsequentlybeen compared by the first author jointly with Kappeler in [6]. Anextension of refined analytic torsion to manifolds with boundary hasbeen undertaken by the second author [44] and Lee and Huang [30] intwo different independent constructions. Recently, Lee and Huang alsocompared the two notions of refined analytic torsion on manifolds withboundary in [32].The fundamental property of the refined analytic and Farber-Turaevtorsions is that they define weakly holomorphic functions on the com-plex representation space (see [24, p. 148] for definition of a weaklyholomorphic function). In particular, its restriction to the regular partof the representation space is holomorphic. The main purpose of thepresent discussion is an extension of this result to the refined analytictorsion on manifolds with boundary, introduced by the second authorin [44]. As a consequence we establish the gluing property of refinedanalytic torsion on connected components of the representation varietythat contain a unitary point.The gluing formula for refined analytic torsion may be used to provea gluing result for the Ray-Singer torsion norm for certain non-unitaryrepresentations path-connected to a unitary element. However we choseto devote the final two sections of the present paper to an alternativeproof of the gluing property for the Ray Singer analytic torsion fornon-unitary representations, which is stronger since we do not singleout connected components without unitary elements.The Ray-Singer theorem has been extended to unimodular represen-tations by M¨uller [36]. In case of a general non-unitary representation,the quotient of the analytic and Reidemeister torsion norms admitsadditional correction terms which have been studied by Bismut and
EFINED ANALYTIC TORSION ON MANIFOLDS WITH BOUNDARY 3
Zhang in [2]. In a separate discussion [9], the authors employ analyt-icity of refined analytic torsion to provide an alternative derivation ofthe Bismut-Zhang correction terms for the connected components ofunitary points in the representation variety.Both the analytic and combinatorial Reidemeister torsions makesense on compact manifolds with boundary after posing relative orabsolute boundary conditions. The gluing property of the analytic tor-sion, which is foremost a spectral invariant, is striking and has beenproved by L¨uck [34], Vishik [46] and generalized by Lesch [33], underthe assumption of product metric structures and unitary representa-tions.The anomaly of analytic torsion on a compact manifold with bound-ary with a non-unitary representation and general metric structuresnear the boundary, has been studied by Br¨uning and Ma in [11]. Re-cently, Br¨uning and Ma established in a follow-up paper [12] the glu-ing formula for analytic torsion in case of a non-unitary representation,generalizing previous results in [34] and [46]. We present an alternativeproof of a result by Br¨uning-Ma [12] by a temporal gauge transforma-tion argument.2.
Refined analytic torsion on manifolds with boundary
This section reviews the construction by the second author [44].2.1.
The flat vector bundle induced by a representation.
Let( M m , g ) be a compact oriented odd-dimensional Riemannian manifoldwith boundary ∂M . Consider a complex representation α of the fun-damental group π = π ( M ) on C n . Let ( E α , ∇ α , h Eα ) be the inducedflat complex vector bundle over M with monodromy equal to α and nocanonical choice of h Eα in case α is not unitary.The flat covariant derivative ∇ α acts on sections Γ( E α ) and ex-tends by Leibniz rule to a twisted differential on E α -valued differentialforms Ω ∗ ( M, E α ), where the lower index refers to compact support inthe open interior of M . This defines the twisted de Rham complex(Ω ∗ ( M, E α ) , ∇ α ). The metrics ( g, h Eα ) induce an L -inner product onΩ ∗ ( M, E α ). We denote the L − completion of Ω ∗ ( M, E α ) by L ∗ ( M, E α ).Throughout this section the representation α is fixed and we omit thelower index α in the notation of ( E α , ∇ α , h Eα ) in most of the discussion.Next we introduce the notion of the dual covariant derivative ∇ ′ . Itis defined by requiring for all u, v ∈ Γ( E ) and X ∈ Γ( T M ) dh E ( u, v )[ X ] = h E ( ∇ X u, v ) + h E ( u, ∇ ′ X v ) . (2.1) MAXIM BRAVERMAN AND BORIS VERTMAN
In the special case that α is unitary, the dual ∇ ′ and the originalcovariant derivative ∇ coincide. As before, the dual ∇ ′ gives rise to atwisted de Rham complex (Ω ∗ ( M, E ) , ∇ ′ ).2.2. Hilbert complexes.
For any differential operator P acting onΩ ∗ ( M, E ), we denote by P min its minimal graph-closed extension in L ∗ ( M, E ). The maximal closed extension is defined by P max := ( P t min ) ∗ .By Br¨uning and Lesch [10, Lemma 3.1], the extensions define Hilbertcomplexes ( D min , ∇ min ), where D min := D ( ∇ min ), and ( D max , ∇ max ),where D max := D ( ∇ max ). The Laplace operators, associated to theseHilbert complexes are respectively defined as △ rel : = ∇ ∗ min ∇ min + ∇ min ∇ ∗ min , △ abs : = ∇ ∗ max ∇ max + ∇ max ∇ ∗ max . Similar definitions hold for the dual connection ∇ ′ and for the Laplaceoperators △ ′ rel and △ ′ abs of the Hilbert complexes ( D ′ min , ∇ ′ min ) and( D ′ max , ∇ ′ max ) respectively. The difference ( ∇ − ∇ ′ ) is a bounded endo-morphism valued operator and hence the equality of domains D min = D ′ min , D max = D ′ max . (2.2)The following theorem, compare [44, Theorem 3.2], summarizes theclassical de Rham theorem on manifolds with boundary, cf. [38, Re-mark after Proposition 4.2] and [10, Theorem 4.1]; strong ellipticity ofthe corresponding Laplace operators follows from [22, Lemma 1.11.1]. Theorem 2.3.
The Hilbert complexes ( D min , ∇ min ) and ( D max , ∇ max ) are Fredholm and the associated Laplacians △ rel and △ abs are stronglyelliptic. The cohomologies H ∗ ( M, ∂M, E ) and H ∗ ( M, E ) of the Fred-holm complexes ( D min , ∇ min ) and ( D max , ∇ max ) , respectively, can becomputed from the following smooth subcomplexes, (Ω ∗ min ( M, E ) , ∇ ) , Ω ∗ min ( M, E ) := { ω ∈ Ω ∗ ( M, E ) | ι ∗ ( ω ) = 0 } , (Ω ∗ max ( M, E ) , ∇ ) , Ω ∗ max ( M, E ) := Ω ∗ ( M, E ) , respectively, where ι : ∂M ֒ → M denotes the natural inclusion of theboundary. Corresponding statement holds also for the complexes asso-ciated to the dual connection ∇ ′ . The chirality operator.
The Riemannian metric g and a fixedorientation on M define the Hodge star operator ∗ and the chiralityoperator ( r := ( m + 1) / i r ( − k ( k +1)2 ∗ : Ω k ( M, E ) → Ω m − k ( M, E ) . (2.3) EFINED ANALYTIC TORSION ON MANIFOLDS WITH BOUNDARY 5
This operator extends to a self-adjoint involution on L ∗ ( M, E ). Thefollowing properties of Γ are essential for the construction below, cf.[44].
Lemma 2.5.
The self-adjoint involution Γ on L ∗ ( M, E ) maps D ( ∇ min ) to D ( ∇ ′∗ max ) , and D ( ∇ max ) to D ( ∇ ′∗ min ) . With Γ restricted to appropriatedomains, we have Γ ∇ min Γ = ∇ ′∗ max , Γ ∇ max Γ = ∇ ′∗ min . Definition 2.6.
We introduce the doubled Hilbert complexes ( e D , D ) := ( D min , ∇ min ) ⊕ ( D max , ∇ max ) , ( e D ′ , D ′ ) := ( D ′ min , ∇ ′ min ) ⊕ ( D ′ max , ∇ ′ max ) . Similar to (2.2), we have the equality of domains D ( D ) = D ( D ′ ) , D ( D ∗ ) = D ( D ′∗ ) . The self-adjoint involution Γ gives rise to the ”chirality operator” G := (cid:18) (cid:19) on L ∗ ( M, E ) ⊕ L ∗ ( M, E ) . (2.4)An immediate consequence of Lemma 2.5 is the following Proposition 2.7.
The chirality operator G acts as G | D ( D ) : D ( D ) → D ( D ∗ ) , G | D ( D ∗ ) : D ( D ∗ ) → D ( D ) . Moreover we have the relation GD = D ′∗ G . The odd signature operator.
We now apply the concepts of [7]to our new setup and define the odd-signature operator of the Hilbertcomplex ( e D , D ) by B := GD + DG , D ( B ) = D ( D ) ∩ D ( D ∗ ) . (2.5)By [44] the odd signature operator B is strongly elliptic with discretespectrum and an Agmon angle θ ∈ ( − π, Spectral decomposition.
Consider for any λ ≥ B onto eigenspaces with eigenvalues of absolute value inthe interval [0 , λ ]: Π B , [0 ,λ ] := i π Z γ ( λ ) ( B − x ) − dx, with γ ( λ ) being a closed counterclockwise circle around the origin sur-rounding eigenvalues of absolute value in [0 , λ ]. By the analytic Fred-holm theorem, the range of the projection lies in D ( B ) and the pro-jection commutes with B . Moreover, Π B , [0 ,λ ] is of finite rank and the MAXIM BRAVERMAN AND BORIS VERTMAN decomposition L ∗ ( M, E ⊕ E ) = Image Π B , [0 ,λ ] ⊕ Image ( − Π B , [0 ,λ ] ) , (2.6)is a direct sum decomposition into closed subspaces of the Hilbert space L ∗ ( M, E ⊕ E ). Note that if α is unitary and hence B is self-adjoint,the projection Π B , [0 ,λ ] is orthogonal. (2.6) induces a decomposition of e D e D = e D [0 ,λ ] ⊕ e D ( λ, ∞ ) . Since D commutes with B , B and hence also with Π B , [0 ,λ ] , we obtaina decomposition of ( e D , D ) into subcomplexes( e D , D ) = ( e D [0 ,λ ] , D [0 ,λ ] ) ⊕ ( e D ( λ, ∞ ) , D ( λ, ∞ ) )where D I := D | e D I for I = [0 , λ ] or ( λ, ∞ ) . (2.7)The chirality operator G commutes with B , B and respects the decom-position (2.7) so that G = G [0 ,λ ] ⊕ G ( λ, ∞ ) , B = B [0 ,λ ] ⊕ B ( λ, ∞ ) . (2.8) Proposition 2.10. [44, Corollary 3.14 and 3.15] . The operator B ( λ, ∞ ) , λ ≥ is bijective. The complex ( e D ( λ, ∞ ) , D ( λ, ∞ ) ) is acyclic and H ∗ ( e D [0 ,λ ] , D [0 ,λ ] ) ∼ = H ∗ ( e D , D ) . The refined torsion element.
Recall the notion of a determi-nant lines of a finite dimensional complex ( C ∗ , ∂ ∗ ) and of its cohomol-ogy. Set Det C ∗ = O k det ( C k ) ( − k , Det H ∗ ( C ∗ , ∂ ∗ ) = O k det H k ( C ∗ , ∂ ∗ ) ( − k , where for a vector space V we denote by det V its top exterior powerand the ( −
1) upper index denotes the dual vector space. We follow [5,Section 1.1] and define the canonical isomorphism φ : Det C ∗ → Det H ∗ ( C ∗ , ∂ ∗ )and the refined torsion element of the complex ( e D [0 ,λ ] , D [0 ,λ ] ) ρ [0 ,λ ] := φ (cid:0) c ⊗ ( c ) − ⊗ · · · ⊗ ( c r ) ( − r ⊗ ( G [0 ,λ ] c r ) ( − r +1 ⊗ · · ·· · · ⊗ ( G [0 ,λ ] c ) ⊗ ( G [0 ,λ ] c ) ( − (cid:1) ∈ Det( H ∗ ( e D [0 ,λ ] , D [0 ,λ ] )) , (2.9)where c k ∈ e D [0 ,λ ] are arbitrary elements of the determinant lines, wedenote the extension of G [0 ,λ ] to a mapping on determinant lines by thesame letter, and for any v ∈ det e D [0 ,λ ] the dual v − ∈ det( e D [0 ,λ ] ) − ≡ det( e D [0 ,λ ] ) ∗ is the unique element such that v − ( v ) = 1. EFINED ANALYTIC TORSION ON MANIFOLDS WITH BOUNDARY 7
By Proposition 2.10 we can view ρ [0 ,λ ] canonically as an element ofDet( H ∗ ( e D , D )), which we do henceforth.2.12. The graded determinant.
The fundamental part of the con-struction is the graded determinant. The operator B ( λ, ∞ ) , λ ≥ I = ( λ, ∞ )to simplify the notation)ker( D I G I ) ∩ ker( G I D I ) = { } . (2.10)Moreover the complex ( e D I , D I ) is acyclic by Proposition 2.10 and dueto G I being an involution on Im(1 − Π B , [0 ,λ ] ) we haveker( D I G I ) = G I ker( D I ) = G I Im( D I ) = Im( G I D I ) , ker( G I D I ) = ker( D I ) = Im( D I ) = Im( D I G I ) . (2.11)We have Im( G I D I ) + Im( D I G I ) = Im( B I ) and by surjectivity of B I we obtain from the last three relations aboveIm(1 − Π B , [0 ,λ ] ) = ker( D I G I ) ⊕ ker( G I D I ) . (2.12)Note that B leaves ker( DG ) and ker( GD ) invariant. Hence, we put B + , ( λ, ∞ )even := B ( λ, ∞ ) ↾ e D even ∩ ker( DG ) , B − , ( λ, ∞ )even := B ( λ, ∞ ) ↾ e D even ∩ ker( GD ) . We arrive at a direct sum decomposition B ( λ, ∞ )even = B + , ( λ, ∞ )even ⊕ B − , ( λ, ∞ )even . By [44], there exists an Agmon angle θ ∈ ( − π,
0) for B , which is clearlyan Agmon angle for the restrictions above, as well. For strongly ellipticboundary value problems ( D, B ) of order ω on M with an Agmon angle θ ∈ ( − π, ζ θ ( s, D B ) := X λ ∈ Spec( D B ) \{ } m ( λ ) · λ − sθ , Re( s ) > dim Mω , where λ − sθ := exp( − s · log θ λ ) and m ( λ ) denotes the multiplicity of theeigenvalue λ . The zeta function is holomorphic for Re( s ) > dim M/ω and admits a meromorphic extension to C with s = 0 being a regularpoint. Consequently, the graded zeta-function ζ gr,θ ( s, B ( λ, ∞ )even ) := ζ θ ( s, B + , ( λ, ∞ )even ) − ζ θ ( s, −B − , ( λ, ∞ )even ) , Re ( s ) ≫ , is regular at s = 0 and we may introduce the following MAXIM BRAVERMAN AND BORIS VERTMAN
Definition 2.13.
Let θ ∈ ( − π, be an Agmon angle for B ( λ, ∞ ) . Thenthe graded determinant associated to B ( λ, ∞ ) and its Agmon angle θ isdefined as follows: Det ′ gr,θ ( B ( λ, ∞ )even ) := exp (cid:18) − dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ζ gr,θ (cid:0) s, B ( λ, ∞ )even (cid:1)(cid:19) . Refined analytic torsion.Proposition 2.15. [7, 44] . The element ρ ( ∇ , g ) := Det ′ gr,θ ( B ( λ, ∞ )even ) · ρ [0 ,λ ] ∈ Det( H ∗ ( e D , D )) is independent of the choice of λ ≥ and choice of Agmon angle θ ∈ ( − π, for the odd-signature operator B ( λ, ∞ ) . The construction of ρ ( ∇ , g ) is in fact independent of the choice of aHermitian metric h E . Indeed, a variation of h E does not change theodd-signature operator B as a differential operator and different Her-mitian metrics give rise to equivalent L − norms over compact mani-folds. Hence D ( B ) is indeed independent of the particular choice of h E .Independence of the choice of a Hermitian metric h E is essential, sincefor non-unitary flat vector bundles there is no canonical choice of h E and a Hermitian metric is fixed arbitrarily.The refined analytic torsion is then obtained by studying the depen-dence of ρ ( ∇ , g ) on the Riemannian metric. We cite the final resultfrom [44]. Theorem 2.16.
Let ( M, g ) be an odd-dimensional oriented compactRiemannian manifold with boundary. Let ( E, ∇ , h E ) be a flat complexvector bundle over M . Consider the trivial vector bundle M × C witha trivial connection d and let B := B ( d ) denote the associated odd-signature operator. η ( B ) denotes the eta invariant of the even part B even . Put b ξ ( d, g ) := 12 m X k =0 ( − k · k · ζ θ ( s = 0 , B ↾ e D k ) . Then the refined analytic torsion of ( M, E, ∇ ) ρ an ( ∇ ) := ρ ( ∇ , g ) · exp h iπ rk( E )( η ( B ) + b ξ ( d, g )) i (2.13) is modulo sign independent of the choice of g in the interior of M . EFINED ANALYTIC TORSION ON MANIFOLDS WITH BOUNDARY 9 Holomorphic structure on the determinant linebundle
In the next step we interpret ρ ( α ) as an analytic section of the de-terminant line bundle over the representation space. This requires aseparate discussion of the analyticity for the refined torsion elementand the graded determinant. The present section studies analyticityof the refined torsion element, while the next deals with analyticity ofthe graded determinant.3.1. The determinant line bundle.
The space R := Rep( π ( M ) , C n )of complex n -dimensional representations of π = π ( M ) has a naturalstructure of a complex analytic space, cf., for example, [7, § α ∈ Rep( π ( M ) , C n ) we denote by E α the flat vector bundle over M whose monodromy is equal to α . Then the disjoint union D et := G α ∈ R Det (cid:0) H • ( M, E α ) (cid:1) ⊗ Det (cid:0) H • ( M, ∂M, E α ) (cid:1) (3.1)has a natural structure of a holomorphic line bundle over R , calledthe determinant line bundle . In this section we describe this structure,using a CW-decomposition of M . Then, we show that the refinedanalytic torsion is a nowhere vanishing holomorphic section of D et .We continue in the notation fixed in § The combinatorial cochain complex.
Fix a CW-decomposition K = { e , . . . , e N } of M . Let e K denote the universal cover of K . Thenthe fundamental group π ( M ) acts on C • ( e K, C ) from the right and C n is a left module over the group ring C [ π ] via the representation α .Then the cochain complex C • ( K, α ) is defined as C • ( K, α ) := C • ( e K, C ) ⊗ C [ π ] C n . (3.2)For each cell e j , fix a lift e e j , a cell of the CW-decomposition of f M ,such that π ( e e j ) = e j . By definition, the pull-back of the bundle E α to f M is the trivial bundle f M × C n → f M . Hence, the choice of the cells e e , . . . , e e N identifies the cochain complex C • ( K, α ) of the CW-complex K with coefficients in E α with the complex0 → C n · k ∂ ( α ) −−−→ C n · k ∂ ( α ) −−−→ · · · ∂ m − ( α ) −−−−−→ C n · k m → , (3.3)where k j ∈ Z ≥ ( j = 0 , . . . , m = dim M ) is equal to the number of j -dimensional cells of K and the differentials ∂ j ( α ) are ( nk j × nk j − )-matrices depending analytically on α ∈ Rep( π ( M ) , C n ). The cohomology of the complex (3.3) is canonically isomorphic to H • ( M, E α ). Let φ C • ( K,α ) : Det (cid:0) C • ( K, α ) (cid:1) −→ Det (cid:0) H • ( M, E α ) (cid:1) (3.4)denote the canonical isomorphism, cf. formula (2.13) of [5] .3.3. A non-zero element of
Det (cid:0) H • ( M, E α ) (cid:1) . The standard basesof C n · k j ( j = 0 , . . . , m ) define an element c ∈ Det (cid:0) C • ( K, α ) (cid:1) , and,hence, an isomorphism ψ α : C −→ Det (cid:0) C • ( K, α ) (cid:1) with ψ α ( z ) = z · c .Then for each α ∈ Rep( π ( M ) , C n ) we define σ ( α ) = φ C • ( K,α ) (cid:0) ψ α (1) (cid:1) ∈ Det (cid:0) H • ( M, E α ) (cid:1) , (3.5)a non-zero element of Det (cid:0) H • ( M, E α ) (cid:1) . Of course, this element de-pends on the choice of the lifts e e , . . . , e e N .3.4. A non-zero element of
Det (cid:0) H • ( M, ∂M, E α ) (cid:1) . Let now K ′ de-note the CW-decomposition of ∂M induced by K . Then K ′ ⊂ K and the choice of the lifts e e i made above identify the cochain com-plex C • ( K ′ , α ) of the CW-complex K ′ with coefficients in E α with thecomplex0 → C n · l ∂ ( α ) −−−→ C n · l ∂ ( α ) −−−→ · · · ∂ m − ( α ) −−−−−→ C n · l m − → , (3.6)As above, the standard bases of C n · l j ( j = 1 , . . . , m −
1) defines acanonical element of Det( C • ( K ′ , α )) and an isomorphism ψ ′ α from C onto Det (cid:0) C • ( K, α ) (cid:1) . Thus we define σ ′ ( α ) = φ C • ( K ′ ,α ) (cid:0) ψ ′ α (1) (cid:1) ∈ Det (cid:0) H • ( ∂M, E α ) (cid:1) , (3.7)where α ′ is the restriction of the representation α to π ( ∂M ) and E ′ α is the restriction of E α to ∂M . Consider the quotient complex C • ( K, K ′ , α ) := C • ( K, α ) /C • ( K ′ , α ) . Using (3.3) and (3.6) we can identify Det (cid:0) C • ( K, K ′ , α ) (cid:1) with C thusconstructing a map ψ ′′ α : C −→ Det (cid:0) C • ( K, K ′ , α ) (cid:1) . (3.8)The cohomology of the complex C • ( K, K ′ , α ) is canonically isomor-phic to the relative cohomology H • ( M, ∂M, E α ). For each α we definea non-zero element of Det (cid:0) H • ( M, ∂M, E α ) (cid:1) by the formula σ ′′ ( α ) = φ C • ( K,K ′ ,α ) (cid:0) ψ ′′ α (1) (cid:1) ∈ Det (cid:0) H • ( M, ∂M, E α ) (cid:1) , (3.9) EFINED ANALYTIC TORSION ON MANIFOLDS WITH BOUNDARY 11
The holomorphic structure on D et . Recall that the elements σ ( α ) and σ ′′ ( α ) are defined in (3.5) and (3.9). Consider the map τ : α σ ( α ) ⊗ σ ′′ ( α ) ∈ Det (cid:0) H • ( M, E α ) (cid:1) ⊗ Det (cid:0) H • ( M, ∂M, E α ) (cid:1) , (3.10)where α ∈ Rep( π ( M ) , C n ), is a nowhere vanishing section of the de-terminant line bundle D et over Rep( π ( M ) , C n ). Definition 3.6.
We say that a section s ( α ) of D et is holomorphic ifthere exists a holomorphic function f ( α ) on Rep( π ( M ) , C n ) , such that s ( α ) = f ( α ) · τ ( α ) . This defines a holomorphic structure on D et , which is independentof the choice of the lifts e e , . . . , e e N of e , . . . , e N , since for a differentchoice of lifts the section τ ( α ) will be multiplied by a constant. Inthe next subsection we show that this holomorphic structure is alsoindependent of the CW-decomposition K of M .3.7. The Farber-Turaev torsion.
The choice of the lifts e e , . . . , e e N of e , . . . , e N determines an Euler structure ε on M , while the order-ing of the cells e , . . . , e N determines a cohomological orientation o ,cf. [43, § ρ ε, o ( α ), corresponding to the pair ( ε, o ), is, by definition, [21, § σ ( α ) defined in (3.5). Since the Farber-Turaev tor-sion is independent of the choice of the CW-decomposition of M , cf.[42, 21], we conclude that the element σ ( α ) is also independent of theCW-decomposition, but only depends on the Euler structure and thecohomological orientation.The lifts e e j and the ordering of the cells also defines an Euler struc-ture ε ′ and a cohomological orientation o ′ of ∂M . The element σ ′ ( α )defined in (3.7) is equal to the Farber-Turaev torsion ρ ε ′ , o ′ ( α ′ ) where α ′ is the restriction of the representation α to π ( ∂M ). Let µ : Det (cid:0) C • ( K ′ , α ) (cid:1) ⊗ Det (cid:0) C • ( K, K ′ , α ) (cid:1) −→ Det (cid:0) C • ( K, α ) (cid:1) (3.11)denote the fusion isomorphism, cf. [5, § ν : Det (cid:0) H • ( ∂M,E α ) (cid:1) ⊗ Det (cid:0) H • ( M, ∂M, E α ) (cid:1) −→ Det (cid:0) H • ( M, E α ) (cid:1) ,ν := φ C • ( K,α ) ◦ µ ◦ (cid:0) φ − C • ( K ′ ,α ) ⊗ φ − C • ( K,K ′ ,α ) (cid:1) . (3.12) It follows from the construction that the elements σ ( α ), σ ′ ( α ) and σ ′′ ( α ) defined in (3.11), (3.5), (3.7), and (3.9) satisfy the equality ν (cid:0) σ ′ ( α ) ⊗ σ ′′ ( α ) (cid:1) = ± σ ( α ) , (3.13)where the sign depends only on the dimensions of the spaces C • ( M, α )and C • ( ∂M, α ) but not on the representation α . Since σ ( α ) and σ ′ ( α )are independent of the CW-decomposition, it follows that σ ′′ ( α ) is in-dependent of CW-decompositon up to a sign. In fact σ ′′ ( α ) can beconsidered as the definition of the relative Farber-Turaev torsion. Itfollows that the section τ ( α ) defined in (3.10) is also an independentof the CW-decomposition up to a sign. Hence so is the holomorphicstructure defined in Definition 3.6.3.8. The acyclic case.
Let Rep ( π ( M ) , C n ) ⊂ Rep( π ( M ) , C n ) de-note the set of representations such that H • ( M, E α ) = 0, H • ( ∂M, E α ′ ) =0, and H • ( M, ∂M, E α ) = 0. Then the determinant lines Det( H • ( M, E α )),Det( H • ( M, ∂M, E α ′ )), and Det( H • ( M, ∂M.E α )) are canonically iso-morphic to C . Hence, the Farber-Turaev torsions ρ ε, o ( α ) and ρ ε ′ , o ′ ( α )can be viewed as a complex-valued functions on Rep ( π ( M ) , C n ). Itis easy to see, cf. [14, Theorem 4.3], that these functions are holomor-phic on Rep ( π ( M ) , C n ). Moreover, they are rational functions onRep( π ( M ) , C n ), all whose poles are inRep( π ( M ) , C n ) \ Rep ( π ( M ) , C n ) . In particular, the holomorphic structure on D et , which we definedabove, coincides, when restricted to Rep ( π ( M ) , C n ), with the nat-ural holomorphic structure obtained from the canonical isomorphism D et | Rep ( π ( M ) , C n ) ≃ Rep ( π ( M ) , C n ) × C . The graded determinant as a holomorphic function
Fix α ∈ Rep( π ( M ) , C n ). Fix a number λ ≥ B α of the odd signature operator B α = B ( ∇ α , g ). Then there is a neighborhood U λ ⊂ Rep( π ( M ) , C n ) of α such that λ is not an eigenvalue of B α for all α ∈ U λ . Denote by B ( λ, ∞ ) α the restriction of B α to the spectral subspace of B α corresponding tothe spectral set ( λ, ∞ ). Then B ( λ, ∞ ) α is an invertible operator. Let θ ∈ ( − π/ ,
0) be an Agmon angle for B (0 ,λ ) α and assume that there areno eigenvalues of B ( λ, ∞ ) α in the solid angles L ( − π/ ,λ ] and L ( π/ ,λ + π/ .Then there exists a neighborhood U λ,θ ⊂ U λ of α such that θ is alsoan Agmon angle for B ( λ, ∞ ) α for all α ∈ U λ,θ . In this section we prove EFINED ANALYTIC TORSION ON MANIFOLDS WITH BOUNDARY 13 that the graded determinant Det ′ gr ,θ ( B ( λ, ∞ ) α, even ) is a holomorphic functionon U λ,θ . Our main result in this section is the following Theorem 4.1.
Let
O ⊂ C be a connected open neighborhood of . Let γ : O → U λ,θ ⊂ Rep( π ( M ) , C n ) be a holomorphic curve such that γ (0) = α . Then the function z Det ′ gr ,θ (cid:0) B ( λ, ∞ ) γ ( z ) , even (cid:1) (4.1) is holomorphic in a neighborhood of . An immediate consequence is the following
Corollary 4.2.
Suppose V ⊂ U λ,θ is an open subset such that all points α ∈ V are regular points of the complex algebraic set Rep( π ( M ) , C n ) .Then the map Det : V −→ C , Det : α Det( α ) := Det ′ gr ,θ ( B ( λ, ∞ ) α, even ) . is holomorphic.Proof. By Hartogs’ theorem (cf., for example, [27, Th. 2.2.8]), a func-tion on a smooth algebraic variety is holomorphic if its restriction toeach holomorphic curve is holomorphic. Hence, the corollary followsimmediately from Theorem 4.1. (cid:3)
The rest of this section is occupied with the proof of Theorem 4.1.4.3.
A germ of connections.
Let us introduce some additional nota-tions. Let E be a vector bundle over M and let C ( E ) denote the affinespace of (not necessarily flat) connections on E . We endow C ( E ) withthe the Fr´echet topology on C ( E ) introduced in Section 13.1 of [7] .Fix a base point x ∗ ∈ M and let E x ∗ denote the fiber of E over x ∗ .We will identify E x ∗ with C n and π ( M, x ∗ ) with π ( M ).For ∇ ∈ C ( E ) and a closed path φ : [0 , → M with φ (0) = φ (1) = x ∗ , we denote by Mon ∇ ( φ ) ∈ End E x ∗ ≃ Mat n × n ( C ) the monodromyof ∇ along φ . Note that, if ∇ is flat then Mon ∇ ( φ ) depends only onthe class [ φ ] of φ in π ( M ). Hence, if ∇ is flat, then the map φ Mon ∇ ( φ ) defines an element of Rep( π ( M ) , C n ), called the monodromyrepresentation of ∇ .Suppose now that O ⊂ C is a connected open neighborhood of 0.For simplicity we also assume that O is convex. Let γ : O →
Rep( π ( M ) , C n )be a holomorphic curve with γ (0) = α . The operator B γ ( z ) is con-structed using a flat connection ∇ γ ( z ) whose monodromy is equal to γ ( z ). Unfortunately, there is no a canonical choice of such connection.Though the graded determinant Det ′ gr ,θ (cid:0) B ( λ, ∞ ) γ ( z ) , even (cid:1) is independent ofthis choice, to study the dependence of this determinant on z ∈ O weneed to choose a family of connections ∇ γ ( z ) . The main difficulty inthe proof of Theorem 4.1 is that it is not clear whether there exists a holomorphic family ∇ γ ( z ) with Mon ∇ γ ( z ) = γ ( z ). We shall now explainhow to circumvent this difficulty.By Proposition 4.5 of [23], all the bundles E γ ( z ) , z ∈ O , are isomor-phic to each other. Moreover, we have the following lemma. Lemma 4.4.
There exists a vector bundle E → M and a real differ-entiable family of flat connections ∇ γ ( z ) , z ∈ O , on E , such that themonodromy representation of ∇ γ ( z ) is equal to γ ( z ) for all z ∈ O .Proof. By Lemma 3.3 of [1] there exists a smooth vector bundle˜ E → M × O and a smooth connection ˜ ∇ on ˜ E , whose restriction ˜ ∇ z to˜ E (cid:12)(cid:12) M ×{ z } → M × { z } is flat for each z ∈ O , and such that the monodromy of ˜ ∇ z is equal to γ ( z ).Set E := ˜ E (cid:12)(cid:12) M ×{ } and letΦ z : E → ˜ E (cid:12)(cid:12) M ×{ z } denote the parallel transport along the intervals (cid:8) ( m, tz ) : m ∈ M, t ∈ [0 , (cid:9) ⊂ M × O . Then ∇ γ ( z ) := Φ − z ◦ ˜ ∇ z ◦ Φ z , z ∈ O , is a smooth family of connections on E , and the monodromy of ∇ γ ( z ) is equal to γ ( z ). (cid:3) Furthermore, Lemma B.6 of [7] shows that the family ∇ γ ( z ) can bechosen so that there exists a one-form ω ∈ Ω ( M, End E ) such that ∇ γ ( z ) = ∇ α + z · ω + o ( z ) , (4.2)where o ( z ) is understood in the sense of the Fr´echet topology on C ( E )introduced in Section 13.1 of [7], and always refers to the behavior as z → EFINED ANALYTIC TORSION ON MANIFOLDS WITH BOUNDARY 15
Remark . Lemma 4.4 asserts that the family ∇ γ ( z ) can be chosen tobe real differentiable at every point z ∈ O . We don’t know whether itcan be chosen to be complex differentiable on the whole set O . How-ever, the equation (4.2) implies that it can be chosen to be complexdifferentiable at 0.4.6. The family of projections.
Let Flat( E ) ⊂ C ( E ) denote the setof flat connections on E and consider a curve ∇ ( z ) ∈ Flat( E ), z ∈ O ,of connections such that Mon ∇ (0) = α . We will assume that it iscomplex differentiable at 0 in the sense that ∇ ( z ) = ∇ α + z · ω + o ( z ) , z → , (4.3)where ω ∈ Ω ( M, End E ). We denote by D ( z ) := ∇ ( z ) min ⊕ ∇ ( z ) max and by B ( z ) the corresponding odd signature operator the sense of § P ( z ) = Π B ( z ) , ( λ, ∞ ) denote the spectral projection of the operator B ( z ) onto the subspace e D ( λ, ∞ ) spanned by eigenforms of B ( z ) witheigenvalues in ( λ, ∞ ).From (2.12) we conclude that the space L ∗ ( M, E ⊕ E ) of L -differentialforms is a direct sum L ∗ ( M, E ⊕ E ) = Im (cid:0) I − P ( z ) (cid:1) ⊕ Im (cid:0) P ( z ) D ( z ) (cid:1) ⊕ Im (cid:0) P ( z ) GD ( z ) (cid:1) . Consider the corresponding orthogonal projections P + ( z ) : L ∗ ( M, E ⊕ E ) −→ Im (cid:0) P ( z ) D ( z ) (cid:1) ,P − ( z ) : L ∗ ( M, E ⊕ E ) −→ Im (cid:0) P ( z ) GD ( z ) (cid:1) . (4.4)Then Im P ( z ) = Im P + ( z ) ⊕ Im P − ( z ) . Lemma 4.7.
There exist bounded operators A ± on L ∗ ( M, E ⊕ E ) with P ± ( z ) = P ± (0) + z (cid:16) P ± (0) A ± (cid:0) Id − P ± (0) (cid:1) + (cid:0) Id − P ± (0) (cid:1) A ± P ± (0) (cid:17) + o ( z ) . (4.5) Proof.
For small enough z ∈ O the operators P ± ( z ) depend smoothlyon z , there exist bounded operators A ± and A such that P ± ( z ) = P ± (0) + z A ± + o ( z ) ,P ( z ) = P (0) + z A + o ( z ) . (4.6) Using the decomposition A = P ± (0) AP ± (0) + (cid:0) Id − P ± (0) (cid:1) AP ± (0)+ P ± (0) A (cid:0) Id − P ± (0) (cid:1) + (cid:0) Id − P ± (0) (cid:1) A (cid:0) Id − P ± (0) (cid:1) (4.7)and the equality P ± ( z ) = P ± ( z ) we obtain P ± ( z ) = P ± ( z ) = P ± (0) + 2 z P ± (0) AP ± (0)+ z (cid:0) Id − P ± (0) (cid:1) AP ± (0) + z P ± (0) A (cid:0) Id − P ± (0) (cid:1) + o ( z ) . (4.8)Comparing (4.6) and (4.8) and using (4.7) we conclude that P ± (0) AP ± (0) = (cid:0) Id − P ± (0) (cid:1) A (cid:0) Id − P ± (0) (cid:1) = 0 . The equality (4.5) follows now from (4.6) and (4.7). (cid:3)
The partial derivatives of the graded determinant.
In termsof the projections P ± ( z ) introduced in the previous section the odd sig-nature operator B ( λ, ∞ ) can be written in the form B ( λ, ∞ ) ( z ) = (cid:0) D ( z ) G + GD ( z ) (cid:1) P ( z )= D ( z ) G P + ( z ) + GD ( z ) P − ( z ) . (4.9)Hence, we may writeDet ′ gr ,θ (cid:0) B ( λ, ∞ )even ( z ) (cid:1) = Det ′ θ (cid:0) D ( z ) G P + ( z ) ↾ L ( M, E ⊕ E ) (cid:1) Det ′ θ (cid:0) − GD ( z ) P − ( z ) ↾ L ( M, E ⊕ E ) (cid:1) Consider a curve κ : ( − , → O such that κ (0) = α . To simplify thenotation, set D + ( z ) := D ( z ) G P + ( z ) ↾ L ( M, E ⊕ E ) ,D − ( z ) := − GD ( z ) P − ( z ) ↾ L ( M, E ⊕ E ) , (4.10)and also D ′ := ddt D ( κ ( t )) (cid:12)(cid:12) t =0 , P ′± := ddt P ± ( κ ( t )) (cid:12)(cid:12) t =0 , D ′± = ddt D ± (cid:0) κ ( t ) (cid:1)(cid:12)(cid:12) t =0 . (4.11)With this notation we haveDet ′ gr ,θ (cid:0) B ( λ, ∞ )even ( z ) (cid:1) = Det ′ θ (cid:0) D + ( z ) (cid:1) Det ′ θ (cid:0) D − ( z ) (cid:1) . (4.12)We consider the function F ( z ) = Det ′ θ (cid:0) D ( z ) G P + ( α ) ↾ L ( M, E ⊕ E ) (cid:1) Det ′ θ (cid:0) − GD ( z ) P − ( α ) ↾ L ( M, E ⊕ E ) (cid:1) . (4.13)Notice that the right hand side of (4.13) is similar to the right handside of (4.12) but P ± ( z ) is replaced by P ± (0) = P ± ( α ). In particular, F (0) = Det ′ gr ,θ (cid:0) B ( λ, ∞ )even (0) (cid:1) . EFINED ANALYTIC TORSION ON MANIFOLDS WITH BOUNDARY 17
Lemma 4.9.
Then for any curve κ : ( − , → O with κ (0) = α wehave ddt (cid:12)(cid:12)(cid:12) t =0 log Det ′ gr ,θ (cid:0) B ( λ, ∞ )even ( κ ( t )) (cid:1) = ddt (cid:12)(cid:12)(cid:12) t =0 log F (cid:0) κ ( t ) (cid:1) . (4.14) Proof.
Using Lemma 4.7 and the equality D ( α ) G P + ( α ) = P + ( α ) D ( α ) G , (4.15)we obtain D ′ + = D ′ G P + ( α ) + P + ( α ) D ( α ) G A + (cid:0) Id − P + ( α ) (cid:1) + (cid:0) Id − P + ( α ) (cid:1) D ( α ) G A + P + ( α ) . (4.16)By the variation formula for the logarithm of the determinant, cf., forexample, Section 3.7 of [13], we have ddt (cid:12)(cid:12)(cid:12) t =0 log Det ′ θ D + ( κ ( t )) = Tr D − s − ( α ) D ′ + (cid:12)(cid:12)(cid:12) s =0 = Tr D − s − ( α ) D ′ G P + (0) (cid:12)(cid:12)(cid:12) s =0 + Tr D − s − ( α ) D α G P ′ + (cid:12)(cid:12)(cid:12) s =0 . (4.17)Using (4.5) and the fact that the operators D + (0) and D α G commutewith P + (0) we conclude thatTr D − s − (0) D α G P ′ + = Tr (cid:0) Id − P + (0) (cid:1) D − s − (0) D α G A + P + (0)+ Tr P + (0) D − s − (0) D α G A + (cid:0) Id − P + (0) (cid:1) = 0 . (4.18)Combining (4.17) and (4.18) we conclude that ddt (cid:12)(cid:12)(cid:12) t =0 log Det ′ θ D + ( κ ( t )) = Tr D − s − ( α ) D ′ G P + (0) (cid:12)(cid:12)(cid:12) s =0 = ddt (cid:12)(cid:12)(cid:12) t =0 Det ′ θ (cid:0) D ( κ ( t )) G P + ( α ) ↾ L ( M, E ⊕ E ) (cid:1) . (4.19)Similarly, ddt (cid:12)(cid:12)(cid:12) t =0 log Det ′ θ D − ( κ ( t ))= ddt (cid:12)(cid:12)(cid:12) t =0 log Det ′ θ (cid:0) − GD ( κ ( t )) P − ( α ) ↾ L ( M, E ⊕ E ) (cid:1) . (4.20)From (4.12), (4.13), (4.19) and (4.20) we obtain (4.14). (cid:3) Proposition 4.10.
Let ∇ ( z ) ( z ∈ O ) be a family of flat connectionssuch that as z → ∇ ( z ) = ∇ α + z · ω + o ( z ) . (4.21) Then the function z f ( z ) := Det ′ gr ,θ (cid:0) B ( λ, ∞ )even ( ∇ γ ( z ) , g ) (cid:1) (4.22) is complex differentiable at zero. In other words, there exists a complexnumber A such that Det ′ gr ,θ (cid:0) B ( λ, ∞ )even ( ∇ γ ( z ) , g ) (cid:1) = Det ′ gr ,θ (cid:0) B ( λ, ∞ )even ( ∇ γ (0) , g ) (cid:1) + A · z + o ( z ) . (4.23) Proof.
By (4.14) it is enough to show that F ( z ) = F (0) + A · z + o ( z ) . (4.24)Set z = x + iy . Using the variation formula for the logarithm of thedeterminant as in the proof of Lemma 4.9, one easily sees that i ∂∂x log F ( z ) = ∂∂y log F ( z ) , which is equivalent to (4.24). (cid:3) Proof of Theorem 4.1.
It follows from Proposition 4.10 thatthe function z Det ′ gr ,θ (cid:0) B ( λ, ∞ )even ( ∇ γ ( z ) , g ) (cid:1) , is complex differentiable at 0.Let a ∈ O be such that γ ( a ) ∈ U λ,θ . By making a change of variables ζ = z − a . Hence, this function is holomorphic in γ − ( U λ,θ ). (cid:3) Refined analytic torsion as a holomorphic section
In this section we show that the refined analytic torsion ρ an is a non-vanishing holomorphic section of D et . More precisely, our main resultis the following5.1. Weakly holomorphic section.
Recall from [24, p. 148] that acontinuous function on a singular space X is called weakly holomor-phic if its restriction to the set of regular points of X is holomorphic.Such functions have many properties of analytic functions on X . Inparticular, they are all meromorphic . We refer to [24], [25, § s ( α ) of D et is weakly holomorphic if thereexists a weakly holomorphic function f ( α ) on Rep( π ( M ) , C n ), suchthat s ( α ) = f ( α ) · τ ( α ). Theorem 5.2.
The refined analytic torsion ρ an is a weakly holomorphicsection of the determinant bundle D et . In particular, the restriction of ρ an to the set Rep ( π ( M ) , C n ) of acyclic representations, viewed as acomplex-valued function via the canonical isomorphism D et | Rep ( π ( M ) , C n ) ≃ Rep ( π ( M ) , C n ) × C , EFINED ANALYTIC TORSION ON MANIFOLDS WITH BOUNDARY 19 is a weakly holomorphic function on
Rep ( π ( M ) , C n ) . Reduction to a finite dimensional complex.
Let α ∈ Rep( π ( M ) , C n ). Fix a Riemannian metric g on M and a number λ ≥ B ( ∇ α , g ) with abso-lute value equal to λ . Let θ be an Agmon angle for B ( ∇ α , g ) andlet U λ,α ⊂ Rep( π ( M ) , C n ) be as in Section 4. By Corollary 4.2 thefunction α Det gr ,θ ( B ( λ, ∞ )even ( ∇ α , g )) is weakly holomorphic on U λ,θ . Itfollows now from the definition of the refined analytic torsion that toprove Theorem 5.2 it is enough to show that α ρ [0 ,λ ] ≡ ρ G [0 ,λ ] ( ∇ α , g )is a weakly holomorphic section of Rep( π ( M ) , C n ). By the definitionof the holomorphic structure on the bundle D et , cf. Definition 3.6, thismeans that the function α ρ G [0 ,λ ] ( ∇ α , g ) τ ( α ) , is continuous at α and is holomorphic at α if α is a regular point ofRep( π ( M ) , C n ). Here, τ ( α ) is defined in (3.10).If α is a regular point of Rep( π ( M ) , C n ), then by Hartog’s theorem,[27, Th. 2.2.8], it is enough to show that for every holomorphic curve γ : O → U λ,θ , where O is a connected open neighborhood of 0 in C ,the function f ( z ) := ρ G [0 ,λ ] ( ∇ γ ( z ) , g ) τ ( γ ( z )) (5.1)is complex differentiable at 0, i.e., there exists a ∈ C , such that as z → f ( z ) = f (0) + z · a + o ( z ) . Choice of a basis.
We use the notation introduced in Subsec-tion 4.3. In particular we have a vector bundle E , a holomorphic curve γ : O →
Rep( π ( M ) , C n ), and a continuous family of flat connection ∇ γ ( z ) ( z ∈ O ) on E such that for each z ∈ O the monodromy of ∇ γ ( z ) is equal to γ ( z ). If α is a regular point of Rep( π ( M ) , C n ) then wealso assume that ∇ γ ( z ) = ∇ γ (0) + z · ω + o ( z ) . (5.2)where o ( z ) is understood in the sense of the Fr´echet topology.Let Π [0 ,λ ] ( z ) ( z ∈ O ) denote the spectral projection of the operator B ( ∇ γ ( z ) , g ) , corresponding to the set of eigenvalues of B ( ∇ γ ( z ) , g ) ,whose absolute value is ≤ λ . Then it follows from the definition of U λ that Π [0 ,λ ] ( z ) depends continuously on z . Moreover, in case when α is a regular point of Rep( π ( M ) , C n ), Π [0 ,λ ] ( z ) is complex differentiable in z .Hence, in this case there exists a bounded operator R on L ∗ ( M, E ⊕ E )such that Π [0 ,λ ] ( z ) = Π [0 ,λ ] (0) + z R + o ( z ) . (5.3)We denote by Ω • ( z ) the image of Π [0 ,λ ] ( z ). Recall that we denote thedimension of M by m = 2 r −
1. For each j = 0 , . . . , r −
1, fix a basis w j = { w j , . . . , w l j j } of Ω j (0) and set w m − j := { G w j , . . . , G w l j j } . To simplify the notationwe will write w m − j = G w j . Then w j is a basis for Ω j (0) for all j =0 , . . . , m .For each z ∈ O , j = 0 , . . . , m , set w j ( z ) = (cid:8) w j ( z ) , . . . , w l j j ( z ) (cid:9) := (cid:8) Π [0 ,λ ] ( z ) w j , . . . , Π [0 ,λ ] ( z ) w l j j (cid:9) . Since Π [0 ,λ ] ( z ) depends continuously on z , there exists a neighborhood O ′ ⊂ O of 0, such that w j ( z ) is a basis of Ω j ( z ) for all z ∈ O ′ , j = 0 , . . . , m . Further, since Π [0 ,λ ] ( z ) commutes with G , we obtain w m − j ( z ) = G w j ( z ) . (5.4)Clearly, w j (0) = w j for all j = 0 , . . . , m .For each z ∈ O ′ , the space Ω • ( z ) is a subcomplex of (cid:0) L ∗ ( M, E ⊕ E ) , D γ ( z ) (cid:1) . Moreover, the embedding Ω • ( z ) ֒ → L ∗ ( M, E ⊕ E ) is aquasi-isomorphism. It follows from Theorem 2.3 that the cohomologyof this compels is canonically isomorphic to H • ( M, E γ ( z ) ) ⊕ H • ( M, ∂M, E γ ( z ) ) . Let φ Ω • ( z ) : Det (cid:0) Ω • ( z ) (cid:1) −→ Det (cid:0) H • ( M, E γ ( z ) ) ⊕ H • ( M, ∂M, E γ ( z ) ) (cid:1) denote the canonical isomorphism, cf. Section 2.4 of [5] . For z ∈O ′ , let w ( z ) ∈ Det (cid:0) Ω • ( z ) (cid:1) be the element determined by the basis w ( z ) , . . . , w m ( z ) of Ω • ( z ). More precisely, we introduce w j ( z ) = w j ( z ) ∧ · · · ∧ w l j j ( z ) ∈ Det (cid:0) Ω j ( z ) (cid:1) , and set w ( z ) := w ( z ) ⊗ w ( z ) − ⊗ · · · ⊗ w m ( z ) − . Then, according to Definition 4.3 of [5], it follows from (5.4) that, for all z ∈ O ′ , the refined torsion of the complex Ω • ( z ) is equal to φ Ω • ( z ) ( w ( z )),i.e., ρ G [0 ,λ ] ( ∇ γ ( z ) ) = φ Ω • ( z ) ( w ( z )) . (5.5) EFINED ANALYTIC TORSION ON MANIFOLDS WITH BOUNDARY 21
Reduction to a family of differentials.
Using the basis w j ( z )we define the isomorphism ψ j ( z ) : C l j −→ Ω j [0 ,λ ] ( z )by the formula ψ j ( z )( x , . . . , x l j ) := l j X k =1 x k w kj ( z ) = l j X k =1 x k Π [0 ,λ ] ( z ) w kj . (5.6)We conclude that for each z ∈ O ′ , the complex (cid:0) Ω • ( z ) , D γ ( z ) (cid:1) is iso-morphic to the complex( W • , d ( z )) : 0 → C l d ( z ) −−−→ C l d ( z ) −−−→ · · · d l − ( z ) −−−−→ C l m → , (5.7)where d j ( z ) := ψ j +1 ( z ) − ◦ D γ ( z ) ◦ ψ j ( z ) , j = 0 , . . . , m. (5.8)It follows from (5.3) and (5.6) that d j ( z ) is continuous family of differ-entials. Moreover, when α is a regular point of Rep( π ( M ) , C n ) it iscomplex differentiable at 0, i.e., there exists a ( l j +1 × l j )-matrix A suchthat d j ( z ) = d j (0) + z A + o ( z ) . Let ψ ( z ) := L dj =0 ψ j ( z ). Since G (cid:0) Ω j ( z ) (cid:1) = Ω m − j ( z ) ( j = 0 , . . . , m ),we conclude that l j = l m − j . From (5.4) we obtain that b Γ := ψ − ( z ) ◦ G ◦ ψ ( z ) (5.9)is independent of z ∈ O ′ and b Γ : ( x , . . . , x l j ) ( x , . . . , x l j ) , j = 0 , . . . , m. (5.10)It follows from (5.8) and (5.9) that ρ b Γ ( z ) = ρ G [0 ,λ ] ( ∇ γ ( z ) ) , (5.11)where ρ b Γ ( z ) denotes the refined torsion of the finite dimensional com-plex ( W • , d ( z )) corresponding to the chirality operator b Γ.Let φ W • ( z ) : Det( W • ) → Det (cid:0) H • ( d ( z )) (cid:1) denote the denote thecanonical isomorphism of Section 2.4 of [5] . The standard bases of C l j ( j = 0 , . . . , m ) define an element ˜ w ∈ Det( W • ). From (5.10) and thedefinition of ρ G ( z ) we conclude that ρ G ( z ) = φ W • ( z )( ˜ w ) . (5.12) The acyclic case.
To illustrate the main idea of the proof letus first consider the case, when both H • ( M, E α ) and H • ( M, ∂M, E α )are trivial. Then there exists a neighborhood O ′′ ⊂ O ′ of 0 such that H • ( M, E γ ( z ) ) = H • ( M, ∂M, E γ ( z ) ) = 0 for all z ∈ O ′′ . Thus the torsion(5.12) is a complex valued function on O ′′ . To finish the proof of Theo-rem 5.2 in this case it remains to show that this function is continuousand, in case when α is a regular point of Rep( π ( M ) , C n ), is complexdifferentiable at 0. In view of (5.8), this follows from the following Lemma 5.7.
Let ( C • , ∂ ( z )) : 0 → C n · k ∂ ( z ) −−−→ C n · k ∂ ( z ) −−−→ · · ·· · · ∂ m − ( z ) −−−−−→ C n · k m → , be a family of acyclic complexes defined for all z in an open set O ⊂ C .For any c ∈ Det( C • ) the function z φ ( C • ,∂ ( z )) ( c ) is continuous if thedifferentials ∂ j ( z ) are continuous, and is complex differentiable at if ∂ j ( z ) are complex differentiable at .Proof. It is enough to prove the lemma for one particular choice of c .To make such a choice let us fix for each j = 0 , . . . , m a complementof Im( ∂ j − (0)) in C j and a basis v j , . . . , v l j j of this complement. Sincethe complex C • is acyclic, for all j = 0 , . . . , m , the vectors ∂ j − (0) v j − , . . . , ∂ j − (0) v l j − j − , v j , . . . , v l j j (5.13)form a basis of C j . Let c ∈ Det( C • ) be the element defined by thesebases. Then, for all z close enough to 0 and for all j = 0 , . . . , m , ∂ j − ( z ) v j − , . . . , ∂ j − ( z ) v l j − j − , v j , . . . , v l j j (5.14)is also a basis of C j . Let A j ( z ) ( j = 0 , . . . , m ) denote the non-degenerate matrix transforming the basis (5.14) to the basis (5.13).Then, by the definition of the isomorphism φ ( C • ,∂ ( z )) , cf. § φ ( C • ,∂ ( z )) ( c ) = ( − N ( C • ) m Y j =0 Det( A ( z )) ( − j , (5.15)where N ( C • ) is the integer defined in formula (2.15) of [5] which isindependent of z . Clearly, the matrix valued functions A j ( z ) and,hence, their determinants are continuous if the differentials ∂ j ( z ) arecontinuous, and are complex differentiable at 0 if ∂ j ( z ) are complexdifferentiable at 0. Thus, so is the function z φ ( C • ,∂ ( z )) ( c ). (cid:3) EFINED ANALYTIC TORSION ON MANIFOLDS WITH BOUNDARY 23
Sketch of the proof of Theorem 5.2 in the non-acyclic case.
We now turn to the proof of Theorem 5.2 in the general case. In thissubsection we sketch the main ideas of the proof. It is enough to showthat the function f ( z ) := ρ G [0 ,λ ] ( ∇ γ ( z ) , g ) τ ( γ ( z )) (5.16)continuous and, if α is a regular point of Rep( π ( M ) , C n ), is complexdifferentiable at 0. Here τ is the map (3.10). To see this we considerthe de Rham integration maps J max z : Ω • max ( M, E γ ( z ) ) −→ C • ( K, γ ( z )) ,J min z : Ω • min ( M, E γ ( z ) ) −→ C • ( K, K ′ , γ ( z )) . (5.17)where the cochain complexes C • ( K, γ ( z )) and C • ( K, K ′ , γ ( z )) are de-fined in § E -valued differential forms is de-fined using a trivialization of E over each cell e j , and, hence, it dependson the flat connection ∇ γ ( z ) , cf. below. More precisely, in the neigh-borhood of any cell e of K , a differential form f ∈ Ω • ( M, E γ ( z ) ) canbe written in the form f = P nj =1 f j ⊗ v j , where f j ∈ Ω • ( M ) is acomplex-valued differential form and v j is a ∇ γ ( z ) -flat section of E γ ( z ) for j = 1 , .., n . The de Rham integration map is then defined by J max z f ( e ) := n X j =1 (cid:18)Z e f j (cid:19) v j . If f ∈ Ω • min ( M, E γ ( z ) ) then J max z f is a well-defined element of the rela-tive cochain complex C • ( K, K ′ , γ ( z )) and we denote the correspondingmap by J min z . Both maps descend to isomorphisms on cohomology [38, § • d ( M, E α ) := Ω • min ( M, E α ) ⊕ Ω • max ( M, E α ) ,C • d ( K, α ) := C • ( K, K ′ , α ) ⊕ C • ( K, α ) . (5.18)Hence we obtain a quasi-isomorphism J z := J max z ⊕ J min z : Ω • d ( M, E γ ( z ) ) → C • d ( K, γ ( z )) . (5.19)The trivialization T e ( z ) : E | e → C n × e induced by the connection ∇ γ ( z ) is continuous. Moreover, if α is a regular point of Rep( π ( M ) , C n ),it is complex differentiable at z = 0 by (4.3). Hence so is the de Rhamintegration map J z .We then consider the restriction J z | Ω • ( z ) of J z to the finite dimen-sional complex Ω • ( z ) and study the cone complex Cone • ( J z | Ω • ( z ) ) of themap J z . This is a finite dimensional acyclic complex with a fixed basis,obtained from the bases of Ω • ( z ), defined in § C • d ( K, γ ( z )). The torsion of this complex is equal to f ( z ). An application of Lemma 5.7to this complex proves Theorem 5.2.In the definition of the integration map J z we have to take intoaccount the fact that the vector bundles E γ ( z ) and E = E γ (0) are iso-morphic but not equal. The integration map J z , cf. Subsection 5.9, isa map from Ω • d ( M, E γ ( z ) ) to the cochain complex C • d ( K, γ ( z )), whichis not equal to the complex C • d ( K, γ (0)). Fix an Euler structure on M . It defines an isomorphism between the complexes C • d ( K, γ (0)) and C • d ( K, γ ( z )) which depends on z . The study of this isomorphism, whichis conducted in Subsection 5.10, is important for the understanding ofthe properties of J z . In particular, it is used to show that in a certainsense J z is complex differentiable at 0, which implies that the conecomplex Cone • ( J z ) satisfies the conditions of Lemma 5.7.5.9. The cochain complex of the bundle E . Fix a CW-decompo-sition K = { e , . . . , e N } of M . For each j = 1 , . . . , N choose a point x j ∈ e j and let E x j denote the fiber of E over x j . Then the cochaincomplex ( C • d ( K, γ ( z )) , ∂ • ) may be naturally identified with the complex( C • d ( K, E ) , ∂ ′• ( z )) where the z -dependence is now fully encoded in thedifferentials. We use the prime in the notation of the differentials ∂ ′ j inorder to distinguish them from the differentials of the cochain complex C • d ( K, γ ( z )). The differentials ∂ ′ j ( z ) are continuous. Moreover, if α is a regular point if Rep( π ( M ) , C n ) then it follows from (5.2) that ∂ ′ j ( z ) are complex differentiable at 0, i.e., there exist linear maps a j : C jd ( K, E ) −→ C j +1 d ( K, E ), s.t. ∂ ′ j ( z ) = ∂ ′ j (0) + z · a j + o ( z ) , j = 1 , . . . , m − . Relationship with the complex C • d ( K, γ ( z )) . Recall that foreach z ∈ O ′ the monodromy representation of ∇ γ ( z ) is equal to γ ( z ).Let π : f M → M denote the universal cover of M and let e E = π ∗ E de-note the pull-back of the bundle E to f M . Recall that in Subsection 4.3we fixed a point x ∗ ∈ M . Let ˜ x ∗ ∈ f M be a lift of x ∗ to f M and fixa basis of the fiber e E ˜ x ∗ of e E over x ∗ . Then, for each z ∈ O ′ , the flatconnection ∇ γ ( z ) identifies e E with the product f M × C n .Recall from Subsection 3.7 that the choice of the Euler structure ε also fixes the lifts e e j ( j = 1 , . . . , N ) of the cell e j fixed in Subsection 3.7.Let ˜ x j ∈ e e j be the lift of x j ∈ e j . Then the trivialization of e E definesa continuous in z family of isomorphisms S z,j : E x j ≃ e E ˜ x j → C n , j = 1 , . . . , N, z ∈ O ′ . EFINED ANALYTIC TORSION ON MANIFOLDS WITH BOUNDARY 25
The isomorphisms S z,j depend on the trivialization of e E , i.e., on theconnection ∇ γ ( z ) . The direct sum S z = L j S z,j defines an isomor-phism S z : C • d ( K, E ) → C • d ( K, γ ( z )). If α is a regular point ofRep( π ( M ) , C n ), it follows from (4.3) that S z is complex differentiableat 0, i.e. for some linear map sS z = S + z · s + o ( z ) . Finally, we consider the morphism of complexes I z := S − z ◦ J z ◦ ψ ( z ) : W • → C • := C • d ( K, E ) , z ∈ O ′ . (5.20)This map is complex differentiable at z = 0 and induces an isomorphismof cohomology.5.11. The cone complex.
The cone complex Cone • ( I z ) of the map I z is given by the sequence of vector spacesCone j ( I z ) := W j ⊕ C j − d (cid:0) K, E ) ≃ C l j ⊕ C n · k j − ,j = 0 , . . . , m , with differentialsˆ ∂ j ( z ) = (cid:18) d j ( z ) 0 I z,j ∂ ′ ( γ ( z )) (cid:19) , where I z,j denotes the restriction of I z to W j . This is a family ofacyclic complexes with differentials ˆ ∂ j ( z ), which are continuous. If α is a regular point of Rep( π ( M ) , C n ) then ˆ ∂ j ( z ) are also complexdifferentiable at 0. The standard bases of C l j ⊕ C n · k j − define an ele-ment c ∈ Det(Cone • ( I z )) which is independent of z ∈ O ′ . Using thecanonical isomorphism of Section 2.4 of [5], we hence obtain for each z ∈ O ′ the number φ Cone • ( I z ) ( c ) ∈ C \{ } . From the discussion in Sub-section 5.8 it follows that this number is equal to the ratio (5.1). Hence,to finish the proof of the Theorem 5.2 it remains to show that the func-tion z φ Cone • ( I z ) ( c ) is continuous and is complex differentiable at 0if α is a regular point of Rep( π ( M ) , C n ). This follows immediatelyfrom Lemma 5.7.6. Gluing formula for refined analytic torsion
Let (
M, g ) be a closed oriented Riemannian manifold and (
N, g N )a separating hypersurface, such that M = M ∪ N M . The metric g restricts to Riemannian metrics on the two compact components M and M . Assume that g is product in an open tubular neighborhoodof N . The transmission complex Ω • ( M M , E ) . A given represen-tation α ∈ Rep( π ( M ) , C n ) induces a connection ∇ α on a vector bundle E ≡ E α , which restricts to well-defined connections on M , . We de-note by ρ i := ρ ( ∇ α , M i ) the refined analytic torsions on M i , i = 1 , ρ = ρ ( ∇ α , M ) for the refined analytic torsion on M , cf. (2.13).Let ι j : N ֒ → M j denote the obvious inclusions, j = 1 , • ( M , E α ) ⊕ Ω • ( M , E α )by specifying transmission boundary conditionsΩ • ( M M , E ) := { ( ω , ω ) ∈ Ω • ( M , E α ) ⊕ Ω • ( M , E α ) | ι ∗ ω = ι ∗ ω } , ∇ α ( ω , ω ) := ( ∇ α ω , ∇ α ω ) . This defines a complex with eigenforms of the corresponding Laplaciangiven by the eigenforms of the Hodge-Laplacian on (Ω • ( M, E α ) , ∇ α ), cf.[45, Theorem 5.2]. In particular their de Rham cohomologies coincide.6.2. The fusion map.
The splitting M = M ∪ N M now gives riseto short exact sequences of the associated complexes0 → Ω • min ( M , E ) α −→ Ω • ( M M , E ) β −→ Ω • max ( M , E ) → , → Ω • min ( M , E ) α −→ Ω • ( M M , E ) β −→ Ω • max ( M , E ) → , (6.1)where α ( ω ) = ( ω, , α ( ω ) = (0 , ω ) and β j ( ω , ω ) = ω j , j = 1 , ( α ) : Det( H • ( M , N, E )) ⊗ Det( H • ( M , E )) → Det( H • ( M, E )) , Φ ( α ) : Det( H • ( M , N, E )) ⊗ Det( H • ( M , E )) → Det( H • ( M, E )) . The fusion isomorphisms, cf. [5, (2.18)] provide canonical identifica-tions µ : Det( H • ( M , N, E )) ⊗ Det( H • ( M , E )) → Det( H • ( e D , D )) ,µ : Det( H • ( M , N, E )) ⊗ Det( H • ( M , E )) → Det( H • ( e D , D )) ,µ : Det( H • ( M, E )) ⊗ Det( H • ( M, E )) → Det( H • ( e D , D )) , where the Hilbert complexes ( e D j , D j ) are defined in Definition 2.6, withthe lower index j referring to the underlying manifold M j , j = 1 ,
2. TheHilbert complex ( e D , D ) is defined over M . We putΦ ≡ Φ( α ) = µ ◦ (Φ ( α ) ⊗ Φ ( α )) ◦ ( µ − ⊗ µ − ) :Det( H • ( e D , D )) ⊗ Det( H • ( e D , D )) → Det( H • ( e D , D )) . If α ∈ Rep( π ( M ) , C n ) is unitary, [45, Theorem 10.6] assertsΦ ( ρ ⊗ ρ ) = K · ρ, (6.2) EFINED ANALYTIC TORSION ON MANIFOLDS WITH BOUNDARY 27 where K ≡ K ( α ) = σ χ ( N ) e πi ( η ( B α ,M ) − η ( B α ,M ) − η ( B α ,M )) . (6.3)Here χ ( N ) stands for the Euler characteristic of ( N, E α ↾ N ) and the η -invariants η ( B α , M ), η ( B α , M ) and η ( B α , M ) are defined in termsof the even parts of the corresponding odd signature operators. Thesign σ is determined by formula (8.4) of [45]. Note that the sign de-pends on the dimensions of various cohomology spaces and is relatedto the sign convention used in defining the fusion isomorphism of de-terminant lines, see [5, §
2] for a detail discussion of the sign conven-tions. Note also that the Euler characteristic χ ( N ) depends only onthe rank n of the representation α but not on the particular choice of α ∈ Rep( π ( M ) , C n ).The η -invariants η ( B α , M ), η ( B α , M ) and η ( B α , M ) are not neces-sarily continuous functions of α : they have integer jumps when someeigenvalues of the odd signature operator cross zero. Hence, K ( α )is a continuous function of α . Moreover, below we show that K ex-tends to a weakly holomorphic function on the space Rep( π ( M ) , C n )of representations.6.3. The gluing formula for some non-unitary representations.
The main result of this section is the following extension of (6.2) tosome class of non-unitary representations.
Theorem 6.4.
Let ( M, g ) be a closed oriented Riemannian manifold ofodd dimension, and N a separating hypersurface such that M = M ∪ N M , and g is product in an open tubular neighborhood of N . Assumethat C ⊂
Rep( π ( M ) , C n ) is a connected component and α ⊂ C is aunitary representation which is a regular point of the complex analyticset C . For any representation α ∈ Rep( π ( M ) , C n ) denote by ρ ( α ) and ρ j ( α ) the refined analytic torsions on M and M j , j = 1 , . Then forany α ∈ C we have Φ ( ρ ( α ) ⊗ ρ ( α )) = ± K ( α ) · ρ ( α ) . The rest of this section is occupied with the proof of Theorem 6.4,which is based on an analytic continuation technique, cf. [9, § Proposition 6.5. K ( α ) is a weakly holomorphic function on the com-plex analytic space Rep( π ( M ) , C n ) .Proof. We need to show that exp (cid:0) iπη ( B α , M i )), ( i = 1 ,
2) andexp (cid:0) iπη ( B α , M ) (cid:1) are weekly holomorphic functions of α . Denote by B α,j ( j = 1 ,
2) the odd signature operator of the complex( e D j , D j ) and by B α the odd signature operator of the comples ( e D , D ).With this notation we have η ( B α , M j ) = η ( B α,j ) , η ( B α , M ) = η ( B α ) . Fix α ∈ Rep( π ( M ) , C n ) and a number λ > B α and B α ,j ( j = 1 ,
2) do not intersect the circle { z ∈ C : | z | = λ } . There exists a neighborhood U λ of α in Rep( π ( M ) , C n )such that for all α ∈ U λ the spectra of the operators B α and B α,j ( j = 1 ,
2) do not intersect the circle { z ∈ C : | z | = λ } . Then (cf.formula (4-1) of [44]) η ( B α,j ) − η ( B ( λ, ∞ ) α,j ) ∈ Z , η ( B α ) − η ( B ( λ, ∞ ) α ) ∈ Z . (6.4)Notice that the functions η ( B ( λ, ∞ ) α,j ) and η ( B ( λ, ∞ ) α ) are continuous on U λ .The functions η ( B α,j ) and η ( B α ) are not necessarily continuous, butmight have integer jumps. Hence, it follows from (6.4) that η ( B α,j ) − η ( B ( λ, ∞ ) α,j ) and η ( B α ) − η ( B ( λ, ∞ ) α ) are constants modulo Z . We concludethat exp (cid:0) iπη ( B α,j ) − exp (cid:0) iπη ( B ( λ, ∞ ) α,j ) , j = 1 , , and exp (cid:0) iπη ( B α ) − exp (cid:0) iπη ( B ( λ, ∞ ) α )are constant functions on U λ . Hence, it suffices to show that the func-tions exp (cid:0) iπη ( B ( λ, ∞ ) α,j ), ( i = 1 ,
2) and exp (cid:0) iπη ( B ( λ, ∞ ) α ) (cid:1) are weaklyholomorphic in a neighborhood of α .Let θ be an Agmon angle for the operators B α and B α ,j . Thenthere exists a neighborhood U λ,θ ⊂ U λ of α , such that for all α ∈ U λ,θ θ is an Agmon angle for B α and B α,j . By [44, (4.6)] we obtainDet ′ gr,θ ( B ( λ, ∞ ) α,j, even ) = exp (cid:16) ξ λ ( α, M j ) − iπξ ′ λ ( α, M j ) − iπη ( B ( λ, ∞ ) α , M j ) (cid:17) , where we have set ξ λ ( α, M j ) := 12 m X k =0 ( − k · k · dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 ζ θ (cid:0) s, B α,j ↾ e D kj, ( λ, ∞ ) (cid:1) ,ξ ′ λ ( α, M j ) := 12 m X k =0 ( − k · k · ζ θ (cid:0) s = 0 , B α,j ↾ e D kj, ( λ, ∞ ) (cid:1) . The graded determinant Det ′ gr,θ ( B α,j, even ) is weakly holomorphic in U λ,θ by Corollary 4.2. The fact that exp (cid:0) ξ ( α, M j ) (cid:1) and EFINED ANALYTIC TORSION ON MANIFOLDS WITH BOUNDARY 29 exp (cid:0) iπξ ′ ( α, M j ) (cid:1) are weakly holomorphic follows similarly from thevariational formula ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 ζ θ (cid:0) s, B κ ( t ) ↾ e D kj, ( λ, ∞ ) (cid:1) = − s · Tr (cid:16) ddt (cid:12)(cid:12)(cid:12) t =0 B κ ( t ) ↾ e D kj, ( λ, ∞ ) (cid:17)(cid:0) B κ (0) ↾ e D kj, ( λ, ∞ ) (cid:1) − s − . As a consequence, exp (cid:0) iπη ( B ( λ, ∞ ) α,j )) ( j = 1 ,
2) are weakly holomor-phic. Similarly, on proves that exp (cid:0) iπη ( B ( λ, ∞ ) α ) (cid:1) is weakly holomor-phic. (cid:3) An analytic continuation.
The set of unitary representationsis the fixed point set of the anti-holomorphic involution τ : Rep( π ( M ) , C n ) → Rep( π ( M ) , C n ) , τ : α α ′ , where α ′ denotes the representation dual to α . Hence, it is a totallyreal submanifold of Rep( π ( M ) , C n ) whose real dimension is equal todim C C , cf. [26, Proposition 3]. In particular there is a holomorphiccoordinates system ( z , . . . , z r ) near α such that the unitary repre-sentations form a real neighborhood of α , i.e. the set Im z = . . . =Im z r = 0. Therefore, cf. [39, p. 21], we obtain the following Proposition 6.7.
If two holomorphic functions coincide on the set ofunitary representations they also coincide on C . The proof of Theorem 6.4.
By Theorem 5.2 the refined an-alytic torsions ρ , ρ and ρ define holomorphic sections on the cor-responding determinant line bundles. The canonical isomorphism Φdefines a bilinear map between holomorphic determinant line bundlesand hence maps holomorphic sections to holomorphic sections. Let usdenote by f ( α ) the unique complex valued function of α such thatΦ ( ρ ⊗ ρ ) = f ( α ) · ρ, holds for all α ∈ C . Since both Φ ( ρ ⊗ ρ ) and ρ are holomorphicsections of the determinant line bundle, f ( α ) is a holomorphic function.It follows from (6.2) that K ( α ) = f ( α ) (6.5)for all unitary representations in C . By Proposition 6.5, K ( α ) is aholomorphic function. Hence, we obtain from Proposition 6.7 that(6.5) holds for all α ∈ C . (cid:3) Remark . The gluing formula for refined analytic torsion may beused to prove a gluing result for the Ray-Singer torsion norm on con-nected components of the representation variety that contain a unitarypoint.7.
Gluing formula for Ray-Singer analytic torsion
We continue in the previously outlined setup of a closed oriented Rie-mannian manifold (
M, g ) and a separating hypersurface (
N, g N ), suchthat M = M ∪ N M . Consider a representation α ∈ Rep( π ( M ) , C n )and the corresponding flat vector bundle ( E, ∇ ). Fix a Hermitian met-ric h on E , of product type near N . Even if the metric structures( g, h ) are product near N , the connection ∇ need not be product near N , so that the resulting Laplacian does not have a product structurenear the separating hypersurface and hence a gluing theorem for Ray-Singer analytic torsion cannot be obtained from the results of [34], [46]and [33].Before we proceed, let us make some chronological remarks on thattopic. Vishik [46] was first to prove the gluing formula for analytictorsion given a unitary representation without using the theorem ofCheeger [17] and M¨uller [35]. Though it was not explicitly statedin [46], the assumption of a unitary representation is obsolete oncea connection is in temporal gauge near N . Under the assumption oftemporal gauge and product metric structures, the Hodge Laplacianis of product type near N and the Vishik’s argument goes through.Recently, Lesch [33] provided an excellent discussion of the gluing for-mula for possibly non-compact spaces, extending the result of Vishik,and stating clearly that the proof requires only product metric struc-tures and a connection in temporal gauge near the cut, rather thanunitariness of the representation.A quite general proof of the gluing formula for general representa-tions and without assuming product metric structures, was providedby Br¨uning and Ma [12]. They derive a gluing formula by relatingthe Ray-Singer and the Milnor torsions, in odd and also in the even-dimensional case. In this section we present a different proof of theirresult in odd dimensions, as a consequence of [33] and the Br¨uning-Maanomaly formula in [11]. To make the main idea of our alternativeargument clear, we restrict ourselves to the situation, when ∂M = ∅ . The notion and properties of temporal gauge are recalled in the Appendix.
EFINED ANALYTIC TORSION ON MANIFOLDS WITH BOUNDARY 31
Proposition 7.1.
Consider two Hermitian metrics h , h on a fixedflat vector bundle ( E, ∇ ) over a compact oriented odd-dimensional Rie-mannian manifold ( K, g ) , j = 1 , . If ∂K = ∅ , assume that h , h coincide over Y = ∂K . Fix either relative or absolute boundary con-ditions at Y for the Hodge Laplacian and denote by k · k RS( g,h j ) , j = 0 , , the corresponding Ray-Singer analytic torsion norms. Then k · k RS( g,h ) = k · k RS( g,h ) . Proof.
The metric anomaly, identified in [11] is expressed in terms ofthe Levi-Civita connections ∇ T K and ∇ T Y on K and its boundary Y ,the respective representatives e ( T K, ∇ T K ) , e ( T Y, ∇ T Y ) of the Eulerclasses of
T K, T Y in Chern-Weil theory, and the quotient k · k h / k · k h between the metrics on det E , induced by h and h .Since dim K is odd, e ( T K, ∇ T K ) = 0. If h , h coincide over Y ,log k · k h / k · k h = 0 over Y , so that the statement follows from [11,Theorem 0.1, (0.5)]. (cid:3) The main idea now is the reduction to the setup of a connection intemporal gauge near N . A connection ∇ is in temporal gauge in anopen neighborhood U = ( − ǫ, ǫ ) × N of N , if ∇ = π ∗ ∇ N for some flatconnection ∇ N on E N , where π : U → N is the natural projectiononto the second factor. Proposition 8.2 below asserts that in fact everyconnection is gauge equivalent to a connection in temporal gauge.We denote the corresponding gauge transformation by γ . The gaugetransformed connection is given by ∇ γ = γ ∇ γ − . We set for any u, v ∈ Γ( M, E ) h γ ( u, v ) := h ( γu, γv ) . This defines a new Hermitian metric on E that coincides with h over N ,since γ acts as identity over N . By construction, γ induces an isometry γ : L ∗ ( M, E ; g, h γ ) → L ∗ ( M, E ; g, h ) . The following theorem is a result by Lesch [33], cf. Vishik [46].
Theorem 7.2.
Let ( M, g ) be a closed oriented Riemannian manifoldof odd dimension, and N a separating hypersurface such that M = M ∪ N M , and g is product in an open tubular neighborhood of N .Consider flat Hermitian vector bundle ( E, ∇ γ , h ) . Denote by k · k RS and k · k RS M i the corresponding Ray-Singer norms on M and M i , i = 1 , ,respectively. The Ray-Singer norm on M is defined with respect to There is no product structure assumption on g and h j , j = 0 , relative boundary conditions, while on M we pose absolute boundaryconditions. Then log k · k RS M Φ (cid:0) k · k RS M ⊗ k · k RS M (cid:1) = 12 χ ( N ) log 2 . Corollary 7.3.
Let the Ray-Singer torsion norms k · k RS and k · k RS M i bedefined with respect to the flat connection ∇ and any Hermitian metric h on E . Then log k · k RS M Φ (cid:0) k · k RS M ⊗ k · k RS M (cid:1) = 12 χ ( N ) log 2 . Proof.
The Laplacian ∆ γ on E γ = ( E, ∇ γ , h ) is related to the Lapla-cian ∆ on E α = ( E, ∇ , h γ ) by the unitary transformation γ with∆ γ = γ ◦ ∆ ◦ γ − . Hence, γ induces a map between the harmonicforms of ∆ and ∆ γ , and hence also between the corresponding deter-minant lines, which we also denote by γ . By construction, we find k γ ( · ) k RS E γ = k · k RS E α , (7.1)where we indicate the dependence on the vector bundle by the subindexand omit the reference to the underlying manifold, since the relationholds both on M and M i , i = 1 ,
2. The isometric identification γ com-mutes with the maps in the (6.1). Hence Φ ◦ ( γ ⊗ γ ) = γ ◦ Φ , and byTheorem 7.2, we findlog k · k RS(
M,E α ) Φ (cid:16) k · k RS( M ,E α ) ⊗ k · k RS( M ,E α ) (cid:17) = 12 χ ( N ) log 2 . (7.2)This is a gluing theorem for any complex representation α , possiblynon-unitary. A priori, however, this relation holds for the specific Her-mitian metric h γ on ( E, ∇ α ). Since γ ↾ N = id, the metrics h , h γ coincide over N and hence, by Proposition 7.1 the gluing formula (7.2)holds for any representation α and any choice of a Hermitian metric h on ( E, ∇ α ). (cid:3) The Riemannian metric g is still assumed to be product near N , sothat variation of g leads to additional anomaly terms, cf. [11, Theorem0.1]. 8. Appendix: Temporal Gauge Transformation
In this section we recall the notion of a connection in temporal gaugeand review some properties of these connections, cf. [45]. In particular,we show that any flat connection is gauge equivalent to a connectionin temporal gauge, cf. Proposition 8.2. Consider a closed oriented
EFINED ANALYTIC TORSION ON MANIFOLDS WITH BOUNDARY 33
Riemannian manifold (
M, g M ) of dimension m and a vector bundle E with structure group G ⊂ GL ( n, C ). Denote the principal G -bundleassociated to E by P , where G acts on P from the right.Consider a hypersurface N ⊂ M and its collar neighborhood U ∼ =( − ǫ, ǫ ) × N . We view the restrictions P | U , P | N as G -bundles, wherethe structure group can possibly be reduced to a subgroup of G . Let π : ( − ǫ, ǫ ) × ∂X → ∂X be the natural projection onto the second com-ponent. Then E | U ∼ = π ∗ E | N and for the associated principal bundles P | U ∼ = π ∗ P | N f −→ P | N , where f is the principal bundle homomorphism,covering π , with the associated homomorphism of the structure groupsbeing the identity automorphism. Definition 8.1.
We call a flat connection ω on P a connection intemporal gauge over U , if there exists a flat connection ω N on P N suchthat ω | U = f ∗ ω N over the collar neighborhood U . Similar conditiondefines a covariant derivative in temporal gauge. We now explain the notion of temporal gauge in local terms. Let ω N denote a flat connection one-form on P | N . Then ω U := f ∗ ω N gives aconnection one-form on P | U which is flat again. Let { e U β , e Φ β } β be a setof local trivializations for P | N . Then P | U ∼ = π ∗ P | N is trivialized overthe local neighborhoods U β := ( − ǫ, ǫ ) × e U β with the induced trivializa-tions Φ β . For any y ∈ e U β , normal variable x ∈ ( − ǫ, ǫ ) and for e ∈ G being the identity matrix we put e s β ( y ) := e Φ − β ( y, e ) , s β ( x, y ) := Φ − β (( x, y ) , e ) . These local sections define local representations for ω U and ω N e ω β := e s ∗ β ω N ∈ Ω ( e U β , G ) ,ω β := s ∗ β ω U ∈ Ω ( U β , G ) , where G denotes the Lie algebra of G . Consider local coordinates y =( y , .., y m − ) on e U β . Then e ω β = n X i =1 ω βi ( y ) dy i , ω β = ω β ( x, y ) dx + n X i =1 ω βi ( x, y ) dy i , with ω β ≡ , and ω βi ( x, y ) ≡ ω βi ( y ) . (8.1) Proposition 8.2.
Any flat connection on the principal bundle P isgauge equivalent to a flat connection in temporal gauge.Proof. By a partition of unity argument it suffices to discuss the prob-lem locally over U β . Let ω be a flat connection on P | U . Let γ ∈ Aut( P | U ) be any gauge transformation on P | U and γ β the correspond-ing local representation. Denote the gauge transform of ω under γ by ω γ . The local G -valued one-forms ω β , ω βγ are related in correspondenceto the transformation law of connections by ω βγ = ( γ β ) − · ω β · γ β + ( γ β ) − dγ β , where the action · is the multiplication of matrices ( G ⊂ GL ( n, C )),after evaluation at a local vector field and a base point in U β .The local one form ω β can be written as ω β = ω β ( x, y ) dx + n X i =1 w βi ( x, y ) dy i . Our task is to identify the correct gauge transformation γ , so that ω is temporal gauge, cf. (8.1). For this reason we consider the followinginitial value problem with parameter y ∈ e U β ∂ x γ β ( x, y ) = − ω β ( x, y ) γ β ( x, y ) ,γ β (0 , y ) = ∈ GL ( n, C ) . (8.2)The solution to (8.2) is given by an integral curve of the time dependentvector field V βx,y on G , parametrized by x ∈ ( − ǫ, ǫ ), such that for any u ∈ G V βx,y u := − ( R u ) ∗ ω β ( x, y ) = − ω β ( x, y ) · u, where R u is the right multiplication on γ and the second equality followsfrom the fact that G ⊂ GL ( n, C ) is a matrix Lie group. The corre-sponding integral curve γ β ( x, y ) with γ β (0 , y ) = ∈ G is G -valued andthe unique solution to (8.2).We now compute for the gauge transformed connection ω γ ω βγ = ( γ β ) − · ω β · γ β + ( γ β ) − dγ β = ( γ β ) − · ω β · γ β dx + n X i =1 ( γ β ) − · ω βi · γ β dy i + ( γ β ) − ∂ x γ β dx + n X i =1 ( γ β ) − ∂ y i γ β dy i = n X i =1 ( γ β ) − · ω βi · γ β dy i + n X i =1 ( γ β ) − ∂ y i γ β dy i . where in the last equality we cancelled two summands due to γ β beingthe solution to (8.2). We now use the fact that ω is a flat connection.A gauge transformation preserves flatness, so ω γ is flat again. Put ω βγ = ω βγ, ( x, y ) dx + n X i =1 ω βγ,i ( x, y ) dy i , EFINED ANALYTIC TORSION ON MANIFOLDS WITH BOUNDARY 35 where by the previous calculation ω βγ, ≡ , ω βγ,i ≡ ( γ β ) − · ω βi · γ β + ( γ β ) − ∂ y i γ β . Flatness of ω γ implies ∂ x ω βγ,i ( x, y ) = ∂ y i ω βγ, ( x, y ) = 0 . Hence the gauge transformed connection is indeed in temporal gauge.This completes the proof. (cid:3)
References [1] Baird, T. and Ramras, D.,
Smoothing maps into algebraic sets and spaces offlat connections , Geom. Dedicata (2015), 359–374. MR 3303057[2] Bismut, J.-M. and Zhang, W.
An extension of a theorem by Cheeger andM¨uller.
With an appendix by Francois Laudenbach. Asterisque No. ,(1992), 235 pp.[3] Bismut, J.-M. and Goette, S.
Families torsion and Morse functions.
AsterisqueNo. , (2001), 293 pp.[4] Braverman, M. and Kappeler, T.
Ray-Singer type theorem for the refined an-alytic torsion.
J. Funct. Anal. (2007), no. 1, 232–256.[5] Braverman, M. and Kappeler, T.
Refined analytic torsion as an elementof the determinant line.
Geom. Topol. (2007), 139–213. MR 2302591(2008a:58031)[6] Braverman, M. and Kappeler, T. Comparison of the refined analytic and theBurghelea-Haller torsions.
Festival Yves Colin de Verdiere. Ann. Inst. Fourier(Grenoble) (2007), no. 7, 2361–2387.[7] Braverman, M. and Kappeler, T. Refined analytic torsion.
J. Differential Geom. (2008), no. 2, 193–267.[8] Braverman, M. and Kappeler, T. A canonical quadratic form on the determi-nant line of a flat vector bundle.
Int. Math. Res. Not. IMRN 2008, no. , 21pp.[9] Braverman, M. and Vertman, B., A new proof of a Bismut-Zhang formulafor some class of representations , Geometric and Spectral Analysis, Contemp.Math., vol. 630, Amer. Math. Soc., Providence, RI, 2014, pp. 1–14.[10] Br¨uning, J. and Lesch, M.
Hilbert complexes
J. Funct. An.
An anomaly formula for Ray-Singer metrics on man-ifolds with boundary.
Geom. Funct. Anal. (2006), no. 4, 767–837.[12] Br¨uning, J. and Ma, X. On the gluing formula for the analytic torsion , Math.Z. (2013), no. 3-4, 1085–1117.[13] D. Burghelea, L. Friedlander, and T. Kappeler,
Meyer-Vietoris type formulafor determinants of elliptic differential operators , Journal of Funct. Anal. (1992), 34–65.[14] Burghelea, D. and Haller, S.,
Euler Structures, the Variety of Representationsand the Milnor-Turaev Torsion , Geom. Topol. (2006), 1185–1238.[15] Burghelea, D. and Haller, S., Complex-valued Ray-Singer torsion.
J. Funct.Anal. (2007), no. 1, 27–78.[16] Burghelea, D. and Haller, S.,
Complex valued Ray-Singer torsion II.
Math.Nachr. (2010), no. 10, 1372–1402. [17] Cheeger, J.
Analytic torsion and the heat equation.
Ann. of Math. (2) (1979), no. 2, 259–322.[18] Farber, M. and Turaev, V.
Absolute torsion.
Tel Aviv Topology Conference:Rothenberg Festschrift (1998), 73–85, Contemp. Math., , Amer. Math.Soc., Providence, RI, 1999.[19] Farber, M. and Turaev, V.
Poincare-Reidemeister metric, Euler structures,and torsion.
J. Reine Angew. Math. (2000), 195–225.[20] Farber, M.
Absolute torsion and eta-invariant.
Math. Z. (2000), no. 2,339–349.[21] Farber, M. and Turaev, V.,
Poincar´e-Reidemeister metric, Euler structures,and torsion , J. Reine Angew. Math. (2000), 195–225.[22] Gilkey, P. ”Invariance Theory, the Heat-equation and the Atiyah-Singer IndexTheorem” , Second Edition, CRC Press (1995)[23] Goldman, W. and Millson, J.,
The deformation theory of representations offundamental groups of compact K¨ahler manifolds , Inst. Hautes ´Etudes Sci.Publ. Math. (1988), no. 67, 43–96.[24] R. C. Gunning,
Lectures on complex analytic varieties: The local parametriza-tion theorem , Mathematical Notes, Princeton University Press, Princeton,N.J.; University of Tokyo Press, Tokyo, 1970. MR 0273060 (42
Lectures on complex analytic varieties: finite analytic map-pings , Princeton University Press, Princeton, N.J.; University of Tokyo Press,Tokyo, 1974, Mathematical Notes, No. 14. MR 0355093 (50
The real locus of an involution map on the moduli space of flatconnections on a Riemann surface , Int. Math. Res. Not. (2004), no. 61, 3263–3285.[27] H¨ormander, L.,
An introduction to complex analysis in several variables , thirded., North-Holland Mathematical Library, vol. 7, North-Holland PublishingCo., Amsterdam, 1990.[28] F. Kamber and Ph. Tondeur,
Flat bundles and characteristic classes of group-representations , Amer. J. Math. (1967), 857–886.[29] Kato, T. Perturbation Theory for Linear Operators , Die Grundlehren der math.Wiss. Volume 132, Springer (1966)[30] Lee, Y. and Huang, R.-T.
The refined analytic torsion and a well-posed bound-ary condition for the odd signature operator , arXiv:1004.1753v1 [math.DG][31] Lee, Y. and Huang, R.-T.
The gluing formula of the refined analytic torsionfor an acyclic Hermitian connection , Manuscripta Math. (2012), no. 1-2,91–122.[32] Lee, Y. and Huang, R.-T.
The comparison of two constructions of the re-fined analytic torsion on compact manifolds with boundary , J. Geom. Phys. (2014), 79–96.[33] Lesch, M. A gluing formula for the analytic torsion on singular spaces , Anal.PDE (2013), no. 1, 221–256.[34] L¨uck, W. Analytic and topological torsion for manifolds with boundary andsymmetry.
J. Differential Geom. (1993), no. 2, 263–322.[35] M¨uller, W. Analytic torsion and R -torsion of Riemannian manifolds. Adv. inMath. , (1978), no. 3, 233–305.[36] M¨uller, W. Analytic torsion and R -torsion for unimodular representations. J.Amer. Math. Soc. (1993), no. 3, 721–753. EFINED ANALYTIC TORSION ON MANIFOLDS WITH BOUNDARY 37 [37] L. Paquet
Probl’emes mixtes pour le syst’eme de Maxwell , Annales Facult. desSciences Toulouse, Volume IV, 103-141 (1982)[38] Ray, D. B. and Singer, I. M. R -torsion and the Laplacian on Riemannianmanifolds. Advances in Math. , 145–210. (1971).[39] Shabat, B. V. , Introduction to complex analysis. Part II , Translations of Math-ematical Monographs, vol. 110, American Mathematical Society, Providence,RI, 1992, Functions of several variables, Translated from the third (1985) Rus-sian edition by J. S. Joel.[40] M. A. Shubin,
Pseudodifferential operators and spectral theory , Springer Verlag,Berlin, New York, 1987.[41] Turaev, V. G.,
Reidemeister torsion in knot theory , Russian Math. Survey (1986), 119–182.[42] Turaev, V. G., Euler structures, nonsingular vector fields, and Reidemeister-type torsions , Math. USSR Izvestia (1990), 627–662.[43] Turaev, V. G., Introduction to combinatorial torsions , Lectures in MathematicsETH Z¨urich, Birkh¨auser Verlag, Basel, 2001, Notes taken by Felix Schlenk.[44] Vertman, B.
Refined analytic torsion on manifolds with boundary.
Geom.Topol. (2009), no. 4, 1989–2027.[45] B. Vertman, Gluing formula for refined analytic torsion , 2008,arXiv:0808.0451v2.[46] Vishik, S. M.
Generalized Ray-Singer conjecture. I. A manifold with a smoothboundary.
Comm. Math. Phys. (1995), no. 1, 1–102.
Department of Mathematics, Northeastern University,Boston, MA 02115, USA
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