Boundary problem for the refined analytic torsion on compact manifolds with boundary

Boundary problem for the refined analytic torsion on compact manifolds with boundary

Rung-Tzung Huang

Department of Mathematics, National Central University, Chung-Li 320, Taiwan

e-mail : rthuang@math.ncu.edu.tw

Yoonweon Lee

Department of Mathematics, Inha University, Incheon 402-751, Korea

e-mail : yoonweon@inha.ac.kr

(2010 Mathematics Subject Classification : 58J52, 58J28.)

Abstract. The refined analytic torsion was introduced by Braverman and Kappeler on an odd dimensional closed Riemannian manifold in 2000’s as an analytic analogue of the Turaev torsion. It is defined by using the graded zeta-determinant of the odd signature operator and is described as an element of the determinant line for cohomologies. In this note we briefly discuss the boundary problem for the refined analytic torsion on a compact Riemannian manifold with boundary by introducing well posed boundary conditions for the odd signature operator. We also discuss the gluing formula for the refined analytic torsion on a closed Riemannian manifold with respect to our chosen boundary conditions.

1 Introduction

The Reidemeister torsion was introduced by Reidemeister and Franz in 1930’s on a finite CW-complex with an orthogonal representation of its fundamental group ([18], [25]). It is a homeomorphic invariant, not a homotopy invariant. They used this invariant to classify the lens spaces up to homeomorphism. Turaev extended the Reidemeister torsion to a torsion for a general representation of a fundamental group, called Turaev torsion, by using new concepts such as Euler structure and homology orientation ([23], [24]). Turaev torsion was developed further by works of Farber and Turaev ([11], [12]).

Key words and phrases: Refined analytic torsion, Zeta-determinant, Eta-invariant, Odd signature operator, Gluing formula.

109

The analytic torsion was introduced in 1970’s by Ray and Singer to recover the Reidemeister torsion by analytic and differential geometric way. They defined the analytic torsion by the zeta-determinants of Hodge Laplacians acting on flat bundle valued differential forms on closed Riemannian manifolds ([21], [22]). They conjectured the equality of the Reidemeister and analytic torsions. Cheeger and M¨uller proved the equality of two torsions on a closed manifold independently ([10], [19]).

On the same line of analytic torsion, the analytic analogues of the Turaev torsion have been studied in two ways. One is the method of Burghelea-Haller ([2], [3]) and the other one is the method of Braverman-Kappeler ([4], [5]). Burghelea and Haller constructed the complex valued analytic torsion by using a Euler structure, Mathai- Quillen form and a non-degenerated bilinear form acting on a flat vector bundle on a closed manifold. They defined non-selfadjoint Laplacians from the non-degenerate bilinear form and constructed a complex valued analytic torsion. On the other hand, Braverman and Kappeler constructed a (sign) refined analytic torsion on a closed Riemannian manifold by using the odd signature operator acting on complex flat bundle valued differential forms. It is described as an element of the determinant line of cohomologies. If all the cohomologies vanish, the refined analytic torsion is a complex number. If the odd signature operator comes from an acyclic Hermitian connection, the refined analytic torsion is a complex number, whose modulus part is the classical Ray-Singer analytic torsion and the phase part is the rho invariant, the difference of two eta invariants ([1]).

In this note we discuss briefly the boundary value problem of the refined analytic torsion on a compact Riemannian manifold with boundary. For this purpose we are going to introduce well posed boundary conditions P₋,L0 and P+,L1 and discuss the refined analytic torsion subject to the boundary conditions P₋,L0 and P+,L1

([13]). We next discuss the gluing problem of the refined analytic torsion with respect to the boundary conditionsP−,L0 and P+,L1 ([14]). Vertman also studied the boundary value problem and gluing problem for the refined analytic torsion in [26] and [27]] but his method is completely different from ours. In this note we restrict ourselves to the case of Hermitian connections for simplicity.

2 Odd signature operator on a compact manifold with boundary Let (M, g^M) be a compact orientedm-dimensional Riemannian manifold with boundary Y ( m = 2r−1, odd integer), where g^M is assumed to be a product metric near the boundary Y. We denote by (E,∇) a flat complex vector bundle E→M with a flat connection∇. We extend ∇to a covariant derivative

∇: Ω^•(M, E)→Ω^•+1(M, E).

We choose a Hermitian inner producth^E onE. Forϕ,ψ∈C^∞(E), if∇ satisfies dh^E(ϕ, ψ) =h^E(∇ϕ, ψ) +h^E(ϕ,∇ψ),

∇ is called a Hermitian connection. All through this note we assume that ∇ is a Hermitian connection for some Hermitian inner producth^E. Using the Hodge star

operator ∗M, we define the involution Γ = Γ(g^M) : Ω^•(M, E)→Ω^m−•(M, E) by

Γω:=i^r(−1)^q(q+1)2 ∗M ω, ω∈Ω^q(M, E),

where r=^m+1₂ . Then Γ²= Id. The odd signature operatorBis defined by (2.1) B = B(∇, g^M) := Γ∇ + ∇Γ : Ω^•(M, E) −→ Ω^•(M, E).

ThenBis an elliptic differential operator of order 1. LetN be a collar neighborhood ofY which is isometric to [0,1)×Y. Then there is a natural isomorphism

(2.2) Ψ : Ω^p(N, E|N)→C^∞([0,1),Ω^p(Y, E|Y)⊕Ω^p−1(Y, E|Y)),

defined by Ψ(ω1+du∧ω2) = (ω1, ω2), where u is a coordinate normal to the boundary Y. Using the product structure we can induce a flat connection ∇^Y : Ω^•(Y, E|Y) → Ω^•(Y, E|Y) from ∇ and a Hodge star operator ∗Y : Ω^•(Y, E|Y)→ Ω^m−1−•(Y, E|Y) from∗M. We define two involutionsβ, Γ^Y by

β : Ω^p(Y, E|Y)→Ω^p(Y, E|Y), β(ω) = (−1)^pω

Γ^Y : Ω^p(Y, E|Y)→Ω^m−1−p(Y, E|Y), Γ^Y(ω) =i^r−1(−1)^p(p+1)2 ∗Y ω.

It is straightforward that

β²= Id, Γ^YΓ^Y = Id.

The odd signature operator Bis expressed onN, under the isomorphism (2.2), by (2.3) B=−iβΓ^Y

{( 1 0 0 1

)

∇∂u+

( 0 −1

−1 0

) (∇^Y + Γ^Y∇^YΓ^Y)}

, where ∂uis the inward normal derivative to the boundaryY onN. We denote

BY :=

( 0 −1

−1 0

) (∇^Y + Γ^Y∇^YΓ^Y)

, H^•(Y, E|Y) := kerB²Y. Then BY is a self-adjoint elliptic operator onY and hence H^•(Y, E|Y) is a finite dimensional vector space. We have

Ω^•(Y, E|Y) = Im∇^Y ⊕Im Γ^Y∇^YΓ^Y ⊕H^•(Y, E|Y).

If∇ϕ= Γ∇Γϕ= 0 forϕ∈Ω^•(M, E), simple computation shows thatϕis expressed, near the boundary Y, by

(2.4) ϕ=∇^Yϕtan+ϕtan,h+du∧(Γ^Y∇^YΓ^Yϕnor+ϕnor,h), where ϕtan,h, ϕnor,h∈H^•(Y, E|Y).

We defineK by

K:={ϕtan,h∈H^•(Y, E|Y)| ∇ϕ= Γ∇Γϕ= 0},

whereϕhas the form (2.4). Then

Γ^YK={ϕnor,h∈H^•(Y, E|Y)| ∇ϕ= Γ∇Γϕ= 0}, and we have

K⊕Γ^YK=H^•(Y, E|Y),

which shows that (H^•(Y, E|Y), ⟨, ⟩Y, −iβΓ^Y) is a symplectic vector space with Lagrangian subspacesKand Γ^YK. We denote by

L0= ( K

K )

, L1=

( Γ^YK Γ^YK

) ,

and define the orthogonal projections P₋,L0, P+,L1 : Ω^•(Y, E|Y)⊕Ω^•(Y, E|Y)→ Ω^•(Y, E|Y)⊕Ω^•(Y, E|Y) by

ImP₋,L0 =

( Im∇^Y ⊕K Im∇^Y ⊕K

)

, ImP+,L1 =

( Im Γ^Y∇^YΓ^Y ⊕Γ^YK Im Γ^Y∇^YΓ^Y ⊕Γ^YK

) . Then P−,L0, P+,L1 are pseudodifferential operators and give well-posed boundary conditions forBand the refined analytic torsion. We denote byBP−,L0 andB²q,P−,L0

the realizations ofBandB²q with respect toP−,L0,i.e.

Dom( BP−,L0

) = {ψ∈Ω^•(M, E)|P−,L0(ψ|Y) = 0} =: Ω^•(M, E)_P−,L0, Dom

(B²q,P−,L0

)

= {ψ∈Ω^q(M, E)|P−,L0(ψ|Y) = 0, P−,L0((Bψ)|Y) = 0}

=: Ω^q(M, E)_P−,L0

We defineBP+,L1,B²q,P+,L1, andB²q,abs,B²q,relin the similar way. Then we have the following result (Lemma 2.11 in [13]).

Lemma 2.1.

kerB²q,P−,L0 = kerB²q,rel=H^q(M, Y;E), kerB²q,P+,L1 = kerB²q,abs=H^q(M;E).

Since the odd signature operator B subject to the boundary condition P−,L0

orP+,L1 is a self-adjoint operator, we can choose ^π₂ andπfor the Agmon angles of Beven,P−,L0/P+,L1 andB²_{q,P−,L0/P+,L1}, respectively. We fix ^π₂ andπfor the Agmon angles. Then, forD=P−,L0 orP+,L1 we define the zeta functionζ_B2

q,D(s) and eta functionη_Beven,D(s) by

ζ_B2

q,D(s) = 1

Γ(s)

∫ _∞

0

t^s−1 (

Tre^−tB2q,D−dim kerB²q,D

) dt η_Beven,D(s) = 1

Γ(^s+1₂ )

∫ _∞

0

t^s−12 Tr

(Be^−tB2even,D )

dt.

(2.5)

It was shown in [13] thatζ_B²

q,D(s) andη_Beven,D(s) have regular values at s= 0. We define the zeta-determinant and eta-invariant by

log DetB²q,D := −ζ_B^′2 q,D(0), η(Beven,D) := 1

2

(η_Beven,D(0) + dim kerBeven,D

). (2.6)

We put

Ω^q₋(M, E) = Im∇ ∩Ω^q(M, E), Ω^q₊(M, E) = Im Γ∇Γ∩Ω^q(M, E), Ω^even_± (M, E) = ∑

q=even

Ω^q_±(M, E),

and denote by B^±even the restriction of Beven to Ω^even_± (M, E). The graded zeta- determinant Det_gr,θ(Beven,D) ofBeven with respect to the boundary conditionDis defined by

Det_gr(Beven,D) = DetB⁺_even,D Det

(−B⁻even,D

).

We next define the projectionseP0,Pe1: Ω^•(Y, E|Y)⊕Ω^•(Y, E|Y)→Ω^•(Y, E|Y)⊕ Ω^•(Y, E|Y) as follows. Forϕ∈Ω^q(M, E)

Pe0(ϕ|Y) =

{P−,L0(ϕ|Y) if q is even

P+,L1(ϕ|Y) if q is odd, Pe1(ϕ|Y) = Id−eP0(ϕ|Y).

Then, log Detgr(Beven,P−,L0/P+,L1) is described as follows ([13]).

(1) log Det_gr(Beven,P−,L0) = 1 2

∑m

q=0

(−1)^q+1·q·log DetB²_q,eP

0

−iπ η(Beven,P−,L0) +πi 2

( 1 4

m∑−1

q=0

ζ_B2 Y,q(0) +

r−2

∑

q=0

(r−1−q)(l⁺_q −l⁻_q) )

,

(2) log Detgr(Beven,P+,L1) = 1 2

∑m

q=0

(−1)^q+1·q·log DetB²_q,eP

1

−iπ η(Beven,P+,L1)−πi 2

( 1 4

m∑−1

q=0

ζ_B2 Y,q(0) +

r−2

∑

q=0

(r−1−q)(l⁺_q −l_q⁻) )

,

where l_q := dim kerB²Y,q, l⁺_q := dimK∩kerB²Y,q, l⁻_q := dim Γ^YK∩kerB²Y,q, and lq =l⁺_q +l⁻_q, l⁻_q =l⁺_m−1−q.

As a final ingredient we consider the trivial connection ∇^trivial acting on the trivial bundle M × C and define the trivial odd signature operator B^trivialeven : Ω^even(M, C) → Ω^even(M, C) in the same way as (2.1). The eta invariant η(B^trivialeven,P−,L0/P+,L1) associated toB^trivialeven and the boundary conditionP−,L0/P+,L1

is defined in the same way as in (2.5) and (2.6) by simply replacingBeven,P−,L0/P+,L1

withB^trivialeven,P−,L0/P+,L1. When∇is acyclic in the de Rham complex, the refined an- alytic torsionsρ_an,P−,L0(M, g^M,∇) andρ_an,P+,L1(M, g^M,∇) subject to the bound- ary conditionP₋,L0/P+,L1 are defined by

logρan,P−,L0(M, g^M,∇) := log Detgr,θ(Beven,P−,L0) +πi

2 (rankE)η_Btrivial

even,P−,L0(0), logρan,P+,L1(M, g^M,∇) := log Detgr,θ(Beven,P+,L1) +πi

2(rankE)η_Btrivial even,P+,L1

(0).

3 Refined analytic torsion on a compact manifold with boundary In this section, we briefly explain the determinant line for cohomologies and the canonical element, following the presentation of [5]. We refer to [5] for more details.

Let (C^•, ∂,Γ) be a finite complex consisting of finite dimensional vector spaces as follows.

(C^•, ∂, Γ) : 0−→C⁰−→^∂ C¹−→ · · ·^∂ −→^∂ C^m−1−→^∂ C^m−→0.

Here Γ :C^•→C^• is an involution, called a chirality operator, satisfying Γ(C^q) = C^m−q for 0≤q≤m. We define the determinant line Det(C^•) of (C^•, ∂,Γ) by

Det(C^q) = ∧^dimCqC^q, Det(C^q)⁽⁻¹⁾ := Hom(

Det(C^q),C) Det(C^•) =

⊗m

q=0

Det(C^q)^(−1)q, m= 2r−1,

We extend the chirality operator Γ to an operator of determinant lines Γ : Det(C^•) →Det(C^•) and choose arbitrary non-zero elements c_q ∈Det(C^q). Then we define the canonical elementc_Γ by

cΓ := (−1)^R(C•)·c0⊗c⁻₁¹⊗ · · · ⊗c⁽_r⁻₋¹⁾₁^r−1⊗

(Γcr−1)^(−1)r⊗(Γcr−2)^(−1)r−1⊗ · · · ⊗(Γc1)⊗(Γc0)⁽⁻¹⁾∈Det(C^•), where (−1)^R(C•) is a normalization factor defined by (cf. (4-2) in [5])

R(C^•) = 1 2

r−1

∑

q=0

dimC^q·(

dimC^q+ (−1)^r+q) ,

andc⁻¹∈Det(C^q)⁻¹is the unique dual element ofc∈Det(C^q) such thatc⁻¹(c) = 1. In fact, cΓ does not depend on the choice of cq’s. We denote by H^•(∂) the

cohomology of the complex (C^•, ∂). Then there is a canonical isomorphism ϕC^• : Det(C^•)→Det(H^•(∂)).

We refer to Subsection 2.4 in [5] for the definition of the isomorphism ϕ_C•. The refined torsion of the pair (C^•,Γ) is the elementρ_Γ in Det(H^•(∂)) defined by (3.1) ρΓ = ρC^•,Γ := ϕC^•(cΓ)∈Det(H^•(∂)).

We now go back to the de Rham complex. For 0 ≤ λ ∈ R we denote by Ω^q_P

−,L0,[0,λ](M, E), Ω^q_P

−,L0,(λ,∞)(M, E) the subspaces of Ω^q_P

−,L0(M, E) spanned by eigenforms of B² whose eigenvalues belong to [0, λ], (λ,∞), respectively. We de- note by B²_{q,P−,L0,[0,λ]}, B²_{q,P−,L0,(λ,∞)} the restrictions of B² to Ω^q_P

−,L0,[0,λ](M, E), Ω^q_P

−,L0,(λ,∞)(M, E), respectively. We define Ω^q_P

+,L1,[0,λ](M, E), Ω^q_P

+,L1,(λ,∞)(M, E) B²_{q,P+,L1,[0,λ]},B²_{q,P+,L1,(λ,∞)}, Ω^even_P

−,L0,(λ,∞)(M, E), Ω^even_P

+,L1,(λ,∞)(M, E),B^even_{P−,L0,(λ,∞)}, B^even_P+,L

1,(λ,∞), in the same way. Then∇maps Ω^q_e

P0/eP1,[0,λ](M, E) to Ω^q_e

P1/eP0,[0,λ](M, E) and

( Ω^•_e

P0/eP1,[0,λ](M, E),∇)

is a subcomplex of (

Ω^•_e

P0/eP1

(M, E),∇)

which com- putes the same cohomologies. Moreover,

( Ω^•_e

P0/eP1,[0,λ](M, E),∇)

is a finite com- plex consisting of finite dimensional vector spaces. Hence we can define the refined torsion elementρ_Pe

0,[0,λ] andρ_Pe

1,[0,λ] as (3.1). We finally define the refined analytic torsionsρan,P−,L0(M, g^M,∇) andρan,P+,L1(M, g^M,∇) as follows.

ρan,P−,L0(M, g^M,∇) := ρ_Pe

0,[0,λ]·Detgr(B^(λ,_even,^∞)_P−,L0)·e

iπ

2(rankE)η

Btrivial even,P−,L0

(0)

, ρ_an,P+,L1(M, g^M,∇) := ρ_Pe

1,[0,λ]·Det_gr(B^(λ,even,^∞)P+,L1)·e

iπ

2(rankE)η_Btrivial even,P+,L1

(0)

. The refined analytic torsionsρan,P−,L0(M, g^M,∇) andρan,P+,L1(M, g^M,∇) sat- isfy the following properties. We refer to [13] for more details.

(1) The right hand side of the definition of ρ_{an,P−,L0/P+,L1}(M, g^M,∇) does not depend on the choice ofλand henceρan,P−,L0/P+,L1(M, g^M,∇) is well defined.

(2) ρ_an,P−,L0(M, g^M,∇) is an element of Det(H_e^•

P0

(M, E)) andρ_an,P+,L1(M, g^M,∇) is an element of Det(H^•_e

P1

(M, E)).

(3) If all cohomologies vanish, then the above definitions coincide with the definition given in Section 2.

(4) ρ_an,P−,L

0/P+,L1(M, g^M,∇) are independent of the choice of the Agmon angles and invariant of the change of metrics in the interior ofM.

(5) ρ_an,P−,L

0/P+,L1(M, g^M,∇) may depend on the choice of metrics on the bound- aryY.

4 Gluing formula for the refined analytic torsion

In this section we discuss briefly the gluing formula for the refined analytic torsion with respect to the boundary conditions P₋,L0/P+,L1, which is shown in [14]. Let (M , gc ^Mc) be a closed Riemannian manifold of dimension m = 2r−1 and Eb → Mc be a flat vector bundle with a flat connection ∇. We denote by Y a hypersurface of Mc such that Mc−Y has two components, whose closures are denoted by M₁ and M₂, i.e. Mc=M₁∪Y M₂. We assume that g^Mc is a product metric nearY and that ∇ is a Hermitian connection. Let ∂u be the unit normal vector field on a collar neighborhood ofY such that∂uis outward onM₁and inward onM2. We denote byB^Mcthe odd signature operator on Mcand denote by B^M1, B^M2 (E1, E2, g^M1, g^M2) the restriction ofB^Mc (E,b g^Mc) to M1, M2. We impose the boundary conditionP+,L1 onM1 andP₋,L0 onM2. In this section we assume that all cohomologies involved vanish. In particular, the tangential operatorBY is invertible. Then computations using the BFK-gluing formula ([7], [16], [17]) and adiabatic limit method ([28], [29]) lead to the following result. We refer to [14] for the proof.

Theorem 4.1. We assume that for each 0 ≤ q ≤ m, i = 1,2, H^q(M ,c E) =b H^q(Mi, Y;Ei) =H^q(Mi;Ei) = 0. Then,

1 2

∑m

q=0

(−1)^q+1·q·(

log Det(B^M_q,eP¹

1

)²+ log Det(B^M_q,eP²

0

)² )

= 1

2

∑m

q=0

(−1)^q+1·q·(

log Det(B^Mq,abs¹ )²+ log Det(B^Mq,rel² )² )

.

We next define a one parameter family of orthogonal projections P(θ) bye Pe(θ) = Π>cosθ+P₋sinθ+1

2(1−cosθ−sinθ) Id, (0≤θ≤ π 2).

Then Pe(θ) is a smooth curve connecting the APS boundary condition Π> and P−. Here Π> is the orthogonal projection to the subspace spanned by positive eigenforms of BY. We denote the Calder´on projector for B^M1 and B^M2 by CM1, CM2. For i= 1,2, we also denote the spectral flow for (B^M_P(θ)eⁱ )θ∈[0,^π₂] and Maslov index for (Pe(θ), CM_i)θ∈[0,^π₂] by SF(B^M_P(θ)eⁱ )θ∈[0,^π₂] and Mas(P(θ),e CM_i)θ∈[0,^π₂]. We refer to [9], [15] and [20] for the definitions of the Calder´on projector, the spectral flow and Maslov index. Then computations following methods in [6] yield the following result. We refer to [14] for the proof.

Theorem 4.2. We assume that for each 0≤q≤m andi= 1,2,H^q(Mi;E|M_i) = H^q(Mi, Y;E|M_i) ={0}. Then :

(1) η(B^M_P−²)−η(B^MΠ_>²) = SF(B^M_P(θ)e² )θ∈[0,^π₂] = Mas(Pe(θ), CM2)θ∈[0,^π₂]. (2) η(B^M_P+¹)−η(B^M_Π<¹) = SF(B^M_Id¹_−P(θ)e )_θ∈[0,^π

2] = Mas(Id−P(θ),e CM1)_θ∈[0,^π

2]. (3) Mas(Pe(θ), CM₂)_θ∈[0,^π

2]= Mas(Id−P(θ),e CM₁)_θ∈[0,^π

2].

We assume that all cohomologies involved vanish. Then the following results are well known ([8], [6], [15]).

(1) 1 2

∑m

q=0

(−1)^q+1·q·log Det2θ(B^Mq^c)²

= 1 2

∑m

q=0

(−1)^q+1·q·(

log Det2θ(B^Mq,abs¹ )²+ log Det2θ(B^Mq,rel² )² )

. (2) η(B^Mˆ) = η(B^MΠ_<¹) +η(B^MΠ_>²).

Theorem 4.1 and 4.2 together with the above results lead to the gluing formula of the refined analytic torsion as follows.

logρan,P+(M1, g^M1,∇) + logρan,P−(M2, g^M1,∇)

= log Detgr(B^Meven,¹ P+) + log Detgr(B^Meven,² P−) + πi

2(rankE) (

η_BM1,trivial

even,P+ (0) +η_BM2,trivial even,P−

(0) )

= 1 2

∑m

q=0

(−1)^q+1·q·(

log Det(B^M_q,Pe¹

1

)²+ log Det(B^M_q,Pe²

0

)² )

+ πi

2 (rankE)· (

ηB^M_even,P+^1,trivial(0) +η

B^M_even,P^2,trivial₋ (0) )

−iπ (

η(B^M_even,¹ _P+,L1) + η(B^M_even,² _P−,L0) )

= 1 2

∑m

q=0

(−1)^q+1·q·(

log Det(B^M_q,abs¹ )²+ log Det(B^M_q,rel² )² )

+ πi

2 (rankE)· (

η_BM1,trivial even,Π<

(0) +η_BM2,trivial even,Π>

(0) )

−iπ (

η(B^Meven,Π¹ _<) + η(B^Meven,Π² _>) )

= 1 2

∑m

q=0

(−1)^q+1·q·log Det(B^Mq^c)²+ πi

2(rankE)·η

B^{M ,trivial}even^c

(0)−iπ η(B^Meven^c )

= logρ_an(M , gc ^Mc,∇) (mod 2πiZ).

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