Equivariant symmetric bilinear torsions ∗
Guangxiang Su †
Abstract
We extend the main result in the previous paper of Zhang and the au-
thor relating the Milnor-Turaev torsion with the complex valued analytic
torsion to the equivariant case.
1 Introduction
Let F be a unitary flat vector bundle on a closed Riemannian manifold X.
In [RS], Ray and Singer defined an analytic torsion associated to (X,F ) and
proved that it does not depend on the Riemannian metric on X. Moreover,
they conjectured that this analytic torsion coincides with the classical Reide-
meister torsion defined using a triangulation on X (cf. [Mi]). This conjecture
was later proved in the celebrated papers of Cheeger [C] and Müller [Mu1].
Müller generalized this result in [Mu2] to the case where F is a unimodular
flat vector bundle on X. In [BZ1], inspired by the considerations of Quillen
[Q], Bismut and Zhang reformulated the above Cheeger-Müller theorem as an
equality between the Reidemeister and Ray-Singer metrics defined on the de-
terminant of cohomology, and proved an extension of it to the case of general
flat vector bundles over X. The method used in [BZ1] is different from those of
Cheeger and Müller in that it makes use of a deformation by Morse functions
introduced by Witten [W] on the de Rham complex.
On the other hand, Turaev generalizes the concept of Reidemeister torsion
to a complex valued invariant whose absolute value provides the original Reide-
meister torsion, with the help of the so-called Euler structure (cf. [T], [FT]). It
is natural to ask whether there exists an analytic interpretation of this Turaev
torsion.
Recently, Burghelea and Haller [BH1, BH2], following a suggestion of Müller,
define a generalized analytic torsion associated to a nondegenerate symmetric
bilinear form on a flat vector bundle over a closed manifold and make an explicit
conjecture between this generalized analytic torsion and the Turaev torsion.
Later this conjecture was proved by Su and Zhang [SZ]. Also Burghelea and
Haller [BH3], up to a sign, proved this conjecture for odd dimensional manifolds,
and comments were made how to derive the conjecture in full generality in their
paper.
∗This work was partially supported by the Qiushi Foundation.
†Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, P.R. China.
(sugx@mail.nankai.edu.cn)
http://arxiv.org/abs/0704.1341v4
In this paper, we will extend the main result in [SZ] to the equivariant case,
which is closer in spirit to the approach developed by Bismut-Zhang in [BZ2].
The rest of this paper is organized as follows. In Section 2, we construct the
equivariant symmetric bilinear torsions associated with equivariant nondegen-
erate symmetric bilinear forms on a flat vector bundle. In Section 3, we state
the main result of this paper. In Section 4, we provide a proof of the main
result. Section 5 is devoted to the proofs of the intermediary results stated in
Section 4.
Since we will make substantial use of the results in [BZ1, BZ2, SZ], we will
refer to [BZ1, BZ2, SZ] for related definitions and notations directly when there
will be no confusion.
2 Equivariant symmetric bilinear torsions associated
to the de Rham and Thom-Smale complexes
In this section, for a G-invariant nondegenerate bilinear symmetric form on
a complex flat vector bundle over an oriented closed manifold, we define two
naturally associated equivariant symmetric bilinear forms on the equivariant
determinant of the cohomology H∗(M,F ) with coefficient F . One constructed
in a combinatorial way through the equivariant Thom-Smale complex associated
to a equivariant Morse function, and the other one constructed in an analytic
way through the equivariant de Rham complex.
2.1 Equivariant symmetric bilinear torsion of a finite dimensional
complex
Let (C, ∂) be a finite cochain complex
(C, ∂) : 0 −→ C0 ∂0−→ C1 ∂1−→ · · · ∂n−1−→ Cn −→ 0,(2.1)
where each Ci, 0 ≤ i ≤ n, is a finite dimensional complex vector space.
Let each Ci, 0 ≤ i ≤ n, admit a nondegenerate symmetric bilinear form bi.
We equip C with the nondegenerate symmetric bilinear form bC =
i=0 bi.
Let G be a compact group. Let ρ : G → End(C) be a representation of G,
with values in the chain homomorphisms of C which preserve the bilinear form
bC . In particular, if g ∈ G, ρ(g) preserves the Ci’s.
Let Ĝ be the set of equivalence classes of complex irreducible representations
of G. An element of Ĝ is specified by a complex finite dimensional vector space
W together with an irreducible representation ρW : G→ End(W ).
For W ∈ Ĝ, set
CiW = HomG(W,C
i)⊗W,(2.2)
CW = HomG(W,C)⊗W.(2.3)
Let ∂W be the map induced by ∂ on CW . Then
(CW , ∂W ) : 0 −→ C0W
∂0,W−→ C1W
∂1,W−→ · · ·
∂n−1,W−→ CnW −→ 0(2.4)
is a chain complex. Thus we obtain the isotypical decomposition,
(C, ∂) =
W∈ bG
(CW , ∂W ),(2.5)
and the decomposition (2.5) is orthogonal.
If E is a complex finite dimensional representation space for G, let χ(E) be
the character of the representation. Put
χ(C) =
(−1)iχ(Ci),
e(C) =
(−1)idimCi,
e(CW ) =
(−1)idim(CiW ).(2.6)
By (2.5), we get
χ(C) =
W∈ bG
e(CW )
χ(W )
rk(W )
.(2.7)
If λ is a complex line, let λ−1 be the dual line. If E is a finite dimensional
complex vector space, set
detE = Λmax(E).(2.8)
detC =
detCi
)(−1)i
detCW =
detCiW
)(−1)i
.(2.9)
By (2.5), we obtain
detC =
W∈ bG
detCW .(2.10)
For 0 ≤ i ≤ n, CiW is a vector subspace of Ci. Let bCiW be the induced
symmetric bilinear form on CiW . let bdetCi
be the symmetric bilinear form on
detCiW induced by bCi
, and let b(detCi
)−1 be the dual symmetric bilinear form
on (detCiW )
−1. Also we have symmetric bilinear forms bdetCW on detCW and
bdetC on detC.
det(C,G) =
W∈ bG
detCW .(2.11)
Definition 2.1. We introduce the formal product
bdet(C,G) =
W∈ bG
(bdetCW )
χ(W )
rk(W ) .(2.12)
For W ∈ Ĝ, let xW , yW ∈ detCW , xW 6= 0, yW 6= 0. Set x = ⊕W∈ bGxW ,
y = ⊕
W∈ bGyW ∈ det(C,G). Then by definition,
bdet(C,G)(x, y) =
W∈ bG
(bdetCW (xW , yW ))
χ(W )
rk(W ) .(2.13)
Tautologically, (2.13) is an identity of characters on G. In particular
bdet(C,G)(x, y)(1) =
W∈ bG
bdetCW (xW , yW ).(2.14)
In fact (2.14) just says that
bdet(C,G)(1) = bdetC .(2.15)
Of course, using the orthogonality of the χW ’s, knowing the formal product
bdet(C,G) is equivalent to knowing the symmetric bilinear forms bdetCW .
Clearly
H(CW , ∂W ) = HomG(W,H(C, ∂)) ⊗W,
H(C, ∂) =
W∈ bG
H(CW , ∂W ).(2.16)
For W ∈ Ĝ, we define detH(CW , ∂W ) as in (2.9). Set
det(H(C, ∂), G) =
W∈ bG
detH(CW , ∂W ).(2.17)
For W ∈ Ĝ, there is a canonical isomorphism (cf. [KM] and [BGS, Section
1a)])
detCW ≃ detH(CW , ∂W ).(2.18)
From (2.18), we get
det(C,G) ≃ det(H(C, ∂), G).(2.19)
Let bdetH(CW ,∂W ) be the symmetric bilinear form on detH(CW , ∂W ) corre-
sponding to bdetCW via the canonical isomorphism (2.18).
Definition 2.2. we introduce the formal product
bdet(H(C,∂),G) =
W∈ bG
bdetH(CW ,∂W )
)χ(W )
rkW .(2.20)
Tautologically, under the identification (2.19),
bdet(C,G) = bdetH((C,∂),G).(2.21)
By an abuse of notation, we will call the formal product bdet(C,G) a sym-
metric bilinear form on det(C,G).
2.2 The Thom-Smale complex of a gradient field
Let M be a closed smooth manifold, with dimM = n. For simplicity, we make
the assumption that M is oriented.
Let (F,∇F ) be a complex flat vector bundle overM carrying the flat connec-
tion ∇F . We make the assumption that F carries a nondegenerate symmetric
bilinear form bF .
Let (F ∗,∇F ∗) be the dual complex flat vector bundle of (F,∇F ) carrying
the dual flat connection ∇F ∗.
Let f : M → R be a Morse function. Let gTM be a Riemannian metric on
TM such that the corresponding gradient vector field −X = −∇f ∈ Γ(TM)
satisfies the Smale transversality conditions (cf. [Sm]), that is, the unstable
cells (of −X) intersect transversally with the stable cells.
B = {x ∈M ;X(x) = 0}.(2.22)
For any x ∈ B, let W u(x) (resp. W s(x)) denote the unstable (resp. stable)
cell at x, with respect to −X. We also choose an orientation O−x (resp. O+x ) on
W u(x) (resp. W s(x)).
Let x, y ∈ B satisfy the Morse index relation ind(y) = ind(x) − 1, then
Γ(x, y) =W u(x)∩W s(y) consists of a finite number of integral curves γ of −X.
Moreover, for each γ ∈ Γ(x, y), by using the orientations chosen above, on can
define a number nγ(x, y) = ±1 as in [BZ1, (1.28)].
If x ∈ B, let [W u(x)] be the complex line generated by W u(x). Set
u, F ∗) =
[W u(x)]⊗ F ∗x ,(2.23)
u, F ∗) =
x∈B, ind(x)=i
[W u(x)]⊗ F ∗x .(2.24)
If x ∈ B, the flat vector bundle F ∗ is canonically trivialized on W u(x). In
particular, if x, y ∈ B satisfy ind(y) = ind(x) − 1, and if γ ∈ Γ(x, y), f∗ ∈ F ∗x ,
let τγ(f
∗) be the parallel transport of f∗ ∈ F ∗x into F ∗y along γ with respect to
the flat connection ∇F ∗.
Clearly, for any x ∈ B, there is only a finite number of y ∈ B, satisfying
together that ind(y) = ind(x)− 1 and Γ(x, y) 6= ∅.
If x ∈ B, f∗ ∈ F ∗x , set
∂(W u(x)⊗ f∗) =
y∈B, ind(y)=ind(x)−1
γ∈Γ(x,y)
nγ(x, y)W
u(y)⊗ τγ(f∗).(2.25)
Then ∂ maps Ci(W
u, F ∗) into Ci−1(W
u, F ∗). Moreover, one has
∂2 = 0.(2.26)
That is, (C∗(W
u, F ∗), ∂) forms a chain complex. We call it the Thom-Smale
complex associated to (M,F,−X).
If x ∈ B, let [W u(x)]∗ be the dual line to W u(x). Let (C∗(W u, F ), ∂) be
the complex which is dual to (C∗(W
u, F ∗), ∂). For 0 ≤ i ≤ n, one has
Ci(W u, F ) =
x∈B, ind(x)=i
[W u(x)]∗ ⊗ Fx.(2.27)
Let G be a compact group acting on M by smooth diffeomorphisms. we
assume that the action of G lifts to F and preserves the flat connection of
F . Then G acts naturally on H∗(M,F ). We assume that f and gTM are G-
invariant. Then −X = −∇f is also G-invariant. We assume that it verifies the
smale transversality conditions.
Clearly B is G-invariant. Also if x ∈ B, g ∈ G,
g (W u(x)) = ǫg(x)W
u(gx),
where ǫg(x) = +1 if g(W
u(x)) has the same orientation as W u(gx), ǫg = −1
if not. Clearly g acts as a chain homomorphism on (C∗(W
u, F ∗), ∂). The
corresponding dual action of g on (C∗(W u, F ), ∂) is such that
g (W u(x)∗) = ǫg(x)W
u(gx)∗.
Then g acts as a chain homomorphism on (C∗(W u, F ), ∂). Therefore g acts on
H∗(C∗(W u, F ), ∂).
2.3 Equivariant Milnor symmetric bilinear torsion
For x ∈ B, let bFx be a nondegenerate symmetric bilinear form on Fx. We
assume that the bFx’s are G-invariant, i.e. for g ∈ G, x ∈ B
= bFg(x).(2.28)
The symmetric bilinear forms bFx ’s determine a G-invariant symmetric bilin-
ear form on C∗(W u, F ) =
x∈B [W
u(x)]∗⊗Fx, such that the various [W u(x)]∗⊗
Fx are mutually orthogonal in C
∗(W u, F ), and that if x ∈ B, f, f ′ ∈ Fx,
W u(x)∗ ⊗ f,W u(x)∗ ⊗ f ′
f, f ′
.(2.29)
We construct the equivariant symmetric bilinear form bdet(C∗(Wu,F ),G) on
det(C∗(W u, F ), G) as in Definition 2.1.
Definition 2.3. The symmetric bilinear form on the determinant line of the
cohomology of the Thom-Smale cochain complex (C∗(W u, F ), ∂), in the sence
of Definition 2.2, is called the equivariant Milnor symmetric bilinear torsion
and is denoted by b
det(H∗(Wu,F ),G).
Take g ∈ G. Set
Mg = {x ∈M, gx = x}.(2.30)
Since G is a compact group, Mg is a smooth compact submanifold of M . Let
N be the normal bundle to Mg in M . By [BZ2, Proposition 1.13], we know
that f |Mg is a Morse function on Mg, and X|Mg is a smooth section of TMg.
For g ∈ G, set
Bg = B ∩Mg.(2.31)
Then Bg is the set of critical points of f |Mg .
Definition 2.4. If x ∈ Bg, let indg(x) be the index of f |Mg at x.
Let now bFx , b′Fx (x ∈ B) be two G-invariant nondegenerate symmetric
bilinear forms on Fx. Let b
det(H∗(Wu,F ),G), b
′M,−X
det(H∗(Wu,F ),G) be the corresponding
equivariant Milnor symmetric bilinear torsions. By [BZ2, Theorem 1.15] and
[SZ, Proposition 2.5], we have the following theorem.
Theorem 2.5. For g ∈ G, the following identity holds
′M,−X
det(H∗(Wu,F ),G)(g) = b
det(H∗(Wu,F ),G)(g)
g log
)])(−1)indg(x)
(2.32)
2.4 Equivariant Ray-Singer symmetric bilinear torsion
We continue the discussion of the previous subsection. However, we do not use
the Morse function and make transversality assumptions.
For any 0 ≤ i ≤ n, denote
Ωi(M,F ) = Γ
Λi(T ∗M)⊗ F
, Ω∗(M,F ) =
Ωi(M,F ).(2.33)
Let dF denote the natural exterior differential on Ω∗(M,F ) induced from ∇F
which maps each Ωi(M,F ), 0 ≤ i ≤ n, into Ωi+1(M,F ).
The group G acts naturally on Ω∗(M,F ). Namely, if g ∈ G, s ∈ Ω∗(M,F ),
gs(x) = g∗s(g
−1x), x ∈M.
Let gF be a G-invariant Hermitian metric on F . The G-invariant Rieman-
nian metric gTM and gF determine a natural inner product 〈 , 〉g (that is, a
pre-Hilbert space structure) on Ω∗(M,F ) (cf. [BZ1, (2.2)] and [BZ2, (2.3)]).
Let dF∗g be the formal adjoint of d
F with respect to 〈 , 〉g and Dg = dF +dF∗g .
On the other hand gTM and the G-invariant symmetric bilinear form bF
determine together a G-invariant symmetric bilinear form on Ω∗(M,F ) such
that if u = αf , v = βg ∈ Ω∗(M,F ) such that α, β ∈ Ω∗(M), f, g ∈ Γ(F ), then
〈u, v〉b =
(α ∧ ∗β)bF (f, g),(2.34)
where ∗ is the Hodge star operator (cf. [Z]).
Consider the de Rham complex
(2.35)
Ω∗(M,F ), dF
: 0 → Ω0(M,F ) d
→ Ω1(M,F ) → · · ·
dF→ Ωn(M,F ) → 0.
Let dF∗b : Ω
∗(M,F ) → Ω∗(M,F ) denote the formal adjoint of dF with
respect to G-invariant the symmetric bilinear form in (2.34). That is, for any
u, v ∈ Ω∗(M,F ), one has
dFu, v
u, dF∗b v
.(2.36)
Db = d
F + dF∗b , D
dF + dF∗b
= dF∗b d
F + dF dF∗b .(2.37)
Then the Laplacian D2b preserves the Z-grading of Ω
∗(M,F ).
As was pointed out in [BH1] and [BH2], D2b has the same principal symbol
as the usual Hodge Laplacian (constructed using the inner product on Ω∗(M,F )
induced from (gTM , gF )) studied for example in [BZ1].
We collect some well-known facts concerning D2b as in [BH2, Proposition
4.1], where the reference [S] is indicated.
Proposition 2.6. The following properties hold for the Laplacian D2b :
(i) The spectrum of D2b is discrete. For every θ > 0 all but finitely many
points of the spectrum are contained in the angle {z ∈ C| − θ < arg(z) < θ};
(ii) If λ is in the spectrum of D2b , then the image of the associated spectral
projection is finite dimensional and contains smooth forms only. We refer to
this image as the (generalized) λ-eigen space of D2b and denote it by Ω
{λ}(M,F ).
There exists Nλ ∈ N such that
D2b − λ
)Nλ∣∣∣
Ω∗{λ}(M,F )
= 0.(2.38)
We have a D2b -invariant 〈 , 〉b-orthogonal decomposition
Ω∗(M,F ) = Ω∗{λ}(M,F )⊕ Ω
{λ}(M,F )
⊥.(2.39)
The restriction of D2b − λ to Ω∗{λ}(M,F )
⊥ is invertible;
(iii) The decomposition (2.39) is invariant under dF and dF∗b ;
(iv) For λ 6= µ, the eigen spaces Ω∗{λ}(M,F ) and Ω
{µ}(M,F ) are 〈 , 〉b-
orthogonal to each other.
For any a ≥ 0, set
Ω∗[0,a](M,F ) =
0≤|λ|≤a
Ω∗{λ}(M,F ).(2.40)
Let Ω∗
[0,a]
(M,F )⊥ denote the 〈 , 〉b-orthogonal complement to Ω∗[0,a](M,F ).
Obviously, each Ω∗{λ}(M,F ) is a G-invariant subspace.
By [BH2, (29)] and Proposition 2.6, one sees that (Ω∗
[0,a]
(M,F ), dF ) forms a
finite dimensional complex whose cohomology equals to that of (Ω∗(M,F ), dF ).
Moreover, the G-invariant symmetric bilinear form 〈 , 〉b clearly induces a
nondegenerate G-invariant symmetric bilinear form on each Ωi
[0,a]
(M,F ) with
0 ≤ i ≤ n. By Definition 2.2 one then gets a symmetric bilinear torsion
det(H∗(Ω∗
[0,a]
(M,F )),G)
on detH∗(Ω∗
[0,a]
(M,F ), dF ) = detH∗(Ω∗(M,F ), dF ).
For any 0 ≤ i ≤ n, let D2b,i be the restriction of D2b on Ωi(M,F ). Then it is
shown in [BH2] (cf. [S, Theorem 13.1]) that for any a ≥ 0, g ∈ G the following
is well-defined,
D2b,(a,+∞),i
(g) = exp
D2b,i
[0,a]
(M,F )⊥
)−s])
.(2.41)
Definition 2.7. If g ∈ G, set
bRSdet(H∗(M,F ),G)(g) = b
det(H∗(Ω∗
[0,a]
(M,F )),G)(g)
D2b,(a,+∞),i
)(−1)ii
(2.42)
by [BH2, Proposition 4.7], we know that bRS
det(H∗(M,F ),G) does not depend on the
choice of a ≥ 0, and is called the equivariant Ray-Singer symmetric bilinear
torsion on detH∗(Ω∗(M,F ), dF ).
2.5 An anomaly formula for the equivariant Ray-Singer symmetric
bilinear torsion
We continue the discussion of the above subsection.
Definition 2.8. Let θg(F, b
F ) be the 1-form on Mg
θg(F, b
F ) = Tr
g(bF )−1∇F bF
.(2.43)
ClearlyMg is a totally geodesic submanifold ofM . Let g
TMg be the Rieman-
nian metric induced by gTM on TMg. Let ∇TMg be the Levi-Civita connection
on (TMg, g
TMg ).
Let e(TMg,∇TMg) be the Chern-Weil representative of the rational Euler
class of TMg, associated to the metric preserving connection ∇TMg . Then
e(TMg,∇TMg) = Pf
if dimMg is even,(2.44)
0 if dimMg is odd.
Let g′TM be another G-invariant metric and let ∇′TMg be the correspond-
ing Levi-Civita connection on TMg. Let ẽ(TMg,∇TMg ,∇′TMg) be the Chern-
Simons class of dimMg − 1 forms on Mg, such that
TMg,∇TMg ,∇′TMg
TMg,∇′TMg
TMg,∇TMg
.(2.45)
Let b′F be another G-invariant nondegenerate symmetric bilinear form on
Let b′RS
det(H∗(M,F ),G) denote the equivariant Ray-Singer symmetric bilinear
torsion associated to g′TM and b′F .
By [SZ, Remark 6.4] and [BZ2, Theorem 2.7], we have the following exten-
sion of the anomaly formula of [SZ, Theorem 2.9].
Theorem 2.9. If bF , b′F lie in the same homotopy class of nondegenerate
symmetric bilinear forms on F , then for g ∈ G the following identity holds,
(2.46)
det(H∗(M,F ),G)
det(H∗(M,F ),G)
(g) = exp
g log
TMg,∇TMg
· exp
θg(F, b
′F )ẽ
TMg,∇TMg ,∇′TMg
Proof. Let bFl is a smooth one-parameter family of fiber wise non-degenerate
symmetric bilinear forms on F and (gTMl , g
l ) be a smooth family of metrics on
TM,F .
By [SZ, (6.4)], we have
(2.47)
b = e−tD
(−1)ktk
e−t1tD
gBb,ge
−t2tD2g · · ·Bb,ge−tk+1tD
gdt1 · · · dtk
+ (−1)n+1tn+1
e−t1tD
gBb,ge
−t2tD2g · · ·Bb,ge−tn+2tD
bdt1 · · · dtn+1,
where ∆k, 1 ≤ k ≤ n+1, is the k-simplex defined by t1+ · · ·+ tk+1 = 1, t1 ≥ 0,
· · · , tk+1 ≥ 0 and Bb,g is defined in [SZ, (6.3)].
Proceeding as in [BZ1, Section 4], we first calculate the asymptotics as t→ 0
of Trs[g(b
−1 ∂bFl
exp(−tD2bl)]. Here the metric g
TM will be fixed.
By the same proof in [SZ, proposition 6.1], we have that as t→ 0+,
g(bFl )
−1 ∂b
e−t1tD
gBb,ge
−t2tD2g · · ·Bb,ge−tn+2tD
dt1 · · · dtn+1 → 0.
(2.48)
Also, by [SZ, (6.22)], we have that for 1 ≤ k ≤ n, (t1, · · · , tk+1) ∈ ∆k,
tkTrs
g(bFl )
e−t1tD
gBb,ge
−t2tD2g · · ·Bb,ge−tk+1tD
= 0.(2.49)
So that by (2.47)-(2.49) we have that
g(bFl )
−1 ∂b
−tD2bl
= lim
g(bFl )
−1 ∂b
−tD2gl
(2.50)
Now we assume that the nondegenerate symmetric bilinear form on F is
fixed, and the metric gTMl on TM depends on l.
Let ∗l be the Hodge star operator associated to gTMl .
By [BZ1, (4.70), (4.74)], analogues of [SZ, (6.5), (6.24), (6.26), (6.27)] re-
placing N by ∗−1l
and [BGV, Chapter 6], we have that
(2.51)
g ∗−1
∂∗TMl
−tD2bl
= lim
g ∗−1
∂∗TMl
−tD2gl
i, j=1
ei ∧ êj
∇ueiω
F (ej)
ωF , ω̂Fg − ω̂F
1≤i,j≤n
)−1 ∂gTMl
ei, ej
ei ∧ êj
 exp
− Ṙl
= lim
g ∗−1l
∂∗TMl
−tD2gl
∇TMl ϕTr
1≤i,j≤n
)−1 ∂gTMl
ei, ej
ei ∧ êj
 exp
− Ṙl
1≤i,j≤n
)−1 ∂gTMl
ei, ej
ei ∧ êj
· exp
− Ṙl
∧ ϕθg
F, bF
From (2.50), (2.51) and the calculations in [BZ1, Section 4], we get (2.46).
The proof of Theorem 2.9 is completed. Q.E.D.
3 A formula relating equivariant Milnor and equiv-
ariant Ray-Singer symmetric bilinear torsions
In this section, we state the main result of this paper, which is an explicit
comparison result between the equivariant Milnor symmetric bilinear torsion
and equivariant Ray-Singer symmetric bilinear torsion.
We assume that we are in the same situation as in Sections 2.2-2.4. By a
simple argument of Helffer-Sjöstrand [HS, Proposition 5.1] (cf. [BZ1, Section
7b)]), we may and we well assume that gTM there satisfies the following property
without altering the Thom-Smale cochain complex (C∗(W u, F ), ∂),
(*): For any x ∈ B, there is a system of coordinates y = (y1, · · · , yn)
centered at x such that near x,
gTM =
∣∣dyi
∣∣2 , f(y) = f(x)− 1
ind(x)∑
∣∣2 + 1
i=ind(x)+1
∣∣2 .(3.1)
By a result of Laudenbach [L], {W u(x) : x ∈ B} form a CW decomposition
of M .
For any x ∈ B, F is canonically trivialized over each cell W u(x).
Let P∞ be the de Rham map defined by
α ∈ Ω∗(M,F ) → P∞α =
W u(x)∗
Wu(x)
α ∈ C∗(W u, F ).(3.2)
By the Stokes theorem, one has
∂P∞ = P∞d
F .(3.3)
Moreover, it is shown in [L] that P∞ is a Z-graded quasi-isomorphism, inducing
a canonical isomorphism
PH∞ : H
∗ (Ω∗(M,F ), dF
→ H∗ (C∗ (W u, F ) , ∂) ,(3.4)
which in turn induces a natural isomorphism between the determinant lines,
P detH∞ : detH
∗ (Ω∗ (M,F ) , dF
→ detH∗ (C∗ (W u, F ) , ∂) .(3.5)
Also by [BZ2, Theorem 1.11], we know that P∞ commutes with G, and P
is the canonical identification of the corresponding cohomology groups as G-
spaces.
Now let hTM be an arbitrary smooth metric on TM .
By Definition 2.7, one has an associated equivariant Ray-Singer symmetric
bilinear torsion bRS
det(H∗(M,F ),G) on detH
∗(Ω∗(M,F ), dF ). From (3.5), one gets
a well-defined equivariant symmetric bilinear form
P detH∞
bRSdet(H∗(M,F ),G)
(3.6)
on detH∗(C∗(W u, F ), ∂).
On the other hand, by Definition 2.3, one has a well-defined equivariant Mil-
nor symmetric bilinear torsion b
det(H∗(M,F ),G) on detH
∗(C∗(W u, F ), ∂), where
X = ∇f is the gradient vector field of f associated to gTM .
Let Mg = ∪mj=1Mg,j be the decomposition of Mg into its connected compo-
nents. Clearly TrF [g] is constant on each Mg,j .
Let N be the normal bundle to Mg in M . We identify N to the orthogonal
bundle to TMg in TM |Mg .
Take x ∈ Bg. Then g acts on TxM as a linear isometry. Also
TMg = {Y ∈ TM |Mg , gY = Y }.
Moreover g acts on N . Let e±iβ1 , · · · , e±iβq (0 < βj ≤ π) be the locally constant
distinct eigenvalues of g|N . Then N splits orthogonally as
Nβj .
For 1 ≤ j ≤ q, g acts on Nβj as an isometry, with eigenvalues e±iβj . In
particular, if e±iβj 6= −1, Nβj is even dimensional.
Take x ∈ Bg. Since f is g-invariant, d2f(x) is also g-invariant. Therefore
the decomposition
TxM = TxMg ⊕
is orthogonal with respect to d2f(x). On TxMg, the index of d
2f(x)|TxMg×TxMg
was already denoted indg(x). Let n+(βj)(x) (resp. n−(βj)(x)) be the num-
ber of positive (resp. negative) eigenvalues of d2f(x)|
βj . Then if e
±iβj 6=
−1, n±(βj)(x) is even.
Let ψ(TMg,∇TMg) be the Mathai-Quillen current ([MQ]) over TMg, asso-
ciated to hTM , defined in [BZ1, Definition 3.6]. As indicated in [BZ1, Remark
3.8], the pull-back current X∗ψ(TMg,∇TMg ) is well-defined over Mg.
The main result of this paper, which generalizes [SZ, Theorem 3.1] to the
equivariant case.
Theorem 3.1. For g ∈ G, the following identity in C holds,
(3.7)
P detH∞
det(H∗(M,F ),G)
det(H∗(Wu,F ),G)
(g) = exp
θg(F, b
F )X∗ψ
TMg,∇TMg
· exp
(−1)indg(x)
(n+(βj)(x)− n−(βj)(x))
1− βj
− 2Γ′(1)
· Tr [g|Fx ]
Remark 3.2. By proceeding similarly as in [BZ2, Section 5b)], in order to prove
(3.7), we may well assume that hTM = gTM . Moreover, we may assume that
bF , as well as the Hermitian metric hF on F , are flat on an open neighborhood
of the zero set B of X. From now on, we will make these assumptions.
4 A proof of Theorem 3.1
We assume that the assumptions in Remark 3.2 hold.
For any T ∈ R, let bFT be the deformed symmetric bilinear form on F defined
bFT (u, v) = e
−2TfbF (u, v).(4.1)
Let dF∗bT be the associated formal adjoint in the sense of (2.36). Set
DbT = d
F + dF∗bT , D
dF + dF∗bT
= dF∗bT d
F + dF dF∗bT .(4.2)
Let Ω∗
[0,1],T
(M,F ) be defined as in (2.40) with respect to D2bT , and let
[0,1],T
(M,F )⊥ be the corresponding 〈 , 〉bT -orthogonal complement.
Let P
[0,1]
T be the orthogonal projection from Ω
∗(M,F ) to Ω∗
[0,1],T
(M,F )
with respect to the inner product determined by gTM and gFT = e
−2TfgF . Set
(1,+∞)
T = Id− P
[0,1]
Following [BZ2, (5.9)-(5.10)], we introduce the notations
χg(F ) =
TrF |Mg,j
x∈Bg∩Mg,j
(−1)indg(x),(4.3)
χ̃′g(F ) =
TrF |Mg,j
x∈Bg∩Mg,j
(−1)indg(x)ind(x),
s [f ] =
TrF |Mg,j
x∈Bg∩Mg,j
(−1)indg(x)f(x).
Let N be the number operator on Ω∗(M,F ) acting on Ωi(M,F ) by multi-
plication by i.
By the technique developed in [SZ] and the corresponding results in [BZ2],
we easily get the following intermediate results. The sketch of the proofs will
be outlined in Section 5.
Theorem 4.1. (Compare with [BZ2, Theorem 5.5] and [SZ, Theorem 3.3]) Let
[0,1]
T be the restriction of P∞ on Ω
[0,1],T
(M,F ), let P
[0,1],detH
T be the induced
isomorphism on cohomology, then the following identity holds,
[0,1],detH
det(H∗(Ω∗
[0,1],T
(M,F )),G)
det(H∗(Wu,F ),G)
χg(F )−eχ′g(F )
s [f ]T
(4.4)
Theorem 4.2. (Compare with [BZ2, Theorem 5.7] and [SZ, Theorem 3.4]) For
any t > 0,
gN exp
−tD2bT
(1,+∞)
= 0.(4.5)
Moreover, for any d > 0 there exist c > 0, C > 0 and T0 ≥ 1 such that for any
t ≥ d and T ≥ T0,
∣∣∣Trs
gN exp
−tD2bT
(1,+∞)
]∣∣∣ ≤ c exp(−Ct).(4.6)
Theorem 4.3. (Compare with [BZ2, Theorem 5.8] and [SZ, Theorem 3.5]) For
T ≥ 0 large enough, then
[0,1]
= χ̃′g(F ).(4.7)
Also,
D2bTP
[0,1]
= 0.(4.8)
For the next results, we will make use the same notation for Clifford multi-
plications and Berezin integrals as in [BZ1, Section 4].
Theorem 4.4. (Compare with [BZ2, Theorem 5.9] and [SZ, Theorem 3.6]) As
t→ 0, the following identity holds,
gN exp
−tD2bT
χg(F ) +O(t) if n is even,(4.9)
TrF [g]
L exp
if n is odd.
Theorem 4.5. (Compare with [BZ2, Theorem A.1] and [SZ, Theorem 3.7])
There exist 0 < α ≤ 1, C > 0 such that for any 0 < t ≤ α, 0 ≤ T ≤ 1
, then
(4.10)
∣∣∣∣∣Trs
gN exp
− (tDb + T ĉ(∇f))2
TrF [g]
L exp (−BT 2)
F, bF
) ∫ B
d̂f exp (−BT 2)−
χg(F )
∣∣∣∣∣ ≤ Ct.
Theorem 4.6. (Compare with [BZ2, Theorem A.2] and [SZ, Theorem 3.8])
For any T > 0, the following identity holds,
(4.11) lim
gN exp
tDb +
ĉ(∇f)
g|F |Mg,j
1− e−2T
(1 + e−2T
x∈B∩Mg,j
(−1)indg(x)indg(x)− dimMg,je−2Tχ(Mg,j)
g|F |Mg,j
sinh(2T )
cosh(2T )− cos(βk)
x∈B∩Mg,j
(−1)indg(x)n−(βk)(x)
g|F |Mg,j
sinh(2T )
cosh(2T ) − cos(βk)
dimNβkχ(Mg,j).
Theorem 4.7. (Compare with [BZ2, Theorem A.3] and [SZ, Theorem 3.9])
There exist α ∈ (0, 1], c > 0, C > 0 such that for any t ∈ (0, α], T ≥ 1, then
∣∣∣∣∣Trs
gN exp
tDb +
ĉ(∇f)
− χ̃′g(F )
∣∣∣∣∣ ≤ c exp(−CT ).(4.12)
Clearly, we may and we will assume that the number α > 0 in Theorems
4.5 and 4.7 have been chosen to be the same.
Next, we use above theorems to give a proof of Theorem 3.1. Since the
process is similar to it in [SZ], so we refer to it for more details.
First of all, by the anomaly formula (2.46), for any T ≥ 0, g ∈ G, one has
(4.13)
[0,1],detH
detH∗(Ω∗
[0,1],T
(M,F ),G)
det(H∗(Wu,F ),G)
[0,1],T
(M,F )⊥∩Ωi(M,F )
)(−1)ii
P detH∞
det(H∗(M,F ),G)
det(H∗(Wu,F ),G)
(g) exp
TrF [g]fe
TMg,∇TMg
From now on, we will write a ≃ b for a, b ∈ C if ea = eb. Thus, we can
rewrite (4.13) as
(4.14)
P detH∞
det(H∗(M,F ),G)
det(H∗(Wu,F ),G)
 ≃ log
[0,1],detH
detH∗(Ω∗
[0,1],T
(M,F ),G)
det(H∗(Wu,F ),G)
(−1)ii log
[0,1],T
(M,F )⊥∩Ωi(M,F )
TrF [g]fe
TMg,∇TMg
Let T0 > 0 be as in Theorem 4.2. For any T ≥ T0 and s ∈ C with
Re(s) ≥ n+ 1, set
θg,T (s) =
ts−1Trs
gN exp
−tD2bT
(1,+∞)
dt.(4.15)
By (4.6), θg,T (s) is well defined and can be extended to a meromorphic function
which is holomorphic at s = 0. Moreover,
(−1)ii log
[0,1],T
(M,F )⊥∩Ωi(M,F )
≃ − ∂θg,T (s)
(4.16)
Let d = α2 with α being as in Theorem 4.7. From (4.15) and Theorems
4.2-4.4, one finds that
(4.17)
∂θg,T (s)
= lim
gN exp
−tD2bT
− a−1√
χg(F )
− 2a−1√
Γ′(1)− log d
χg(F )− χ̃′g(F )
To study the first term in the right hand side of (4.17), we observe first that
for any T ≥ 0, one has
e−TfD2bT e
Tf = (Db + T ĉ(∇f))2 .(4.18)
Thus, one has
N exp
−tD2bT
= Trs
N exp
−t (Db + T ĉ(∇f))2
.(4.19)
By (4.19), one writes
(4.20)
gN exp
−tD2bT
− a−1√
χg(F )
∫ √dT
gN exp
Db + t
T ĉ(∇f)
a−1 −
χg(F )
gN exp
− (tDb + tT ĉ(∇f))2
− a−1
χg(F )
In view of Theorem 4.5, we write
(4.21)
gN exp
− (tDb + tT ĉ(∇f))2
− a−1
χg(F )
gN exp
− (tDb + tT ĉ(∇f))2
TrF [g]
L exp
(tT )2
F, bF
) ∫ B
d̂f exp
−B(tT )2
χg(F )
TrF [g]
L exp
(tT )2
− a−1
F, bF
) ∫ B
d̂f exp
−B(tT )2
By [BZ1, Definitions 3.6, 3.12 and Theorem 3.18], one has, as T → +∞,
(4.22)
F, bF
) ∫ B
d̂f exp
−B(tT )2
F, bF
(∇f)∗ψ
TMg,∇TMg
From [BZ1, (3.54)], [SZ, (3.35)] and integration by parts, we have
(4.23)
TrF [g]
L exp
(tT )2
− a−1
TrF [g]
L exp (−BT ) +
Ta−1 − T
TrF [g]f
exp (−BT )
TrF [g]f
exp (−B0) .
From Theorems 4.5, 4.6, [BZ1, Theorem 3.20], [BZ1, (7.72) and (7.73)] and the
dominate convergence, one finds that as T → +∞,
(4.24)
gN exp
− (tDb + tT ĉ(∇f))2
TrF [g]
L exp
(tT )2
F, bF
) ∫ B
d̂f exp
−B(tT )2
χg(F )
gN exp
Db + t
T ĉ(∇f)
TrF [g]
L exp
F, bF
) ∫ B
d̂f exp
χg(F )
g|F |Mg,j
1− e−2t2
1 + e−2t
x∈B∩Mg,j
(−1)indg(x)indg(x)− dimMg,je−2t
χ(Mg,j)
g|F |Mg,j
sinh(2t2)
cosh(2t2)− cos(βk)
x∈B∩Mg,j
(−1)indg(x)n−(βk)(x)
g|F |Mg,j
sinh(2t2)
cosh(2t2)− cos(βk)
dimNβkχ(Mg,j)
g|F |Mg,j
x∈B∩Mg,j
(−1)indg(x) (dimMg,j − 2indg(x))−
χg(F )
g|F |Mg,j
x∈B∩Mg,j
(−1)indg(x)indg(x)−
χ(Mg,j)dimMg,j
1 + e−2t
1− e−2t −
g|F |Mg,j
dimNβkχ(Mg,j)−
x∈B∩Mg,j
(−1)indg(x)n−(βk)(x)
sinh(2t)
cosh(2t)− cos(βk)
On the other hand, by Theorems 4.6, 4.7 and the dominate convergence, we
have that as T → +∞,
(4.25)
∫ √Td
gN exp
Db + t
T ĉ(∇f)
a−1 −
χg(F )
∫ √Td
gN exp
Db + t
T ĉ(∇f)
− χ̃′g(F )
χ̃′g(F ) log (Td) + a−1
χg(F ) log (Td)
g|F |Mg,j
1− e−2t2
1 + e−2t
x∈B∩Mg,j
(−1)indg(x)indg(x)− dimMg,je−2t
χ(Mg,j)
g|F |Mg,j
sinh(2t2)
cosh(2t2)− cos(βk)
x∈B∩Mg,j
(−1)indg(x)n−(βk)(x)
g|F |Mg,j
sinh(2t2)
cosh(2t2)− cos(βk)
dimNβkχ(Mg,j)− χ̃′g(F )
χ̃′g(F ) log (Td) + a−1
χg(F ) log (Td) + o(1)
g|F |Mg,j
x∈B∩Mg,j
(−1)indg(x)indg(x)−
χ(Mg,j)dimMg,j
1− e−2t
g|F |Mg,j
dimNβkχ(Mg,j)−
x∈B∩Mg,j
(−1)indg(x)n−(βk)(x)
sinh(2t)
cosh(2t)− cos(βk)
χ̃′g(F )−
χg(F )
log(Td) +
Ta−1 + o(1).
Combining (4.4), (4.14) and (4.20)-(4.25), one deduces, by setting T → +∞,
(4.26) log
P detH∞
det(H∗(M,F ),G)
det(H∗(Wu,F ),G)
− 2TrBgs [f ]T +
χ̃′g(F )−
χg(F )
log T −
χ̃′g(F )−
χg(F )
log π
F, bF
(∇f)∗ψ
TMg,∇TMg
TrF [g]
L exp (−BT )−2
Ta−1+2T
TrF [g]f
exp (−BT )
TrF [g]f
exp (−B0)
g|F |Mg,j
x∈B∩Mg,j
(−1)indg(x)indg(x)−
χ(Mg,j)dimMg,j
1 + e−2t
1− e−2t −
2e−2t
1− e−2t
g|F |Mg,j
dimNβkχ(Mg,j)−
x∈B∩Mg,j
(−1)indg(x)n−(βk)(x)
sinh(2t)
cosh(2t)− cos(βk)
sinh(2t)
cosh(2t)− cos(βk)
χ̃′g(F )−
χg(F )
log(Td)− 2a−1√
TrF [g]fe
TMg,∇TMg
2a−1√
Γ′(1)− log d
χ̃′g(F )−
χg(F )
+o(1).
By [BZ1, Theorem 3.20] and [BZ1, (7.72)], one has
(4.27) lim
TrF [g]f
exp(−BT )− 2TTr
s [f ]
g|F |Mg,j
x∈B∩Mg,j
(−1)indg(x)indg(x)−
χ(Mg,j)dimMg,j
(4.28) lim
TrF [g]
L exp(−BT )
g|F |Mg,j
x∈B∩Mg,j
(−1)indg(x)indg(x)−
χ(Mg,j)dimMg,j
On the other hand, by [BZ1, (7.93)] and [BZ2, (5.55)], one has
1 + e−2t
1− e−2t −
2 e−2t
1− e−2t
= 1− log π − Γ′(1),(4.29)
(4.30)
sinh(2t)
cosh(2t)− cos(βk)
sinh(2t)
cosh(2t)− cos(βk)
= − log(π)− 1
1− βk
Also, by [BZ2, (5.64)], if x ∈ B ∩Mg,
dimNβk
− n−(βk)(x)
[n+(βk)(x)− n−(βk)(x)] .(4.31)
From (4.26)-(4.31), we get (3.7), which completes the proof of Theorem 3.1.
5 Proofs of the intermediary Theorems
The purpose of this section is to give a sketch of the proofs of the intermediary
Theorems. Since the methods of the proofs of these theorems are essentially the
same as the corresponding theorem in [SZ], so we will refer to [SZ] for related
definitions and notations directly when there will be no confusion, such as Bb,g,
Ab,t,T , Ag,t,T , Ct,T , · · · .
5.1 Proof of Theorem 4.1
From Theorem 2.5 and [SZ, (4.44)] which in our situation we also have that
P∞,T commutate with g ∈ G, one finds
[0,1],detH
det(H∗(Ω∗
[0,1],T
(M,F )),G)
det(H∗(Wu,F ),G)
(g) =
∞,TP∞,T
[0,1],T
(M,F )
)(−1)i+1
(5.1)
From [SZ, Propositions 4.4 and 4.5], one deduces that as T → +∞,
(5.2) det
∞,TP∞,T
[0,1],T
(M,F )
= det
∞,TP∞,T
[0,1],T
(M,F )
(g) · det−1
[0,1],T
(M,F )
= det
(P∞,T eT )
P∞,T eT
Ci(Wu,F )
(g) · det−1
Ci(Wu,F )
= det
))# ( π
)N−n/2
))∣∣∣∣
Ci(Wu,F )
· det−1
Ci(Wu,F )
From (5.1) and (5.2), one gets (4.4) immediately.
The proof of Theorem 4.1 is completed. Q.E.D.
5.2 Proof of Theorem 4.2
The proof of Theorem 4.2 is the same as the proof of [SZ, Theorem 3.4] given
in [SZ, Section 5].
5.3 Proof of Theorem 4.3
Recall that the operator eT : C
∗(W u, F ) → Ω∗
[0,1],T
(M,F ) has been defined
in [SZ, (4.38)], and in the current case, we also have that eT commute with
G. So by [SZ, Proposition 4.4], we have that for T ≥ 0 large enough, eT :
C∗(W u, F ) → Ω∗
[0,1],T
(M,F ) is an identification of G-spaces. So (4.7) follows.
Also (4.8) was already proved in [SZ, Theorem 3.5].
5.4 Proof of Theorem 4.4
In this section, we provide a proof of Theorem 4.4, which computes the asymp-
totic of Trs[gN exp(−tD2bT )] for fixed T ≥ 0 as t → 0. The method is the
essentially same as it in [SZ].
By [SZ, (6.4)], we have
(5.3)
b = e−tD
(−1)ktk
e−t1tD
gBb,ge
−t2tD2g · · ·Bb,ge−tk+1tD
gdt1 · · · dtk
+ (−1)n+1tn+1
e−t1tD
gBb,ge
−t2tD2g · · ·Bb,ge−tn+2tD
bdt1 · · · dtn+1,
where ∆k, 1 ≤ k ≤ n+1, is the k-simplex defined by t1+ · · ·+ tk+1 = 1, t1 ≥ 0,
· · · , tk+1 ≥ 0. Also, by the same proof of [SZ, Proposition 6.1], we have the
following result.
Proposition 5.1. As t→ 0+, one has
gNe−t1tD
gBb,ge
−t2tD2g · · ·Bb,ge−tn+2tD
dt1 · · · dtn+1 → 0.
(5.4)
By [SZ, (6.22) and (6.23)], we have that for any 1 < k ≤ n, (t1, · · · , tk+1) ∈
tkTrs
gNe−t1tD
gBb,ge
−t2tD2g · · ·Bb,ge−tk+1tD
= 0,(5.5)
while for k = 1, 0 ≤ t1 ≤ 1,
(5.6) lim
gNe−t1tD
gBb,ge
−(1−t1)tD2g
= lim
gNBb,ge
−tD2g
i, j=1
ei ∧ êj
∇ueiω
F (ej)
ωF , ω̂Fg − ω̂F
· L exp
So by [BZ2, (2.13)], and proceed as in [SZ, (6.26)-(6.28)], we have
gNe−t1tD
gBb,ge
−(1−t1)tD2g
= 0.(5.7)
From (5.3), (5.4), (5.5), (5.7) and [BZ2, Theorem 5.9], one gets (4.9).
The proof of Theorem 4.4 is completed. Q.E.D.
5.5 Proof of Theorem 4.5
In order to prove (4.10), one need only to prove that under the conditions of
Theorem 4.5, there exists constant C ′′ > 0 such that
(5.8)∣∣∣Trs
gN exp
− (tDb + T ĉ(∇f))2
− Trs
gN exp
− (tDg + T ĉ(∇f))2
F, bF
F, gF
)) ∫ B
d̂f exp (−BT 2)
∣∣∣∣∣ ≤ C
By [SZ, (7.8)], we have
(5.9) e
b,t,T = e
g,t,T
(−1)k
−t1A2g,t,TCt,T e
−t2A2g,t,T · · ·Ct,T e−tk+1A
g,t,T dt1 · · · dtk
+ (−1)n+1
−t1A2g,t,TCt,T e
−t2A2g,t,T · · ·Ct,T e−tn+2A
b,t,T dt1 · · · dtn+1.
By the same proof of [SZ, (7.21)], we have that there exists C1 > 0 such that
for any t > 0 small enough and T ∈ [0, 1
∣∣∣∣∣
−t1A2g,t,TCt,T e
−t2A2g,t,T · · ·Ct,T e−tn+2A
b,t,T
dt1 · · · dtn+1
∣∣∣∣∣ ≤ C1t.
(5.10)
Also by the same proof of [SZ, (7.23)], we have that there exists C2 > 0,
0 < d < 1 such that for any 1 < k ≤ n, 0 < t ≤ d, T ≥ 0 with tT ≤ 1,
−t1A2g,t,TCt,T e
−t2A2g,t,T · · ·Ct,T e−tk+1A
g,t,T
dt1 · · · dtk
∣∣∣∣ ≤ C2t,
(5.11)
while for k = 1 one has for any 0 < t ≤ d, T ≥ 0 with tT ≤ 1 and 0 ≤ t1 ≤ 1,
by [BZ2, Proposition 9.3], we have
(5.12)∣∣∣∣∣Trs
−t1A2g,t,TCt,T e
−(1−t1)A2g,t,T
gωF (∇f)
L exp (−BT 2)
∣∣∣∣∣
≤ C2t.
Now similar as [SZ, (7.25)], we have
(5.13)
gωF (∇f)
L exp (−BT 2)
F, gF
F, bF
)) ∫ B
∇̂f exp (−BT 2) .
From (5.9)-(5.13), we get (5.8), which completes the proof of Theorem 4.5.
Q.E.D.
5.6 Proof of Theorem 4.6
In order to prove Theorem 4.6, we need only to prove that for any T > 0,
gN exp
b,t,T
gN exp
g,t,T
= 0.(5.14)
By [SZ, (8.2) and (8.4)], there exists 0 < C0 ≤ 1, such that when 0 < t ≤ C0,
one has the absolute convergent expansion formula
(5.15) e
b,t, T
t − e
g,t, T
(−1)k
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
t dt1 · · · dtk,
and that
(−1)k
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
t dt1 · · · dtk(5.16)
is uniformly absolute convergent for 0 < t ≤ C0.
Proceed as in [SZ, Section 8], one has that for any (t1, · · · , tk+1) ∈ ∆k \
{t1 · · · tk+1 = 0},
(5.17)
∣∣∣∣Trs
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
]∣∣∣∣
≤ C3tk (t1 · · · tk)−
g,t, T
∥∥∥∥∥ψe
g,t, T
∥∥∥∥∥
for some positive constant C3 > 0.
Also, by [SZ, (8.4)], (5.17) and the same assumption in [SZ] that tk+1 ≥ 1k+1 ,
one gets
(5.18)
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
dt1 · · · dtk
≤ C4tk−n
∥∥∥∥ψe
2(k+1)
g,t, T
for some constant C4 > 0.
From (5.15), (5.16), (5.18), [SZ, (8.9) and (8.10)] and the dominate conver-
gence, we get (5.14), which completes the proof of Theorem 4.6. Q.E.D.
5.7 Proof of Theorem 4.7
In order to prove Theorem 4.7, we need only to prove that there exist c > 0,
C > 0, 0 < C0 ≤ 1 such that for any 0 < t ≤ C0, T ≥ 1,
∣∣∣Trs
gN exp
b,t,T
− Trs
gN exp
g,t,T
)]∣∣∣ ≤ c exp(−CT ).(5.19)
First of all, one can choose C0 > 0 small enough so that for any 0 < t ≤ C0,
T > 0, by (5.15), we have the absolute convergent expansion formula
(5.20) e
b,t, T
t − e
g,t, T
(−1)k
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
t dt1 · · · dtk,
from which one has
(5.21) Trs
gN exp
b,t,T
− Trs
gN exp
g,t,T
(−1)k
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
dt1 · · · dtk.
Thus, in order to prove (5.19), we need only to prove
(5.22)
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
dt1 · · · dtk
−(t1+tk+1)A2
g,t, T
−t2A2
g,t, T
t · · ·C
dt1 · · · dtk
≤ c exp(−CT ).
By [SZ, (8.6)], we have for any t > 0, T ≥ 1, (t1, · · · , tk+1) ∈ ∆k \
{t1 · · · tk+1 = 0},
(5.23) Trs
−(t1+tk+1)A2
g,t, T
−t2A2
g,t, T
t · · ·C
= Trs
−(t1+tk+1)A2
g,t, T
t Ct,T
−t2A2
g,t, T
t Ct,T
· · ·ψe
−tkA2
g,t, T
t Ct,T
From (5.23), [SZ, (9.18) and (9.19)], one sees that there exists C5 > 0,
C6 > 0 and C7 > 0 such that for any k ≥ 1,
(5.24)
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
dt1 · · · dtk
≤ C5 (C6t)k
from which one sees that there exists 0 < c1 ≤ 1, C8 > 0, C9 > 0 such that for
any 0 < t ≤ c1 and T ≥ 1, one has
(5.25)∣∣∣∣∣
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
dt1 · · · dtk
∣∣∣∣∣
≤ C8 exp (−C9T ) .
On the other hand, for any 1 ≤ k < n, by proceeding as in (5.18), one has
that for any 0 < t ≤ c1, T ≥ 1,
(5.26)
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
dt1 · · · dtk
≤ C10tk−n
∥∥∥∥ψe
2(k+1)
g,t, T
for some constant C10 > 0.
From (5.26) and [SZ, (9.23)], one sees immediately that there exists C11 > 0,
C12 > 0 such that for any 1 ≤ k ≤ n− 1, 0 < t ≤ c1 and T ≥ 1, one has
(5.27)
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
dt1 · · · dtk
≤ C11e−C12T .
From (5.21), (5.25) and (5.27), one gets (5.19).
The proof of Theorem 4.7 is completed. Q.E.D.
References
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http://arxiv.org/abs/math/0610577
	Introduction
	Equivariant symmetric bilinear torsions associated to the de Rham and Thom-Smale complexes
	Equivariant symmetric bilinear torsion of a finite dimensional complex
	The Thom-Smale complex of a gradient field
	Equivariant Milnor symmetric bilinear torsion
	Equivariant Ray-Singer symmetric bilinear torsion 
	An anomaly formula for the equivariant Ray-Singer symmetric bilinear torsion 
	A formula relating equivariant Milnor and equivariant Ray-Singer symmetric bilinear torsions
	A proof of Theorem 3.1
	Proofs of the intermediary Theorems
	Proof of Theorem ??
	Proof of Theorem ??
	Proof of Theorem ??
	Proof of Theorem ??
	Proof of Theorem ??
	Proof of Theorem ??
	Proof of Theorem ??

  We extend the main result in the previous paper of Zhang and the author
relating the Milnor-Turaev torsion with the complex valued analytic torsion to
the equivariant case.

Introduction
Let F be a unitary flat vector bundle on a closed Riemannian manifold X.
In [RS], Ray and Singer defined an analytic torsion associated to (X,F ) and
proved that it does not depend on the Riemannian metric on X. Moreover,
they conjectured that this analytic torsion coincides with the classical Reide-
meister torsion defined using a triangulation on X (cf. [Mi]). This conjecture
was later proved in the celebrated papers of Cheeger [C] and Müller [Mu1].
Müller generalized this result in [Mu2] to the case where F is a unimodular
flat vector bundle on X. In [BZ1], inspired by the considerations of Quillen
[Q], Bismut and Zhang reformulated the above Cheeger-Müller theorem as an
equality between the Reidemeister and Ray-Singer metrics defined on the de-
terminant of cohomology, and proved an extension of it to the case of general
flat vector bundles over X. The method used in [BZ1] is different from those of
Cheeger and Müller in that it makes use of a deformation by Morse functions
introduced by Witten [W] on the de Rham complex.
On the other hand, Turaev generalizes the concept of Reidemeister torsion
to a complex valued invariant whose absolute value provides the original Reide-
meister torsion, with the help of the so-called Euler structure (cf. [T], [FT]). It
is natural to ask whether there exists an analytic interpretation of this Turaev
torsion.
Recently, Burghelea and Haller [BH1, BH2], following a suggestion of Müller,
define a generalized analytic torsion associated to a nondegenerate symmetric
bilinear form on a flat vector bundle over a closed manifold and make an explicit
conjecture between this generalized analytic torsion and the Turaev torsion.
Later this conjecture was proved by Su and Zhang [SZ]. Also Burghelea and
Haller [BH3], up to a sign, proved this conjecture for odd dimensional manifolds,
and comments were made how to derive the conjecture in full generality in their
paper.
∗This work was partially supported by the Qiushi Foundation.
†Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, P.R. China.
(sugx@mail.nankai.edu.cn)
http://arxiv.org/abs/0704.1341v4
In this paper, we will extend the main result in [SZ] to the equivariant case,
which is closer in spirit to the approach developed by Bismut-Zhang in [BZ2].
The rest of this paper is organized as follows. In Section 2, we construct the
equivariant symmetric bilinear torsions associated with equivariant nondegen-
erate symmetric bilinear forms on a flat vector bundle. In Section 3, we state
the main result of this paper. In Section 4, we provide a proof of the main
result. Section 5 is devoted to the proofs of the intermediary results stated in
Section 4.
Since we will make substantial use of the results in [BZ1, BZ2, SZ], we will
refer to [BZ1, BZ2, SZ] for related definitions and notations directly when there
will be no confusion.
2 Equivariant symmetric bilinear torsions associated
to the de Rham and Thom-Smale complexes
In this section, for a G-invariant nondegenerate bilinear symmetric form on
a complex flat vector bundle over an oriented closed manifold, we define two
naturally associated equivariant symmetric bilinear forms on the equivariant
determinant of the cohomology H∗(M,F ) with coefficient F . One constructed
in a combinatorial way through the equivariant Thom-Smale complex associated
to a equivariant Morse function, and the other one constructed in an analytic
way through the equivariant de Rham complex.
2.1 Equivariant symmetric bilinear torsion of a finite dimensional
complex
Let (C, ∂) be a finite cochain complex
(C, ∂) : 0 −→ C0 ∂0−→ C1 ∂1−→ · · · ∂n−1−→ Cn −→ 0,(2.1)
where each Ci, 0 ≤ i ≤ n, is a finite dimensional complex vector space.
Let each Ci, 0 ≤ i ≤ n, admit a nondegenerate symmetric bilinear form bi.
We equip C with the nondegenerate symmetric bilinear form bC =
i=0 bi.
Let G be a compact group. Let ρ : G → End(C) be a representation of G,
with values in the chain homomorphisms of C which preserve the bilinear form
bC . In particular, if g ∈ G, ρ(g) preserves the Ci’s.
Let Ĝ be the set of equivalence classes of complex irreducible representations
of G. An element of Ĝ is specified by a complex finite dimensional vector space
W together with an irreducible representation ρW : G→ End(W ).
For W ∈ Ĝ, set
CiW = HomG(W,C
i)⊗W,(2.2)
CW = HomG(W,C)⊗W.(2.3)
Let ∂W be the map induced by ∂ on CW . Then
(CW , ∂W ) : 0 −→ C0W
∂0,W−→ C1W
∂1,W−→ · · ·
∂n−1,W−→ CnW −→ 0(2.4)
is a chain complex. Thus we obtain the isotypical decomposition,
(C, ∂) =
W∈ bG
(CW , ∂W ),(2.5)
and the decomposition (2.5) is orthogonal.
If E is a complex finite dimensional representation space for G, let χ(E) be
the character of the representation. Put
χ(C) =
(−1)iχ(Ci),
e(C) =
(−1)idimCi,
e(CW ) =
(−1)idim(CiW ).(2.6)
By (2.5), we get
χ(C) =
W∈ bG
e(CW )
χ(W )
rk(W )
.(2.7)
If λ is a complex line, let λ−1 be the dual line. If E is a finite dimensional
complex vector space, set
detE = Λmax(E).(2.8)
detC =
detCi
)(−1)i
detCW =
detCiW
)(−1)i
.(2.9)
By (2.5), we obtain
detC =
W∈ bG
detCW .(2.10)
For 0 ≤ i ≤ n, CiW is a vector subspace of Ci. Let bCiW be the induced
symmetric bilinear form on CiW . let bdetCi
be the symmetric bilinear form on
detCiW induced by bCi
, and let b(detCi
)−1 be the dual symmetric bilinear form
on (detCiW )
−1. Also we have symmetric bilinear forms bdetCW on detCW and
bdetC on detC.
det(C,G) =
W∈ bG
detCW .(2.11)
Definition 2.1. We introduce the formal product
bdet(C,G) =
W∈ bG
(bdetCW )
χ(W )
rk(W ) .(2.12)
For W ∈ Ĝ, let xW , yW ∈ detCW , xW 6= 0, yW 6= 0. Set x = ⊕W∈ bGxW ,
y = ⊕
W∈ bGyW ∈ det(C,G). Then by definition,
bdet(C,G)(x, y) =
W∈ bG
(bdetCW (xW , yW ))
χ(W )
rk(W ) .(2.13)
Tautologically, (2.13) is an identity of characters on G. In particular
bdet(C,G)(x, y)(1) =
W∈ bG
bdetCW (xW , yW ).(2.14)
In fact (2.14) just says that
bdet(C,G)(1) = bdetC .(2.15)
Of course, using the orthogonality of the χW ’s, knowing the formal product
bdet(C,G) is equivalent to knowing the symmetric bilinear forms bdetCW .
Clearly
H(CW , ∂W ) = HomG(W,H(C, ∂)) ⊗W,
H(C, ∂) =
W∈ bG
H(CW , ∂W ).(2.16)
For W ∈ Ĝ, we define detH(CW , ∂W ) as in (2.9). Set
det(H(C, ∂), G) =
W∈ bG
detH(CW , ∂W ).(2.17)
For W ∈ Ĝ, there is a canonical isomorphism (cf. [KM] and [BGS, Section
1a)])
detCW ≃ detH(CW , ∂W ).(2.18)
From (2.18), we get
det(C,G) ≃ det(H(C, ∂), G).(2.19)
Let bdetH(CW ,∂W ) be the symmetric bilinear form on detH(CW , ∂W ) corre-
sponding to bdetCW via the canonical isomorphism (2.18).
Definition 2.2. we introduce the formal product
bdet(H(C,∂),G) =
W∈ bG
bdetH(CW ,∂W )
)χ(W )
rkW .(2.20)
Tautologically, under the identification (2.19),
bdet(C,G) = bdetH((C,∂),G).(2.21)
By an abuse of notation, we will call the formal product bdet(C,G) a sym-
metric bilinear form on det(C,G).
2.2 The Thom-Smale complex of a gradient field
Let M be a closed smooth manifold, with dimM = n. For simplicity, we make
the assumption that M is oriented.
Let (F,∇F ) be a complex flat vector bundle overM carrying the flat connec-
tion ∇F . We make the assumption that F carries a nondegenerate symmetric
bilinear form bF .
Let (F ∗,∇F ∗) be the dual complex flat vector bundle of (F,∇F ) carrying
the dual flat connection ∇F ∗.
Let f : M → R be a Morse function. Let gTM be a Riemannian metric on
TM such that the corresponding gradient vector field −X = −∇f ∈ Γ(TM)
satisfies the Smale transversality conditions (cf. [Sm]), that is, the unstable
cells (of −X) intersect transversally with the stable cells.
B = {x ∈M ;X(x) = 0}.(2.22)
For any x ∈ B, let W u(x) (resp. W s(x)) denote the unstable (resp. stable)
cell at x, with respect to −X. We also choose an orientation O−x (resp. O+x ) on
W u(x) (resp. W s(x)).
Let x, y ∈ B satisfy the Morse index relation ind(y) = ind(x) − 1, then
Γ(x, y) =W u(x)∩W s(y) consists of a finite number of integral curves γ of −X.
Moreover, for each γ ∈ Γ(x, y), by using the orientations chosen above, on can
define a number nγ(x, y) = ±1 as in [BZ1, (1.28)].
If x ∈ B, let [W u(x)] be the complex line generated by W u(x). Set
u, F ∗) =
[W u(x)]⊗ F ∗x ,(2.23)
u, F ∗) =
x∈B, ind(x)=i
[W u(x)]⊗ F ∗x .(2.24)
If x ∈ B, the flat vector bundle F ∗ is canonically trivialized on W u(x). In
particular, if x, y ∈ B satisfy ind(y) = ind(x) − 1, and if γ ∈ Γ(x, y), f∗ ∈ F ∗x ,
let τγ(f
∗) be the parallel transport of f∗ ∈ F ∗x into F ∗y along γ with respect to
the flat connection ∇F ∗.
Clearly, for any x ∈ B, there is only a finite number of y ∈ B, satisfying
together that ind(y) = ind(x)− 1 and Γ(x, y) 6= ∅.
If x ∈ B, f∗ ∈ F ∗x , set
∂(W u(x)⊗ f∗) =
y∈B, ind(y)=ind(x)−1
γ∈Γ(x,y)
nγ(x, y)W
u(y)⊗ τγ(f∗).(2.25)
Then ∂ maps Ci(W
u, F ∗) into Ci−1(W
u, F ∗). Moreover, one has
∂2 = 0.(2.26)
That is, (C∗(W
u, F ∗), ∂) forms a chain complex. We call it the Thom-Smale
complex associated to (M,F,−X).
If x ∈ B, let [W u(x)]∗ be the dual line to W u(x). Let (C∗(W u, F ), ∂) be
the complex which is dual to (C∗(W
u, F ∗), ∂). For 0 ≤ i ≤ n, one has
Ci(W u, F ) =
x∈B, ind(x)=i
[W u(x)]∗ ⊗ Fx.(2.27)
Let G be a compact group acting on M by smooth diffeomorphisms. we
assume that the action of G lifts to F and preserves the flat connection of
F . Then G acts naturally on H∗(M,F ). We assume that f and gTM are G-
invariant. Then −X = −∇f is also G-invariant. We assume that it verifies the
smale transversality conditions.
Clearly B is G-invariant. Also if x ∈ B, g ∈ G,
g (W u(x)) = ǫg(x)W
u(gx),
where ǫg(x) = +1 if g(W
u(x)) has the same orientation as W u(gx), ǫg = −1
if not. Clearly g acts as a chain homomorphism on (C∗(W
u, F ∗), ∂). The
corresponding dual action of g on (C∗(W u, F ), ∂) is such that
g (W u(x)∗) = ǫg(x)W
u(gx)∗.
Then g acts as a chain homomorphism on (C∗(W u, F ), ∂). Therefore g acts on
H∗(C∗(W u, F ), ∂).
2.3 Equivariant Milnor symmetric bilinear torsion
For x ∈ B, let bFx be a nondegenerate symmetric bilinear form on Fx. We
assume that the bFx’s are G-invariant, i.e. for g ∈ G, x ∈ B
= bFg(x).(2.28)
The symmetric bilinear forms bFx ’s determine a G-invariant symmetric bilin-
ear form on C∗(W u, F ) =
x∈B [W
u(x)]∗⊗Fx, such that the various [W u(x)]∗⊗
Fx are mutually orthogonal in C
∗(W u, F ), and that if x ∈ B, f, f ′ ∈ Fx,
W u(x)∗ ⊗ f,W u(x)∗ ⊗ f ′
f, f ′
.(2.29)
We construct the equivariant symmetric bilinear form bdet(C∗(Wu,F ),G) on
det(C∗(W u, F ), G) as in Definition 2.1.
Definition 2.3. The symmetric bilinear form on the determinant line of the
cohomology of the Thom-Smale cochain complex (C∗(W u, F ), ∂), in the sence
of Definition 2.2, is called the equivariant Milnor symmetric bilinear torsion
and is denoted by b
det(H∗(Wu,F ),G).
Take g ∈ G. Set
Mg = {x ∈M, gx = x}.(2.30)
Since G is a compact group, Mg is a smooth compact submanifold of M . Let
N be the normal bundle to Mg in M . By [BZ2, Proposition 1.13], we know
that f |Mg is a Morse function on Mg, and X|Mg is a smooth section of TMg.
For g ∈ G, set
Bg = B ∩Mg.(2.31)
Then Bg is the set of critical points of f |Mg .
Definition 2.4. If x ∈ Bg, let indg(x) be the index of f |Mg at x.
Let now bFx , b′Fx (x ∈ B) be two G-invariant nondegenerate symmetric
bilinear forms on Fx. Let b
det(H∗(Wu,F ),G), b
′M,−X
det(H∗(Wu,F ),G) be the corresponding
equivariant Milnor symmetric bilinear torsions. By [BZ2, Theorem 1.15] and
[SZ, Proposition 2.5], we have the following theorem.
Theorem 2.5. For g ∈ G, the following identity holds
′M,−X
det(H∗(Wu,F ),G)(g) = b
det(H∗(Wu,F ),G)(g)
g log
)])(−1)indg(x)
(2.32)
2.4 Equivariant Ray-Singer symmetric bilinear torsion
We continue the discussion of the previous subsection. However, we do not use
the Morse function and make transversality assumptions.
For any 0 ≤ i ≤ n, denote
Ωi(M,F ) = Γ
Λi(T ∗M)⊗ F
, Ω∗(M,F ) =
Ωi(M,F ).(2.33)
Let dF denote the natural exterior differential on Ω∗(M,F ) induced from ∇F
which maps each Ωi(M,F ), 0 ≤ i ≤ n, into Ωi+1(M,F ).
The group G acts naturally on Ω∗(M,F ). Namely, if g ∈ G, s ∈ Ω∗(M,F ),
gs(x) = g∗s(g
−1x), x ∈M.
Let gF be a G-invariant Hermitian metric on F . The G-invariant Rieman-
nian metric gTM and gF determine a natural inner product 〈 , 〉g (that is, a
pre-Hilbert space structure) on Ω∗(M,F ) (cf. [BZ1, (2.2)] and [BZ2, (2.3)]).
Let dF∗g be the formal adjoint of d
F with respect to 〈 , 〉g and Dg = dF +dF∗g .
On the other hand gTM and the G-invariant symmetric bilinear form bF
determine together a G-invariant symmetric bilinear form on Ω∗(M,F ) such
that if u = αf , v = βg ∈ Ω∗(M,F ) such that α, β ∈ Ω∗(M), f, g ∈ Γ(F ), then
〈u, v〉b =
(α ∧ ∗β)bF (f, g),(2.34)
where ∗ is the Hodge star operator (cf. [Z]).
Consider the de Rham complex
(2.35)
Ω∗(M,F ), dF
: 0 → Ω0(M,F ) d
→ Ω1(M,F ) → · · ·
dF→ Ωn(M,F ) → 0.
Let dF∗b : Ω
∗(M,F ) → Ω∗(M,F ) denote the formal adjoint of dF with
respect to G-invariant the symmetric bilinear form in (2.34). That is, for any
u, v ∈ Ω∗(M,F ), one has
dFu, v
u, dF∗b v
.(2.36)
Db = d
F + dF∗b , D
dF + dF∗b
= dF∗b d
F + dF dF∗b .(2.37)
Then the Laplacian D2b preserves the Z-grading of Ω
∗(M,F ).
As was pointed out in [BH1] and [BH2], D2b has the same principal symbol
as the usual Hodge Laplacian (constructed using the inner product on Ω∗(M,F )
induced from (gTM , gF )) studied for example in [BZ1].
We collect some well-known facts concerning D2b as in [BH2, Proposition
4.1], where the reference [S] is indicated.
Proposition 2.6. The following properties hold for the Laplacian D2b :
(i) The spectrum of D2b is discrete. For every θ > 0 all but finitely many
points of the spectrum are contained in the angle {z ∈ C| − θ < arg(z) < θ};
(ii) If λ is in the spectrum of D2b , then the image of the associated spectral
projection is finite dimensional and contains smooth forms only. We refer to
this image as the (generalized) λ-eigen space of D2b and denote it by Ω
{λ}(M,F ).
There exists Nλ ∈ N such that
D2b − λ
)Nλ∣∣∣
Ω∗{λ}(M,F )
= 0.(2.38)
We have a D2b -invariant 〈 , 〉b-orthogonal decomposition
Ω∗(M,F ) = Ω∗{λ}(M,F )⊕ Ω
{λ}(M,F )
⊥.(2.39)
The restriction of D2b − λ to Ω∗{λ}(M,F )
⊥ is invertible;
(iii) The decomposition (2.39) is invariant under dF and dF∗b ;
(iv) For λ 6= µ, the eigen spaces Ω∗{λ}(M,F ) and Ω
{µ}(M,F ) are 〈 , 〉b-
orthogonal to each other.
For any a ≥ 0, set
Ω∗[0,a](M,F ) =
0≤|λ|≤a
Ω∗{λ}(M,F ).(2.40)
Let Ω∗
[0,a]
(M,F )⊥ denote the 〈 , 〉b-orthogonal complement to Ω∗[0,a](M,F ).
Obviously, each Ω∗{λ}(M,F ) is a G-invariant subspace.
By [BH2, (29)] and Proposition 2.6, one sees that (Ω∗
[0,a]
(M,F ), dF ) forms a
finite dimensional complex whose cohomology equals to that of (Ω∗(M,F ), dF ).
Moreover, the G-invariant symmetric bilinear form 〈 , 〉b clearly induces a
nondegenerate G-invariant symmetric bilinear form on each Ωi
[0,a]
(M,F ) with
0 ≤ i ≤ n. By Definition 2.2 one then gets a symmetric bilinear torsion
det(H∗(Ω∗
[0,a]
(M,F )),G)
on detH∗(Ω∗
[0,a]
(M,F ), dF ) = detH∗(Ω∗(M,F ), dF ).
For any 0 ≤ i ≤ n, let D2b,i be the restriction of D2b on Ωi(M,F ). Then it is
shown in [BH2] (cf. [S, Theorem 13.1]) that for any a ≥ 0, g ∈ G the following
is well-defined,
D2b,(a,+∞),i
(g) = exp
D2b,i
[0,a]
(M,F )⊥
)−s])
.(2.41)
Definition 2.7. If g ∈ G, set
bRSdet(H∗(M,F ),G)(g) = b
det(H∗(Ω∗
[0,a]
(M,F )),G)(g)
D2b,(a,+∞),i
)(−1)ii
(2.42)
by [BH2, Proposition 4.7], we know that bRS
det(H∗(M,F ),G) does not depend on the
choice of a ≥ 0, and is called the equivariant Ray-Singer symmetric bilinear
torsion on detH∗(Ω∗(M,F ), dF ).
2.5 An anomaly formula for the equivariant Ray-Singer symmetric
bilinear torsion
We continue the discussion of the above subsection.
Definition 2.8. Let θg(F, b
F ) be the 1-form on Mg
θg(F, b
F ) = Tr
g(bF )−1∇F bF
.(2.43)
ClearlyMg is a totally geodesic submanifold ofM . Let g
TMg be the Rieman-
nian metric induced by gTM on TMg. Let ∇TMg be the Levi-Civita connection
on (TMg, g
TMg ).
Let e(TMg,∇TMg) be the Chern-Weil representative of the rational Euler
class of TMg, associated to the metric preserving connection ∇TMg . Then
e(TMg,∇TMg) = Pf
if dimMg is even,(2.44)
0 if dimMg is odd.
Let g′TM be another G-invariant metric and let ∇′TMg be the correspond-
ing Levi-Civita connection on TMg. Let ẽ(TMg,∇TMg ,∇′TMg) be the Chern-
Simons class of dimMg − 1 forms on Mg, such that
TMg,∇TMg ,∇′TMg
TMg,∇′TMg
TMg,∇TMg
.(2.45)
Let b′F be another G-invariant nondegenerate symmetric bilinear form on
Let b′RS
det(H∗(M,F ),G) denote the equivariant Ray-Singer symmetric bilinear
torsion associated to g′TM and b′F .
By [SZ, Remark 6.4] and [BZ2, Theorem 2.7], we have the following exten-
sion of the anomaly formula of [SZ, Theorem 2.9].
Theorem 2.9. If bF , b′F lie in the same homotopy class of nondegenerate
symmetric bilinear forms on F , then for g ∈ G the following identity holds,
(2.46)
det(H∗(M,F ),G)
det(H∗(M,F ),G)
(g) = exp
g log
TMg,∇TMg
· exp
θg(F, b
′F )ẽ
TMg,∇TMg ,∇′TMg
Proof. Let bFl is a smooth one-parameter family of fiber wise non-degenerate
symmetric bilinear forms on F and (gTMl , g
l ) be a smooth family of metrics on
TM,F .
By [SZ, (6.4)], we have
(2.47)
b = e−tD
(−1)ktk
e−t1tD
gBb,ge
−t2tD2g · · ·Bb,ge−tk+1tD
gdt1 · · · dtk
+ (−1)n+1tn+1
e−t1tD
gBb,ge
−t2tD2g · · ·Bb,ge−tn+2tD
bdt1 · · · dtn+1,
where ∆k, 1 ≤ k ≤ n+1, is the k-simplex defined by t1+ · · ·+ tk+1 = 1, t1 ≥ 0,
· · · , tk+1 ≥ 0 and Bb,g is defined in [SZ, (6.3)].
Proceeding as in [BZ1, Section 4], we first calculate the asymptotics as t→ 0
of Trs[g(b
−1 ∂bFl
exp(−tD2bl)]. Here the metric g
TM will be fixed.
By the same proof in [SZ, proposition 6.1], we have that as t→ 0+,
g(bFl )
−1 ∂b
e−t1tD
gBb,ge
−t2tD2g · · ·Bb,ge−tn+2tD
dt1 · · · dtn+1 → 0.
(2.48)
Also, by [SZ, (6.22)], we have that for 1 ≤ k ≤ n, (t1, · · · , tk+1) ∈ ∆k,
tkTrs
g(bFl )
e−t1tD
gBb,ge
−t2tD2g · · ·Bb,ge−tk+1tD
= 0.(2.49)
So that by (2.47)-(2.49) we have that
g(bFl )
−1 ∂b
−tD2bl
= lim
g(bFl )
−1 ∂b
−tD2gl
(2.50)
Now we assume that the nondegenerate symmetric bilinear form on F is
fixed, and the metric gTMl on TM depends on l.
Let ∗l be the Hodge star operator associated to gTMl .
By [BZ1, (4.70), (4.74)], analogues of [SZ, (6.5), (6.24), (6.26), (6.27)] re-
placing N by ∗−1l
and [BGV, Chapter 6], we have that
(2.51)
g ∗−1
∂∗TMl
−tD2bl
= lim
g ∗−1
∂∗TMl
−tD2gl
i, j=1
ei ∧ êj
∇ueiω
F (ej)
ωF , ω̂Fg − ω̂F
1≤i,j≤n
)−1 ∂gTMl
ei, ej
ei ∧ êj
 exp
− Ṙl
= lim
g ∗−1l
∂∗TMl
−tD2gl
∇TMl ϕTr
1≤i,j≤n
)−1 ∂gTMl
ei, ej
ei ∧ êj
 exp
− Ṙl
1≤i,j≤n
)−1 ∂gTMl
ei, ej
ei ∧ êj
· exp
− Ṙl
∧ ϕθg
F, bF
From (2.50), (2.51) and the calculations in [BZ1, Section 4], we get (2.46).
The proof of Theorem 2.9 is completed. Q.E.D.
3 A formula relating equivariant Milnor and equiv-
ariant Ray-Singer symmetric bilinear torsions
In this section, we state the main result of this paper, which is an explicit
comparison result between the equivariant Milnor symmetric bilinear torsion
and equivariant Ray-Singer symmetric bilinear torsion.
We assume that we are in the same situation as in Sections 2.2-2.4. By a
simple argument of Helffer-Sjöstrand [HS, Proposition 5.1] (cf. [BZ1, Section
7b)]), we may and we well assume that gTM there satisfies the following property
without altering the Thom-Smale cochain complex (C∗(W u, F ), ∂),
(*): For any x ∈ B, there is a system of coordinates y = (y1, · · · , yn)
centered at x such that near x,
gTM =
∣∣dyi
∣∣2 , f(y) = f(x)− 1
ind(x)∑
∣∣2 + 1
i=ind(x)+1
∣∣2 .(3.1)
By a result of Laudenbach [L], {W u(x) : x ∈ B} form a CW decomposition
of M .
For any x ∈ B, F is canonically trivialized over each cell W u(x).
Let P∞ be the de Rham map defined by
α ∈ Ω∗(M,F ) → P∞α =
W u(x)∗
Wu(x)
α ∈ C∗(W u, F ).(3.2)
By the Stokes theorem, one has
∂P∞ = P∞d
F .(3.3)
Moreover, it is shown in [L] that P∞ is a Z-graded quasi-isomorphism, inducing
a canonical isomorphism
PH∞ : H
∗ (Ω∗(M,F ), dF
→ H∗ (C∗ (W u, F ) , ∂) ,(3.4)
which in turn induces a natural isomorphism between the determinant lines,
P detH∞ : detH
∗ (Ω∗ (M,F ) , dF
→ detH∗ (C∗ (W u, F ) , ∂) .(3.5)
Also by [BZ2, Theorem 1.11], we know that P∞ commutes with G, and P
is the canonical identification of the corresponding cohomology groups as G-
spaces.
Now let hTM be an arbitrary smooth metric on TM .
By Definition 2.7, one has an associated equivariant Ray-Singer symmetric
bilinear torsion bRS
det(H∗(M,F ),G) on detH
∗(Ω∗(M,F ), dF ). From (3.5), one gets
a well-defined equivariant symmetric bilinear form
P detH∞
bRSdet(H∗(M,F ),G)
(3.6)
on detH∗(C∗(W u, F ), ∂).
On the other hand, by Definition 2.3, one has a well-defined equivariant Mil-
nor symmetric bilinear torsion b
det(H∗(M,F ),G) on detH
∗(C∗(W u, F ), ∂), where
X = ∇f is the gradient vector field of f associated to gTM .
Let Mg = ∪mj=1Mg,j be the decomposition of Mg into its connected compo-
nents. Clearly TrF [g] is constant on each Mg,j .
Let N be the normal bundle to Mg in M . We identify N to the orthogonal
bundle to TMg in TM |Mg .
Take x ∈ Bg. Then g acts on TxM as a linear isometry. Also
TMg = {Y ∈ TM |Mg , gY = Y }.
Moreover g acts on N . Let e±iβ1 , · · · , e±iβq (0 < βj ≤ π) be the locally constant
distinct eigenvalues of g|N . Then N splits orthogonally as
Nβj .
For 1 ≤ j ≤ q, g acts on Nβj as an isometry, with eigenvalues e±iβj . In
particular, if e±iβj 6= −1, Nβj is even dimensional.
Take x ∈ Bg. Since f is g-invariant, d2f(x) is also g-invariant. Therefore
the decomposition
TxM = TxMg ⊕
is orthogonal with respect to d2f(x). On TxMg, the index of d
2f(x)|TxMg×TxMg
was already denoted indg(x). Let n+(βj)(x) (resp. n−(βj)(x)) be the num-
ber of positive (resp. negative) eigenvalues of d2f(x)|
βj . Then if e
±iβj 6=
−1, n±(βj)(x) is even.
Let ψ(TMg,∇TMg) be the Mathai-Quillen current ([MQ]) over TMg, asso-
ciated to hTM , defined in [BZ1, Definition 3.6]. As indicated in [BZ1, Remark
3.8], the pull-back current X∗ψ(TMg,∇TMg ) is well-defined over Mg.
The main result of this paper, which generalizes [SZ, Theorem 3.1] to the
equivariant case.
Theorem 3.1. For g ∈ G, the following identity in C holds,
(3.7)
P detH∞
det(H∗(M,F ),G)
det(H∗(Wu,F ),G)
(g) = exp
θg(F, b
F )X∗ψ
TMg,∇TMg
· exp
(−1)indg(x)
(n+(βj)(x)− n−(βj)(x))
1− βj
− 2Γ′(1)
· Tr [g|Fx ]
Remark 3.2. By proceeding similarly as in [BZ2, Section 5b)], in order to prove
(3.7), we may well assume that hTM = gTM . Moreover, we may assume that
bF , as well as the Hermitian metric hF on F , are flat on an open neighborhood
of the zero set B of X. From now on, we will make these assumptions.
4 A proof of Theorem 3.1
We assume that the assumptions in Remark 3.2 hold.
For any T ∈ R, let bFT be the deformed symmetric bilinear form on F defined
bFT (u, v) = e
−2TfbF (u, v).(4.1)
Let dF∗bT be the associated formal adjoint in the sense of (2.36). Set
DbT = d
F + dF∗bT , D
dF + dF∗bT
= dF∗bT d
F + dF dF∗bT .(4.2)
Let Ω∗
[0,1],T
(M,F ) be defined as in (2.40) with respect to D2bT , and let
[0,1],T
(M,F )⊥ be the corresponding 〈 , 〉bT -orthogonal complement.
Let P
[0,1]
T be the orthogonal projection from Ω
∗(M,F ) to Ω∗
[0,1],T
(M,F )
with respect to the inner product determined by gTM and gFT = e
−2TfgF . Set
(1,+∞)
T = Id− P
[0,1]
Following [BZ2, (5.9)-(5.10)], we introduce the notations
χg(F ) =
TrF |Mg,j
x∈Bg∩Mg,j
(−1)indg(x),(4.3)
χ̃′g(F ) =
TrF |Mg,j
x∈Bg∩Mg,j
(−1)indg(x)ind(x),
s [f ] =
TrF |Mg,j
x∈Bg∩Mg,j
(−1)indg(x)f(x).
Let N be the number operator on Ω∗(M,F ) acting on Ωi(M,F ) by multi-
plication by i.
By the technique developed in [SZ] and the corresponding results in [BZ2],
we easily get the following intermediate results. The sketch of the proofs will
be outlined in Section 5.
Theorem 4.1. (Compare with [BZ2, Theorem 5.5] and [SZ, Theorem 3.3]) Let
[0,1]
T be the restriction of P∞ on Ω
[0,1],T
(M,F ), let P
[0,1],detH
T be the induced
isomorphism on cohomology, then the following identity holds,
[0,1],detH
det(H∗(Ω∗
[0,1],T
(M,F )),G)
det(H∗(Wu,F ),G)
χg(F )−eχ′g(F )
s [f ]T
(4.4)
Theorem 4.2. (Compare with [BZ2, Theorem 5.7] and [SZ, Theorem 3.4]) For
any t > 0,
gN exp
−tD2bT
(1,+∞)
= 0.(4.5)
Moreover, for any d > 0 there exist c > 0, C > 0 and T0 ≥ 1 such that for any
t ≥ d and T ≥ T0,
∣∣∣Trs
gN exp
−tD2bT
(1,+∞)
]∣∣∣ ≤ c exp(−Ct).(4.6)
Theorem 4.3. (Compare with [BZ2, Theorem 5.8] and [SZ, Theorem 3.5]) For
T ≥ 0 large enough, then
[0,1]
= χ̃′g(F ).(4.7)
Also,
D2bTP
[0,1]
= 0.(4.8)
For the next results, we will make use the same notation for Clifford multi-
plications and Berezin integrals as in [BZ1, Section 4].
Theorem 4.4. (Compare with [BZ2, Theorem 5.9] and [SZ, Theorem 3.6]) As
t→ 0, the following identity holds,
gN exp
−tD2bT
χg(F ) +O(t) if n is even,(4.9)
TrF [g]
L exp
if n is odd.
Theorem 4.5. (Compare with [BZ2, Theorem A.1] and [SZ, Theorem 3.7])
There exist 0 < α ≤ 1, C > 0 such that for any 0 < t ≤ α, 0 ≤ T ≤ 1
, then
(4.10)
∣∣∣∣∣Trs
gN exp
− (tDb + T ĉ(∇f))2
TrF [g]
L exp (−BT 2)
F, bF
) ∫ B
d̂f exp (−BT 2)−
χg(F )
∣∣∣∣∣ ≤ Ct.
Theorem 4.6. (Compare with [BZ2, Theorem A.2] and [SZ, Theorem 3.8])
For any T > 0, the following identity holds,
(4.11) lim
gN exp
tDb +
ĉ(∇f)
g|F |Mg,j
1− e−2T
(1 + e−2T
x∈B∩Mg,j
(−1)indg(x)indg(x)− dimMg,je−2Tχ(Mg,j)
g|F |Mg,j
sinh(2T )
cosh(2T )− cos(βk)
x∈B∩Mg,j
(−1)indg(x)n−(βk)(x)
g|F |Mg,j
sinh(2T )
cosh(2T ) − cos(βk)
dimNβkχ(Mg,j).
Theorem 4.7. (Compare with [BZ2, Theorem A.3] and [SZ, Theorem 3.9])
There exist α ∈ (0, 1], c > 0, C > 0 such that for any t ∈ (0, α], T ≥ 1, then
∣∣∣∣∣Trs
gN exp
tDb +
ĉ(∇f)
− χ̃′g(F )
∣∣∣∣∣ ≤ c exp(−CT ).(4.12)
Clearly, we may and we will assume that the number α > 0 in Theorems
4.5 and 4.7 have been chosen to be the same.
Next, we use above theorems to give a proof of Theorem 3.1. Since the
process is similar to it in [SZ], so we refer to it for more details.
First of all, by the anomaly formula (2.46), for any T ≥ 0, g ∈ G, one has
(4.13)
[0,1],detH
detH∗(Ω∗
[0,1],T
(M,F ),G)
det(H∗(Wu,F ),G)
[0,1],T
(M,F )⊥∩Ωi(M,F )
)(−1)ii
P detH∞
det(H∗(M,F ),G)
det(H∗(Wu,F ),G)
(g) exp
TrF [g]fe
TMg,∇TMg
From now on, we will write a ≃ b for a, b ∈ C if ea = eb. Thus, we can
rewrite (4.13) as
(4.14)
P detH∞
det(H∗(M,F ),G)
det(H∗(Wu,F ),G)
 ≃ log
[0,1],detH
detH∗(Ω∗
[0,1],T
(M,F ),G)
det(H∗(Wu,F ),G)
(−1)ii log
[0,1],T
(M,F )⊥∩Ωi(M,F )
TrF [g]fe
TMg,∇TMg
Let T0 > 0 be as in Theorem 4.2. For any T ≥ T0 and s ∈ C with
Re(s) ≥ n+ 1, set
θg,T (s) =
ts−1Trs
gN exp
−tD2bT
(1,+∞)
dt.(4.15)
By (4.6), θg,T (s) is well defined and can be extended to a meromorphic function
which is holomorphic at s = 0. Moreover,
(−1)ii log
[0,1],T
(M,F )⊥∩Ωi(M,F )
≃ − ∂θg,T (s)
(4.16)
Let d = α2 with α being as in Theorem 4.7. From (4.15) and Theorems
4.2-4.4, one finds that
(4.17)
∂θg,T (s)
= lim
gN exp
−tD2bT
− a−1√
χg(F )
− 2a−1√
Γ′(1)− log d
χg(F )− χ̃′g(F )
To study the first term in the right hand side of (4.17), we observe first that
for any T ≥ 0, one has
e−TfD2bT e
Tf = (Db + T ĉ(∇f))2 .(4.18)
Thus, one has
N exp
−tD2bT
= Trs
N exp
−t (Db + T ĉ(∇f))2
.(4.19)
By (4.19), one writes
(4.20)
gN exp
−tD2bT
− a−1√
χg(F )
∫ √dT
gN exp
Db + t
T ĉ(∇f)
a−1 −
χg(F )
gN exp
− (tDb + tT ĉ(∇f))2
− a−1
χg(F )
In view of Theorem 4.5, we write
(4.21)
gN exp
− (tDb + tT ĉ(∇f))2
− a−1
χg(F )
gN exp
− (tDb + tT ĉ(∇f))2
TrF [g]
L exp
(tT )2
F, bF
) ∫ B
d̂f exp
−B(tT )2
χg(F )
TrF [g]
L exp
(tT )2
− a−1
F, bF
) ∫ B
d̂f exp
−B(tT )2
By [BZ1, Definitions 3.6, 3.12 and Theorem 3.18], one has, as T → +∞,
(4.22)
F, bF
) ∫ B
d̂f exp
−B(tT )2
F, bF
(∇f)∗ψ
TMg,∇TMg
From [BZ1, (3.54)], [SZ, (3.35)] and integration by parts, we have
(4.23)
TrF [g]
L exp
(tT )2
− a−1
TrF [g]
L exp (−BT ) +
Ta−1 − T
TrF [g]f
exp (−BT )
TrF [g]f
exp (−B0) .
From Theorems 4.5, 4.6, [BZ1, Theorem 3.20], [BZ1, (7.72) and (7.73)] and the
dominate convergence, one finds that as T → +∞,
(4.24)
gN exp
− (tDb + tT ĉ(∇f))2
TrF [g]
L exp
(tT )2
F, bF
) ∫ B
d̂f exp
−B(tT )2
χg(F )
gN exp
Db + t
T ĉ(∇f)
TrF [g]
L exp
F, bF
) ∫ B
d̂f exp
χg(F )
g|F |Mg,j
1− e−2t2
1 + e−2t
x∈B∩Mg,j
(−1)indg(x)indg(x)− dimMg,je−2t
χ(Mg,j)
g|F |Mg,j
sinh(2t2)
cosh(2t2)− cos(βk)
x∈B∩Mg,j
(−1)indg(x)n−(βk)(x)
g|F |Mg,j
sinh(2t2)
cosh(2t2)− cos(βk)
dimNβkχ(Mg,j)
g|F |Mg,j
x∈B∩Mg,j
(−1)indg(x) (dimMg,j − 2indg(x))−
χg(F )
g|F |Mg,j
x∈B∩Mg,j
(−1)indg(x)indg(x)−
χ(Mg,j)dimMg,j
1 + e−2t
1− e−2t −
g|F |Mg,j
dimNβkχ(Mg,j)−
x∈B∩Mg,j
(−1)indg(x)n−(βk)(x)
sinh(2t)
cosh(2t)− cos(βk)
On the other hand, by Theorems 4.6, 4.7 and the dominate convergence, we
have that as T → +∞,
(4.25)
∫ √Td
gN exp
Db + t
T ĉ(∇f)
a−1 −
χg(F )
∫ √Td
gN exp
Db + t
T ĉ(∇f)
− χ̃′g(F )
χ̃′g(F ) log (Td) + a−1
χg(F ) log (Td)
g|F |Mg,j
1− e−2t2
1 + e−2t
x∈B∩Mg,j
(−1)indg(x)indg(x)− dimMg,je−2t
χ(Mg,j)
g|F |Mg,j
sinh(2t2)
cosh(2t2)− cos(βk)
x∈B∩Mg,j
(−1)indg(x)n−(βk)(x)
g|F |Mg,j
sinh(2t2)
cosh(2t2)− cos(βk)
dimNβkχ(Mg,j)− χ̃′g(F )
χ̃′g(F ) log (Td) + a−1
χg(F ) log (Td) + o(1)
g|F |Mg,j
x∈B∩Mg,j
(−1)indg(x)indg(x)−
χ(Mg,j)dimMg,j
1− e−2t
g|F |Mg,j
dimNβkχ(Mg,j)−
x∈B∩Mg,j
(−1)indg(x)n−(βk)(x)
sinh(2t)
cosh(2t)− cos(βk)
χ̃′g(F )−
χg(F )
log(Td) +
Ta−1 + o(1).
Combining (4.4), (4.14) and (4.20)-(4.25), one deduces, by setting T → +∞,
(4.26) log
P detH∞
det(H∗(M,F ),G)
det(H∗(Wu,F ),G)
− 2TrBgs [f ]T +
χ̃′g(F )−
χg(F )
log T −
χ̃′g(F )−
χg(F )
log π
F, bF
(∇f)∗ψ
TMg,∇TMg
TrF [g]
L exp (−BT )−2
Ta−1+2T
TrF [g]f
exp (−BT )
TrF [g]f
exp (−B0)
g|F |Mg,j
x∈B∩Mg,j
(−1)indg(x)indg(x)−
χ(Mg,j)dimMg,j
1 + e−2t
1− e−2t −
2e−2t
1− e−2t
g|F |Mg,j
dimNβkχ(Mg,j)−
x∈B∩Mg,j
(−1)indg(x)n−(βk)(x)
sinh(2t)
cosh(2t)− cos(βk)
sinh(2t)
cosh(2t)− cos(βk)
χ̃′g(F )−
χg(F )
log(Td)− 2a−1√
TrF [g]fe
TMg,∇TMg
2a−1√
Γ′(1)− log d
χ̃′g(F )−
χg(F )
+o(1).
By [BZ1, Theorem 3.20] and [BZ1, (7.72)], one has
(4.27) lim
TrF [g]f
exp(−BT )− 2TTr
s [f ]
g|F |Mg,j
x∈B∩Mg,j
(−1)indg(x)indg(x)−
χ(Mg,j)dimMg,j
(4.28) lim
TrF [g]
L exp(−BT )
g|F |Mg,j
x∈B∩Mg,j
(−1)indg(x)indg(x)−
χ(Mg,j)dimMg,j
On the other hand, by [BZ1, (7.93)] and [BZ2, (5.55)], one has
1 + e−2t
1− e−2t −
2 e−2t
1− e−2t
= 1− log π − Γ′(1),(4.29)
(4.30)
sinh(2t)
cosh(2t)− cos(βk)
sinh(2t)
cosh(2t)− cos(βk)
= − log(π)− 1
1− βk
Also, by [BZ2, (5.64)], if x ∈ B ∩Mg,
dimNβk
− n−(βk)(x)
[n+(βk)(x)− n−(βk)(x)] .(4.31)
From (4.26)-(4.31), we get (3.7), which completes the proof of Theorem 3.1.
5 Proofs of the intermediary Theorems
The purpose of this section is to give a sketch of the proofs of the intermediary
Theorems. Since the methods of the proofs of these theorems are essentially the
same as the corresponding theorem in [SZ], so we will refer to [SZ] for related
definitions and notations directly when there will be no confusion, such as Bb,g,
Ab,t,T , Ag,t,T , Ct,T , · · · .
5.1 Proof of Theorem 4.1
From Theorem 2.5 and [SZ, (4.44)] which in our situation we also have that
P∞,T commutate with g ∈ G, one finds
[0,1],detH
det(H∗(Ω∗
[0,1],T
(M,F )),G)
det(H∗(Wu,F ),G)
(g) =
∞,TP∞,T
[0,1],T
(M,F )
)(−1)i+1
(5.1)
From [SZ, Propositions 4.4 and 4.5], one deduces that as T → +∞,
(5.2) det
∞,TP∞,T
[0,1],T
(M,F )
= det
∞,TP∞,T
[0,1],T
(M,F )
(g) · det−1
[0,1],T
(M,F )
= det
(P∞,T eT )
P∞,T eT
Ci(Wu,F )
(g) · det−1
Ci(Wu,F )
= det
))# ( π
)N−n/2
))∣∣∣∣
Ci(Wu,F )
· det−1
Ci(Wu,F )
From (5.1) and (5.2), one gets (4.4) immediately.
The proof of Theorem 4.1 is completed. Q.E.D.
5.2 Proof of Theorem 4.2
The proof of Theorem 4.2 is the same as the proof of [SZ, Theorem 3.4] given
in [SZ, Section 5].
5.3 Proof of Theorem 4.3
Recall that the operator eT : C
∗(W u, F ) → Ω∗
[0,1],T
(M,F ) has been defined
in [SZ, (4.38)], and in the current case, we also have that eT commute with
G. So by [SZ, Proposition 4.4], we have that for T ≥ 0 large enough, eT :
C∗(W u, F ) → Ω∗
[0,1],T
(M,F ) is an identification of G-spaces. So (4.7) follows.
Also (4.8) was already proved in [SZ, Theorem 3.5].
5.4 Proof of Theorem 4.4
In this section, we provide a proof of Theorem 4.4, which computes the asymp-
totic of Trs[gN exp(−tD2bT )] for fixed T ≥ 0 as t → 0. The method is the
essentially same as it in [SZ].
By [SZ, (6.4)], we have
(5.3)
b = e−tD
(−1)ktk
e−t1tD
gBb,ge
−t2tD2g · · ·Bb,ge−tk+1tD
gdt1 · · · dtk
+ (−1)n+1tn+1
e−t1tD
gBb,ge
−t2tD2g · · ·Bb,ge−tn+2tD
bdt1 · · · dtn+1,
where ∆k, 1 ≤ k ≤ n+1, is the k-simplex defined by t1+ · · ·+ tk+1 = 1, t1 ≥ 0,
· · · , tk+1 ≥ 0. Also, by the same proof of [SZ, Proposition 6.1], we have the
following result.
Proposition 5.1. As t→ 0+, one has
gNe−t1tD
gBb,ge
−t2tD2g · · ·Bb,ge−tn+2tD
dt1 · · · dtn+1 → 0.
(5.4)
By [SZ, (6.22) and (6.23)], we have that for any 1 < k ≤ n, (t1, · · · , tk+1) ∈
tkTrs
gNe−t1tD
gBb,ge
−t2tD2g · · ·Bb,ge−tk+1tD
= 0,(5.5)
while for k = 1, 0 ≤ t1 ≤ 1,
(5.6) lim
gNe−t1tD
gBb,ge
−(1−t1)tD2g
= lim
gNBb,ge
−tD2g
i, j=1
ei ∧ êj
∇ueiω
F (ej)
ωF , ω̂Fg − ω̂F
· L exp
So by [BZ2, (2.13)], and proceed as in [SZ, (6.26)-(6.28)], we have
gNe−t1tD
gBb,ge
−(1−t1)tD2g
= 0.(5.7)
From (5.3), (5.4), (5.5), (5.7) and [BZ2, Theorem 5.9], one gets (4.9).
The proof of Theorem 4.4 is completed. Q.E.D.
5.5 Proof of Theorem 4.5
In order to prove (4.10), one need only to prove that under the conditions of
Theorem 4.5, there exists constant C ′′ > 0 such that
(5.8)∣∣∣Trs
gN exp
− (tDb + T ĉ(∇f))2
− Trs
gN exp
− (tDg + T ĉ(∇f))2
F, bF
F, gF
)) ∫ B
d̂f exp (−BT 2)
∣∣∣∣∣ ≤ C
By [SZ, (7.8)], we have
(5.9) e
b,t,T = e
g,t,T
(−1)k
−t1A2g,t,TCt,T e
−t2A2g,t,T · · ·Ct,T e−tk+1A
g,t,T dt1 · · · dtk
+ (−1)n+1
−t1A2g,t,TCt,T e
−t2A2g,t,T · · ·Ct,T e−tn+2A
b,t,T dt1 · · · dtn+1.
By the same proof of [SZ, (7.21)], we have that there exists C1 > 0 such that
for any t > 0 small enough and T ∈ [0, 1
∣∣∣∣∣
−t1A2g,t,TCt,T e
−t2A2g,t,T · · ·Ct,T e−tn+2A
b,t,T
dt1 · · · dtn+1
∣∣∣∣∣ ≤ C1t.
(5.10)
Also by the same proof of [SZ, (7.23)], we have that there exists C2 > 0,
0 < d < 1 such that for any 1 < k ≤ n, 0 < t ≤ d, T ≥ 0 with tT ≤ 1,
−t1A2g,t,TCt,T e
−t2A2g,t,T · · ·Ct,T e−tk+1A
g,t,T
dt1 · · · dtk
∣∣∣∣ ≤ C2t,
(5.11)
while for k = 1 one has for any 0 < t ≤ d, T ≥ 0 with tT ≤ 1 and 0 ≤ t1 ≤ 1,
by [BZ2, Proposition 9.3], we have
(5.12)∣∣∣∣∣Trs
−t1A2g,t,TCt,T e
−(1−t1)A2g,t,T
gωF (∇f)
L exp (−BT 2)
∣∣∣∣∣
≤ C2t.
Now similar as [SZ, (7.25)], we have
(5.13)
gωF (∇f)
L exp (−BT 2)
F, gF
F, bF
)) ∫ B
∇̂f exp (−BT 2) .
From (5.9)-(5.13), we get (5.8), which completes the proof of Theorem 4.5.
Q.E.D.
5.6 Proof of Theorem 4.6
In order to prove Theorem 4.6, we need only to prove that for any T > 0,
gN exp
b,t,T
gN exp
g,t,T
= 0.(5.14)
By [SZ, (8.2) and (8.4)], there exists 0 < C0 ≤ 1, such that when 0 < t ≤ C0,
one has the absolute convergent expansion formula
(5.15) e
b,t, T
t − e
g,t, T
(−1)k
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
t dt1 · · · dtk,
and that
(−1)k
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
t dt1 · · · dtk(5.16)
is uniformly absolute convergent for 0 < t ≤ C0.
Proceed as in [SZ, Section 8], one has that for any (t1, · · · , tk+1) ∈ ∆k \
{t1 · · · tk+1 = 0},
(5.17)
∣∣∣∣Trs
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
]∣∣∣∣
≤ C3tk (t1 · · · tk)−
g,t, T
∥∥∥∥∥ψe
g,t, T
∥∥∥∥∥
for some positive constant C3 > 0.
Also, by [SZ, (8.4)], (5.17) and the same assumption in [SZ] that tk+1 ≥ 1k+1 ,
one gets
(5.18)
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
dt1 · · · dtk
≤ C4tk−n
∥∥∥∥ψe
2(k+1)
g,t, T
for some constant C4 > 0.
From (5.15), (5.16), (5.18), [SZ, (8.9) and (8.10)] and the dominate conver-
gence, we get (5.14), which completes the proof of Theorem 4.6. Q.E.D.
5.7 Proof of Theorem 4.7
In order to prove Theorem 4.7, we need only to prove that there exist c > 0,
C > 0, 0 < C0 ≤ 1 such that for any 0 < t ≤ C0, T ≥ 1,
∣∣∣Trs
gN exp
b,t,T
− Trs
gN exp
g,t,T
)]∣∣∣ ≤ c exp(−CT ).(5.19)
First of all, one can choose C0 > 0 small enough so that for any 0 < t ≤ C0,
T > 0, by (5.15), we have the absolute convergent expansion formula
(5.20) e
b,t, T
t − e
g,t, T
(−1)k
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
t dt1 · · · dtk,
from which one has
(5.21) Trs
gN exp
b,t,T
− Trs
gN exp
g,t,T
(−1)k
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
dt1 · · · dtk.
Thus, in order to prove (5.19), we need only to prove
(5.22)
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
dt1 · · · dtk
−(t1+tk+1)A2
g,t, T
−t2A2
g,t, T
t · · ·C
dt1 · · · dtk
≤ c exp(−CT ).
By [SZ, (8.6)], we have for any t > 0, T ≥ 1, (t1, · · · , tk+1) ∈ ∆k \
{t1 · · · tk+1 = 0},
(5.23) Trs
−(t1+tk+1)A2
g,t, T
−t2A2
g,t, T
t · · ·C
= Trs
−(t1+tk+1)A2
g,t, T
t Ct,T
−t2A2
g,t, T
t Ct,T
· · ·ψe
−tkA2
g,t, T
t Ct,T
From (5.23), [SZ, (9.18) and (9.19)], one sees that there exists C5 > 0,
C6 > 0 and C7 > 0 such that for any k ≥ 1,
(5.24)
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
dt1 · · · dtk
≤ C5 (C6t)k
from which one sees that there exists 0 < c1 ≤ 1, C8 > 0, C9 > 0 such that for
any 0 < t ≤ c1 and T ≥ 1, one has
(5.25)∣∣∣∣∣
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
dt1 · · · dtk
∣∣∣∣∣
≤ C8 exp (−C9T ) .
On the other hand, for any 1 ≤ k < n, by proceeding as in (5.18), one has
that for any 0 < t ≤ c1, T ≥ 1,
(5.26)
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
dt1 · · · dtk
≤ C10tk−n
∥∥∥∥ψe
2(k+1)
g,t, T
for some constant C10 > 0.
From (5.26) and [SZ, (9.23)], one sees immediately that there exists C11 > 0,
C12 > 0 such that for any 1 ≤ k ≤ n− 1, 0 < t ≤ c1 and T ≥ 1, one has
(5.27)
−t1A2
g,t, T
−t2A2
g,t, T
t · · ·C
−tk+1A2
g,t, T
dt1 · · · dtk
≤ C11e−C12T .
From (5.21), (5.25) and (5.27), one gets (5.19).
The proof of Theorem 4.7 is completed. Q.E.D.
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	Introduction
	Equivariant symmetric bilinear torsions associated to the de Rham and Thom-Smale complexes
	Equivariant symmetric bilinear torsion of a finite dimensional complex
	The Thom-Smale complex of a gradient field
	Equivariant Milnor symmetric bilinear torsion
	Equivariant Ray-Singer symmetric bilinear torsion 
	An anomaly formula for the equivariant Ray-Singer symmetric bilinear torsion 
	A formula relating equivariant Milnor and equivariant Ray-Singer symmetric bilinear torsions
	A proof of Theorem 3.1
	Proofs of the intermediary Theorems
	Proof of Theorem ??
	Proof of Theorem ??
	Proof of Theorem ??
	Proof of Theorem ??
	Proof of Theorem ??
	Proof of Theorem ??
	Proof of Theorem ??
