Ray-Singer Torsion and the Rumin Laplacian on Lens Spaces
We express explicitly the analytic torsion functions associated with the Rumin complex on lens spaces in terms of the Hurwitz zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions. Moreover, we have a formula between this torsion and the Ra...
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| description | We express explicitly the analytic torsion functions associated with the Rumin complex on lens spaces in terms of the Hurwitz zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions. Moreover, we have a formula between this torsion and the Ray-Singer torsion.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 091, 16 pages
Ray–Singer Torsion and the Rumin Laplacian
on Lens Spaces
Akira KITAOKA
Graduate School of Mathematical Sciences, The University of Tokyo, Japan
E-mail: akira5kitaoka@gmail.com
URL: https://akira5kitaoka.github.io/Akira5Kitaoka-en.github.io/
Received May 19, 2022, in final form November 14, 2022; Published online November 28, 2022
https://doi.org/10.3842/SIGMA.2022.091
Abstract. We express explicitly the analytic torsion functions associated with the Rumin
complex on lens spaces in terms of the Hurwitz zeta function. In particular, we find that
the functions vanish at the origin and determine the analytic torsions. Moreover, we have
a formula between this torsion and the Ray–Singer torsion.
Key words: analytic torsion; Rumin complex; CR geometry; contact geometry
2020 Mathematics Subject Classification: 58J52; 32V20; 53D10; 43A85
1 Introduction
Let (M,H) be a compact contact manifold of dimension 2n + 1 and E be the flat vector bun-
dle with unitary holonomy on M . Rumin [12] introduced a complex (E•(M,E), d•R), which
is a subquotient of the de Rham complex of E. A specific feature of the complex is that
the operator D = dnR : En(M,E) → En+1(M,E) in ‘middle degree’ is a second-order, while
dkR : Ek(M,E) → Ek+1(M,E) for k ̸= n are first order which are induced by the exterior deriva-
tives. Let ak = 1/
√
|n− k| for k ̸= n and an = 1. Then, (E•(M,E), d•E), where dkE = akd
k
R, is
also a complex. We call (E•(M,E), d•E) the Rumin complex. In virtue of the rescaling, d•E satis-
fies Kähler-type identities on Sasakian manifolds [13, equation (34)], which include the case of
lens spaces.
Let θ be a contact form ofH and J be an almost complex structure onH. Then we may define
a Riemann metric gθ,J on TM by extending the Levi metric dθ(−, J−) on H (see Section 2.1).
Following [12], we define the Rumin Laplacians ∆E associated with (E•(M,E), d•E) and the
metric gθ,J by
∆k
E :=
(dEdE
†)2 + (dE
†dE)
2, k ̸= n, n+ 1,
(dEdE
†)2 +D†D, k = n,
DD† + (dE
†dE)
2, k = n+ 1.
Rumin showed that ∆E has discrete eigenvalues with finite multiplicities.
We next introduce the analytic torsion and metric of the Rumin complex (E•(M,E),d•E)
by following [2, 8, 14]. We define the contact analytic torsion function associated with (E•(M,E),
d•E) by
κE(M,E, gθ,J)(s) :=
n∑
k=0
(−1)k+1(n+ 1− k)ζ
(
∆k
E
)
(s), (1.1)
where ζ
(
∆k
E
)
(s) is the spectral zeta function of ∆k
E . Since the Rumin Laplacians ∆E satisfies the
Rockland condition by [12, p. 300], the spectral zeta function ζ
(
∆k
E
)
(s) extends to a meromorphic
akira5kitaoka@gmail.com
https://akira5kitaoka.github.io/Akira5Kitaoka-en.github.io/
https://doi.org/10.3842/SIGMA.2022.091
2 A. Kitaoka
function on C which is holomorphic at zero by [9, Section 4]. Here, we use 0s = 1. We define
the contact analytic torsion TE by
2 log TE(M,E, gθ,J) = κE(M,E, gθ,J)
′(0).
Let H•(E•,d•E) be the cohomology of the Rumin complex. We define the contact metric on
detH•(E•,d•E) by
∥ ∥E(M,E, gθ,J) = T−1
E (M,E, gθ,J)| |L2(E•),
where the metric | |L2(E•) is induced by L2 metric on E•(M,E) via identification of the coho-
mology classes by the harmonic forms on E•(M,E).
Rumin and Seshadri [14] defined another analytic torsion function κR from (E•(M,E),d•R),
which is different from κE except in dimension 3. In dimension 3, they showed that κR(M,E,
gθ,J)(0) is a contact invariant, that is, independent of the metric gθ,J . Moreover, on 3-dimensional
Sasakian manifolds with S1-action, κR(M,E, gθ,J)(0) = 0. Here, Sasakian manifolds with S1-
action means Sasakian manifolds whose Reeb vector filed generates the circle action S1. 3-dimen-
sional Sasakian manifolds with S1-action are CR Seifert manifolds. Furthermore, they showed
that this analytic torsion and the Ray–Singer torsion TdR(M,E, gθ,J) equal for flat bundles with
unitary holonomy on 3-dimensional Sasakian manifolds with S1 action.
To extend the coincidence, with dE instead of dR, the author [8] showed that TE
(
S2n+1,C, gθ,J
)
= n!TdR
(
S2n+1,C, gθ,J
)
on the standard CR spheres S2n+1
(
⊂Cn+1
)
, where C is the trivial line
bundle. Here the standard CR sphere is triple
(
S2n+1, θ, J
)
, where θ is given the contact form
by θ =
√
−1
(
∂̄ − ∂
)
|z|2 and J is an almost complex structure J induced from the complex
structure of Cn+1. It is simply denoted by S2n+1. Moreover, Albin and Quan [1, Corollary 3
and equation (4)] showed the difference between the Ray–Singer torsion and the contact analytic
torsion is given by some integrals of universal polynomials in the local invariants of the metric
on contact manifolds.
In this paper, we extend this coincidence on lens spaces and determine explicitly the analytic
torsion functions associated with the Rumin complex in terms of the Hurwitz zeta function.
Let gstd be the standard metric on S2n+1 and we note that gθ,J = 4gstd. Let µ, ν1, . . . , νn+1 be
integers such that the νj are coprime to µ. Let Γ be the subgroup of
(
S1
)n+1
generated by
γ = (γ1, . . . , γn+1) :=
(
exp
(
2π
√
−1ν1/µ
)
, . . . , exp
(
2π
√
−1νn+1/µ
))
.
We denote the lens space by
K := S2n+1/Γ.
Let C be the trivial line bundle on K. Fix u ∈ Z and consider the unitary representation
αu : π1(K) = Γ → U(1), defined by
αu
(
γℓ
)
:= exp
(
2π
√
−1uℓ/µ
)
for ℓ ∈ Z.
Let Eα be the flat vector bundle associated with the unitary representation α : π1(K) = Γ→U(r),
and Eαu = Eu. The sections of this bundle correspond to αu-equivariant functions on S
2n+1.
Our main result is
Theorem 1.1. Let K be the lens space with the contact form and the almost complex structure
which are induced by the action Γ on the standard CR sphere S2n+1.
(1) The contact analytic torsion function of (K,C) is given by
κE(K,C, gθ,J)(s) = −(n+ 1)
(
1 + 22s+1µ−2sζ(2s)
)
, (1.2)
Ray–Singer Torsion and the Rumin Laplacian on Lens Spaces 3
where ζ is the Riemann zeta function. In particular, we have
κE(K,C, gθ,J)(0) = 0, (1.3)
TE(K,C, gθ,J) =
(
4π
µ
)n+1
. (1.4)
(2) The contact analytic torsion function of (K,Eu) for u ∈ {1, . . . , µ− 1} is given by
κE(K,Eu, gθ,J)(s) = −22sµ−2s
n+1∑
j=1
(
ζ(2s,Aµ(uτj)/µ) + ζ(2s,Aµ(−uτj)/µ)
)
, (1.5)
where ζ(s, a) :=
∑∞
q=0(q + a)−s is the Hurwitz zeta function for 0 < a ≤ 1, Aµ(w) is the
integer between 1 and µ such that Aµ(w) ≡ w mod µ and τjνj ≡ 1 mod µ. In particular,
we have
κE(K,Eu, gθ,J)(0) = 0, (1.6)
TE(K,Eu, gθ,J) =
n+1∏
j=1
∣∣e2π√−1uτj/µ − 1
∣∣. (1.7)
The equations (1.2) and (1.5) extend the following results of κE the spheres to on lens spaces.
Rumin and Seshadri [14, Theorem 5.4] showed (1.2) in the case of 3-dimensional lens spaces.
The author [8] showed (1.2) in the case of
(
S2n+1,C
)
for arbitrary n.
From (1.3) and (1.6), we see that the metric ∥ ∥E on (K,Eu, gθ,J) is invariant under the
constant rescaling θ 7→ Cθ. The argument is exactly the same as the one in [14].
In the same way as [8], the fact that the representations determine the eigenvalues of ∆E
cause several cancellations in the linear combination (1.1), which significantly simplifies the
computation of κE(s). We cannot get such a simple formula for the contact analytic torsion
function κR of (E•(M,E),d•R) for dimensions higher than 3.
Let us compare the contact analytic torsion with the Ray–Singer torsion on lens spaces.
Ray [10] showed that for u(= 1, . . . , µ− 1)
TdR(K,Eu, 4gstd) =
n+1∏
j=1
∣∣e2π√−1uτj/µ − 1
∣∣.
Weng and You [15] calculate the Ray–Singer torsion on spheres. We extend their results for the
trivial bundle on lens spaces:
Proposition 1.2. In the setting of Theorem 1.1, we have
TdR(K,C, 4gstd) =
(4π)n+1
n!µn+1
.
The metric 4gstd agrees with the metric gθ,J defined from the contact form θ =
√
−1
(
∂̄−∂
)
|z|2.
Since the cohomology of (E•(M,E),d•E) coincides with that of (Ω•(M,E), d) (e.g., [12, p. 286]),
there is a natural isomorphism detH•(E•(M,E), d•E)
∼= detH•(Ω•(M,E),d), which turns out
to be isometric for the L2 metrics. Therefore (1.4) and (1.7) give
Corollary 1.3. In the setting of Theorem 1.1, for all unitary holonomy α : π1(K) → U(r), we
have
TE(K,Eα, gθ,J) = n!dimH0(K,Eα)TdR(K,Eα, gθ,J),
∥ ∥E(K,Eα, gθ,J) = n!−dimH0(K,Eα)∥ ∥dR(K,Eα, gθ,J),
via the isomorphism detH•(E•(M,Eα), d
•
E)
∼= detH•(Ω•(M,Eα), d).
4 A. Kitaoka
The paper is organized as follows. In Section 2, we recall the definition and properties of the
Rumin complex on S2n+1. In Section 3, we calculate the contact analytic torsion function κE of
flat vector bundles on lens spaces. In Section 4, we compute the Ray–Singer torsion TdR of the
trivial vector bundle. In Section 5, we compare the Ray–Singer torsion and the contact analytic
torsion.
2 The Rumin complex
2.1 The Rumin complex on contact manifolds
We call (M,H) an orientable contact manifold of dimension 2n+1 if H is a subbundle of TM of
codimension 1 and there exists a 1-form θ, called a contact form, such that Ker(θ : TM→R) = H
and θ ∧ (dθ)n ̸= 0. The Reeb vector field of θ is the unique vector field T satisfying θ(T ) = 1
and IntT dθ = 0, where IntT is the interior product with respect to T .
ForH and θ, we call J ∈ End(TM) an almost complex structure associated with θ if J2 = − Id
on H, JT = 0, and the Levi form dθ(−, J−) is positive definite on H. Given θ and J , we define
a Riemannian metric gθ,J on TM by
gθ,J(X,Y ) := dθ(X, JY ) + θ(X)θ(Y ) for X,Y ∈ TM.
Let ∗ be the Hodge star operator on ∧•T ∗M with respect to gθ,J .
Let M be a manifold, M̃ be the universal covering of M , π1(M) be the fundamental group
of M . For each unitary representation α : π1(M) → U(r), we denote the flat vector bundle
associated with α by
Eα := M̃ ×α Cr →M.
Let ∇α be the flat connection on Eα induced from the trivial connection on M̃ × Cr, and d∇α
be the exterior covariant derivative of ∇α.
The Rumin complex [12] is defined on contact manifolds as follows. We set L := dθ∧ and
Λ := ∗−1L∗, which is the adjoint operator of L with respect to the metric gθ,J at each point.
We set∧k
primH
∗ :=
{
v ∈
∧kH∗ ∣∣Λv = 0
}
,∧k
LH
∗ :=
{
v ∈
∧kH∗ ∣∣Lv = 0
}
,
Ek(M,Eα) :=
{
C∞(M,
∧k
primH
∗ ⊗ Eα
)
, k ≤ n,
C∞(M, θ ∧
∧k−1
L H∗ ⊗ Eα
)
, k ≥ n+ 1.
We embed H∗ into T ∗M as the subbundle {ϕ ∈ T ∗M |ϕ(T ) = 0} so that we can regard
Ωk
H(M,Eα) := C∞(M,
∧kH∗ ⊗ Eα
)
as a subspace of Ωk(M), the space of k-forms. We define db : Ω
k
H(M,Eα) → Ωk+1
H (M,Eα) by
dbϕ := d∇αϕ− θ ∧
(
IntT d∇αϕ
)
,
and then D : En(M,Eα) → En+1(M,Eα) by
D = θ ∧
(
LT + dbL
−1db
)
,
where LT is the Lie derivative with respect to T , and we use the fact that L :
∧n−1H∗ →∧n+1H∗ is an isomorphism.
Ray–Singer Torsion and the Rumin Laplacian on Lens Spaces 5
Let P :
∧kH∗ →
∧k
primH
∗ be the fiberwise orthogonal projection with respect to gθ,J , which
also defines a projection P : Ωk(M,Eα) → Ek(M,Eα). We set
dkR :=
P ◦ d∇α on Ek(M,Eα), k ≤ n− 1,
D on En(M,Eα),
d∇α on Ek(M,Eα), k ≥ n+ 1.
Then (E•(M,Eα), d
•
R) is a complex. Let dkE = akd
k
R, where ak = 1/
√
|n− k| for k ̸= n and
an = 1. We call (E•(M,Eα),d
•
E) the Rumin complex.
We define the L2-inner product on Ωk(M,Eα) by
(ϕ, ψ) :=
∫
M
gθ,J(ϕ, ψ) d volgθ,J
and the L2-norm on Ωk(M,Eα) by ∥ϕ∥ :=
√
(ϕ, ϕ). Since the Hodge star operator ∗ induces
a bundle isomorphism from
∧k
primH
∗ to θ ∧
∧2n−k
L H∗, it also induces a map Ek(M,Eα) →
E2n+1−k(M,Eα). We note that
Ek(M,Eα) =
{
ϕ ∈ Ek
(
M̃,Cr
)
| t∗ϕ = α(t)−1ϕ for t ∈ π1(M)
}
.
Let d†E and D† denote the formal adjoint of dE and D, respectively for the L2-inner product.
We define the fourth-order Laplacians ∆E on Ek(M,Eα) by
∆k
E :=
(
dk−1
E dk−1
E
†)2 + (dkE†dkE)2, k ̸= n, n+ 1,(
dn−1
E dn−1
E
†)2 +D†D, k = n,
DD† +
(
dn+1
E
†dn+1
E
)2
, k = n+ 1.
We call it the Rumin Laplacian [12]. Since ∗ and ∆E commute, to determine the eigenvalue on
E•(M,Eα), it is enough to compute them on Ek(M,Eα) for k ≤ n.
2.2 The Rumin complex on the CR spheres
Let S := {z ∈ Cn+1 | |z|2 = 1} and θ :=
√
−1
(
∂̄−∂
)
|z|2. (We will omit the dimension from S2n+1
for the simplicity of the notation.) Let gstd be the standard metric on S. Then, gθ,J coincides
with 4gstd. With respect to the standard almost complex structure J , we decompose the bundles
defined in the previous subsection as follows:
H∗1,0 :=
{
v ∈ CH∗ | Jv =
√
−1v
}
,
H∗0,1 :=
{
v ∈ CH∗ | Jv = −
√
−1v
}
,∧i,j H∗ :=
∧iH∗1,0 ⊗
∧j H∗0,1,∧i,j
primH
∗ :=
{
ϕ ∈
∧i,j H∗ ∣∣Λϕ = 0
}
,
E i,j := C∞(S,∧i,j
primH
∗).
We decompose E i,j into a direct sum of irreducible representations of the unitary group U(n+1).
Recall that irreducible representations of U(m) are parametrized by the highest weight λ =
(λ1, . . . , λm) ∈ Zm with λ1 ≥ λ2 ≥ · · · ≥ λm; the representation corresponding to λ will be
denoted by V (λ). To simplify the notation, we introduce the following notation: for a1, . . .,
al ∈ Z and k1, . . . , kl ∈ Z, (a1k1 , . . . , alkl) denotes the k1 + · · · + kl-tuple whose first k1 entries
are a1, whose next k2 entries are a2, etc. For example,
(13, 02,−12) = (1, 1, 1, 0, 0,−1,−1).
We note that a1 is a, and a0 is the zero tuple.
6 A. Kitaoka
In [7], it is shown that the multiplicity of V (q, 1j , 0n−1−i−j ,−1i,−p) in Es,t is at most one.
Thus we may set
Ψ
(s,t)
(q,j,i,p) := Es,t ∩ V (q, 1j , 0n−1−i−j ,−1i,−p).
Proposition 2.1 ([7, Section 4(b)]). Given (q, j, i, p), we list up all (s, t) such that s + t ≤ n
and Ψ
(s,t)
(q,j,i,p) ̸= {0} as the following:
Case I: For i = j = 0 and p = q = 0, the space is
Ψ
(0,0)
(0,0,0,0).
Case II: For i+ j ≤ n− 2, p ≥ 1 and q ≥ 1, the spaces are
Ψ
(i,j)
(q,j,i,p), Ψ
(i+1,j)
(q,j,i,p), Ψ
(i,j+1)
(q,j,i,p), Ψ
(i+1,j+1)
(q,j,i,p) .
Case III: For 0 ≤ i ≤ n− 1, j = 0, p ≥ 1 and q = 0, the spaces are
Ψ
(i,0)
(0,0,i,p), Ψ
(i+1,0)
(0,0,i,p).
Case IV: For i = 0, 0 ≤ j ≤ n− 1, p = 0 and q ≥ 1, the spaces are
Ψ
(0,j)
(q,j,0,0), Ψ
(0,j+1)
(q,j,0,0).
Case V: For i+ j = n− 1, p ≥ 1 and q ≥ 1, the spaces are
Ψ
(i,j)
(q,j,i,p), Ψ
(i+1,j)
(q,j,i,p), Ψ
(i,j+1)
(q,j,i,p).
Case VI: i = n− 1, j = 0, p ≥ 1 and q = −1, the space is
Ψ
(n,0)
(−1,0,n−1,p).
Case VII: i = 0, j = n− 1, p = −1 and q ≥ 1, the space is
Ψ
(0,n)
(q,n−1,0,−1).
Remark 2.2. About Case VI, we substitute s = n, t = 0, q = −1, j = 0, i = n−1, for Ψ
(s,t)
(q,j,i,p),
Ψ
(n,0)
(−1,0,n−1,p) = En,0 ∩ V (−1, 10, 00,−1n−1,−p) = En,0 ∩ V (−1n,−p).
As the same way, about VII we obtain
Ψ
(0,n)
(q,n−1,0,−1) = E0,n ∩ V (q, 1n−1, 00,−10,−(−1)) = E0,n ∩ V (q, 1n).
In [8], ∆E acts as a scalar on these spaces, and eigenvalues of ∆E on the standard CR spheres
are explicitly determined.
Proposition 2.3 ([8, Theorem 0.1]). Let S be the standard CR sphere with the contact form
θ =
√
−1
(
∂̄ − ∂
)
|z|2. Then, on the subspaces of the complexification of E•(S) corresponding to
the representations Ψ
(•,•)
(q,j,i,p), the eigenvalue of ∆E is(
(p+ i)(q + n− i) + (q + j)(p+ n− j)
)2
4(n− i− j)2
.
Ray–Singer Torsion and the Rumin Laplacian on Lens Spaces 7
3 Contact analytic torsion of flat vector bundles
Let µ, ν1, . . . , νn+1 be integers such that the νj are coprime to µ. Let Γ be the subgroup
of
(
S1
)n+1
generated by
γ = (γ1, . . . , γn+1) :=
(
exp
(
2π
√
−1ν1/µ
)
, . . . , exp
(
2π
√
−1νn+1/µ
))
.
We denote the lens space by
K := S2n+1/Γ.
Fix u ∈ {1, . . . , µ} and consider the unitary representation αu : π1(K) = Γ → U(1), defined
by
αu
(
γℓ
)
:= exp
(
2π
√
−1uℓ/µ
)
for ℓ ∈ Z,
where
γℓ :=
(
γℓ1, . . . , γ
ℓ
n+1
)
=
(
exp
(
2π
√
−1ν1ℓ/µ
)
, . . . , exp
(
2π
√
−1νn+1ℓ/µ
))
.
Let Eα be the flat vector bundle associated with the unitary representation α : π1(K) = Γ→U(r),
and Eu := Eαu , which can be considered as αu-equivariant functions on S
2n+1.
For each unitary representation (V, ρ) of U(n + 1), we define the vector subspace V αu of V
by
V αu :=
{
ϕ ∈ V
∣∣ ρ(γ)ϕ = α(γ)−1ϕ
}
.
Proposition 3.1. We have
κE(K,Eu, gθ,J)(s) = κ1(K,Eu, gθ,J)(s) + κ2(K,Eu, gθ,J)(s) + κ3(K,Eu, gθ,J)(s), (3.1)
where
κ1(K,Eu, gθ,J)(s) := −(n+ 1) dimV αu(0n+1) =
{
−(n+ 1), u = 0,
0, u ̸= 0,
(3.2)
κ2(K,Eu, gθ,J)(s) := (−1)1(n+ 1)
∑
q≥1
dimV αu(q, 0n)(
q/2
)2s
+
n∑
j=1
(−1)j+1(n+ 1− j)
∑
q≥1
(
dimV αu(q, 1j , 0n−j)(
(q + j)/2
)2s +
dimV αu(q, 1j−1, 0n−j+1)(
(q + j − 1)/2
)2s
)
,
κ3(K,Eu, gθ,J)(s) := κ2(K,E−u, gθ,J)(s), (3.3)
Proof. From Proposition 2.3, we check that the terms of κE(K,Eu, g)(s) in Cases II and V in
Proposition 2.1 cancel each other. “The sum of the terms of κE(K,Eu, gθ,J)(s) in Case II” is
n−2∑
a=0
∑
i+j=a
∑
p≥1,
q≥1
(
(−1)i+j+1(n+ 1− i− j)
+ 2(−1)i+j+2(n− i− j) + (−1)i+j+3(n− 1− i− j)
)
×
dimV αu(q, 1j , 0n−1−i−j ,−1i,−p)(
((p+ i)(q + n− i) + (q + j)(p+ n− j))/2(n− i− j)
)2s
8 A. Kitaoka
=
n−2∑
a=0
∑
i+j=a
∑
p≥1,
q≥1
(
0
) dimV αu(q, 1j , 0n−1−i−j ,−1i,−p)(
((p+ i)(q + n− i) + (q + j)(p+ n− j))/2(n− i− j)
)2s = 0.
Similarly, “the sum of the terms of κE(K,Eu, gθ,J)(s) in Case V” is∑
i+j=n−1
∑
p≥1,
q≥1
(
(−1)n(n+ 1− (n− 1))) + 2(−1)n+1(n+ 1− n)
)
×
dimV αu(q, 1j ,−1i,−p)(
((p+ i)(q + n− i) + (q + j)(p+ n− j))/2(n+ 1− i− j)
)2s
=
∑
i+j=n−1
∑
p≥1,
q≥1
(
0
) dimV αu(q, 1j ,−1i,−p)(
((p+ i)(q + n− i) + (q + j)(p+ n− j))/2(n− i− j)
)2s = 0.
The function κ1(K,Eu, gθ,J)(s) is the sum of the terms of κE(K,Eu, gθ,J)(s) in Case I.
Next we consider the sum of the terms of κE in Cases III and VI. For j, “the sum of the
terms of κE(K,Eu, gθ,J) in E0,j , Cases III and VI” is
(−1)1(n+ 1)
∑
q≥1
dimV αu(q, 0n)(
q/2
)2s , j = 0,
(−1)j+1(n+ 1− j)
∑
q≥1
(
dimV αu(q, 1j , 0n−j)(
(q + j)/2
)2s +
dimV αu(q, 1j−1, 0n−j+1)(
(q + j − 1)/2
)2s
)
, 1 ≤ j ≤ n.
“The sum of the terms of κE in Cases III and VI” is
(−1)1(n+ 1)
∑
q≥1
dimV αu(q, 0n)(
q/2
)2s
+
n∑
j=1
(−1)j+1(n+ 1− j)
∑
q≥1
(
dimV αu(q, 1j , 0n−j)(
(q + j)/2
)2s +
dimV αu(q, 1j−1, 0n−j+1)(
(q + j − 1)/2
)2s
)
= κ2(K,Eu, gθ,J)(s). (3.4)
Finally, we consider the sum of the terms of κE in Cases IV and VII. As the same way (3.4)
in Cases III and VI, “the sum of the terms of κE in Cases IV and VII” is given by
(−1)1(n+ 1)
∑
q≥1
dimV αu(0n,−q)(
q/2
)2s (3.5)
+
n∑
j=1
(−1)j+1(n+ 1− j)
∑
q≥1
(
dimV αu(0n−j ,−1j ,−q)(
(q + j)/2
)2s +
dimV αu(0n−j+1,−1j−1,−q)(
(q + j − 1)/2
)2s
)
.
Let (V, ρ) be the unitary representation of U(n+ 1). We define the representation (V , ρ) by
V := V, ρ(U) := ρ(U) for U ∈ U(n+ 1).
Since (V (q, 1j , 0n−j), ρ) is the unitary representation, its conjugate representation is isomor-
phic to its dual representation as U(n + 1)-module. From [6, Theorem 3.2.13], the conjugate
representation of (V (q, 1j , 0n−j), ρ) is isomorphic to (V (0n−j ,−1j ,−q), ρ) as U(n + 1)-module.
Therefore, we have
V α−u(q, 1j , 0n−j) =
{
ϕ ∈ V (q, 1j , 0n−j)
∣∣α−u(−γ)ϕ = ρ(γ)ϕ
}
Ray–Singer Torsion and the Rumin Laplacian on Lens Spaces 9
=
{
ϕ ∈ V (q, 1j , 0n−j)
∣∣αu(−γ)−1ϕ = ρ(γ)ϕ
}
=
{
ϕ ∈ V (q, 1j , 0n−j)
∣∣αu(γ)ϕ = ρ(γ)ϕ
}
∼=
{
ϕ ∈ V (0n−j ,−1j ,−q)
∣∣αu(γ)ϕ = ρ(γ)ϕ
}
=
{
ϕ ∈ V (0n−j ,−1j ,−q)
∣∣αu(γ)
−1ϕ = ρ(γ)ϕ
}
= V (0n−j ,−1j ,−q)
αu ,
where ∼= means isomorphic as real vector spaces via the complex conjugate. Then, from (3.4)
and (3.5), “the sum of the terms of κE in Cases IV and VII” is given by
(−1)1(n+ 1)
∑
q≥1
dimV αu(0n,−q)(
q/2
)2s
+
n∑
j=1
(−1)j+1(n+ 1− j)
∑
q≥1
(
dimV αu(0n−j ,−1j ,−q)(
(q + j)/2
)2s +
dimV αu(0n−j+1,−1j−1,−q)(
(q + j − 1)/2
)2s
)
= (−1)1(n+ 1)
∑
q≥1
dimV α−u(q, 0n)(
q/2
)2s
+
n∑
j=1
(−1)j+1(n+ 1− j)
∑
q≥1
(
dimV α−u(q, 1j , 0n−j)(
(q + j)/2
)2s +
dimV α−u(q, 1j−1, 0n−j+1)(
(q + j − 1)/2
)2s
)
= κ2(K,E−u, gθ,J)(s). ■
We set for q ≥ 1,
V (q, 1−1, 0n+1) := {0}.
We have
κ2(K,Eu, gθ,J)(s) = (−1)1(n+ 1)
∑
q≥1
dimV αu(q, 0n)(
q/2
)2s
+
n∑
j=1
(−1)j+1(n+ 1− j)
∑
q≥1
(
dimV αu(q, 1j , 0n−j)(
(q + j)/2
)2s +
dimV αu(q, 1j−1, 0n−j+1)(
(q + j − 1)/2
)2s
)
=
n∑
j=0
(−1)j+1(n+ 1− j)
∑
q≥1
(
dimV αu(q, 1j , 0n−j)(
(q + j)/2
)2s +
dimV αu(q, 1j−1, 0n−j+1)(
(q + j − 1)/2
)2s
)
=
n∑
j=0
(−1)j+1(n+ 1− j)
∑
q≥1
dimV αu(q, 1j , 0n−j) + dimV αu(q + 1, 1j−1, 0n−j+1)(
(q + j)/2
)2s
+
n∑
j=1
(−1)j+1(n+ 1− j)
dimV αu(1j , 0n−j+1)(
j/2
)2s .
Let χV be the character of the representation (V, ρ) of U(n + 1). We note that for each
representation (V, ρ),
dimV αu =
∑
t∈Γ
χV (t)αu(t)/#Γ,
(cf. [5, equation (2.9)]). By Littlewood–Richardson’s rule (cf. [4, Corollary 3]), we have
χV (1j ,0n−j+1)
χV (q,0n)
= χV (q,1j ,0n−j)
+ χV (q+1,1j−1,0n−j+1)
. (3.6)
10 A. Kitaoka
From (3.6),
κ2(K,Eu, gθ,J)(s) =
1
µ
n∑
j=0
(−1)j+1(n+ 1− j)
∑
q≥1
µ−1∑
ℓ=0
χV (1j ,0n−j+1)
(
γℓ
)
χV (q,0n)
(
γℓ
)
αu
(
γℓ
)(
(q + j)/2
)2s
+
1
µ
n∑
j=1
(−1)j+1(n+ 1− j)
µ−1∑
ℓ=0
χV (1j ,0n−j+1)
(
γℓ
)
αu
(
γℓ
)(
j/2
)2s
=
22s
µΓ(2s)
n∑
j=0
(−1)j+1(n+ 1− j)
×
∑
q≥1
µ−1∑
ℓ=0
∫ ∞
0
χV (1j ,0n−j+1)
(
γℓ
)
χV (q,0n)
(
γℓ
)
αu
(
γℓ
)
e−(j+q)xx2s−1 dx
+
22s
µΓ(2s)
n∑
j=1
(−1)j+1(n+ 1− j)
µ−1∑
ℓ=0
∫ ∞
0
χV (1j ,0n−j+1)
(
γℓ
)
αu
(
γℓ
)
e−jxx2s−1 dx
=
22s
µΓ(2s)
µ−1∑
ℓ=0
∫ ∞
0
(
n∑
j=0
(−1)j+1(n+ 1− j)χV (1j ,0n−j+1)
(
γℓ
)
e−jx
∑
q≥1
χV (q,0n)
(
γℓ
)
e−qx
+
n∑
j=1
(−1)j+1(n+ 1− j)χV (1j ,0n−j+1)
(
γℓ
)
e−jx
)
αu
(
γℓ
)
x2s−1 dx. (3.7)
We consider the contents of the integral for the last equation of (3.7). It is known that for
t = (t1, . . . , tn+1) ∈
(
S1
)n+1
,
χV (1j ,0n−j+1)
(t) =
∑
β1+···+βn+1=j,
0≤β1,...,βn+1≤1
tβ1
1 · · · tβn+1
n+1 , (3.8)
χV (q,0n)
(t) =
∑
α1+···+αn+1=q
α1,...,αn+1≥0
tα1
1 · · · tαn+1
n+1 , (3.9)
(cf. [5, equations (6.1) and (6.2)]). We set X := e−x and
F1(t,X) :=
n+1∑
j=0
(−1)jχV (1j ,0n−j+1)
(t)Xj .
Then (3.8) gives
F1(t,X) =
n+1∏
j=1
(1− tjX), (3.10)
X
∂F1
∂X
(t,X) =
n+1∑
j=0
(−1)jjχV (1j ,0n−j+1)
(t)Xj = −
n+1∑
i=1
tiX
n+1∏
j=1,j ̸=i
(1− tjX). (3.11)
From the definition of F1 and (3.11), we have
n∑
j=0
(−1)j+1(n+ 1− j)χV (1j ,0n−j+1)
(t)Xj = −(n+ 1)F1(t,X) +X
∂F1
∂X
(t,X). (3.12)
Ray–Singer Torsion and the Rumin Laplacian on Lens Spaces 11
We set
F2(t,X) :=
∑
q≥1
χV (q,0n)
(t)Xq.
From (3.9) and (3.10), we can rewrite F2 as
F2(t,X) =
n+1∏
j=1
1
1− tjX
− 1 =
1
F1(t,X)
− 1. (3.13)
From (3.10)–(3.13), we can deduce that
n∑
j=0
(−1)j+1(n+ 1− j)χV (1j ,0n−j+1)
(t)Xj
∑
q≥1
χV (q,0n)
(t)Xq
+
n∑
j=1
(−1)j+1(n+ 1− j)χV (1j ,0n−j+1)
(t)Xj
= −
(
(n+ 1)F1(t,X)−X
∂F1
∂X
(t,X)
)(
1
F1(t,X)
− 1
)
−
(
(n+ 1)(F1(t,X)− 1)−X
∂F1
∂X
(t,X)
)
=
X ∂F1
∂X (t,X)
F1(t,X)
= −
n+1∑
j=1
tjX
1− tjX
. (3.14)
From (3.7) and (3.14), we see
κ2(K,Eu, gθ,J)(s) = − 22s
µΓ(2s)
µ−1∑
ℓ=0
n+1∑
j=1
∫ ∞
0
γℓje
−x
1− γℓje
−x
e2π
√
−1uℓ/µx2s−1 dx
= −
n+1∑
j=1
22s
µΓ(2s)
µ−1∑
ℓ=0
∫ ∞
0
∞∑
q=1
exp
(
2π
√
−1(qνj + u)ℓ/µ
)
e−qxx2s−1 dx
= −22s
n+1∑
j=1
µ−1∑
ℓ=0
∞∑
q=1
exp
(
2π
√
−1(qνj + u)ℓ/µ
)
µ
q−2s. (3.15)
Let τj be the integers in {1, . . . , µ} such that τjνj ≡ 1 mod µ. Since the multiplication of
νj ∈ (Z/µZ)× induced the bijective map from Z/µZ to Z/µZ, We have
µ−1∑
ℓ=0
exp
(
2π
√
−1(qνj + u)ℓ/µ
)
=
µ−1∑
ℓ=0
exp
(
2π
√
−1(q + uτj)νjℓ/µ
)
=
µ−1∑
ℓ=0
exp
(
2π
√
−1(q + uτj)ℓ/µ
)
=
{
0, q ̸≡ −uτj mod µ,
µ, q ≡ −uτj mod µ.
For w ∈ Z let Aµ(w) be the integer between 1 and µ which is congruent to w modulo µ, then
from (3.15), we can rewrite κ2 as
κ2(K,Eu, gθ,J)(s) = −22s
n+1∑
j=1
∞∑
q>0,
q≡−uτj mod µ
q−2s = −22s
n+1∑
j=1
∞∑
q=0
(
qµ+Aµ(−uτj)
)−2s
12 A. Kitaoka
= −22sµ−2s
n+1∑
j=1
ζ
(
2s,Aµ(−uτj)/µ
)
, (3.16)
where for 0 < a ≤ 1, ζ(s, a) :=
∑∞
q=0(q + a)−s is the Hurwitz zeta function.
Next, we calculate κ3. As the same way in calculating κ2, from (3.3), we can rewrite κ3 as
κ3(K,Eu, gθ,J)(s) = −22sµ−2s
n+1∑
j=1
ζ
(
2s,Aµ(uτj)/µ
)
. (3.17)
From (3.1), (3.2), (3.16) and (3.17), we have
κE(K,Eu, gθ,J)(s) =
−(n+ 1)
(
1 + 22s+1µ−2sζ(2s)
)
, u = 0.
−22sµ−2s
n+1∑
j=1
(
ζ
(
2s,Aµ(uτj)/µ
)
+ ζ
(
2s, 1−Aµ(uτj)/µ
))
, u ̸= 0.
It is known that ζ(0) = −1/2 and ζ ′(0) = − log(2π)/2 and for 0 < a < 1,
ζ(0, a) + ζ(0, 1− a) = 0,
ζ ′(0, a) + ζ ′(0, 1− a) = − log
∣∣e2π√−1a − 1
∣∣.
Using the above equations, we conclude for u = 0
κE(K,C, gθ,J)(0) = −(n+ 1)
(
1 + 2ζ(0)
)
= 0,
κE(K,C, gθ,J)′(0) = 2(n+ 1) log
4π
µ
,
and for u ̸= 0
κE(K,Eu, gθ,J)(0) = 0,
κE(K,Eu, gθ,J)
′(0) = 2
n+1∑
j=1
log
∣∣e2π√−1uτj/µ − 1
∣∣
as claimed.
4 Ray–Singer torsion of the trivial bundle
We compute the analytic torsion of the trivial bundle on lens spaces. We define the Ray–Singer
torsion function associated with
(
Ω•(M,E),d∇
)
by
κdR(M,E, g)(s) :=
2n+1∑
k=0
(−1)kkζ
(
∆k
dR,g
)
(s),
where ζ
(
∆k
dR,g
)
(s) is the spectral zeta function of the k-th Hodge–de Rham Laplacian ∆k
dR,g
with respect to g. We define the Ray–Singer torsion TdR by
2 log TdR(M,E, g) := κdR(M,E, g)′(0).
Following the derivation of [10, equation (3)], we have
2 log TdR(K,C, gstd) =
1
µ
µ−1∑
ℓ=0
2n∑
j=0
(−1)j+1ζ ′
(
0; j, γℓ
)
,
Ray–Singer Torsion and the Rumin Laplacian on Lens Spaces 13
where λm is the m-th eigenvalue of d†d and
ζ
(
s; j, γℓ
)
=
∞∑
m=0
λ−s
m Tr
(
γℓ
∣∣
Xj,m
)
,
Xj,m :=
{
ϕ ∈ Ωj(K) | d†dϕ = λmϕ
}
.
We recall from [10, p. 123]
2n∑
j=0
(−1)j+1ζ
(
s; j, γℓ
)
=
(
Γ(s)
)−2
∫ ∞
0
t2s−1
∫ 1
0
(
u(1− u)
)s−1(
f0(t, u) + f1(t, ℓν/µ)
)
dudt+O
(
s2
)
,
where ν = (ν1, . . . , νn+1) and for σ = (σ1, . . . , σn+1) ∈ Rn+1,
f0(t, u) =
n+1∑
j=0
(−1)j
(
n
j
)(
2 sinh tu sinh t(1− u)
sinh t
)j
− e−(2n+1)tu
2 sinh tu
− e−(2n+1)t(1−u)
2 sinh t(1− u)
,
f1(t, σ) =
n+1∑
k
(
1− sinh t
cosh t− cos 2πσk
)
.
We set
h0(s) :=
(
Γ(s)
)−2
∫ ∞
0
t2s−1
∫ 1
0
(
u(1− u)
)s−1
f0(t, u) dudt,
h1
(
s, γℓ
)
:=
(
Γ(s)
)−2
∫ ∞
0
t2s−1
∫ 1
0
(
u(1− u)
)s−1
f1(t, ℓν/µ) dudt.
From [10, p. 125], it is seen that
h1
(
s, γℓ
)
= −1
2
µ−1∑
j=0
Tr
(
γjℓ
)(
ζ(2s, j/µ) + ζ(2s, 1− j/µ)
)
− 2(n+ 1)µ−2sζ(2s),
where
Tr
(
γℓ
)
=
n+1∑
j=1
(
e2π
√
−1ℓνj/µ + e−2π
√
−1ℓνj/µ
)
.
Taking the average of h1(s, γ
ℓ), we have
1
µ
µ−1∑
ℓ=0
h1
(
s, γℓ
)
= −2(n+ 1)µ−2sζ(2s)
Using ζ ′(0) = − log(2π)/2, we get
1
µ
µ−1∑
ℓ=0
h′1
(
0, γℓ
)
= 2(n+ 1) log
(
2π
µ
)
. (4.1)
We recall the Ray–Singer torsion on spheres,
14 A. Kitaoka
Proposition 4.1 ([15]).
TdR(S,C, gstd) =
2πn+1
n!
Remark 4.2. The Ray–Singer torsion of spheres can easily be determined using the Cheeger–
Müller theorem, a result which predates [15]. Details can be found in [3].
To put µ = 1, ν = (1, . . . , 1), from (4.1), it follows that
h′0(0) = 2 log TdR(S,C, gstd)−
1
µ
µ−1∑
ℓ=0
h′1
(
0, γℓ
)
= 2 log
(
2πn+1
n!
)
− 2(n+ 1) log(2π) = 2 log
(
2−n
n!
)
. (4.2)
By (4.1) and (4.2), we conclude
2 log TdR(K,C, gstd) =
1
µ
µ−1∑
ℓ=0
h′0(0) +
1
µ
µ−1∑
ℓ=0
h′1
(
0, γℓ
)
= 2 log
(
2−n
n!
)
+ 2(n+ 1) log
(
2π
µ
)
= 2 log
(
2πn+1
n!µn+1
)
. (4.3)
For the metric g on E over M , we set gρ := e2ρg for ρ ∈ R. Then, the Hodge–de Rham
Laplacian ∆dR,gρ with respect to gρ is given by
∆dR,gρ = e−2ρ∆dR,g (4.4)
(e.g., see [11, equation (5.4)]). From (4.4), we see
κdR(M,E, gρ)(s) =
2n+1∑
k=0
(−1)kk dimHk
dR(M,E)
+ e2ρs
2n+1∑
k=0
(−1)kk
(
ζ
(
∆k
dR,g
)
(s)− dimHk
dR(M,E)
)
.
To derivate the above equation,
κdR(M,E, gρ)
′(s) = 2ρe2ρs
2n+1∑
k=0
(−1)kk
(
ζ
(
∆k
dR,g
)
(s)− dimHk
dR(M,E)
)
+ e2ρs
2n+1∑
k=0
(−1)kk
(
ζ
(
∆k
dR,g
)′
(s)
)
.
To substitute s = 0, since
ζ
(
∆k
dR,g
)
(0) = 0
on manifolds with dimension 2n+ 1, the Ray–Singer torsion is given by
log TdR(M,E, gρ) = log TdR(M,E, g)− ρ
2n+1∑
k=0
(−1)kk dimHdR(M,E). (4.5)
To substitute M = K, E is the trivial bundle, ρ = log 2 and g = gstd, we obtain
TdR(K,C, 4gstd) = 22n+1TdR(K,C, gstd). (4.6)
By (4.3) and (4.6), we conclude Proposition 1.2.
Ray–Singer Torsion and the Rumin Laplacian on Lens Spaces 15
5 Proof of Corollary 1.3
Since α(γ) ∈ U(r) is diagonalizable by a unitary matrix, we have
Eα = Eu1 ⊕ · · · ⊕ Eur .
We recall the Ray–Singer torsion on lens spaces,
Proposition 5.1 ([10]). For u (= 1, . . . , µ− 1),
TdR(K,Eu, 4gstd) =
n+1∏
j=1
∣∣e2π√−1uτj/µ − 1
∣∣.
From Proposition 5.1, Theorem 1.1, and Proposition 1.2, we conclude
TdR(K,Eα, gθ,J) =
r∏
j=1
TdR(K,Euj , gθ,J) =
r∏
j=1
n!− dimH0(K,Euj )TE(K,Euj , gθ,J)
= n!− dimH0(K,Eα)TE(K,Eα, gθ,J).
A Alternative shorter derivation of Proposition 1.2
The decomposition
Ω•(S2n+1,C
)
=
µ−1⊕
u=0
Ω•(K,Eu)
gives
κdR
(
S2n+1,C, 4gstd
)
=
µ−1∑
u=0
κdR(K,Eu, 4gstd).
In particular,
TdR
(
S2n+1,C, 4gstd
)
= TdR(K,C, 4gstd)
µ−1∏
u=1
TdR(K,Eu, 4gstd).
Combining this with (4.5), (4.6), Propositions 4.1 and 5.1 and [3], we obtain
(4π)n+1
n!
= TdR(K,C, 4gstd)
µ−1∏
u=1
n+1∏
j=1
∣∣e2π√−1uτj/µ − 1
∣∣ = TdR(K,C, 4gstd)µn+1,
whence Proposition 1.2.
Acknowledgement
The author is grateful to his supervisor Professor Kengo Hirachi for introducing this subject and
for helpful comments. This work was supported by the program for Leading Graduate Schools,
MEXT, Japan. The author also thanks the referees for their valuable comments.
16 A. Kitaoka
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1 Introduction
2 The Rumin complex
2.1 The Rumin complex on contact manifolds
2.2 The Rumin complex on the CR spheres
3 Contact analytic torsion of flat vector bundles
4 Ray–Singer torsion of the trivial bundle
5 Proof of Corollary 1.3
A Alternative shorter derivation of Proposition 1.2
References
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| id | nasplib_isofts_kiev_ua-123456789-211813 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-16T12:21:46Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kitaoka, Akira 2026-01-12T10:16:09Z 2022 Ray-Singer Torsion and the Rumin Laplacian on Lens Spaces. Akira Kitaoka. SIGMA 18 (2022), 091, 16 pages 1815-0659 2020 Mathematics Subject Classification: 58J52; 32V20; 53D10; 43A85 arXiv:2009.03276 https://nasplib.isofts.kiev.ua/handle/123456789/211813 https://doi.org/10.3842/SIGMA.2022.091 We express explicitly the analytic torsion functions associated with the Rumin complex on lens spaces in terms of the Hurwitz zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions. Moreover, we have a formula between this torsion and the Ray-Singer torsion. The author is grateful to his supervis or, Professor Kengo Hirachi, for introducing this subject and for helpful comments. This work was supported by the program for Leading Graduate Schools, MEXT, Japan. The author also thanks the referees for their valuable comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Ray-Singer Torsion and the Rumin Laplacian on Lens Spaces Article published earlier |
| spellingShingle | Ray-Singer Torsion and the Rumin Laplacian on Lens Spaces Kitaoka, Akira |
| title | Ray-Singer Torsion and the Rumin Laplacian on Lens Spaces |
| title_full | Ray-Singer Torsion and the Rumin Laplacian on Lens Spaces |
| title_fullStr | Ray-Singer Torsion and the Rumin Laplacian on Lens Spaces |
| title_full_unstemmed | Ray-Singer Torsion and the Rumin Laplacian on Lens Spaces |
| title_short | Ray-Singer Torsion and the Rumin Laplacian on Lens Spaces |
| title_sort | ray-singer torsion and the rumin laplacian on lens spaces |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211813 |
| work_keys_str_mv | AT kitaokaakira raysingertorsionandtheruminlaplacianonlensspaces |