Weighted Analytic Torsions for Weighted Digraphs

Home , John Milnor

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Theorem 1.1. Let G = (V, E) be a digraph such that there exists a positive integer N satisfying Ωn(G)=0 for any n N +1. Let f and g be two non-vanishing functions ≥ on V . Let T (G,f,g) be the (f,g)-weighted analytic torsion of G. Then

(i). T (G,f,f)= T (G) for any non-vanishing weight f;

(ii). T (G,f,cg)= t(G, c)T (G,f,g) and T (G,cf,g)= t(G, c)−1T (G,f,g) for any non- vanishing weights f and g and any non-zero real constant c. Here t(G, c)= c s(G) n | | only depends on G and c where s(G)= ( 1) dim ∂nΩn(G). Pn≥0 − The next corollary follows immediately by combining Theorem 1.1 (i) and (ii).

Corollary 1.2. Let G be a digraph such that there exists a positive integer N satisfying Ωn(G)=0 for any n N +1. For any non-vanishing weight f on V and any real ≥ number c =0, we have 6 T (G,cf,cf)= T (G,f,f)= t(G, c)T (G,cf,f)= t(G, c)−1T (G,f,cf).

As a by-product, we obtain in Corollary 3.5 that for any digraph G = (V, E) and any non-vanishing weight f on V , if we represent the homology classes by harmonic chains, then the (f,f)-weighted path homology (with the f-twisted boundary operators and the inner products rescaled by f) of G is isometrically isomorphic to the usual path homology of G. We point out that Corollary 3.5 holds without the assumption of a positive integer N satisfying Ωn(G)=0 for any n N +1. ≥ The remaining part of this paper is organized as follows. In Section 2, we review the R-torsion and the analytic torsion for chain complexes given in [3, Section 3]. In Section 3, we prove Theorem 1.1. In Section 4, we calculate some examples of weighted analytic torsions for weighted digraphs.

2 Preliminaries

In this section, we review the R-torsion and the analytic torsion theory of chain complexes (cf. [3, Section 3]). We prove some lemmas.

Let C∗ = Cn, ∂n n≥ be a chain complex where for each n 0, Cn is a (ﬁnite- { } 0 ≥ dimensional) real vector space and ∂n : Cn Cn−1 is the boundary operator. Suppose −→ ∗ each Cn is equipped with an inner product , . Let ∂ : Cn− Cn be the adjoint h i n 1 −→ operator of ∂n with respect to the inner products on Cn and Cn−1. The Hodge-Laplace operator ∆n : Cn Cn is deﬁned by −→ ∗ ∗ ∆nu = ∂n∂nu + ∂n+1∂n+1u.

2 An element u Cn is called harmonic if ∆nu = 0. Let n be the set of all harmonic ∈ H chains in Cn. It can be proved that (cf. [3, Section 3.1])

∗ n = Ker∂n Ker∂ H ∩ n+1 and the space Cn is an orthogonal sum of three subspaces

∗ Cn = ∂n Cn ∂ Cn− n. +1 +1 ⊕ n 1 ⊕ H Moreover, there is a natural isomorphism

Hn(C∗) = n (2.1) ∼ H where Hn(C∗) is the n-th homology of C∗. It is direct to check that ∆n is a self-adjoint semi-positive deﬁnite operator on Cn. We denote its eigenvalues as λi 0 i { | ≤ ≤ dim Cn 1 . The zeta function ζn(s) is deﬁned by − } 1 ζn(s)= s . X λi λi>0

Now we suppose that there exists a positive integer N such that Cn = 0 for any n N. We prove the next lemma. ≥ Lemma 2.1. For each n 0, let Hn be the n-th homology of C∗. Then ≥ N N n n ( 1) n dim Cn dim Hn = ( 1) dim Im∂n. X − − X − n=0 n=0

Proof. For each n 0, we note that dim Hn = dim Ker∂n dim Im∂n and dim Cn = ≥ − +1 dim Ker∂n + dim Im∂n. By a direct calculation, it follows that

N N n n ( 1) n dim Cn dim Hn = ( 1) n dim Im∂n + dim Im∂n X − − X − +1 n=0 n=0 N n n−1 = ( 1) n + ( 1) (n 1) dim Im∂n X − − − n=0 N n = ( 1) dim Im∂n. X − n=0 We obtain the lemma.

The analytic torsion T (C∗, , ) of the chain complex C∗ with an inner product , h i h i is deﬁned by

N 1 n ′ log T (C∗, , )= ( 1) nζ (0). h i 2 X − n n=0 On the other hand, we can use the transition matrices of certain bases to define the R-torsion τ (cf. [3, Section 3]), which is proved in [3, Theorem 3.14] to be the same as the analytic torsion T . For completeness, we describe the R-torsion as follows. For each 0 n N, let cn be a basis in Cn and let hn be a basis in Hn(C∗). Let ≤ ≤ Bn = ∂n+1Cn+1 and bn be any basis in Bn. Let Zn = Ker(∂n). Then Hn(C∗)= Zn/Bn. −1 For each w bp− , choose one element v ∂ w such that ∂nv = w. Let ˜bn be the ∈ 1 ∈ n collection of all these chains v. Then ˜bn is a linearly independent set in Cn. Similarly, for each element of hn we choose its representative in Zn and denote the resulting independent set by hñ. The union (bn, hñ) is a basis in Zn. And the union (bn, hñ, ˜bn) is

3 a basis in Cn. By using the notation [bn, hñ, ˜bn/cn] to denote the absolute value of the determinant of the transition matrix from (bn, hñ, ˜bn) to cn, the R-torsion τ(C∗,c,h) of the chain complex C∗ with the preferred bases c and h is a positive real number defined by (cf. [3, Definition 3.6])

N n log τ(C∗,c,h)= ( 1) log[bn, h˜n, ˜bn/cn]. X − n=0

It is proved in [3, Lemma 3.7] that the value of τ(C∗,c,h) does not depend on the choice ′ ′ of the bases bn, the representatives in ˜bn and the representatives in h˜n; and if c and h are other collections of bases in C∗ and H∗ respectively, then

N ′ ′ n ′ ′ log τ(C∗,c ,h ) = log τ(C∗,c,h)+ ( 1) log[cn/c ] + log[h /hn]). (2.2) X − n n n=0

Let us ﬁx an inner product , in Cn for each 0 n N. Then we have the induced h i ≤ ≤ inner product in the subspaces Bn, Zn and n. By (2.1), we represent each homology H class by the corresponding harmonic chain. Thus we can transfer the inner product on the harmonic chains to Hn(C∗). We choose the bases cn and hn to be orthonormal and deﬁne the R-torsion of (C∗, , ) by h i

τ(C∗, , )= τ(C∗,c,h). h i By (2.2), the right-hand side does not depend on the choice of orthonormal bases c and h. It is proved in [3, Theorem 3.14] that

τ(C∗, , )= T (C∗, , ). (2.3) h i h i

For convenience, we also use ι∗ to denote the inner products , on the graded Euclidean h i space C∗ for short. Now we consider two diﬀerent inner products ι1 and ι2 on C∗. Assume that there are positive real numbers cn, 0 n N, such that for all chains u, w Cn, ≤ ≤ ∈

u, w = cn u, w . (2.4) h i2 h i1 Then by [3, Corollary 3.8] and (2.3),

N (−1)n (dim C −dim H (C∗)) T (C , ι )= T (C , ι ) c 2 n n . (2.5) ∗ 2 ∗ 1 Y n n=0

The following lemma which plays a key role in this paper. For solidarity, we give a detailed proof.

′ Lemma 2.2. Let C∗ and C∗ be two chain complexes. For each n 0, let ιn be an ′ ′ ≥ inner product on Cn and ιn an inner product on Cn. If there is a chain isomorphism ′ ϕ : C∗ C s.t. for any n 0 and any a,b Cn, −→ ∗ ≥ ∈ ′ ιn(ϕ(a), ϕ(b)) = ιn(a,b), then for each n 0, ≥ −1 ′ ′ ∆n(C∗, ι∗)= ϕ ∆n(C , ι ) ϕ (2.6) ◦ ∗ ∗ ◦ where ∆n(C∗, ι∗) is the Hodge-Laplace operator of the chain complex C∗ with respect to ′ ′ the inner products ι∗ and ∆n(C∗, ι∗) is the Hodge-Laplace operator of the chain complex

4 ′ ′ C∗ with respect to the inner products ι∗. As consequences, by choosing orthonormal bases properly, the matrix representatives of the Hodge-Laplace operators are equal

′ ′ [∆n(C∗, ι∗)] = [∆n(C∗, ι∗)], which implies

′ ′ T (C∗, ι∗)= T (C∗, ι∗).

′ ′ ′ Proof. We write ∆n(C∗, ι∗) as ∆n and write ∆n(C∗, ι∗) as ∆n for short. We denote ιn as , an denote ι′ as , ′. h i n h i Firstly, we prove that for any n 0 and any a Cn, ≥ ∈ ′ ∗ ∗ (∂n+1) ϕ(a)= ϕ(∂n+1(a)). (2.7)

Note that since ϕ is a chain isomorphism, we have that ϕ is a linear isomorphism and for any n 0, ≥ ′ ∂nϕ = ϕ∂n. (2.8)

Thus for any c Cn , we have ∈ +1 (∂′ )∗ϕ(a), ϕ(c) ′ = ϕ(a), ∂′ ϕ(c) ′ h n+1 i h n+1 i ′ = ϕ(a), ϕ∂n (c) h +1 i = a, ∂n (c) h +1 i = ∂∗ (a),c h n+1 i = ϕ∂∗ (a), ϕ(c) ′. h n+1 i ′ Letting c run over Cn+1, we note that ϕ(c) runs over Cn+1. Thus (2.7) follows by the last equation. Secondly, we prove that for any n 0 and any a Cn, ≥ ∈ −1 ′ ϕ ∆nϕ(a) = ∆n(a). (2.9)

Let b Cn. By a straight-forward calculation and with the help of (2.8) and (2.7), ∈ ϕ−1∆′ ϕ(a),b = ∆′ ϕ(a), ϕ(b) ′ h n i h n i = (∂′ )∗∂′ (ϕ(a)), ϕ(b) ′ + ∂′ (∂′ )∗(ϕ(a)), ϕ(b) ′ h n n i h n+1 n+1 i ′ ∗ ′ ′ ∗ ′ = (∂ ) ϕ∂n(a), ϕ(b) + ∂ ϕ(∂n ) (a), ϕ(b) h n i h n+1 +1 i ∗ ′ ∗ ′ = ϕ(∂n) ∂n(a), ϕ(b) + ϕ∂n (∂n ) (a), ϕ(b) h i h +1 +1 i ∗ ∗ = (∂n) ∂n(a),b + ∂n (∂n ) (a),b h i h +1 +1 i = ∆n(a),b . h i

Letting b run over Cn, (2.9) follows from the last equation. Thirdly, we prove the lemma. By (2.9), we have (2.6). We observe that both ∆n and ′ ∆n are semi-positive defniite and self-adjoint. We let 0 λ0 < λ1 < < λk be the ′ ′ ′ ≤ ··· ′ eigenvalues of ∆n and let 0 λ < λ < λ ′ be the eigenvalues of ∆ . Let E(λ) ≤ 0 1 ···≤ k n and E′(λ′) be the corresponding eigenspaces of λ and λ′ respectively. Then

k ′ k′ ′ ′ Cn = E(λi), C = E (λ ). ⊕i=0 n ⊕i=0 i Let c E′(λ′ ). Then by (2.6), ∈ i −1 −1 ′ −1 ′ ′ −1 ∆n(ϕ (c)) = ϕ ∆n(c)= ϕ (λic)= λiϕ (c).

5 ′ −1 ′ Thus λi is also an eigenvalue of ∆n and ϕ (c) E(λi). Conversely, it can be proved in ∈ ′ ′ a similar way that for any c E(λi), λi is also an eigenvalue of ∆n and ϕ(c) E (λi). ∈ ′ ∈ Hence the sets of eigenvalues λi i = 0, 1,... and λ i = 0, 1,... are equal, and { | } { i | } E(λ) is linearly isomorphic to E′(λ) for each such eigenvalue λ. Therefore, as multi-sets, ′ ′ λi i =0, 1,... and λ i =0, 1,... are equal. The analytic torsions of ∆n and ∆ { | } { i | } n must be equal as well. The lemma follows.

3 Vertex-Weighted Digraphs

Let G = (V, E) be a digraph. Let g : V R× be a non-vanishing real-valued weight −→ function on G assigning a non-zero number to each vertex of G. Let n 0. Let Λn(V ) ≥ be the vector space of n-paths on V . Then Λn(V ) has a basis consisting of all the elementary n-paths

v v ...vn, vi V, 0 i n. 0 1 ∈ ≤ ≤

The weight g on V induces a weight g :Λn(V ) R by −→ g x v v ...v = x g(v )g(v ) ...g(v ). (3.1) X v0v1...vn 0 1 n X v0v1...vn 0 1 n We note that

g(v )g(v ) ...g(vn) =0 0 1 6 for any elementary n-path v0v1 ...vn on V . The weight g induces an inner product

ιg :Λn(V ) Λn(V ) R × −→ by

n ι v v ...v ,u u ...u = g(v )g(u )δ(v ,u ) g 0 1 n 0 1 n Y i i i i i=0 which extends bilinearly over R. Here for any two vertices v,u V , we use the notation ∈ δ(v,u)=0 if v = u and δ(v,u)=1 if v = u. Particularly, we let ι be the usual 6 (un-weighted) inner product on Λn(V ) given by the constant weight g =1. Let f : V R× be another non-vanishing real-valued weight function on V (here −→ the choices of f and g do not depend on each other). The f-weighted boundary map

f ∂ :Λn(V ) Λn− (V ) n −→ 1 is given by

n f i ∂ (v v ...vn)= ( 1) f(vi)v ... vi ...vn n 0 1 X − 0 i=0 b f ∗ which extends linearly over the real numbers. The adjoint operator (∂n)g :Λn−1(V ) f −→ Λn(V ) of ∂n with respect to ιg is given by

f f ∗ ιg(∂n(a),b)= ιg(a, (∂n)g(b)) for any a Λn(V ) and any b Λn− (V ). The n-th (f,g)-weighted Hodge-Laplace ∈ ∈ 1 operator on V is a linear map

f,g ∆ :Λn(V ) Λn(V ) n −→

6 given by

f,g f ∗ f f f ∗ ∆n = (∂n)g(∂n) + (∂n+1)(∂n+1)g.

f For diﬀerentiation, we use Λ∗(V,f,g) to denote the chain complex Λn(V ), ∂ n≥ with { n} 0 the inner product ιg. Particularly when f = g, we have the next lemma. Lemma 3.1. For any non-vanishing weight f on V , there is a canonical chain isomorphism

ϕ :Λ∗(V ) Λ∗(V,f,f) (3.2) −→ given by

xv0v1...vn ϕ xv0 v1...vn v0v1 ...vn = v0v1 ...vn (3.3) X X f(v )f(v ) f(vn) 0 1 ··· such that

ιf (ϕ(a), ϕ(b)) = ι(a,b) for any a,b Λn(V ). ∈ Proof. Firstly, let xv v ...v v0v1 ...vn be in Λn(V ). We have P 0 1 n f ∂ ϕ xv v ...v v v ...vn n ◦ X 0 1 n 0 1 f xv0v1...vn = ∂n v0v1 ...vn X f(v0)f(v1) f(vn) ··· n xv0v1...vn i = ( 1) f(vi)v0 ... vi vn X f(v0)f(v1) f(vn) X − ··· ··· i=1 b n v ... vi vn = x ( 1)i 0 ··· X v0v1...vn X [ − f(v ) ... f(bvi) f(vn) i=0 0 ··· = ϕ ∂n xv v ...v v v ...vn . ◦ X 0 1 n 0 1 Thus ϕ is a chain map. Secondly, it is direct that ϕ is a linear isomorphism. Thirdly, let xv v ...v v0v1 ...vn and yu u ...u u0u1 ...un be in Λn(V ). Then P 0 1 n P 0 1 n ι ϕ x v v ...v , ϕ y u u ...u f X v0v1...vn 0 1 n X u0u1...un 0 1 n

xv0v1...vn yu0u1...un = ιf (v0v1 ...vn,u0u1 ...un) XX f(v0)f(v1) f(vn) · f(u0)f(u1) f(un) ··· ··· n xv0v1...vn yu0u1...un = f(vi)f(ui)δ(vi,ui) XX f(v )f(v ) f(vn) · f(u )f(u ) f(un) Y 0 1 ··· 0 1 ··· i=0 n = x y δ(v ,u ) XX v0v1...vn u0u1...un Y i i i=0 = ι x v v ...v , y u u ...u . X v0v1...vn 0 1 n X u0u1...un 0 1 n Thus ϕ preserves the inner products.

We notice that ϕ in (3.2) sends a regular n-path to a regular n-path. The next lemma follows from Lemma 3.1 which transits the assertion to the quotient spaces R∗(V ). Lemma 3.2. For any non-vanishing weight f on V , there is a canonical chain isomorphism

ϕ : ∗(V ) ∗(V,f,f) (3.4) R −→ R

7 given by (3.3) such that

ιf (ϕ(a), ϕ(b)) = ι(a,b) for any a,b n(V ). ∈ R Let n 0. The space n(G) consisting of all the allowed n-paths on G is a subspace ≥ A of n(V ) spanned by all the allowed elementary n-paths R

v v ...vn, (vi− , vi) E for 1 i n. 0 1 1 ∈ ≤ ≤ For any two non-vanishing weights f and g on V , the (f,g)-weighted ∂-invariant complex of G is a chain complex

f,g f −1 Ω (G)= n(G) (∂ ) n− (G) , n 0, n A ∩ n A 1 ≥ f with the weighted boundary operators ∂ , n 0, and the inner products ιg. The n ≥ (f,g)-weighted Hodge-Laplace operator is a linear map

∆f,g(G):Ωf,g(G) Ωf,g(G) n n −→ n given by

f,g f ∗ f f f ∗ ∆n (ω) = (∂n f,g ) ∂n(ω)+ ∂ (∂ f,g ) (ω) |Ωn (G) n+1 n+1 |Ωn (G) f,g f,g for any ω Ωn (G). We denote the analytic torsion of ∆n (G) n≥0 as T (G,f,g). ∈ { } f,f Consider the case f = g. We notice that the map (3.4) sends Ωn(G) to Ωn (G) for each n 0. We can prove this observation by a similar veriﬁcation with the proof of ≥ Lemma 3.1. Consequently, the next lemma follows from Lemma 3.2 which restricts the assertion to the sub-chain complex Ω∗(G).

Lemma 3.3. For any non-vanishing weight f on V , there is a canonical chain isomorphism

f,f ϕ :Ω∗(G) Ω (G) (3.5) −→ ∗ given by (3.3) such that

ιf (ϕ(a), ϕ(b)) = ι(a,b) for any a,b Ω∗(G). ∈ By Lemma 2.2 and Lemma 3.3, we have

Proposition 3.4. For any digraph G and any non-vanishing weight f on V , we have

−1 f,f ∆n(G)= ϕ ∆ (G) ϕ ◦ n ◦ where ϕ is the isomorphism given in Lemma 3.1. Consequently, by choosing orthonormal bases properly, the matrix representatives satisfy

f,f [∆n(G)] = [∆n (G)] (3.6) for each n 0. In addition, if there exists a positive integer N such that Ωn(G)=0 for ≥ any n N +1, then it follows from (3.6) that ≥ T (G, 1, 1) = T (G,f,f).

8 Remark 1: For a digraph G, if there are inﬁnitely many n such that Ωn(G) =0, 6 then the analytic torsion T is not well-deﬁned. In this case, we cannot get the last assertion in Proposition 3.4. Following from (3.6) in Proposition 3.4, we have

Corollary 3.5. For any digraph G and any non-vanishing weights f and g on V , ϕ induces a canonical linear isomorphism

f,g f ϕ : Hn( Ω (G), ∂ k≥ ) Hn(G), n 0. (3.7) { k k } 0 −→ ≥

Here Hn(G) is the usual (un-weighted) path homology of G with coeﬃcients in real numbers. Moreover, if f = g, then representing the homology classes by harmonic chains, we have that (3.7) is an isometry.

We ﬁx f and multiply g by a non-zero scalar c. By applying [3, Corollary 3.8] to the R-torsion, we have

Proposition 3.6. Let G be a digraph. Suppose there exists a positive integer N such that Ωn(G)=0 for any n N +1. Let f and g be a non-vanishing weights on V . For ≥ any real number c =0, we use cg to denote the weight on V with the value cg(v) at any 6 v V . Then ∈ T (G,f,cg)= t(G, c)T (G,f,g) (3.8) where t(G, c) is a real function whose variables are G and c (which does not depend on f) given by

t(G, c)= c s(G) (3.9) | | where

N n s(G)= ( 1) dim Im ∂n . (3.10) X − |Ωn(G) n=0

Proof. For any n 0 and any chains ω,ω′ Ωf,cg(G), we have ≥ ∈ n ′ 2n ′ ιcg(ω,ω )= c ιg(ω,ω ).

By Proposition 3.4 and Corollary 3.5, for each n 0 we have ≥ f,cg f,g dim Ωn (G) = dim Ωn (G)

= dim Ωn(G), f,cg f f,g f dim Hn( Ω (G), ∂ k≥ ) = dim Hn( Ω (G), ∂ k≥ ) { k k } 0 { k k } 0 = dim Hn(G).

Thus by [3, Corollary 3.8], we have (3.8) where

N n t(G, c) = c (−1) n dimΩn(G)−dim Hn(G) Y | | n=0 PN (−1)nn dimΩ (G)−dim H (G) = c n=0 n n . | | By Lemma 2.1, the right-hand side of the last equality equals to c s(G). | | We ﬁx g and multiply f by a non-zero scalar c. By a straight-forward calculation of the analytic torsion, we have

9 Proposition 3.7. Let G be a digraph. Suppose there exists a positive integer N such that Ωn(G)=0 for any n N +1. Let f and g be a non-vanishing weights on V . For ≥ any real number c =0, we use cf to denote the weight on V with the value cf(v) at any 6 v V . Then ∈ T (G,cf,g)= t(G, c)−1T (G,f,g). (3.11)

Proof. Let n 0. It follows directly that ≥ cf f ∂n = c∂n. (3.12)

By a similar argument in the proof of Lemma 2.2, we have

cf ∗ f ∗ (∂ ) u, w g = c (∂ ) u, w g h n+1 i h n+1 i for any chains u, w Ωn(G), which implies that ∈ cf ∗ f ∗ (∂n+1) = c(∂n+1) . (3.13)

By (3.12) and (3.13), we have

cf,g 2 f,g ∆n = c ∆n .

f,g Consequently, if we denote the non-zero eigenvalues of ∆n as 0 < λ1 < λ2 < . . ., then cf,g 2 2 2 the non-zero eigenvalues of ∆n are 0 < c λ1 < c λ2 < . . . where c λi has the same multiplicity with λi. Therefore, with the help of [3, (3.21)], we have

N 1 d 1 log T (G,cf,g) = ( 1)nn 2 s 2 X − ds s=0 X (c λi) n=0 λi>0 N 1 n 2 = ( 1) n log(c λi) 2 X − − X n=0 λi>0 N 1 n = ( 1) n log(λi) 2 X − − X n=0 λi>0 N 1 n ( 1) n(2 log c ) dim Ωn(G) dim Hn(G) −2 X − | | − n=0 N 1 d 1 = ( 1)nn (log c )s(G) X X s 2 − ds s=0 λi − | | n=0 λi>0 = log T (G,f,g) (log c )s(G). − | | Taking the exponential map on both sides of the equations, we have (3.11).

Finally, summarizing Proposition 3.4, Proposition 3.6 and Proposition 3.7, we obtain Theorem 1.1.

4 Examples

We give some examples.

Example 4.1. Consider the line digraph G = (V, E) given in [3, Figure 1, Example 3.9] where V = 0, 1, 2, 3, 4 and E consists of 4 directed edges where the i-th directed edge { } has the form either (i,i+1) or (i+1,i), for 0 i 3. Let f and g be two non-vanishing ≤ ≤ real-valued weight functions on V . As in [3, (3.17)], we denote e¯i(i+1) for the directed

10 edge (which is called 1-path in our setting) (i,i + 1) or (i +1,i) in E, and denote ei for the 0-path consisting of a single vertex i. Let σi = 1 if (i,i + 1) E and 1 if ∈ − (i +1,i) E. Then ∈ f ∂ e¯ = σi(f(i)ei f(i + 1)ei) 1 i(i+1) +1 − and

ei,ej g = g(i)g(j)δ(i, j), h i e¯ , e¯ g = g(i)g(i + 1)g(j)g(j + 1)δ(i, j). h i(i+1) j(j+1)i Note that the last equality holds because each edge is assigned with exactly one direction. Choose the following ιg-orthonormal bases e ω = i 0 i 3 0 ng(i) | ≤ ≤ o

f,g in Ω0 (G) and

e¯i i ω = ( +1) 0 i 2 1 n g(i)g(i + 1) | ≤ ≤ o in Ωf,g(G). We note Ωf,g(G)=0 for n 2. In ∂f (Ωf,g(G)) choose the basis 1 n ≥ 1 1 f(i) ei+1 f(i + 1) ei b = σi 0 i 2 0 n g(i) g(i + 1) − g(i + 1) g(i) | ≤ ≤ o and set

e¯i i ˜b = ( +1) 0 i 2 . 1 ng(i)g(i + 1) | ≤ ≤ o

f f,g f f,g It is clear that Ker∂0 = Ω0 (G). Thus the ιg-orthogonal complement of ∂1 Ω1 (G) in f Ker∂0 is f(0) f(1) f(2) f(3) = Span e + e + e + e H0 n g(0)2 0 g(1)2 1 g(2)2 2 g(3)2 3o so that

f(0) e0 f(1) e1 f(2) e2 f(3) e3 g(0) g(0) + g(1) g(1) + g(2) g(2) + g(3) g(3) h0 = . n f(0)2 f(1)2 f(2)2 f(3)2 o q g(0)2 + g(1)2 + g(2)2 + g(3)2 Similar with [3, (3.19)], by taking the absolute value of the determinant of the transition matrix, we see that

3 ˜ 1 k+1 f(k) σj f(j + 1) σj f(j) [b0,h0, b0/ω0] = ( 1) − 3 f(i) 2 X − g(k) Y g(j + 1) Y g(j) ( ) k=0 0≤j≤k−1 k≤j≤2 qPi=0 g(i) 3 1 2k+1 f(k) σj f(j + 1) σj f(j) = ( 1) 3 f(i) 2 X − g(k) Y g(j + 1) Y g(j) ( ) k=0 0≤j≤k−1 k≤j≤2 qPi=0 g(i) 3 1 f(k) σj f(j + 1) σj f(j) = . 3 f(i) 2 X g(k) Y g(j + 1) Y g(j) ( ) k=0 0≤j≤k−1 k≤j≤2 qPi=0 g(i)

Moreover, since b1 is the empty-set, we have e¯ h = 34 . 1 g(3)g(4)

11 Consequently, by taking the absolute value of the determinant of the transition matrix, we have

[b1,h1, ˜b1/ω1]=1.

It follows that the weighted analytic torsion is

1 n T (G,f,g) = [b ,h , ˜b /ω ](−1) Y n n n n n=0 3 1 f(k) σj f(j + 1) σj f(j) = . 3 f(i) 2 X g(k) Y g(j + 1) Y g(j) ( ) k=0 0≤j≤k−1 k≤j≤2 qPi=0 g(i) In particular, consider the following cases:

(i). f = g. Then T (G,f,f) reduces to T (G, 1, 1) in [3, Example 3.9], which takes value √3 and does not depend on the choice of f.

(ii). g takes a constant value c =0. 6 Then by a direct calculation, we have t(f,c)= c −3 and T (G,f,c)= c −3T (G, f, 1) | | | | for any non-vanishing weight f. On the other hand, we note that

s(G) = ( 1)1 dim ∂ (Ω (G)) = 3. − 1 1 − (iii). f takes a constant value c =0. 6 Then by a direct calculation, we have T (G,c,g) = c 3T (G, 1,g) for any non- | | vanishing weight g.

Example 4.2. Consider the triangle G = (V, E) where V = 0, 1, 2 and E = 01, 12, 02 { } { } (cf. [3, Example 3.10]). Let f and g be two non-vanishing real functions on V . We have

Ωf,g(G) = Span e ,e ,e , Ωf,g(G) = Span e ,e ,e , Ωf,g(G) = Span e , 0 { 0 1 2} 1 { 01 12 02} 2 { 012} ∂f Ωf,g(G) = Span f(0)e f(1)e ,f(1)e f(2)e ,f(0)e f(2)e , 1 1 { 1 − 0 2 − 1 2 − 0} ∂f Ωf,g(G) = Span f(0)e f(1)e + f(2)e . 2 2 { 12 − 02 01} f,g And Ω (G)=0 for any n 3. We choose the following ιg-orthonormal bases n ≥ e e e ω = 0 , 1 , 2 0 ng(0) g(1) g(2)o

f,g in Ω0 (G), e e e ω = 01 , 12 , 02 1 ng(0)g(1) g(1)g(2) g(0)g(2)o

f,g in Ω1 (G), and e ω = 012 2 ng(0)g(1)g(2)o

f,g in Ω2 (G). Choose also the bases f(0)e f(1)e f(1)e f(2)e b = 1 − 0 , 2 − 1 0 n g(0)g(1) g(1)g(2) o

12 f f,g in ∂1 Ω1 (G) and f(0)e f(1)e + f(2)e b = 12 − 02 01 1 n g(0)g(1)g(2) o

f f,g in ∂2 Ω2 (G). Then their lifts are e e ˜b = 01 , 12 1 ng(0)g(1) g(1)g(2)o and e ˜b = 012 . 2 n g(0)g(1)g(2)o Hence representing the homology classes by harmonic chains and taking the orthogonal f,g f,g f complement of ∂1 Ω1 (G) in Ker∂0 with respect to ιg, we have f(0) f(1) f(2) = Span e + e + e H0 ng(0)2 0 g(1)2 1 g(2)2 2o so that 1 f(0) f(1) f(2) h0 = e0 + e1 + e2 . n 2 f(i)2 g(0)2 g(1)2 g(2)2 o qPi=0 g(i)2 ˜ f,g Moreover, note that b1 and b1 spans Ω1 (G). Thus h1 is the empty-set. Similarly, both b2 and h2 are empty-sets. By taking the absolute values of the determinants of the transition matrices, it follows that 1 f(1) f(0)2 f(1)2 f(2)2 [b0,h0, ˜b0/ω0]= + + , 2 2 2 2 2 f(i) g(1) · g(0) g(1) g(2) qPi=0 g(i)2

f(1) [b ,h , ˜b /ω ]= | |, 1 1 1 1 g(1) | | and

[b2,h2, ˜b2/ω2]=1.

Hence we obtain

2 n T (G,f,g) = [b ,h , ˜b /ω ](−1) Y n n n n n=0 1 f(0)2 f(1)2 f(2)2 = + + . 2 2 2 2 2 f(i) g(0) g(1) g(2) qPi=0 g(i)2 In particular, consider the following cases: (i). f = g. Then T (G,f,f)= √3 which is the same as [3, Example 3.10] and does not depend on the choice of f.

(ii). g takes a constant value c =0. 6 Then by a direct calculation, we have t(f,c)= c −1 and T (G,f,c)= c −1T (G, f, 1) | | | | for any non-vanishing weight f. On the other hand, we note that

s(G) = ( 1)1 dim ∂ Ω (G) + ( 1)2 dim ∂ Ω (G)= 2+1= 1. − 1 1 − 2 2 − −

13 (iii). f takes a constant value c =0. 6 Then by a direct calculation, we have T (G,c,g) = c T (G, 1,g) for any non- | | vanishing weight g.

Acknowledgements. The authors would like to express their deep gratitude to Professor Yong Lin for his kind instruction and helpful guidance.

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Shiquan Ren (for correspondence) Address: Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, P. R. China E-mail: [email protected]

Chong Wang Address: School of Mathematics, Renmin University of China, Beijing, 100872, P. R. China; School of Mathematics and Statistics, Cangzhou Normal University, Cangzhou, Hebei, 061000, P. R. China E-mail: [email protected]