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Determinants of Laplacians, the Ray-Singer Torsion on Lens Spaces

arXiv:hep-th/9212022v1 3 Dec 1992Determinants of Laplacians, the Ray-Singer Torsion on Lens Spacesand the Riemann zeta functionbyCharles Nash and D. J. O’ ConnorDepartment of Mathematical PhysicsSchool of Theoretical PhysicsSt. Patrick’s CollegeDublin Institute for Advanced StudiesMaynoothand10 Burlington RoadIrelandDublin 4IrelandAbstract: We obtain explicit expressions for the determinants of the Laplacians on zeroand one forms for an infinite class of three dimensional lens spaces L(p, q).

These ex-pressions can be combined to obtain the Ray-Singer torsion of these lens spaces. As aconsequence we obtain an infinite class of formulae for the Riemann zeta function ζ(3).The value of these determinants (and the torsion) grows as the size of the fundamentalgroup of the lens space increases and this is also computed.

The triviality of the torsionfor just the three lens spaces L(6, 1), L(10, 3) and L(12, 5) is also noted.§ 1. IntroductionTopological phenomena are now known to play an important part in many quantum fieldtheories.This is especially true of gauge theories.

There are also topological quantumfield theories in which the excitations are purely topological and the classical phase spacesof these theories usually reduce to a finite dimensional space: these spaces can be zerodimensional discrete sets, or whole moduli spaces. The semi-classical, or stationary phase,approximation to the functional integral of such a theory is then a weighted sum, orintegral, over the finite dimensional phase space.

In addition, for some of these theories,this approximation is exact providing thereby a reduction of the functional integral to afinite dimensional integral.If a topological quantum field theory contains a gauge field the reduced functionalintegral mentioned above often consists of sums or integrals over flat connections; thenon-triviality of such connections is determined purely by their holonomy, and, if A is aflat connection over a manifold M, its holonomy is an element of the fundamental groupπ1(M). This means that an ideal laboratory within which to study such theories is providedby taking the manifold M to have a non-trivial fundamental group but to be otherwisetopologically rather simple.

An ideal way to do this is to take M to be the quotient of asphere Sn by a finite cyclic group G. This quotient Sn/G, described in more detail below,is what is called a lens space, written as L(p, q). In this paper we take M to be a lensspace on which is placed a topological field theory whose classical phase space consists offlat connections.1

Our approach is to take the model given by the field theory and analyse it in detailon a whole infinite class of lens spaces. We work in three dimensions and realise M as thequotient of the manifold S3 by the action of a discrete group Zp.

The resulting partitionfunction on this manifold is a combinatorial invariant of the manifold known as the Ray-Singer torsion of the manifold. However the field theory gives this partition function asthe ratio of a set of determinants.

A standard technique in field theory has been to definethese functional determinants through the analytic continuation of the zeta functions ofthe associated operators.In this work we investigate the individual determinants that arise and obtain highlyexplicit expressions for them. Our expressions have an intriguing structure of their ownFor example, on the lens space L(2, 1), we find thatln Det d∗d0 = −3ζ(3)2π2 + ln 2ln Det d∗d1 = −3ζ(3)π2−2 ln 2(1.1)Similar, though more complicated, expressions occur for each of the lens spaces L(p, 1) forp = 3, 4, .

. ..

This in turn leads to non-trivial formulae for ζ(3): to give two examples wefind thatζ(3) = 2π27ln(2) −87Z π/20dz z2 cot(z)ζ(3) = 2π213 ln 3 −913Zπ30dz z(z + π3 ) cot(z) −913Z2π30dz z(z −π3 ) cot(z)(1.2)these being the formulae that come from L(2, 1) and L(3, 1) respectively.The structure of the paper is as follows. In Section 2 we describe the topological fieldtheory under consideration.

In section 3 we define the Ray-Singer torsion and describe thelens spaces with which we work; we also carry out the non-trivial task of obtaining theeigenvalues and degeneracies for the Laplacians on p-forms acting on these spaces. Section4 deals with the lens space L(2, 1) (SO(3)) and is a construction of the analytic continua-tion of the appropriate p-form zeta functions followed by a calculation of their associateddeterminants.Sections 5 and 6 describe the analogous calculation and expressions forthe infinite classes of manifolds corresponding to L(p, 1) for p odd and even respectively.Finally in section 7 we present our conclusions, some comments on the torsion of L(p, q)for general q, and some graphical data for the resulting determinants and torsion.§ 2.

Topological Field TheoryThe torsion studied in this paper has its origins in the 1930’s, cf. Franz [1], where itwas combinatorially defined and used to distinguish various lens spaces from one another.Given a manifold M and a representation of its fundamental group π1(M) in a flat bundleE, this Reidemeister-Franz torsion is a real number which is defined as a particular productof ratio’s of volume elements V i constructed from the cohomology groups Hi(M; E).2

Since volume elements are essentially determinants then, for any alternative definitionof a determinant, an alternative definition of the torsion can be given. Now if one uses deRham cohomology to compute Hi(M; E) then these determinants become determinants ofLaplacians ∆Ep on p-forms with coefficients in E. But zeta functions for elliptic operatorscan be used to give finite values to such infinite dimensional determinants and so an analyticdefinition of the torsion results and this is the analytic torsion of Ray and Singer [2,3,4]given in the 1970’s; furthermore this torsion was proved by them to be independent of theRiemannian metric used to define the Laplacian’s ∆Ep .This analytic torsion coincided, for the case of lens spaces, with the combinatorially de-fined Reidemeister-Franz torsion.

Finally Cheeger and M¨uller [5,6] independently provedthat the analytic Ray-Singer torsion coincides with the combinatorial Reidemeister-Franztorsion in all cases.Infinite dimensional determinants also occur naturally in quantum field theories whencomputing correlation functions and partition functions. In 1978 Schwarz [7] showed howto construct a quantum field theory on a manifold M whose partition function is a powerof the Ray-Singer torsion on M.Schwarz’s construction uses an Abelian gauge theory but in three dimensions a non-Abelian gauge theory—the SU(2) Chern-Simons theory—can be constructed and has deepand important properties established by Witten in 1988: Its partition function is theWitten invariant for the three manifold M and the correlation functions of Wilson loopsgive the Jones polynomial invariant for the link determined by the Wilson loops—cf.

[8,9].Finally the weak coupling limit of the partition function is a power of the Ray-Singertorsion.To define the Ray-Singer torsion, or simply torsion, we take a closed compact Rieman-nian manifold M over which we have a flat bundle E. Let M have a non-trivial fundamentalgroup π1(M) which is represented on E—this latter property arises very naturally in thephysical gauge theory context where it corresponds simply to the space of flat connectionsall of whose content resides in their holonomy—In any case the torsion is then the realnumber T(M, E) whereln T(M, E) =nX0(−1)qq ln Det ∆Eq ,n = dim M(2.1)The metric independence of the torsion requires that we assume, in the above definition,that the cohomology ring H∗(M; E) is trivial; this means that the Laplacians ∆Eq haveempty kernels and so are strictly positive definite.Given this fact one may use zetafunctions to define Det ∆Eq in the standard way. Recall that if P is a positive ellipticdifferential or pseudo-differential operator with spectrum {µn} and degeneracies {Γn} thenits associated zeta function ζP (s) is a meromorphic function of s, regular at s = 0, whichis given byζP (s) =XµnΓnµsn(2.2)and its determinant Det P is defined byln Det P = −dζP (s)dss=0(2.3)3

Using this we haveln T(M, E) = −nX0(−1)qqdζ∆Eq (s)dss=0(2.4)Quantum field theories of the type alluded to above are usually referred to as topo-logical quantum field theories or simply topological field theories.It turns out that more than one topological field theory can be used to give the torsion,for an excellent review of this question cf. Birmingham et al.

[10]. For example one cantake the actionS[ω] = iZMωndωn,dim M = 2n + 1(2.5)where ωn is an n-form.

The partition function is thenZ[M] =ZDωµ[ω] exp[S[ω]](2.6)S[ω] has a gauge invariance whereby S[ω] = S[ω +dλ] and therefore to define the partitionfunction it is necessary to integrate over only inequivalent field configurations. The measureDωµ[ω] thus contains functional delta functions which constrain the integration and playthe role of gauge fixing, together with their associated determinants.

This measure can beconstructed using, for example, the Batalin-Vilkovisky BRST construction [11,12].We shall be concerned here with the special situation of three dimensions and with thecase where the three manifold M is a lens space. The topological field theory of interestto us in this paper is given by the actionS[ω] = iZMω1dω1(2.7)where ω1 is now a 1-form.

To construct the integration measure we will follow the Batalin-Vilkovisky BRST construction [11,12]. The essential element of this construction is whatis termed a “gauge Fermion” whose BRST variation gives the gauge fixing and ghostportion of the BRST invariant action.

Integrating out these fields yields the contributionµ[ω] to the measure.The gauge Fermion is constructed by choosing a gauge fixing for the field ω1 (whichwe take to be d∗ω = 0), and multiplying the condition by an anti-ghost c¯0, which is a3-form denoted by its conjugated Poincar´e dual label, this indicates its anti-ghost naturealso. Thus the gauge Fermion is given byΨ = c¯0d∗ω1(2.8)The associated BRST variations of these fields areδω1 = −dc0,δc0 = 0δc¯0 = iω¯0δω¯0 = 0(2.9)4

With these definitions it is easy to check that δ2 = 0. The BRST gauge fixed action isthenL = iω1dω1 + δΨwhich expands toL = iω1dω1 −c¯0d∗dc0 + iω¯0d∗ω1(2.10)If we integrate out all fields as they appear the resulting partition function isZ = (Det L−)−12 Det d∗d0(2.11)where the operator L−is obtained by integrating out the ω1 fields and is a linear operatoracting on odd forms.

The partition function is thereforeZ =Det ∆0Det ∆114 Det ∆314Using Poincar´e duality the logarithm of this partition function is then given byln Z = 14 (3 ln Det ∆0 −ln Det ∆1)and we see it is proportional to the logarithm of the Ray-Singer torsion.Our task in what follows is to evaluate the individual components of this expressionboth for their usefulness in their own right and to verify that the combined result agreeswith the Ray-Singer torsion. We do this in the restricted setting where M belongs to aclass of three dimensional lens spaces.

In the next section we specify the lens spaces thatwe work with and obtain the eigenvalues and their degeneracies of the Laplacians on thesespaces.§ 3. Lens SpacesWe now want to turn to field theories defined on lens spaces—for general background onlens spaces cf.

[3,4] and references therein—briefly, a lens space can be constructed asfollows: Take an odd dimensional sphere S2n+1, considered as a subset of Cn, on whicha finite cyclic group of rotations G, say, acts. The quotient S2n+1/G of the sphere underthis action is a lens space.

More precisely, suppose that G is of order p, (z1, . .

., zn) ∈Cnand the group action takes the form(z1, . .

., zn) 7−→(exp(2πiq1/p)z1, . .

., exp(2πiqn/p)zn)with q1, . .

., qnintegers relatively prime to p(3.1)then the quotient S2n+1/G is a lens space often denoted by L(p; q1, . .

., qn). A formulafor the torsion of these spaces was first worked out by Ray [2].

To our knowledge howeverthere is no computation of the individual determinants of Laplacians on these spaces inthe literature. Since these are of independent field theoretic significance and from thesethe torsion is constructed it is instructive to examine these separately and construct the5

torsion from them. This we will proceed to do in the next sections focusing on the situationthat obtains when n = 2 and G is the group Zp ≡Z/pZ.

For simplicity we shall denotethe resulting lens space S3/Zp = L(p; 1, 1) by L(p), we shall also denote the lens spaceL(p, 1, q) by L(p, q); in passing we note that when p = 2 we have L(2) = RP 3 ≃SO(3).The group action above defines a representation V , say, of π1(L(p)) and also deter-mines a flat bundle F = (V × S3)/Zp, over L(p). It is the determinants of Laplacians andthe resulting torsion of this F over L(p) with which we are concerned here.

Using zetafunctions the torsion of these lens spaces is therefore given byln T(L(p), F) = −3X0(−1)qqdζ∆Fq (s)dss=0(3.2)As an aid to the calculation of ln T(L(p), F) it is useful to introduce the notationτ(p, s) = −3X0(−1)qqζ∆Fq (s)T(p) = T(L(p), F)The relationship between the two functions being clearlyln T(p) = dτ(p, s)dss=0(3.3)For τ(p, s) itself we now haveτ(p, s) = ζ∆F1 (s) −2ζ∆F2 (s) + 3ζ∆F3 (s)= 3ζ∆F0 (s) −ζ∆F1 (s),using Poincar´e duality(3.4)Combining the standard decomposition ∆p = (d∗d + dd∗)p, with the fact that ker d∗∩ker d = ∅, we further obtain the formulaτ(p, s) = 2ζd∗d0(s) −ζd∗d1(s)(3.5)We now simplify our notation by labellingτ+(p, s) = 2ζd∗d0(s),τ−(p, s) = ζd∗d1(s)For the individual zeta functions we denote the eigenvalues and their degeneracies byλn(q, p) and Γn(q, p) respectively giving the expressionsτ+(p, s) = 2XnΓn(0, p)λsn(0, p),τ−(p, s) =XnΓn(1, p)λsn(1, p)(3.6)It remains to compute these eigenvalues and degeneracies. The former are actually inde-pendent of p and are fairly easily calculated by the technique of starting with harmonic6

forms in R2n and then restricting successively to S2n+1 and L(p). In any case they aregiven byλn(0, p) = n(n + 2), n = 1, 2, .

. .λn(1, p) = (n + 1)2, n = 1, 2, .

. .

(3.7)To calculate the degeneracies is more difficult; we make use of the fact that S3 is agroup manifold and proceed as follows: Consider the Laplacians d∗dq on S3, and d∗dFq onL(p) also, if λ is an eigenvalue, denote the corresponding eigenspaces by Λq(λ) and ΛFq (λ)respectively. Letv(z) ∈Λq(λ), with z ∈S3 ⊂C2,and g ∈Zp, where g ≡exp[2πij/p], 0 ≤j ≤(p −1)(3.8)The element g acts on v(z) to give g · v(z) whereg · v(z) = v(gz)gz = (exp[2πij/p]z1, exp[2πij/p]z2)(3.9)The above definitions allow us to define the projection P(λ) on Λq(λ) byP(λ)v = 1pXg∈Zpexp[−2πij/p]g · v(3.10)Evidently[P(λ), d∗dq] = 0(3.11)and so P(λ) projects the space Λq(λ) onto the space ΛFq (λ).

Finally this means that weobtain a formula for the degeneracy Γn(q, p), namelyΓn(q, p) = trP|ΛFq (λ)= 1p(p−1)Xj=0exp[−2πij/p] trg|ΛFq (λ)(3.12)To actually apply this formula we now add in the fact that S3 is the group manifoldfor SU(2). The Peter–Weyl theorem tells us, in this case where all representations areself-conjugate, thatL2(S3) = L2(SU(2)) = LµcµDµ = LµDµ ⊗Dµ(3.13)where cµ measures the multiplicity of the representation µ which must therefore be dim Dµ.But Hodge theory gives us the alternative decompositionL2(S3) = LλΛ0(λ)(3.14)7

In addition the Casimir operator for SU(2) is a multiple of the Laplacian and, if therepresentation label µ is taken to be the usual half-integer j, then we know that thisCasimir has eigenvalues j(j + 1), and also that dim Dj = 2j + 1. These facts identify theLaplacian ∆0 = d∗d0 as four times the Casimir and identify Λ0(λ) as dim Dj copes of Dj.Thus if we set n = 2j, so that n is always integral, then we have the degeneracy formulaΓn(0, p) = (n + 1)p(p−1)Xj=0exp[−2πij/p] χn/2(2πj/p)(3.15)where χj(θ) denotes the SU(2) character, on Dj, for rotation through the angle θ; i.e.χj(θ) = sin((2j + 1)θ)sin(θ)(3.16)Hence our explicit degeneracy formula for 0-forms on L(p) isΓn(0, p) = (n + 1)p(p−1)Xj=0exp[−2πij/p] sin(2π(n + 1)j/p)sin(2πj/p)(3.17)We now have to find the analogous formula for the 1-forms.

The formula that resultsisΓn(1, p) = 1p(p−1)Xj=0exp[−2πij/p]nnχ(n+1)/2(2πj/p) + (n + 2)χ(n−1)/2(2πj/p)o(3.18)or, more explicitly,Γn(1, p) = 1p(p−1)Xj=0exp[−2πij/p]nsin(2π(n + 2)j/p)sin(2πj/p)+ (n + 2)sin(2πnj/p)sin(2πj/p)(3.19)To simplify the notation we introduce the ‘p-averaged character’χjp which we definebyχjp = 1p(p−1)Xj=0exp[−2πij/p] χj(2pij/p)(3.20)Finally this gives us a concrete expression for τ(p, s), i.e.τ(p, s) =Xn(2(n + 1)χn/2p{n(n + 2)}2s−nχ(n+1)/2p + (n + 2)χ(n−1)/2p(n + 1)2s)= τ+(p, s) −τ−(p, s)whereτ+(p, s) =Xn2(n + 1)χn/2p{n(n + 2)}s,τ−(p, s) =Xnnχ(n+1)/2p + (n + 2)χ(n−1)/2p(n + 1)2s(3.21)8

To make further progress towards a computation of the determinants and torsion weneed to be able to evaluate these p-averaged characters. This is a somewhat non-trivialcombinatorial task but this task is eased if we use for χj(θ), the alternative expressionχj(θ) =jXm=−jexp[2imθ](3.22)It is also necessary to divide n up into its conjugacy classes mod p by writingn = pk −j, k ∈Z, j = 0, 1, .

. ., (p −1)(3.23)We eventually discover thatDχ(pk−j)/2Ep =kfor j = 0, 2, .

. ., (p −1)kfor j = 1(k −1)for j = 3, 5, .

. ., (p −2)if p is odd0for j = 0, 2, .

. ., (p −2)2kfor j = 1(2k −1)for j = 3, 5, .

. ., (p −1)if p is even(3.24)We now lack only one ingredient among those necessary for a calculation of the deter-minants and the resulting torsion: this is the construction of the analytic continuation ofthe series for τ(p, s).

We shall construct this in the next section. The technique we shalluse will be more easily followed if we first use it in a more simple case.

Thus, to beginwith, we set p = 2 and then construct the continuation.§ 4. The Analytic Continuation for p = 2The series to be continued areτ+(p, s) =∞Xn=12(n + 1)χn/2p{n(n + 2)}sτ−(p, s) =∞Xn=1nχ(n+1)/2p + (n + 2)χ(n−1)/2p(n + 1)2sand their difference which leads to the torsionτ(p, s) =Xn(2(n + 1)χn/2p{n(n + 2)}s−nχ(n+1)/2p + (n + 2)χ(n−1)/2p(n + 1)2s)(4.1)These already converges for Re s > 3/2; however a calculation of the determinants andthe torsion requires us to work at s = 0, hence we see the need for, and the extent of, theanalytic continuation.9

Our interest in this section for illustrative purposes is in the case p = 2 where we haveτ(2, s) = τ+(2, s) −τ−(2, s)=Xn(2(n + 1)χn/22{n(n + 2)}s−nχ(n+1)/22 + (n + 2)χ(n−1)/22(n + 1)2s)(4.2)But using 3.24 we find thatDχ(n+1)/2E2 =Dχ(2k−j+1)/2E2 ,(n = 2k −j)= 0,j = 12k + 2,j = 0≡0,n odd2k + 2,n even(4.3)SimilarlyDχ(n−1)/2E2 =Dχ(2k−j)/2E2 ,(n = 2k −j)=2k,j = 10,j = 0≡(n + 1),n odd0,n even(4.4)Thus τ(2, s) becomesτ(2, s) = τ+(2, s) −τ−(2, s)=Xn odd2(n + 1)2{n(n + 2)}s −Xn even2n(n + 2)(n + 1)2s(4.5)Setting n = (2m −1) in τ+(2, s) and n = 2m in τ−(2, s) we haveτ+(2, s) =∞Xm=18m2(4m2 −1)s ,τ−(2, s) =∞Xm=04m(2m + 2)(2m + 1)2sandτ(2, s) =∞Xm=18m2(4m2 −1)s −∞Xm=02(2m + 1)(2s−2) +∞Xm=02(2m + 1)2s(4.6)Now if we use the fact thatXn=1,3,5,...1ns = (1 −2−s)ζ(s)(4.7)where ζ(s) is the usual Riemann zeta function then we getτ−(2, s) = 2(1 −2−(2s−2))ζ(2s −2) −2(1 −2−2s)ζ(2s)(4.8)10

and denotingA2(m, 0, s) =(2m)2n(2m)2 −1os ,andA2(0, s) =∞Xm=1A2(m, 0, s)(This notation is used to agree with the general case to be discussed in the next section.See also Appendix A.) Thus we haveτ+(2, s) = 2A2(0, s)and these combine to giveτ(2, s) = 2A2(0, s) −2(1 −2−(2s−2))ζ(2s −2) + 2(1 −2−2s)ζ(2s)(4.9)Since the terms involving the Riemann zeta function already have a well defined continu-ation it remains to continue A2(0, s).

NowA2(m, 0, s) =4m2(4m2 −1)s =4m2(4m2)s1 −14m2−s=1(4m2)(s−1)n1 +s4m2 + · · ·o=1(4m2)(s−1) +s(4m2)s + R(m, s),(def. of R(m, s))(4.10)So that the remainder term R(m, s) is given byR(m, s) = A2(0, m, s) −1(4m2)(s−1) −s(4m2)s=4m2(4m2 −1)s −1(4m2)(s−1) −s(4m2)s(4.11)The definition of the remainder term is chosen to ensure that|R(m, s)| ≤(ln m)αm2(4.12)and this has the vital consequence that the operations d/ds (at s = 0) and Pm commutewhen applied to R(m, s).DefiningR(s) =∞Xm=0R(m, s)(4.13)allows us to tidy our expressions up somewhat.

Collecting our regulated expressions wetherefore haveτ+(2, s) = 84s ζ(2s −2) + 2s4s ζ(2s) + 2R(s)andτ−(2, s) = 2(1 −2−(2s−2))ζ(2s −2) −2(1 −2−2s)ζ(2s)(4.14)11

In fact the expression for τ(2, s) can be further tidied up to giveτ(2, s) = 2 84(s) −1ζ(2s −2) + 21 + (s −1)4sζ(2s) + 2R(s)(4.15)The series for R(s) is guaranteed to be convergent and the analytic continuation is nowcomplete.Evaluating our expressions at s = 0 we findτ+(2, 0) = 8ζ(−2) + 2R(0)τ−(2, 0) = −6ζ(−2)andτ(2, 0) = 14ζ(−2)(4.16)Observe that with our continuation R(0) is automatically zero. Thus noting also thatζ(−2) = 0, we concludeτ+(2, 0) = 0 ,τ−(2, 0) = 0 ,andτ(2, 0) = 0(4.17)That τ±(p, 0) = 0 is quite generally true for arbitrary p; we shall see this in the next sectionand this agrees with general considerations for generalised zeta functions of second orderoperators on compact odd dimensional manifolds.We can now take the final step which is to differentiate 4.14 and obtain τ ′+(2, 0) andτ ′−(2, 0), which we denote by τ ′+(2), τ ′−(2) respectively, and hence the torsion T(2).The resulting expressions areτ ′+(2) = 16ζ′(−2) + 2ζ(0) + 2R′(0)τ ′−(2) = −12ζ′(−2) −2 ln 4ζ(0)and for the torsionln T(2) = dτ(2, 0)ds= 28ζ′(−2) + 2(1 + ln 4)ζ(0) + 2R′(0)(4.18)Butζ(0) = −1/2,andζ′(−2) = −ζ(3)4π2from the functional relation(4.19)and by our remark above concerning the motive for our choice of definition for R(m, s) wehaveR′(0) = ddsXmR(m, s)|s=0⇒R′(0) =XmdR(m, s)dss=0=Xm4m2 ln(4m2) −ln(4m2 −1)−1= −Xm4m2 ln(1 −1/4m2) + 1(4.20)12

Henceτ ′+(2) = −4π2 ζ(3) −1 −2Xm4m2 ln(1 −1/4m2) + 1τ ′−(2) = 3π2 ζ(3) + 2 ln 2andln T(2) = −7π2 ζ(3) −1 −2 ln(2) −2Xm4m2 ln(1 −1/4m2) + 1(4.21)However the series for R′(0) can be expressed as a trigonometric integral; in fact, as aspecial case of more general results which will be derived below, we have∞Xm=14m2 ln(1 −1/4m2) + 1= −12 + 4π2Z π/20dz z2 cot(z)(4.22)which means thatτ ′+(2) = −4π2 ζ(3) −8π2Zπ20dzz2 cot(z)τ ′−(2) = 3π2 ζ(3) + 2 ln 2andln T(2) = −7π2 ζ(3) −2 ln(2) −8π2Z π/20dz z2 cot(z)(4.23)The formula 4.23 above for T(2) can be pushed even further: By using it with Ray’sexpression [2] for the torsion we can deduce thatln T(p) = −4p(p−1)Xj=1pXk=1cos(2jkπp) ln(2 sin(2kπp )) exp[2kπip]= −4 ln2 sin(πp )(4.24)which, for p = 2, becomes simplyln T(2) = −4 ln(2)(4.25)Hence we straightaway have the identity−4 ln(2) = −7π2 ζ(3) −2 ln(2) −8π2Z π/20dz z2 cot(z)(4.26)Orζ(3) = 2π27ln(2) −87Z π/20dz z2 cot(z)(4.27)13

In other words our computation of the torsion has given us a formula for the zeta functionat its first odd argument. Equivalently we can use this relation to eliminate the integral andobtain quite simple expressions for the logarithms of the determinants of the Laplacianson zero and one forms respectively.

We conclude this section by quoting these resultsNoting first of all that the expressions for τ ′+(2) and τ ′−(2) in their simplest form nowbecomeτ ′+(2) = 3π2 ζ(3) −2 ln 2andτ ′−(2) = 3π2 ζ(3) + 2 ln 2(4.28)and from the definitions of these objects we have at once thatln Det d∗d0 = −32π2 ζ(3) + ln 2ln Det d∗d1 = −3π2 ζ(3) −2 ln 2(4.29)It is interesting to note the role that the Riemann zeta function ζ(3) plays in these ex-pressions. Since these are expressions for volume elements on the discrete moduli spacesassociated with the Laplacians, we expect that there are deeper things to be learned froma further study of such expressions.In the next section we tackle the continuation for arbitrary p.§ 5.

The Determinants and the Torsion for p Odd.The analytic continuation for a general value of p naturally divides into two cases: p oddand p even; in fact we shall see below that the case for p even further divides into twosubcases which correspond to p = 0, 2 mod4. Due to the size of the expressions it is nowmuch more convenient to continue τ+(p, s) and τ−(p, s) separately and then combine theminto expressions for the torsion.

We deal first with τ+(p, s) and begin with p odd.§§ 5.1 The Function τ+(p, s) and ln Det d∗d0Let us recall that τ+(p, s) is given byτ+(p, s) =∞Xn=12(n + 1)χn/2p{n(n + 2)}s(5.1)Reference to the general character formula 3.24 shows that we must resolve n into itsconjugacy classes mod p by writingn = pk −j, j = 0, . .

., (p −1)(5.2)and that we must distinguish the two parities of j. To implement these requirements wesetp = 2r + 1,and parametrisejodd by j = 2l + 1, l = 0, 1, .

. ., (r −1)jeven by j = 2l, l = 0, 1, .

. ., r(5.3)14

This givesτ+(p, s) = 2p−1Xj=0∞Xk=1(pk −j + 1)χ(pk−j)/2p{(pk −j)(pk −j + 2)}s= 2∞Xk=1"r−1Xl=0(pk −2l)χ(pk−2l−1)/2p{(pk −2l −1)(pk −2l + 1)}s +rXl=0(pk −2l + 1)χ(pk−2l)/2p{(pk −2l)(pk −2l + 2)}s#(5.4)Then when we use 3.24 for p odd we getτ+(p, s) =∞Xk=12pk2{(pk −1)(pk + 1)}s +r−1Xl=12(k −1)(pk −2l){(pk −2l −1)(pk −2l + 1)}s+rXl=02k(pk −2l + 1){(pk −2l)(pk −2l + 2)}s(5.5)To aid in marshalling the combinatorics of τ+(p, s) we define Hp(k, s, x) byHp(k, s, x) =pk(pk + x){(pk + x −1)(pk + x + 1)}s(5.6)The point being that each of the three summands in 5.5 is of the form Hp(k, s, λ) forappropriate λ. To see this we introduce precisely p constants of the type λ defined byλ0 = 0λl = −2l + 1, l = 0, .

. ., r¯λl = −2l + p, l = 1, .

. ., (r −1)(5.7)With this notation it can be checked that τ+(p, s) is given byτ+(p, s) = 2p∞Xk=1Hp(k, s, λ0) +(r−1)Xl=1Hp(k, s, ¯λl) +rXl=0Hp(k, s, λl)(5.8)Also if we denote the entire set of λ’s by {λ} i.e.

{λ} ≡{λ0, λl, ¯λl} = {−(p −2), . .

., −5, −3, −1, 0, 1, 3, 5 . .

. (p −2)}(5.9)then we have the even more concise expressionτ+(p, s) = 2pX{λ}∞Xk=1Hp(k, λ, s)(5.10)The functions Hp(λ, s) obtained by summing over k then form a set of p functions whosederivative at s = 0 can be viewed as living on the appropriate space of sections for the15

Laplacian acting on 0 forms on the lens space L(p); taking the trace over these functionsviewed as forming a matrix then gives the analytic continuation of the determinant of theLaplacian.Next observingHp(k, x, s) =(pk + x)2{(pk + x)2 −1}s −x(pk + x){(pk + x)2 −1}s(5.11)we see that there are therefore two additional functions of interest here i.e.Ap(k, x, s) =(pk + x)2n(pk + x)2 −1os(5.12)andBp(k, x, s) =(pk + x)n(pk + x)2 −1os(5.13)and we haveHp(k, x, s) = Ap(k, x, s) −xBp(k, x, s)(5.14)and we can equally writeτ+(p, s) = 2pX{λ}[Ap(λ, s) −λBp(λ, s)](5.15)Let us further note that in the set {λ} the non zero elements come in pairs of the form{λ, −λ}. Thus we can further writeτ+(p, s) = 2pAp(0, s) + 2pr−1Xl=1A+p (2l + 1, s) −(2l + 1)B−p (2l + 1, s)where the ∓superscripts refer to the symmetric or anti-symmetric combination withrespect to the first argument:i.e.A+p (x, s) = Ap(x, s) + Ap(−x, s) and B−p (x, s) =Bp(x, s) −Bp(−x, s).

We relegate the details of the computation of these functions andtheir analytic continuation to appendices A and B—the calculations are generalisationsof those performed for p = 2. Quoting here from appendices A and B we have that therelevant functions and their derivatives at s = 0 are given byAp(x, 0) = p2ζ(−2, 1 + xp ) = −x(x + p)(2x + p)6pBp(x, 0) = pζ(−1, 1 + xp ) + 12p = −p12 −xp + x2 + 12pHp(x, 0) = −px12 −x2p + x36p(5.16)16

Which immediately implies that H+p (x, 0) = 0 and we have our first result that for p oddτ+(p, 0) = 0The significance of this is that the analytic continuation of the scaling dimension of thedeterminant is zero.Next we note again from appendices A and B thatA+p (x) = −p2π2 ζ(3) −p2π2Z(x+1)πp0dz z(z −2πxp ) cot(z) −p2π2Z(x−1)πp0dz z(z −2πxp ) cot(z)−x2 ln2 sin ((x + 1)π/p)x + 1−x2 ln2 sin ((x −1)π/p)x −1(5.17)andB−p (x) = pπZ(x+1)πp0dz z cot(z) + pπZ(x−1)πp0dz z cot(z)−x ln2 sin ((x + 1)π/p)x + 1−x ln2 sin ((x −1)π/p)x −1(5.18)Now combining our expressions 5.17, 5.18 and summing over {λ} we obtain for τ ′+(p)the expressionτ ′+(p) = −2p"p32π2 ζ(3) + p2π2Zπp0dz z2 cot(z) + p2π2Z(p−1)πp0dz z(z −(p −2)πp) cot(z)+2(p−3)2Xl=1p2π2Z2lπp0dz z(z −2lπp ) cot(z)(5.19)for p odd. But the expression τ ′+(p) above is 2 ln Det d∗d0: i.e.

twice the logarithm of theLaplacian on 0-forms for the lens space L(p). More precisely our analytic continuation hasshown us thatln Det d∗d0 = 1p"p32π2 ζ(3) + p2π2Zπp0dzz2 cot(z) + p2π2Z(p−1)πp0dzz(z −(p −2)πp) cot(z)+2(p−3)2Xl=1p2π2Z2lπp0dzz(z −2lπp ) cot(z)(5.20)§§ 5.2 The Function τ−(p, s) and ln Det d∗d1Let us now turn to τ−(p, s) for p odd.τ−(p, s) =∞Xn=1nDχ(n+1)2Ep + (n + 2)Dχ(n−1)2Ep(n + 1)2s(5.21)17

which on decomposing n over the conjugacy classes mod p n = pk −j,j = 0, . .

., (p −1)as for τ+(p, s), distinguishing the two parities as in 5.3 yieldsτ−(p, s) =∞Xk=1kpDχkp+12Ep + (kp + 2)Dχkp−12Ep(kp + 1)2s+∞Xk=1(kp −2)Dχkp−12Ep + kpDχkp−32Ep(kp −1)2s+∞Xk=1(p−3)2Xl=2(kp −2l)Dχkp−2l+12Ep + (kp −2l + 2)Dχkp−2l−12Ep(kp −2l + 1)2s+∞Xk=1(kp −p + 1)Dχkp−p2Ep + (kp −p + 3)Dχkp−p−22Ep(kp −p + 2)2s+∞Xk=1r−1Xl=0(kp −2l −1)Dχkp−2l2Ep + (kp −2l + 1)Dχkp−2l−22Ep(kp −2l)2s(5.22)Using our degeneracy formula 3.24 we have with a little re-arrangementτ−(p, s) =∞Xk=12(kp + 1)k + kp(kp + 1)2s+∞Xk=12(kp −1)k −kp(kp −1)2s+(p−3)2Xl=−(p−3)2∞Xk=12(kp + 2l)k(kp + 2l)2s(5.23)We now observe that this is of the formτ−(p, s) = 2pX{ν}∞Xk=1kp(kp + ν)(kp + ν)2s+∞Xk=1kp(kp + 1)2s −∞Xk=1kp(kp −1)2s(5.24)where we have denoted by {ν} the set{ν} ≡{ν0, νl, ¯νl} = {−(p −3), . .

., −6, −4, −2, −1, 0, 1, 2, 4, 6 . .

. (p −3)}(5.25)The sums occuring in 5.24 are naturally expressible in terms of Hurwitz zeta functions andthis provides the natural analytic continuation of this expression.

We also note thatX{ν}∞Xk=11(kp + ν)(2s−2) = ζ(2s −2) −1 −(p−3)2Xl=1(2l)(2−2s)(5.26)18

On utilising these relations we find thatτ−(p, s) = 2pζ(2s −2) −1 −(p−3)2Xl=1(2l)2−2s(p −2)p−2sζ(2s −1, 1 + 1p) −ζ(2s −1, 1 −1p)−2p−2s(p−3)2Xl=−(p−3)2(2l)ζ(2s −1, 1 + 2lp )−p−2sζ(2s, 1 + 1p) + ζ(2s, 1 −1p)(5.27)If we analytically continue the RHS of this expression to s = 0, and observe that ζ(−1, 1+x) −ζ(−1, 1 −x) = −x, then we obtainτ−(p, 0) = −2p1 +(p−3)2Xl=1(2l)2+ (p −2)(−1p) −2(p−3)2Xl=1(2l)(−2lp )(5.28)which immediately givesτ−(p, 0) = 0(5.29)Thus again the scaling dimension of the associated determinant is zero.Passing to τ ′−(p) by taking the derivative at s = 0 givesτ ′−(p) = 4pζ′(−2) −ln p +(p−3)2Xl=1(2l)2 ln[2lp ]+ 2(p −2)ζ′(−1, 1 + 1p) −ζ′(−1, 1 −1p)−4(p−3)2Xl=1(2l)ζ′(−1, 1 + 2lp ) −ζ′(−1, 1 −2lp )−2ζ′(0, 1 + 1p) + ζ′(0, 1 −1p)(5.30)A useful identity for Hurwitz zeta functions derived in appendix C isζ′(−1, 1 + x) −ζ′(−1, 1 −x) = −x ln2 sin(πx)x+ 1πZ πx0dzz cot(z)19

Using this and the expression ζ′(−2) = −ζ(3)/4π2 we conclude thatτ ′−(p) = −1pπ2 ζ(3) + 4p ln2 sin(πp )+ 4p(p−3)2Xl=1(2l)2 ln2 sin(2πlp )+ 2(p −2)πZπp0dzz cot(z) −4π(p−3)2Xl=1(2l)Z2πlp0dzz cot(z)(5.31)But −τ ′−(p) is the analytic continuation which gives ln Det d∗d1. Hence we find thatln Det d∗d1 =1pπ2 ζ(3) + 4p ln2 sin(πp )+ 4p(p−3)2Xl=1(2l)2 ln2 sin(2πlp )+ 2(p −2)πZπp0dzz cot(z) −4π(p−3)2Xl=1(2l)Z2πlp0dzz cot(z)(5.32)§§ 5.3 The Torsion T(p) and τ(p, s).We are therefore now in a position to put these together and obtain an expression for thetorsion.

The first observation is that since τ+(p, 0) and τ−(p, 0) are both zero we haveτ(p, 0) = 0(5.33)This vanishing of τ(p, 0) is related to the metric independence of the torsion somethingwhich has been established quite generally by Ray and Singer [3].The torsion T(p) itself is given by the difference of 5.19 and 5.31. Combining these twoexpressions we find, upon a little simplification, that T(p) is determined by the equationln T(p) = −2p(p3 −1)2π2ζ(3) + 2 ln2 sin(πp )+ 2(p−3)2Xl=14l2 ln2 sin(2lπp )+ p2π2Zπp0dz z(z + (p −2)πp) cot(z) + p2π2Z(p−1)πp0dz z(z −(p −2)πp) cot(z)+2(p−3)2Xl=1p2π2Z2lπp0dz z(z −4lπp ) cot(z)(5.34)Now the expression for the torsion from Ray’s calculation gave the alternative expression4.24 i.e.ln T(p) = −4 ln2 sin(πp )(5.35)20

We have verified that these two expressions 5.34 and 4.24 agree numerically, yet it isnot transparent by inspection that this should be so; also using C.41 of appendix C wecan reduce 5.34 to Ray’s expression. One may conclude that, by following two alternatederivations, we have arrived at what is a sequence of non-trivial identities.

As we saw inthe case of p = 2 utilising these identities one can obtain non-trivial formulae for ζ(3) andalso can be used to further simplify the expressions for the individual determinants. Theresulting expressions for ζ(3) areζ(3) =2π2(p3 −1)2(p −1) ln2 sin(πp )−2(p−3)2Xl=14l2 ln2 sin(2lπp )−p2π2Zπp0dz z(z + (p −2)πp) cot(z) −p2π2Z(p−1)πp0dz z(z −(p −2)πp) cot(z)−2(p−3)2Xl=1p2π2Z2lπp0dz z(z −4lπp ) cot(z)(5.36)For the sake of illustration let us quote the implications of these formulae for thesimplest odd case: p = 3.

On utilising all of the information at our disposal we find thatτ ′+(3) = −ζ(3)3π2 −43 ln 3 + 2πZπ30dz z cot(z)τ ′−(3) = −ζ(3)3π2 + 23 ln 3 + 2πZπ30dz z cot(z)andln T(3) = −2 ln 3The relation between the expressions we have obtained which we expressed in terms of aformula for ζ(3) (the analog of 4.27 for p = 2) becomesζ(3) = 2π213 ln 3 −913Zπ30dz z(z + π3 ) cot(z) −913Z2π30dz z(z −π3 ) cot(z)We will now turn to the case of even p.§ 6. Determinants and the Torsion for p Even.When p is even we follow a slightly different route to that used in the previous section butwe arrive at expressions of a similar general form for the respective determinants and theircorresponding torsion.§§ 6.1 The Function τ+(p, s) and ln Det d∗d0Let us first obtain series expressions for τ+(p, s) for p even, and observe that the samefunctions as those encountered for p odd enter these also.

Recalling our expression forτ+(p, s)τ+(p, s) =∞Xn=12(n + 1)χn2 pn(n + 1)2 −1os(6.1)21

We again resolve n into its conjugacy classes mod p by writingn = pk −j, j = 0, . .

., (p −1)(6.2)and distinguish the two parities of j by settingp = 2r,and parametrisejodd by j = 2l + 1, l = 0, 1, . .

., (r −1)jeven by j = 2l, l = 0, 1, . .

., (r −1)(6.3)Reference to the general character formula 3.24 shows that only j odd contributes and weobtainτ+(p, s) =∞Xk=14pkn(pk)2 −1os +(p−2)2Xl=0∞Xk=12(pk −2l)(2k −1)n(pk −2l)2 −1os(6.4)which givesτ+(p, s) = 4p∞Xk=1(p−2)2Xl=0(pk −2l)2n(pk −2l)2 −1os +(p−2)2Xl=1(2l −p2)(pk −2l)n(pk −2l)2 −1os(6.5)We recognise the expressions arising as the functions from the preceding analysis in thecase of p odd and which are analysed in appendices A and B. We can therefore write 6.5asτ+(p, s) = 4pr−1Xl=0[Ap(−2l, s) + (2l −r)Bp(−2l, s)](6.6)Some further rearrangement will allow us to write these again in terms of the symmetricand anti-symmetric parts of Ap(x, s) and Bp(x, s) respectively.

Note first of all, howeverthat the term involving Ap is a sum over all even conjugacy classes i.e.r−1Xl=1A(−2l, s) =r−1Xl=1∞Xk=1(2rk −2l)2n(2rk −2l)2 −1os=∞Xm=1(2m)2n(2m)2 −1os(6.7)and is therefore τ+(2, s)/2; our expression 6.6 for τ+(p, s) can hence be written asτ+(p, s) = 2p"τ+(2, s) + 2r−1Xl=1(2l −r)Bp(−2l, s)#(6.8)Now when l ranges from 1 to r −1, (2l −r) ranges over the set−(r −2), (r −4), . .

., (r −4), (r −2)(6.9)22

This allows us to divide the range up into a sum from 1 up to the integer part of (r −1)/2which we denote by [(r −1)/2] Thusτ+(p, s) = 2pτ+(2, s) + 2 (r−1)2Xl=1(r −2l) [Bp(−p + 2l, s) −Bp(−2l, s)](6.10)Observing thatBp(−p + x, s) =x{x2 −1}s + Bp(x, s)This givesτ+(p, s) = 2pτ+(2, s) + 2 (p−2)4Xl=1(p2 −2l)B−p (2l, s) +2l{x2 −1}s(6.11)Noting that τ+(2, 0) = 0 and that B−p (x) = −x we see again immediately thatτ+(p, 0) = 0(6.12)Again as we expect the scaling dimension of the associated determinant is zero.Proceeding now to the expression for the determinant itself, we find the resultingexpression from 6.11 for the derivative at s = 0 isτ ′p(p, 0) = 2pτ ′+(2) + (p−2)4Xl=1(p −4l)B−p (2l) −2l ln[2l2 −1](6.13)Substituting for B−p (2l) from 5.18 givesτ ′+(p) = 2pτ ′+(2) + pπ (p−2)4Xl=1(p −4l)"Z(2l+1)πp0z cot(z) +Z(2l−1)πp0dzz cot(z)−2lπp ln2 sin(2l + 1)πp−2lπp ln2 sin(2l −1)πp(6.14)where from 4.28 τ ′+(2) =3π2 ζ(3) −2 ln 2 . As we see the case of even p divides naturallyinto two classes p = 2 mod 4 and p = 0 mod 4.

Making this division we can further simplifythings to obtainτ ′+(p) = 2r32π2 ζ(3) −2(r−3)2Xl=1[(2l + 1)(r −2l −1) −1] ln2 sin(2l + 1)π2r+2r(r −2)πZπ2r0dz z cot(z) + 4rπ(r−2)2Xl=1(r −2l −1)Z(2l+1)π2r0dz z cot(z)p = 2 mod 423

andτ ′+(p) = 2r32π2 ζ(3) −ln 2 −2(r−2)2Xl=1[(2l + 1)(r −2l −1) −1] ln2 sin(2l + 1)π2r+2r(r −2)πZπ2r0dz z cot(z) + 4rπ(r−2)2Xl=1(r −2l)Z(2l+1)π2r0dz z cot(z)p = 0 mod 4We note that these expressions agree with the result for p = 2 and, for p = 4, we notein passing that inspection of the series shows that τ ′±(4, 0) = τ ′±(2, 0)/2; this turns outto be also a property of (the logarithm of) the torsion itself, i.e. T(p) satisifies ln T(4) =(ln T(2))/2.We therefore have from 6.14 an expression for the appropriate logarithmic determinanton the lens space L(p) namelyln Det d∗d0 = −3pπ2 ζ(3) + 2p ln 2−2p (p−2)4Xl=1(p2 −2l)"pπZ(2l+1)πp0z cot(z) + pπZ(2l−1)p0dzz cot(z)−2l ln2 sin(2l + 1)πp−2l ln2 sin(2l −1)πpp = 2r(6.15)§§ 6.2 The Function τ−(p, s) and ln Det d∗d1Let us now turn to the evaluation of τ−(p, s).τ−(p, s) =∞Xn=1nDχn+12Ep + (n + 2)Dχn−12Ep(n + 1)2s(6.16)Decomposing the sum over n into the different conjugacy classes and using our generalcharacter formula we haveτ−(p, s) =∞Xk=1pk(pk + 1) + (pk + 2)2k(pk + 1)2s+∞Xk=1(pk −2)2k + pk(2k −1)(pk −1)2s+r−1Xl=22(pk −2l + 1)(2k −1)(pk −2l + 1)2s(6.17)24

After some rearrangement we arrive atτ−(p, s) =∞Xk=1(4pr−1Xl=01(pk −2l + 1)2s−2+ (1 −4p)"1(pk + 1)2s−1 −1(pk −1)2s−1#−"1(pk + 1)2s +1(pk −1)2s#+ 4pr−1Xl=2(2l −1 −r)1(pk −2l + 1)2s−1)(6.18)Since the first term involves a sum over all odd conjugacy classes we have∞Xk=1r−1Xl=01(pk −2l + 1)2s−2 =∞Xm=11(2m + 1)2s−2= (1 −122s−2 )ζ(2s −2) −1(6.19)Thusτ−(p, s) =4p(1 −122s−2 )ζ(2s −2) −1+ (p −4) 1p2s [ζ(2s −1, 1 + 1p) −ζ(2s −1, 1 −1p)]−1p2s [ζ(2s, 1 + 1p) + ζ(2s, 1 −1p)]−2p2sr−1Xl=2(p −(2l −1)) ζ(2s −1, 1 −2l −1p)(6.20)Noting that p −(2l −1) ranges from −(p −6) to (p −6) in steps of 2 when l ranges from2 to r −1 the final sum in 6.20 is therefore of the formr−1Xl=2(p −(2l −1)) ζ(2s −1, 1 −2l −1p) =[ r−12]Xl=2(p −2(2l −1))ζ(2s −1, 1 −2l −1p)−ζ(2s −1, 2l −1p)(6.21)25

Substituting back we obtainτ−(p, s) = 4p(1 −122s−2 )ζ(2s −2) −1+ (p −4) 1p2s [ζ(2s −1, 1 + 1p) −ζ(2s −1, 1 −1p)]−1p2s [ζ(2s, 1 + 1p) + ζ(2s, 1 −1p)]+ 2p2s[ r−12]Xl=2(p −(2l −1))ζ(2s −1, 2l −1p) −ζ(2s −1, 1 −2l −1p)(6.22)Our first observation is that using ζ(−2) = 0,ζ(−1, 1+a)−ζ(−1, 1−a) = −a,andζ(0, 1 + a) + ζ(0, 1 −a) = −1.τ−(p, 0) = 0(6.23)Now differentiation of 6.22 with respect to s and evaluating the expression at s = 0, andusing some of our relations from Appendix C givesτ ′−(p) = −24p ζ′(−2) + 2(p −4)[ζ′(−1, 1p) −ζ′(−1, 1 −1p)] −2[ζ′(0, 1p) + ζ′(0, 1 −1p)]+ 4[ r−32]Xl=1(p −(2l −1))ζ′(−1, 2l −1p) −ζ′(−1, 1 −2l −1p)(6.24)Further use or our the relations derived in Appendix C allows us to express the resultin terms of integrals over trigonometric functions as in the preceding sections to yieldτ ′−(p) = 4 ln(2 sin πp ) +6pπ2 ζ(3) + 2(p −4)[ 1πZπp0dzz cot(z) −1p ln(2 sin πp )]+ 4[ r−32]Xl=1(p −(2l + 1))"1πZπ(2l+1)p0dzz cot(z) −(2l + 1)pln(2 sin π(2l + 1)p)#(6.25)This again decomposes into the two cases p = 0, 2 mod 4 and these yield the expressionsτ ′−(p) = 4 ln(2 sin πp ) +6pπ2 ζ(3) + 2(p −4)[ 1πZπp0dzz cot(z) −1p ln(2 sin πp )]+ 4t−1Xl=1(p −(2l + 1))"1πZπ(2l+1)p0dzz cot(z) −(2l + 1)pln(2 sin π(2l + 1)p)#p = 2 mod 4(6.26)26

andτ ′−(p) = 4 ln(2 sin πp ) +6pπ2 ζ(3) + 2(p −4)[ 1πZπp0dzz cot(z) −1p ln(2 sin πp )]+ 4t−1Xl=1(p −(2l + 1))"1πZπ(2l+1)p0dzz cot(z) −(2l + 1)pln(2 sin π(2l + 1)p)#p = 0 mod 4(6.27)§§ 6.3 The Torsion T(p) and τ(p, s).We can now combine our results for τ+(p, 0) and τ−(p, 0) to obtain expressions for thetorsion in the present case where p is even. Note again that since τ±(p, 0) = 0 we haveτ(p, 0) = 0(6.28)for the case of p even, and again this ensures that the torsion is metric independent.Combining the expressions 6.14 and 6.25 for τ ′+(p) and τ ′−(p) we obtain two expressionsfor ln T(p): one for each conjugacy class; these areln T(p) = −4 ln(2 sin πp ) + [2(p −4)pln(2 sin πp )]+ 4p(p−6)/4Xl=1h(2l + 1)2 −2iln2 sin(2l + 1)πp−8π(p−6)/4Xl=1(2l + 1)Z(2l+1)πp0dz z cot(z)p = 2 mod 4(6.29)andln T(p) = −4 ln(2 sin πp ) −4p ln 2 + 2(p −4)[1p ln(2 sin πp )]+ 4p(p−4)/4Xl=1[(2l + 1)(2l + 1) −2] ln2 sin(2l + 1)πp−4π(p−4)/4Xl=1(2l −1)Z(2l+1)πp0dz z cot(z)p = 0 mod 4(6.30)We note that for the above two formulae the torsion is already given by the first term ontheir RHS’s and so we obtain somewhat non-trivial integration formulae for the integralstherein.

On using the relations derived at the end of appendix B we can also reduce theabove expressions to Ray’s expression for the torsion.We have therefore now obtained a complete list of the determinants of Laplacians for0 and 1 forms on the lens spaces L(p) for all integer p ≥2, as well as the torsion T(p) forall p ≥2.27

§ 7. ConclusionIn the preceeding sections we analysed by direct computation the the determinants ofLaplacians on 0 and 1-forms on the lens spaces L(p), defined via the derivatives of theirassociated zeta functions.In this concluding section we collect our results and present in graphical form thebehaviour of the sequence of determinants we have analysed and their related torsion.For 0-forms we found thatln Det d∗d0 =12pπ2 ζ(3) + ln2 sin(πp )+ (p −2)πZ π/p0ln [2 sin(z)]−2π(p−3)/2Xl=1(2l)Z 2lπ/p0ln [2 sin(z)]p = 3, 5, .

. .

(7.1)in the case of p odd andln Det d∗d0 = −3pπ2 ζ(3) + (p −4)πZπp0dz ln [2 sin(z)]+ 2t−1Xl=1(p −(2l + 1)) 1πZπ(2l+1)p0dz ln [2 sin(z)]p = 2 mod 4(7.2)andln Det d∗d0 = −6pπ2 ζ(3)+ (p −4)πZπp0dz ln [2 sin(z)]+ 2t−1Xl=1(p −(2l + 1)) 1πZπ(2l+1)p0dz ln [2 sin(z)]p = 0 mod 4(7.3)These are plotted in figure 1.While for 1-forms we foundln Det d∗d1 = 1pπ2 ζ(3) −2 ln2 sin(πp )+ 2(p −2)πZ π/p0ln [2 sin(z)]−4π(p−3)/2Xl=1(2l)Z 2lπ/p0ln [2 sin(z)]p = 3, 5, . .

. (7.4)while when p is even we found thatln Det d∗d1 = −4 ln(2 sin πp ) −6pπ2 ζ(3) + 2(p −4) 1πZπp0dz ln [2 sin(z)]+ 4t−1Xl=1(p −(2l + 1)) 1πZπ(2l+1)p0dz ln [2 sin(z)]p = 2 mod 4(7.5)28

andln Det d∗d1 = −4 ln(2 sin πp ) −6pπ2 ζ(3)+ 2(p −4) 1πZπp0dz ln [2 sin(z)]+ 4t−1Xl=1(p −(2l + 1)) 1πZπ(2l+1)p0dz ln [2 sin(z)]p = 0 mod 4(7.6)and these results are displayed in figure 2. The differenceln T(p) = ln Det d∗d1 −2 ln Det d∗d0= −4 ln2 sin(πp )(7.7)which gives the torsion itself, is plotted in figure 3.A perusal of figure 3 shows that ln T(p) is negative for small p and large and positivefor large p. This raises the question as to whether ln T(p) crosses the p axis at an integervalue or not.

If so this corresponds to a trivial value for the torsion. In fact this clearlydoes happen for the value p = 6 i.e.

we haveln T(6) = 0(7.8)We show the more detailed behaviour of the torsion for small p in figure 4.Further interesting results are that that if we work with L(p, q) rather than L(p) ≡L(p, 1) then the torsion, now denoted by T(p, q), is trivial for only two other three dimen-sional lens spaces, namely L(10, 3) and L(12, 5): we find thatln T(p, q) = −2 ln4 sinπpsinπq∗pwhere q∗satisfies qq∗= 1 mod p(7.9)It is then possible to prove that, for p > 12, ln T(p, q) is strictly positive; while for p ≤12a check of the finite number of cases yields triviality in just the three cases given above.We conjecture that this may be understandable using some form of supersymmetry. Theseformulae have yet to be elucidated further.The precise meaning of our formulae such asζ(3) = 2π27ln(2) −87Z π/20dz z2 cot(z)ζ(3) = 2π213 ln 3 −913Zπ30dz z(z + π3 ) cot(z) −913Z2π30dz z(z −π3 ) cot(z)(7.10)29

and the more generalζ(3) =2π2(p3 −1)2(p −1) ln2 sin(πp )−2(p−3)2Xl=14l2 ln2 sin(2lπp )−p2π2Zπp0dz z(z + (p −2)πp) cot(z) −p2π2Z(p−1)πp0dz z(z −(p −2)πp) cot(z)−2(p−3)2Xl=1p2π2Z2lπp0dz z(z −4lπp ) cot(z),p = 3, 5, . .

. (7.11)is, as yet, unclear.

There may be some number theoretic matters underlying them as seemsto be the case in other work on lens spaces, cf. [15].

A thought provoking fact is that ζ(3)occurs in a recent paper of Witten [16] where, after multiplication by a known constant, itgives the volume of the symplectic space of flat connections over a non-orientable Riemannsurface. The corresponding calculation for orientable surfaces (where the volume elementis a rational cohomology class) allows a cohomological rederivation of the irrationality ofζ(2), ζ(4), .

. ..

This paper also involves the torsion but in two dimensions rather than three.The proof that ζ(3) is irrational was only obtained in 1978 cf. [17] and the rationality ofζ(5), ζ(7), .

. .

is at present open. However there are now other proofs [18], one of whichuses the characters of conformal quantum field theory.Finally, our technique, applied in five dimensions instead of three, would yield formulaefor ζ(5) but their nature has not yet been explored.30

FIGURES: The DeterminantsFigure 1: 2 ln d∗d0 versus p.Figure 2: ln d∗d1 versus p.31

FIGURES: The TorsionFigure 3: ln T(p) versus p.Figure 4: ln T(p) versus p for small p.32

Appendix A. The function Ap(x, s)In this appendix we analyze the functionAp(x, s) =∞Xk=1(pk + x)2n(pk + x)2 −1os(A.1)We are interested in particular in the value of this function and its derivative with respectto s at s = 0.

For this purpose we denoteAp(k, x, s) =(pk + x)2n(pk + x)2 −1os(A.2)which has the expansionAp(k, x, s) =1(pk)2s−2"1 + 2(1 −s) xpk +s + (s −1)(2s −1)x21(pk)2−2sxs + (s −1)(2s −1)3x21(pk)3 + . .

.#(A.3)Summing over k leads toAp(x, s) =1p2s−2 ζ(2s −2) +2xp2s−1 (1 −s)ζ(2s −1) + 1p2ss + (1 −s)(1 −2s)x2ζ(2s)−1p2s+1s + (1 −s)(1 −2s)3x2x2sζ(2s + 1) + ˆAp(x, s)(A.4)whereˆAp(x, s) =∞Xk=1(pk + x)2n(pk + x)2 −1os −1(pk)2s−21 + 2(1 −s) xpk+s + (s −1)(2s −1)x21(pk)2 −2sxs + (s −1)(2s −1)3x21(pk)3#) (A.5)Now the function ˆAp(x, s) is such that the processes of summation and differentiation withrespect to s at s = 0 commute. Also it is such that ˆAp(x, 0) = 0, which yieldsAp(x, 0) = −x(x + p)(2x + p)6p(A.6)33

Next evaluating the derivative at s = 0 we haveA′p(x, 0) = −Ap(x, 0) lnp2 + 2p2ζ′(−2) + ζ(0) + 2px [2ζ′(−1) −ζ(−1)]+ [2ζ′(0) −3ζ(0)] x2 −xp + 13p [3 −2γ] x3 + ˆA′p(x, 0)(A.7)which on using ζ(0) = −12, ζ(−1) = −112, 2sζ(2s + 1) = 1 + 2γs + . .

. and ζ′(0) = −12 ln 2πwe haveA′p(x, 0) = −Ap(x, 0) lnp2 + 2p2ζ′(−2) −12 +4pζ′(−1) + p6 −1px+32 −ln [2π]x2 + 13p [3 −2γ] x3 + ˆA′p(x, 0)(A.8)Since the processes of differentiation with respect to s and performing the sum over kcommute for ˆAp(x, s) we analyze this function by first taking the derivative and thenperforming the sum.ˆA′p(k, x, 0) = −(pk + x)2ln1 + (x + 1)pk+ ln1 + (x −1)pk+ (pk)2"2xpk −(1 −3x2)(pk)2+2x33(pk)3#(A.9)We note thatln (1 + (x + 1)pk) + ln (1 + (x −1)pk) =∞Xm=1(−1)(m+1) [(x + 1)m + (x −1)m]m (pk)m(A.10)Hence combining this with ˆA′p(k, x, 0) givesˆA′p(k, x, 0) =∞Xm=4(−1)m [(1 + x)m + (x −1)m]m (pk)(m−2)+ 2x∞Xm=3(−1)m [(1 + x)m + (x −1)m]m (pk)(m−1)+ x2∞Xm=2(−1)m [(1 + x)m + (x −1)m]m (pk)m(A.11)which givesˆA′p(x, 0) =∞Xm=2(−1)m h(1 + x)(m+2) + (x −1)(m+2)i(m + 2) pmζ(m)−2x∞Xm=2(−1)m h(1 + x)(m+1) + (x −1)(m+1)i(m + 1) pmζ(m)+ x2∞Xm=2(−1)m [(1 + x)m + (x −1)m]m pmζ(m)(A.12)34

We now observe the identity∞Xm=2(−1)mam−1ζ(m, α) = ψ(α + a) −ψ(α)(A.13)which yields∞Xm=2(−1)mam+νm + νζ(a, α) =Z a0dy yν [ψ(y + α) −ψ(α)](A.14)for ν ≥0. Now using this identity we haveˆA′p(x, 0) = p2Z (x+1)/p0dy (y −xp )2[ψ(1 + y) −ψ(1)]+ p2Z (x−1)/p0dy (y −xp )2[ψ(1 + y) −ψ(1)](A.15)Finally using ψ(1) = −γ we arrive at at our expression for the desired analytic con-tinuation of the derivative at zero,A′p(x, 0) = −Ap(x, 0) lnp2 −p22π2 ζ(3) −12 + px624ζ′(−1) + 1 −6p2+ x2(32 −ln[2π])+ x3p + p2Z(x+1)p0(y −xp )2ψ(1 + y) + p2Z(x−1)p0(y −xp )2ψ(1 + y)(A.16)We are interested in particular in the symmetric sumA+p (x) = A′p(x, 0) + A′p(−x, 0)(A.17)which from A.16 on usingΓ(1 + x)Γ(1 −x) =πxsin(πx)(A.18)we find to beA+p (x) = −p2π2 ζ(3) −p2π2Z(x+1)πp0dz z(z −2πxp ) cot(z) −p2π2Z(x−1)πp0dz z(z −2πxp ) cot(z)−x2 ln2 sin(x+1)πpx + 1−x2 ln2 sin(x−1)πpx −1(A.19)Alternatively one can see this directly from the expansion of ˆA+p in the series representationand the identities∞Xl=11l + 11a2l ζ(2l) = 12 −a2π2Zπa0dzz2 cot(z)(A.20)35

∞Xl=112l + 11a2l ζ(2l) = 12 −a2πZπa0dzz cot(z)(A.21)∞Xl=11l1a2l ζ(2l) = −lnsin( πa)πa(A.22)A useful expression is obtained by summing over all conjugacy classes to obtain anexpression independent of p. Hence note thatp−1Xj=0∞Xk=1Ap(k, −j, s) =X{νp}∞Xk=1(pk −j)2{(pk −j)2 −1}s=∞Xn=1(n + 1)2{(n + 1)2 −1}s(A.23)is independent of p since we have merely decomposed it as a sum over conjugacy classesmod p. We label this sum A(s) and its derivative at s = 0 simply A.Now denotingAp = A′p(0, 0) +(p−3)2Xl=0A+p (2l + 1)As a consequence of the above we have thatA = Ap −(p−3)2Xl=1(2l + 1)2 ln[(2l + 1)2 −1](A.24)is independent of p. This invariant can be written asA = −p32π2 ζ(3) −(p−3)2Xl=1(2l + 1)2 ln[(2l + 1)2 −1] −p2π2Zπp0dzz2 cot(z)−p2π2(p−3)2Xl=0"Z(2l+2)πp0dzz(z −2π(2l + 1)p) cot(z) +Z2lπp0dzz(z −2π(2l + 1)p) cot(z)#−(p−3)2Xl=0(2l + 1)2ln2 sin(2l+2)πp2l + 2+ ln2 sin2lπp2l(A.25)36

Which on simplifying givesA = −p32π2 ζ(3) −(p −2)2 ln[2 sin(πp )] −ln[πp ] −p2π2Zπp0dzz2 cot(z)−p2π2Z(p−1)πp0dzz(z −2(p −2)πp) cot(z)−2(p−3)2Xl=1"p2π2Z2lπp0dzz(z −4lπp ) cot(z) + (4l2 + 1) ln[2 sin(2lπp )]#(A.26)This constant has the value A = −1.20563One can use this together with the expression for B−p (x) obtained in the next appendixto give an alternative form of the expression for τ ′+(p). Explicitlyτ ′+(p) =2pA′p(0, 0) + 2p(p−3)/2Xl=0A+p (2l + 1) −(2l + 1)B−p (2l + 1)=2pA −(p−3)/pXl=1(2l + 1)hBp(2l + 1) −(2l + 1) lnh(2l + 1)2 −1iiwhich works out to beτ ′+(p) = 2p"A + ln[πp ] + (p −2)2 ln[2 sin(πp )] −(p −2)pπZ(p−1)πp0dzz cot(z)−2 pπ(p−3)2Xl=12lZ2lπp0dzz cot(z) +(p−3)2Xl=1(4l2 + 1) ln2 sin(2lπp )(A.27)Appendix B.

The function Bp(x, s)In this appendix we analyze the functionBp(x, s) =∞Xk=1(pk + x)n(pk + x)2 −1os(B.1)We are again interested in the value of this function and its derivative with respect to s ats = 0. For this we denoteBp(k, x, s) =(pk + x)n(pk + x)2 −1os(B.2)37

The continuation now requires us to extract the appropriate large k behaviour fromBp(x, s). Proceeding to do this we findBp(k, x, s) =(pk)1−2s −(2s −1)x(pk)−2s + s(1 + (2s −1)x2)(pk)−2s−1 + ˆBp(k, x, s)(B.3)Hence summing over k yieldsBp(x, s) =1p2s"pζ(2s −1) −x(2s −1)ζ(2s) + s1 + (2s −1)x2pζ(2s + 1)#+ ˆBp(s, x)(B.4)whereˆBp(x, s) =∞Xk=1(pk + x)n(pk + x)2 −1os −1(pk)2s−11 −x(2s −1)pk+s1 + (2s −1)x2(pk)2−x3s(2s + 1)3 + (2s −1)x2(pk)3)#(B.5)ˆBp(x, s) has the property that the process of taking the limit differentiating with respectto s and taking s →0 commutes with the sum over k.The second property is thatˆBp(x, 0) = 0.

Hence this expression has the analytic continuation to s = 0Bp(x, 0) = pζ(−1, 1 + xp ) + 12p= −p12 −x2 + (1 −x2)2p(B.6)Differentiating with respect to s and evaluating at s = 0 yieldsB′p(x, 0) = −Bp(x, 0) lnp2 + 2pζ′(−1) + 2x [ζ′(0) −ζ(0)] + x2p + γ(1 −x2)p+ ˆB′p(x, 0)(B.7)We choose to do the evaluate ˆB′p(x, 0) by doing the derivative first and the sum last.ThereforeˆB′p(k, x, 0) =∞Xm=2(−1)(m+1) h(1 + x)(m+1) + (x −1)(m+1)i(m + 1) pkm+ x∞Xm=2(−1)m [(1 + x)m + (x −1)m]m pkm(B.8)andˆB′p(x, 0) =∞Xm=2(−1)(m+1) h(1 + x)(m+1) + (x −1)(m+1)i(m + 1) pmζ(m)+ x∞Xm=2(−1)m [(1 + x)m + (x −1)m]m pmζ(m)(B.9)38

Again using the summation A.14 as in appendix A we haveˆB′p(x, 0) = −pZ (x+1)/p0dy (y −xp ) (ψ(1 + y) −ψ(1))−pZ (x−1)/p0dy (y −xp ) (ψ(1 + y) −ψ(1))(B.10)Thus we arrive atB′p(x, 0) = −Bp(x, 0) lnp2 + 2pζ′(−1) + [1 −ln[2π]] x + x2p−pZ (x+1)/p0dy (y −xp )ψ(1 + y) −pZ (x−1)/p0dy (y −xp )ψ(1 + y)(B.11)For the function Bp it is the anti-symmetric part that contributes to the quantities ofinterest thus definingB−p (x) = B′p(x, 0) −B′p(−x, 0)= x ln p2 + [1 −ln[2π]] 2x + ˆB−p (x)(B.12)we haveB−p (x) = pπZ (x+1)π/p0dz z cot(z) + pπZ (x−1)π/p0dz z cot(z)−x ln2 sin(x+1)πpx + 1−x ln2 sin(x−1)πpx −1(B.13)B.1 The function Hp(x, s)One can combine the results above with those in Appendix A to obtain an expression forH′p(x, 0), where we have definedHp(x, s) = Ap(x, s) −xBp(x, s)(B.14)Explicitly we haveH′p(x, 0) = −Hp(x, 0) lnp2 −p22π2 ζ(3) −12 + px612ζ′(−1) + 1 −6p2+ x22 + p2Z(x+1)p0y(y −xp )ψ(1 + y) + p2Z(x−1)p0dyy(y −xp)ψ(1 + y)(B.15)whereHp(x, 0) = −px12 −x2p + x36p(B.16)39

By analogy with A+p (x) we can defineH+p (x) = A+p (x) −xB−p (x)(B.17)which is given byH+p (x) = −p2π2 ζ(3) −p2π2Z (x+1)π/p0dz z(z −πxp ) cot(z) −p2π2Z (x−1)π/p0dz z(z −πxp ) cot(z)(B.18)Appendix C. Expressions involving the Hurwitz zeta function.The task of this appendix is to obtain an expressions for ζ′(0, 1 + a), ζ′(−1, 1 + a) andζ′(−2, 1 + a). We begin with ζ′(0, 1 + a).

Note first of all thatζ(s, 1 + a) =∞Xn=11(n + a)s(C.1)has the series expansionζ(s, 1 + a) = ζ(s) + sζ(s + 1)a + ˆζ(s, 1 + a)(C.2)whereˆζ(s, 1 + a) =∞Xn=11(n + a)s −1ns +sans+1=∞Xk=2(−1)kΓ(s + k)Γ(s)k!ζ(s + k)ak(C.3)Which gives the analytic continuation for ζ(s, 1 + a) to s = 0. We therefore deduceζ(0, 1 + a) = ζ(0) −a(C.4)andζ′(0, 1 + a) = −12 ln[2π] −γa + ˆζ′(0, 1 + a)where we used ζ′(0) = −12 ln[2π] and sζ(s + 1) = 1 + γs + .

. .. Differentiation with respectto s at s = 0 and summation over n commute for ˆζ(s, 1 + a) we therefore haveˆζ′(0, 1 + a) = −∞Xn=1hln[1 + an] −ani=∞Xk=2(−1)kakkζ(k)(C.5)40

Thus using A.14 we haveˆζ′(0, 1 + a) =Z a0dy [ψ(1 + y) −ψ(1)](C.6)We therefore obtain thatζ′(0, 1 + a) = −12 ln[2π] + ln Γ(1 + a)(C.7)The symmetric part of this function on using A.18 can be expressed as is expressed asζ′(0, 1 + a) + ζ′(0, 1 −a) = −ln2 sin(πa)a(C.8)Finally by noting thatζ(s) = p−sp−1Xj=0ζ(s, 1 −jp)is a decomposition over conjugacy classes of ζ(s) Differentiating and evaluating at s = 0we obtainζ′(0) = −p−1Xj=0ζ(0, 1 −jp) ln p +p−1Xj=0ζ′(0, 1 −jp)Now the first term is just a decomposition of ζ(0) over conjugacy classes therefore C.9gives the identityp−1Xj=0ζ′(0, 1 −jp) = ζ′(0) + ζ(0) ln pwhich for p odd gives(p−1)/2Xl=1ln2 sin(2lπp )= 12 ln por(p−1)/2Xl=0ln2 sin2l + 1)πp= 12 ln pNow for p even C.9 takes the formr−1Xl=0ζ′(0, 1 −lr) + ζ′(0, 1 −2l + 12r)= ζ′(0) + ζ(0) ln[2r]The first term in C.9 is of the form of the original expresson C.9 and therefore the resultingexpression isr−1Xl=0ζ′(0, 1 −2l + 12r) = −12 ln 241

For p = 4t + 2 we divide the l-summation into the two ranges, 0 to t −1, and, t to 2t −1;rearanging the latter sum and using C.8 then makes then gives us−t−1Xl=0ln2 sin2l + 1)πp+ ζ′(0, 12) = −12 ln 2Now noting that ζ′(0, 12) = −12 ln 2 we have the identity. (p−6)/4Xl=1ln2 sin(2l + 1)πp= 0We find for p = 4t, by a similar procedure, that(p−4)/2Xl=0ln2 sin(2l + 1)πp= 12 ln 2These are the first set of identities we obtain by decomposing over conjugacy classes.C.1 Relations involving ζ′(−1, 1 + a)Next we turn to ζ′(−1), thus note thatζ(s −1, 1 + a) =∞Xn=11(n + a)(s−1)(C.9)has the series expansion in terms of aζ(s −1, 1 + a) =∞Xk=0(−)kΓ(s + k −1) akΓ(s −1)k!ζ(k + s −1)(C.10)which on extracting the divergent part givesζ(s −1, 1 + a) = ζ(s −1) −(s −1)aζ(s) + s(s −1)a22ζ(s + 1) + ˆζ(s −1, 1 + a)(C.11)whereˆζ(s −1, 1 + a) =∞Xn=11(n + a)s−1 −1ns−1 + (s −1)ans−s(s −1)a2ns+1(C.12)This is now expressed in terms of our prescription for analytic continuation.

Hence wehaveζ(−1, 1 + a) = −112 −a(1 −a)2(C.13)42

where ζ(−1) = −1/2, ζ(0) = −1/2 and sζ(s + 1) = 1 + γs + . .

. have been used.

Differen-tiation with respect to s and setting it to zero yieldsζ′(−1, 1 + a) = ζ′(−1) + [1 −ln[2π]] a2 + (γ −1)a22+ ˆζ′(−1, 1 + a)(C.14)Since differentiation with respect to s at s = 0 and summation over n commute we haveˆζ′(−1, 1 + a) = −(n + a) ln(1 + an) + a + a22n= −∞Xk=2(−1)kak+1k + 1ζ(k) + a∞Xk=2(−)kakkζ(k)(C.15)Now using the identity∞Xk=2(−)kak+νk + νζ(k) =Z a0dy yν (ψ(1 + y) −ψ(1))(C.16)valid for a < 1 to remain away from the poles of ψ(y) which occur at y = −n for n =0, 1, 2, . .

. we haveˆζ′(−1, 1 + a) = −Z a0dy (y −a) [ψ(1 + y) −ψ(1)](C.17)and thereforeζ′(−1, 1 + a) = ζ′(−1) + [1 −ln(2π)] a2 −a22 −Z a0dy (y −a)ψ(1 + y)(C.18)Finally we conclude that the anti-symmetrised sum is given byζ′(−1, 1 + a) −ζ′(−1, 1 −a) = [1 −ln[2π]] a −Z a0dy (y −a) [ψ(1 + y) −ψ(1 −y)] (C.19)Now usingψ(1 + y) −ψ(1 −y) = −ddy lnsin(πy)πy(C.20)we obtainζ′(−1, 1 + a) −ζ′(−1, 1 −a) = −a ln2 sin(πa)a+ 1πZ πa0dz z cot(z)(C.21)An alternative way of expressing this which is useful in the main text and simplifies someof the expressions isζ′(−1, a) −ζ′(−1, 1 −a) = −1πZ πa0ln [2 sin(z)](C.22)43

Let us now examine the consequences of decomposing over conjugacy classes. Proceedingas for ζ′(0), on decomposing over conjugacy classes we obtain the identityζ′(−1) = 112 ln p + pp−1Xj=0ζ′(−1, 1 −jp)For p = 2r + 1 this yields the identityζ′(−1) =1p2 −1p(p−1)/2Xl=1"2lp ln2lp+2lp2+Z 2l/p0dy [ψ(1 + y) + ψ(1 −y)]#−112 ln p#i.e.ζ′(−1) = 16 −14 ln p+1p2 −1(p−1)/2Xl=1"2l ln[2l] + pZ 2l/p0dy [ψ(1 + y) + ψ(1 −y)]#−112 ln pThis can be used numerically to verify that ζ′(−1) = 0.16791.

Now noting thatZ π(1−a)0dz ln [2 sin(z)] = −Z πa0dz ln [2 sin(z)]We findp−1Xj=1Z jπ/p0dz ln [2 sin(z)] = 0For p even we find that by observing the two decompositionsζ′(−1, 1 + α) = −ζ′(−1, 1 + α) ln p + pp−1Xl=0ζ′(−1, 1 −l −αp)for α not necessarily integer andζ′(−1, 1 + α) = −ζ(−1, 1 + α) ln p + ppXl=1ζ′(−1, l + αp)valid for arbitrary p can be used to simplify the expression for(p−2)/2Xl=1Z (2l+1)π/p0dz ln [2 sin(z)] = 2 ln 2p−12 ln 2 −4p ln[p/2]44

These are used in simplifying the expression for the torsion in the case of p even.C.2 Relations involving ζ′(−2, 1 + a)We now turn to ζ′(−2, 1 + a) which we obtain fromζ(s −2, 1 + a) =∞Xn=11(n + a)s−2(C.23)We again extract the divergent parts from the sum to obtainζ(s −2, 1 + a) = ζ(s −2) −(s −2)aζ(s −1) + (s −1)(s −2)a22ζ(s)−s(s −1)(s −2)ζ(s + 1)a36+ ˆζ(s −2, 1 + a)(C.24)whereˆζ(s −2, 1 + a) =∞Xn=1"1(n + a)s−2 −1ns−2 + (s −2)ans−1−(s −1)(s −2)a22ns+ s(s −1)(s −2)a36ns+1(C.25)Thusζ(−2, 1 + a) = −a6 −a22 −a33(C.26)and differentiating with respect to s and evaluating at s = 0 we findζ′(−2, 1 + a) = ζ′(−2) +2ζ′(−1) + 112a +34 −12 ln[2π]a2−γ3 −12a3 + ˆζ′(−1, 1 + a)(C.27)We evaluate ˆζ′(−2, 1+a) by differentiating with respect to s first and then performing thesum over n obtainingˆζ′(−2, 1 + a) = −∞Xn=1(n + a)2 ln[1 + an] −n a −3a22−a33n=∞Xk=2(−1)kak+2k + 2ζ(k) −2a∞Xk=2(−1)kak+1k + 1ζ(k) + a2∞Xk=2(−1)kakkζ(k)(C.28)Thus using the summation formula C.16 we haveˆζ′(−2, 1 + a) =Z a0(y −a)2 [ψ(1 + y) −ψ(1)](C.29)45

We therefore obtainζ′(−2, 1 + a) = ζ′(−2) + 112 [24ζ′(−1) + 1] a + 14 [3 −2 ln[2π]] a2+ 12a3 +Z a0dy (y −a)2ψ(1 + y)(C.30)We note that our expression for ζ′(−2, 1 + a) impliesζ′(−2, 1 + a) + ζ′(−2, 1 −a) = 2ζ′(−2) + 12 [3 −2 ln[2π]]a2+Z a0dy(y −a)2 [ψ(1 + y) −ψ(1 −y)](C.31)Differentiating the functional relationξ(s) = ξ(1 −s),where ξ(s) = Γ(s/2)π−s/2ζ(s)(C.32)with respect to s we find for s = −2ζ′(−2) = −14π2 ζ(3)(C.33)Finally using C.20 and C.33 we haveζ′(−2, 1 + a) + ζ′(−2, 1 −a) = −12π2 ζ(3) −a2 ln2 sin(πa)a−1π2Z πa0dz z(z −2πa) cot(z)(C.34)We can obtain an identity by decomposing over conjugacy classes mod p, by notingthatζ(s −2) = p2−sp−1Xj=0ζ(s −2, 1 −jp)(C.35)impliesζ′(−2) = p2p−1Xj=0ζ(−2, 1 −jp) ln p + p2p−1Xj=0ζ′(−2, 1 −jp)(C.36)The first sum is a decomposition of ζ(−2) and therefore zero. If we decompose j into oddand even elements, by setting j = 2l + 1 and j = 2l respectively, we find for p odd i.e.p = 2r + 1ζ′(−2) = p2ζ′(−2) + p2rXl=1ζ′(−2, 1 −2lp ) + ζ′(−2, 1 −2l + 1p)(C.37)46

which can be rewritten as eitherζ′(−2) = p2ζ′(−2) + p2rXl=1ζ′(−2, 1 −2lp ) + ζ′(−2, 2lp )(C.38)orζ′(−2) = p2ζ′(−2) + p2r−1Xl=0ζ′(−2, 2l + 1p) + ζ′(−2, 1 −2l + 1p)(C.39)Therefore we have thatζ′(−2) = p3ζ′(−2) −rXl=1"4l2 ln2 sin(2lπp )+ p2π2Z 2lπ/p0dz z(z −4lπp ) cot(z)#(C.40)which on using C.33 gives(1 −p3)4π2ζ(3) =rXl=1"4l2 ln2 sin(2lπp )+ p2π2Z 2lπ/p0dz z(z −4lπp ) cot(z)#(C.41)orζ′(−2) = p3ζ′(−2) −r−1Xl=0(2l + 1)2 ln2 sin((2l + 1)πp)+ p2π2Z (2l+1)π/p0dz z(z −2(2l + 1)πp) cot(z)#(C.42)which gives(1 −p3)4π2ζ(3) =r−1Xl=0(2l + 1)2 ln2 sin((2l + 1)πp)+ p2π2Z (2l+1)π/p0dz z(z −2(2l + 1)πp) cot(z)#(C.43)We can equally establish that(1 −p3)4π2ζ(3) =rXj=1"j2 ln2 sin(jπp )+ p2π2Z jπ/p0dz z(z −2jπp ) cot(z)#(C.44)Finally we tabulate here for future reference some useful identities regarding the Hur-witz and Riemann zeta functionsζ(s, a) = a−s + ζ(s, 1 + a)(C.45)47

ζ(−2) = 0ζ(−1, 1 + a) −ζ(−1, 1 −a) = −aζ(0, a) + ζ(0, 1 −a) = 0ζ(0, 1 + a) + ζ(0, 1 −a) = −1ζ′(0, 1 + a) = ζ′(0, a) + ln aζ′(−1, 1 + a) = ζ′(−1, a) + a ln aζ′(0, 1 + a) + ζ′(0, 1 −a) = −ln2 sin(πa)a(C.46)ζ′(−1, 1 + a) −ζ′(−1, 1 −a) = −a ln2 sin(πa)a+ 1πZ πa0dz z cot(z)ζ′(−1, a) −ζ′(−1, 1 −a) = −a ln(2 sin πa) + 1πZ πa0dz z cot(z)(C.47)andζ′(−2, 1 + a) + ζ′(−2, 1 −a) = −12π2 ζ(3) −a2 ln2 sin(πa)a−1π2Z πa0dz z(z −2πa) cot(z)(C.48)ζ′(−2, a) + ζ′(−2, 1 −a) = −12π2 ζ(3) −a2 ln [2 sin(πa)] −1π2Z πa0dz z(z −2πa) cot(z)(C.49)References1.Franz W., ¨Uber die Torsion einer ¨Uberdeckung, J. Reine Angew. Math., 173, 245–254, (1935).2.Ray D. B., Reidemeister torsion and the Laplacian on lens spaces, Adv.

in Math., 4,109–126, (1970).3.Ray D. B. and Singer I. M., R-torsion and the Laplacian on Riemannian manifolds,Adv. in Math., 7, 145–201, (1971).4.Ray D. B. and Singer I. M., Analytic Torsion for complex manifolds, Ann.

Math.,98, 154–177, (1973).5.Cheeger J., Analytic torsion and the heat equation, Ann.Math., 109, 259–322,(1979).6.M¨uller W., Analytic torsion and the R-torsion of Riemannian manifolds, Adv. Math.,28, 233–, (1978).7.Schwarz A. S., The partition function of degenerate quadratic functional and Ray-Singer invariants, Lett.

Math. Phys., 2, 247–252, (1978).8.Witten E., Quantum field theory and the Jones polynomial, I.

A. M. P. Congress,Swansea, 1988, edited by: Davies I., Simon B. and Truman A., Institute of Physics,(1989).9.Nash C., Differential Topology and Quantum Field Theory, Academic Press, (1991).10.Birmingham D., Blau M., Rakowski M. and Thompson G., Topological field theory,Phys. Rep., 209, 129–340, (1991).48

11.Batalin I. A. and Vilkovisky G. A., Quantisation of Gauge Theories with linearlyindependent generators, Phys.

Rev., D28, 2567–, (1983).12.Batalin I. A. and Vilkovisky G. A., Existence theorem for gauge algebras, Jour.

Math.Phys., 26, 172–, (1985).13.Rolfsen D., Knots and Links, Publish or Perish, (1976).14.Bott R. and Tu L. W., Differential Forms in Algebraic Topology, Springer-Verlag,New York, (1982).15.Jeffrey L., Chern-Simons-Witten invariants of lens spaces and torus bundles, and thesemiclassical approximation, Commun. Math.

Phys., 147, 563–604, (1992).16.Witten E., On quantum gauge theories in two dimensions, Commun. Math.

Phys.,141, 153–209, (1991).17.Poorten Alfred van der, A proof that Euler missed.... Ap´ery’s proof of the irrationalityof ζ(3). An informal report., The Mathematical Intelligencer, 1, 195–203, (1979).18.Beukers, unpublished, private communication from W. Nahm.49

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