Starting with topological field theories we investigate the Ray-Singer
analytic torsion in three dimensions. For the lens Spaces L(p;q) an explicit
analytic continuation of the appropriate zeta functions is contructed and
implemented. Among the results obtained are closed formulae for the individual
determinants involved, the large $p$ behaviour of the determinants and the
torsion, as well as an infinite set of distinct formulae for zeta(3): the
ordinary Riemann zeta function evaluated at s=3.
  The torsion turns out to be trivial for the cases L(6,1), L((10,3) and
L(12,5) and is, in general, greater than unity for large p and less than unity
for a finite number of p and q.Comment: 10 page

Connor, Denjoe O'

Nash, Charles

English

arXiv

Starting with topological field theories we investigate the Ray-Singer analytic torsion in three dimensions. For the lens Spaces L(p; q) an explicit analytic continuation of the appropriate zeta functions is constructed and implemented. Among the results obtained are closed formulae for the individual determinants involved, the large p behaviour of the determinants and the torsion, as well as an infinite set of distinct formulae for ζ(3): the ordinary Riemann zeta function evaluated at s = 3. The torsion turns out to be trivial for the cases L(6, 1), L((10, 3) and L(12, 5) and is, in general, greater than unity for large p and less than unity for a finite number of p and q

O'Connor, Denjoe

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DIAS-STP-92-39Ray-Singer Torsion, Topological field theories and the Riemann zeta function at s = 3.byCharles Nash and D. J. 0’ ConnorDepartment of Mathematical Physics School of Theoretical PhysicsSt. Patrick’s College Dublin Institute for Advanced StudiesMaynooth and 10 Burlington RoadIreland Dublin 4IrelandAbstract: Starting with topological field theories we investigate the Ray-Singer analytictorsion in three dimensions. For the lens Spaces L(p; q) an explicit analytic continuation ofthe appropriate zeta functions is contructed and implemented. Among the results obtainedare closed formulae for the individual determinants involved, the large p behaviour of thedeterminants and the torsion, as well as an infinite set of distinct formulae for C(3): theordinary Riemann zeta function evaluated at s = 3. The torsion turns out to be trivial forthe cases L(6, 1), L((10, 3) and L(12, 5) and is, in general, greater than unity for large pand less than unity for a finite number of p and q.§ 1. IntroductionThe 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 7r1 (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 constructed from the cohomology groups W(M; B).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 Hz(M; B) then these determinants become determinants ofLaplacians Li on p-forms with coefficients in B. But zeta functions for elliptic operatorscan be used to give finite values to such infinite dimensional determinaüts 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 thçm to be independent of theRiemannian metric used to define the Laplacian’s .This analytic torsion coincided, for the case of lens spaces, with the combinatorially defined Reidemeister-Franz torsion. Finally Cheeger and Muller [5,6] independently provedthat the analytic Ray-Singer torsion coincides with the combinatorial Reidemeister-Franztorsion in all cases.1Infinite 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 manfold 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 nonAbelian 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.We shall be concerned here with the special situation of three dimensions and with thecase where the three manifold M is a lens space. In the next two sections we describe theprecise setting and the analytic continuation while the final section contains our concludingremarks.§ 2. Topological field theories, analytic torsion and lens spacesQuantum field theories of the type alluded to in the previous section are usually referredto as topological 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[w]= I wdw, dimM = 2n + 1 (2.1)JMwhere w is an n-form. The partition function is thenZ{M] = fvw[w]exp{_s[w]] (2.2)S[wj has a gauge invariance whereby S[wj = S[w + d] and therefore to define the partitionfunction it is necessary to integrate over only inequivalent field configurations. The measureDw0u[ ] 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]. Wewish to devote more space here to lens spaces and the computation of their torsion and sowe turn to that now.To define the Ray-Singer torsion, or simply torsion, we take a closed compact Riemannian manifold M over which we have a flat bundle E’. Let M have a non-trivial fundamentalgroup 7t1 (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) wherelnT(M, E) = (_1)qln det , n = dimM (2.3)2The 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 LI haveempty kernels and so are strictly positive definite. Given this fact one may use zetafunctions to define det in the standard way. Recall that if P is a positive ellipticdifferential or pseudo-differential operator with spectrum {,u4 and degeneracies F thenits associated zeta function Cp(s) is a meromorphic function of s, regular at s = 0, whichis given by(2.4)and its determinant det P is defined bylndetP=—dp(s) (2.5)ds.5=0Using this we have•d(,E(s)lnT(M,E) = _(_1)q (2.6)Next we turn to lens spaces. For general background on lens spaces cf. [13,14]and references therein—briefly, a lens space can be constructed as follows: Take an odddimensional sphere 521, considered as a subset of C, on which a finite cyclic groupof rotations G, say, acts. The quotient S2’/G of the sphere under this action is a lensspace. More precisely, suppose that G is of order p, (z1,. . . , z) E C and the group actiontakes the form(zi,... ,z) —* (exp(27riq1/p)zi,... ,exp(2iriq/p)z) (2 7)with q,. . ., qn integers relatively prime to pthen the quotient 521 /G is a lens space often denoted by L(p; qi,... , qn). A formulafor the torsion of these spaces was first worked out by Ray [2]. We wish to focus on thesituation that obtains when n = 2 and G is the group Z, Z/pZ. For the most part weshall deal with the lens space L(2; 1, 1) which, for simplicity, we shall denote by L(p); weshall also use the notation L(p; q) to denote the Lens space L(p; 1, q). In passing we notethat when p = 2 we have L(2) = RP3 50(3).The group action above defines a representation V, say, of iri(L(p)) and also determines a flat bundle F = (V x S3)/Z, over L(p). It is the torsion of this F over L(p)with which we are concerned here. Using zeta functions the torsion of these lens spaces istherefore given byd(F(s)lnT(L(p),F) = _(_1)q (2.8)As an aid to the calculation of lnT(L(p), F) it is useful to introduce the notationr(p,s) = _(_1)qF(s) T(p) = T(L(p),F) (2.9)3For r(p, s) itself we now haver(p, s) = — 2(,F(S) + 3F(s) = 3F(3) — CF(s), using Poincaré duality (2.10)Making use of the triviality of the kernels of we further obtain [2] the formulaT(p,S) = 2Cd*d0(s) — (d*d1s) (2.11)For the individual zeta functions we denote the eigenvalues and their degeneracies by)n(q,p) and I’n(q,p) respectively giving the expressions.s) = (d*di(s) = (2.12)It remains to compute these eigenvalues and degeneracies cf. [15]. The eigenvalues are\n(O,p) =n(m+2), L(l,p) = (m+ 1)2, n = 1,2,... (2.13)To calculate the degeneracies is more difficult; we make use of the fact that 53 is agroup mamfold and proceed as follows: Consider the Laplacians d*dq on S3, and d*d onL(p) also, if .\ is an eigenvaiue, denote the corresponding eigenspaces by Aq(.\) andrespectively. Letv(z) Aq(), with z e c C2, and g e Z,, where g exp[2irij/p], 0 i (p — 1)(2.14)The element g acts on v(z) to give g v(z) whereg v(z) = v(gz) where gz = (exp[2irij/p]z1,exp[2irij/pjz2) (2.15)The above definitions allow us to define the projection F(A) on Aq(.A) by= exp[—2ij/pjg. v (2.16)gEZ,Evidently [P(\), d*dqj = 0 and so P(s) projects the space Aq(.A) onto the spaceFinally this means that we obtain a formula for the degeneracy F(q,p), namely(p—i)F(q,p) = tr (PAF()) = exp[—2ij/p]tr (gIAF()) (2.17)To actually apply this formula we now add in the fact that 53 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)) = c,J,D,L = e D, 0 (2.18)4where measures the multiplicity of the representation which must therefore be dim D,h.But lodge theory gives us the alternative decompositionL2(S3)= A(.A) (2.19)AIn addition the Casimir operator for SU(2) is a multiple of the Laplacian and, if therepresentation label ,u is taken to be the usual half-integer j, then we know that thisCasimir has eigenvalues j(j + 1), and also that dimD3 = 2j + 1. These facts identify theLaplacian o d*do as four times the Casimir and identify A0X) as dimD, copes of D,.Thus if we set n = 2j, so that n is always integral, then we have the degeneracy formula(p—i)(n+1 exp[_2ij/p}Th/2(2j/ ) (2.20)where (8) denotes the SU(2) character, on D3, for rotation through the angle 6; i.e.x(6) = sin((2j + 1)6) (2.21)sin(8)Hence our explicit degeneracy formula for 0-forms on L(p) is(p—i)F(Op)= (n+1) exp[_2ij/p] 51fl m±1)j/) (2.22)p sin(2irj/p)•j=OWe now have to find the analogous formula for the 1-forms. The formula that resultsis(p—i)r(1,) = exp[—2ij/pj {n +i)/2(2/ + (m + 2)x(Th_12(2j/p)} (2.23)To simplify the notation we introduce the ‘p-averaged character’ Kx) which we defineby(p—i)Kx = exp[_2ij/p](2pij/p) (2.24)Finally this gives us a concrete expression for r(p, s), i.e.—f mKx()/2) + (n ± 2) Kx’ 2(m + 1) Kx2 1r(p,- (n + 1)25 - {m(n + 2)}25 J (2.25)= T+(p, s) — r(p, s)with the obvious definition for r(p, s) and T(p, s).5To actually compute the torsion we need to be able to evaluate these p-averagedcharacters. This is a somewhat non-trivial combinatorial task and it is necessary to divideii up into its conjugacy classes mod p by writing ii = pk—j, k Z, j = 0, 1, ...,(p — 1)We eventually discover that1k forj=0,2,...,(p—1)k for j = 1 if p is odd(k—1) forj=3,5,...,(p—2)(x_2).(2.26)(0 forj=0,2,...,(p—2)2k for j = 1 if p is even1(2k—i) forj=3,5,...,(p—1)We must now construct the analytic continuation of the series for T(p, s). For thedetails we refer the reader to [15j. We shall just describe here the p = 2 case.§ 3. The Analytic ContinuationThe series to be cohtinued isI (x(12) + (n + 2) ((1)/2) 2(n + 1) Kx’2T(p, ) = (m + 1)2s - {m(n + 2)}2s J (3.1)and it already converges for Re s > 3/2; however a calculation of the torsion reqres usto work at .s = 0, hence we see the need for, and the extent of, the continuation.With p = 2 we haveT(2 ) = f 2(n + 1)(122 — m (x’’2)± (m + 2) (x’’ (3.2)I {m(n+2)}8 (n±1)2s JBut using 2.26 we find that(x+1)/2) = (x(2k_i+1)/2), (m = 2k—j)(3.3)fo, j=i _fo, modd2k+2, j=0 2k+2, mevenSimilarlyKx_12)2= (x2k_i/2 (n = 2k —j)(3.4)f2k, j1 f(n+i), nodd— 10, j=0 — 0, neven6Thus r(2, s) becomesT(2, s) = r+(2, s) — r_(2, s) = 2(n +1)2 2n(n + 2) (3.5)nodd{m(n+2)}3 (n+1)2n evenSetting ii (2m — 1) in r+(2, s) and n = 2m in r(2, s) we have28m 2m(2m+2)r(2,s) = (4m2—1) — (2m + 1)23m=1 rn=O (3.6)2_____1 1= Z (4m2— 1) — (2m + 1)(23-2) + (2m + 1)23m= 1 m= 0 m 0Now if <(s) is the usual Riemann zeta function we can use the fact that1—= (1 — 2)(s) (3.7)Ti3n=1 ,3,5...then we get8m2T(2, s)= :z 2(1 — 2_(23_2))(2s — 2) + 2(1 —2)(2s) (3.8)m=14m2 — 1) —The only term in 3.8 without a well defined continuation is the first term. To this end wedefine the quantity A(m, s) by4m2 4m2 ( 1 \S 1A(m,= (4m2 — 1) = (4m2)—= (4m2)(3_1){i + 42 + ()1—(4m2)(31) + (4m2) + R(m, s), (def. of R(m, s))So that the remainder term R(m, s) is given by1R(m,s) = A(m,s) -__________(4m2)(1)—(4m2) (3.10)4m2 1 s= (4m2 — 1) — (4m2)(3_1) — (4m2)The definition of the remainder term is chosen to ensure that(lnm) (3.11)R(m,s)Im2and this has the vital consequence that the operations d/ds (at s = 0) and Zm commutewhen applied to R(m, s).7Returning to r(s, 2) itself we haver(2, s) = 2 A(m, s) — 2(1 — 225_2))((2s — 2) + 2(1 —2)((2s)1 1 (3.12)r(2, s) = 2(4m2)(s_1) +2s (4m2) + 2 R(m, s)rn—i rn—i rn—i— 2(1 — 2_(232(2s — 2) + 2(1 — 2_2(2s)DefiningR(s) = R(m,s) (3.13)rn=0gives a series for R(s) which is guaranteed to be convergent and the analytic continuationis now complete; thus we can now take the final step which is to differentiate and obtainthe torsion T(2). The result that we get is thatlnT(2) = dr(2,O) = 28’(—2) ± 2(1 +ln4)(O) + 2R’(O) (3.14)But it is easy to check that (O) = —1/2 and ‘(—2) = —((3)/4ir2 and by our remarkabove concerning the motive for our choice of definition for R(m, s) we haveR’(O) = R(m,R’(O) = dR(m,s)ds= [4mg {ln(4m2)— ln(4m2 — i)} — 1] = — [4m ln(1 — 1/4m2)+ 1]rn rn (3.15)HencelnT(2) = _!((3) —1— 2ln(2)—2Z [4m2ln(1 — 1/4m2)+ i] (3.16)However the series for R’(O) can be expressed as a trigonometric integral cf. [15]. In factwe have[4m2ln(1 —1/4m2)+1] =—+ dzz2cot( ) (3.17)rn=i 0which means thatir/2lnT(2) = —-(3) — 21n(2) — f dz z2 cot(z) (3.18)8This formula 3.18 above for T(2) can be pushed even further; by using Ray’s expression[2] for the torsion we can deduce that(p—i) plnT(p) = — cos(1)ln(2sin())exp{2Z] = —41n(2sin()) (3.19)j=ikj. p p pwhich, for p = 2, becomes simplylnT(2) = —41n(2) (3.20)Hence we straightaway have that7 8 /2/ dzzcot( ) (3.21)irOrV22 8 7r/2C(3) = —f.- ln(2) —fdz z2 cot(z) (3.22)in other words our computation of the torsion has given us a formula for (3).In [15] we construct the continuation for arbitrary p but here we limit ourselves toquoting the torsion formula for p odd which (recall that lnT(p) = —4ln(2sin(7r/p))) is(p..-1)irlnT(p) =_ (p3—1)((3) (p—2) fdzz cot(z)+ (p—2)fdzz cot(z)(p—1)r . it.2 2 I 2 4 ln(2 sin())— dzz cot(z) ——-dzz cot(z)—___________16 (p—3)/22 (p—3)/2 iLi+ — if dzz cot(z) — f dzz2 cot(z)(p—3)/22 . 2l-ir——41 ln(2 sin(—)), for p oddp p(3.23)§ 4. Concluding remarks VThese formulae have yet to be elucidated further. VA thought provoking fact is that (3) occurs in a recent paper of Witten [16] where,after multiplication by a known constant, it gives the volume of the symplectic space offlat connections over a mon-orientable Riemann surface. The corresponding calculationfor orientable surfaces (where the volume element is a rational cohomology class) allows acohomological rederivation of the irrationality of ((2), ((4) This paper also involvesthe torsion but in two dimensions rather than three. The proof that ((3) is irrational wasonly obtained in 1978 cf. [17] and the rationality of ((5), ((7),... is at present open.9Our technique, applied in five dimensions instead of three would yield formulae forC(S) but their nature is as yet unclear.Further interesting results are that the T(p) is trivial (i.e unity) when p = 6; andthat if we work with L(p, q) rather than L(p) then the only other three dimensional lensspaces for which the torsion is trivial are L(1O, 3) and L(12, 5). The large p behaviour ofthe determinants is also computable: T(p) grows asp4/(2ir)+p/6(2r) for large p, whilethe determinants grow much faster than quartically.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 0. 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. Muller W., Analytic torsion and the R-torsion of Riemannian manifolds, Adv. Math.,28, 233—, (1978).7. 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).8. Witten E., Quantum field theory and the Jones polynomial, Commun. Math. Phys.,121, 351—400, (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).11. Batalin I. A. and Vilkovisky G. A., Quantisation of Gauge Theories with linearlyindependent generators, Phys. Rev., D28, 2567—, (1983).12. Bata]in I. A. and Vilkovisky G. A., Existence theorem for gauge algebras, Jour. Math.Phys., 26, 172—, (1985).13. Rolfsen 0., 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. Nash C. and 0’ Connor 0. J., in preparation.16. Witten E., On quantum gauge theories in two dimensions, Commun. Math. Phys.,,153—209, (1991).17. Poorten Alfred van der, A proof that Euler missed.... Apéry’s proof of the irrationalityof ((3). An informal report., The Mathematical Intelligencer, 1, 195—203, (1979).10

Ray-Singer Torsion, Topological field theories and the Riemann zeta function at s = 3

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Ray-Singer Torsion, Topological field theories and the Riemann zeta function at s=3

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