---

Effective action in spherical domains

arXiv:hep-th/9306154v2 30 Jun 1993MUTP/93/15Effective action in spherical domainsJ.S.DowkerDepartment of Theoretical Physics,The University of Manchester, Manchester, England.AbstractThe effective action on an orbifolded sphere is computed for minimally coupledscalar fields. The results are presented in terms of derivatives of Barnes ζ–functionsand it is shown how these may be evaluated.

Numerical values are shown. Ananalytical, heat-kernel derivation of the Ces`aro-Fedorov formula for the number ofsymmetry planes of a regular solid is also presented.1

1. Introduction.In earlier work [1] we have shown that the ζ–function, ζΓ(s), on orbifold-factoredspheres, Sd/Γ, for a conformally coupled scalar field, is given by a Barnes ζ–function,[2], ζd(s, a | d), where the di are the degrees associated with the tiling group Γ. Thefree-field Casimir energy on the space-time R×Sd/Γ was given as the value of the ζ–function at a negativ e integer which evaluated to a generalised Bernoulli function.In the present work we wish to consider the effective action on orbifolds Sd/Γ whichthis time are to be looked upon as Euclidean space-times.

In particular we willdiscuss d = 2 and d = 3, concentrating on the former.The simplifying assumption in our previous work was that of conformal cou-pling on R×Sd/Γ. This made the relevant eigenvalues perfect squares and allowedus to use known generating functions to incorporate the degeneracies.

From thepoint of view of field theories on the space-times Sd/Γ, retaining this assumptionwould be rather artificial. A more appropriate choice would be minimal coupling,or possibly conformal coupling, on Sd/Γ.

(These coincide for d = 2. )The quantities in which we are interested are ζ′Γ(0) and ζΓ(0).The latterdetermines the divergence in the effective action and the former is, up to a factorand a finite addition, the renormalised effective action (i.e.

half the logarithm ofthe functional determinant).2. Eigenvalues, degeneracies and zeta functions.For the aforementioned conformal coupling, the eigenvalues of the second orderoperator −∆2 + ξR (ξ = (d −1)/4d) areλn = 14(n + d −2)2(1)with degeneracies that we shall leave unspecified here.In our previous work [1] we showed that the corresponding Neumann andDirichlet ζ–functions on Sd/Γ were,ζ(C)N(s) = ζd (2s, (d −1)/2 | d) ,(2)ζ(C)D(s) = ζd (2s, Pdi −(d −1)/2 | d) ,(3)where the general definition of the Barnes ζ–function isζd(s, a | d) = iΓ(1 −s)2πZLdτexp(−aτ)(−τ)s−1Qdi=11 −exp(−diτ)=∞Xm=01(a + m.d)s ,Re s > d.(4)This shows that the eigenvalues are given specifically byλn = (a + m.d)2(5)2

the degeneracies coming from coincidences. The parameter a is (d −1)/2 in theNeumann case and comparison with the previous form shows that the integer n =2m.d + 1, m = 0 upwards.

For Dirichlet conditions, a = P di −(d −1)/2 and thenn = 2m.d −1 with m = (1, 1) upwards. The interpretation in two dimensions isthat the angular momentum is L = m.d for Neumann and m.d −1 for Dirichletconditions.Turning to minimal coupling, (ξ = 0), the eigenvalues of the Laplacian areλn = (a + m.d)2 −(d −1)24.

(6)and the corresponding ζ–function isζ(s) =Xm1(a + m.d)2 −(d −1)2/4s . (7)The origin m = 0 is to be omitted for Neumann conditions, when the ζ–function isdenoted by ¯ζ(s).Consider a sum of the formζ(s) =Xm1(a + m.d)2 −α2s(8)so that¯ζ(s) = ζ(s) −(a2 −α2)−s.

(9)For minimal coupling, α = (d −1)/2, while for conformal coupling in d–dimensions,α = 1/2. We concentrate on minimal coupling.A standard way of obtaining information about an expression such as (8) isto perform a binomial expansion to produce a sum of known ζ–functions, in thepresent case a sum of Barnes ζ–functions,ζ(s) =∞Xr=0α2r s(s + 1) .

. .

(s + r −1)r!ζd(2s + 2r, a | d). (10)From this, the value of ζ(s) at a nonpositive integer is easily found.

For examplethe important value ζ(0) is given byζ(0) = ζd(0, a | d) + 12uXr=1α2rr N2r(d)(11)and, more generally, we haveζ(−n) =nXr=0(−α2)r nrζd(2r −2n, a | d) + (−1)n2uXr=1n! (r −1)!

(r + n)! α2n+2rN2r, (12)3

where u = d/2 if d is even and u = (d −1)/2 if d is odd.Nr(d) is the residue defined byζd(s + r, a | d) →Nr(d)s+ Rr(d)as s →0,(13)where 1 ≤r ≤d. Expressions for the residue and remainder involve generalisedBernoulli functions and can be found in Barnes [2].

For shortness, their dependenceon the parameter a is not indicated.The form of the residues given by Barnes [2] isNr(d) = (−1)r+d(r −1)!dS(r+1)1(a)wheredS(r+1)1(a) is the (r +1)-th derivative of Barnes’ generalised Bernoulli poly-nomialdS1(a). The general relation with the more usual polynomials, [5], will notbe given here.

Specific forms aredS(d+1)1(a) =1Q di,dS(d)1 (a) = 2a −P di2 Q di,dS(d−1)1(a) =112 Q di6a2 −6aXdi +Xd2i + 3Xi

(d + n)!B(d)d+n(a | d). (15)From (11), (14) and (15) we find, for two dimensions,ζN(0) = ζD(0) =112d1d23 −3(d1 + d2) + (d1 + d2)2 + d1d2.

(16)This corrects our previous expression [1] obtained by an incorrect manipulationof the heat-kernel.3. The derivative of the zeta function.The derivative at s = 0 is a little more difficult to find.

From (10) a first step isζ′(0) = 2ζ′d(0, a | d) +uXr=1α2rr R2r + 12N2rr−1X11k!+∞Xr=u+1α2rr ζd(2r, a | d). (17)4

The integral representation of the Barnes ζ–function allows the final sum in(17) to be written asXr=u+1α2rrΓ(2r)Z ∞0dττ 2r−1 exp(−aτ)Qi1 −exp(−diτ) =2Z ∞0exp(−aτ) cosh ατ −uXr=0(ατ)2r(2r)! !dττ Qi1 −exp(−diτ).

(18)In the Neumann case a = α and there is an infra-red, logarithmic divergence atinfinity caused by the zero mode which will be taken care of by the transition to ¯ζ,(9).Although the integral converges nicely at τ = 0, the individual terms of the in-tegrand do not. It is enough to introduce another ultra-violet analytic regularisationand define the intermediate quantity,2Z ∞0exp(−aτ) cosh ατ −uXr=0(ατ)2r(2r)!

!τ s−1dτQi1 −exp(−diτ)(19)whose s = 0 limit gives (18).After continuation, (19) integrates toΓ(s)ζd(s, a −α | d) + ζd(s, a + α | d)−2uXr=0α2r(2r)!Γ(s + 2r)ζd(s + 2r, a | d). (20)As s tends to zero, each term in (20) yields a pole and a finite remainder.

Thepoles must cancel and soζd(0, a −α | d) + ζd(0, a + α | d) −2ζd(0, a | d) =uXr=1α2rr N2r(d). (21)This condition is an identity between generalised Bernoulli functions.

Combininedwith (11) it produces the symmetrical expressionζ(0) = 12ζd(0, a −α | d) + ζd(0, a + α | d). (22)The finite remainder in (20) isζ′d(0, a −α | d) + ζ′d(0, a + α | d) −2ζ′d(0, a | d) −uXr=1α2rr R2r(d)−5

γζd(0, a −α | d) + ζd(0, a + α | d) −2ζd(0, a | d)−uXr=1α2rr ψ(2r)N2r(d)(23)which, in view of (21), can be writtenζ′d(0, a −α | d) + ζ′d(0, a + α | d) −2ζ′d(0, a | d)−uXr=1α2rr R2r(d) −uXr=1α2rrψ(2r) + γN2r(d). (24)Combining this with (17) we have finallyζ′(0) = ζ′d(0, a−α | d)+ζ′d(0, a+α | d)−uXr=1α2r2r2ψ(2r)−ψ(r−1)+γN2r(d).

(25)The fact that the remainders have cancelled, suggests that there is a more rapidroute to this result.Apart from the final term, (25) is the expression that would have been obtainedby a naive application of the ‘surrogate’ ζ–function method which is based on theproduct nature of the eigenvalues, (a −α + m.d) (a + α + m.d), in (8) followed byan application of the rule ln det (AB) = ln det A+ln det B. This method is suspect,as discussed by Allen [3] and by Chodos and Myers [4].

Allen [3] derives a particular‘correction’ term as in (25). He also points out that (22) could be expected on thebasis of the eigenvalue factorisation, being the average of the regularised dimensionsof the operator factors.The final term in (25) can be rewrittenuXr=1α2r2r2ψ(2r) −ψ(r −1) + γN2r(d) =uXr=1α2rr N2r(d)r−1X012k + 1.In order to evaluate the effective action we must substitute the appropriatevalues of a and α for Neumann and Dirichlet conditions into (25).

In the formercase it is also necessary to remove the zero mode i.e. to use ¯ζ.

The relevant quantitythen is the Γ-modular form ρ, defined by, [2],limǫ→0 ζ′r(0, ǫ | d) = −ln ǫ −ln ρr(d). (26)We find the following basic expressionsζ′N(0) = −ln ρd(d) + ζ′d(0, d −1 | d) −ln(d −1) −uXr=1α2rr N2r(d)r−1X012k + 1 (27)6

andζ′D(0) = ζ′d(0, d0 | d) + ζ′d(0, d0 + d −1 | d) −uXr=1α2r2r N2r(d)r−1X012k + 1,(28)where d0 = Pi di −d + 1 is the number of reflecting planes in Γ. We recall thatα = (d−1)/2 for minimal coupling and that in (27), N is evaluated at a = (d−1)/2and in (28) at a = d0 + (d −1)/2.In the case of the two-sphere, (27) and (28) becomeζ′N(0) = −ln ρ2(d) + ζ′2(0, 1 | d) −14g(29)andζ′D(0) = ζ′2(0, d0 | d) + ζ′2(0, d0 + 1 | d) −14g(30)where we have set g = d1d2, the order of the rotational part of Γ.For the three-sphereζ′N (0) = −ln ρ3(d) + ζ′3(0, 2 | d) −ln 2 + d02g(31)andζ′D(0) = ζ′3(0, d0 | d) + ζ′3(0, d0 + 2 | d) −d02g(32)where g = d1d2d3.

We note the change of sign in the last term.Equations (29) to (32) are the calculational formulae we shall use in the rest ofthis paper. It is also possible to evaluate the derivative of the ζ–function at negativeintegers, ζ′(−n).

This would be relevant if we were interested in the effective actionon product spaces like R × Rk × Sd/Γ. A few details are presented in the appendix.Although our main interest is in minimal coupling, it should be mentionedthat the result (25) can be used immediately for massive fields, assuming that theappropriate value of α is real.

This means that the mass κ is restricted to the region0 ≤κ ≤(d −1)/2. For larger masses a slightly different continuation is needed.4.

The derivative of the Barnes zeta function.We turn now to the evaluation of the derivatives needed in (29) and (30). A pre-liminary step is to remove any common factors of the degrees d1 and d2 by settingdi = cei with e1 and e2 coprime so that the denominator function in (4) equalsc(b + m.e) where b = a/c.The summation in (4) is rewitten by introducing the residue classes with respectto e. On settingm1 = n1e2 + p2,m2 = n2e1 + p1,7

where 0 ≤pi ≤ei −1, the denominator function in (4) equals cb + e1e2(n1 + n2) +p2e1 + p1e2and the sum over m becomesζ2(s, a | d) = c−s Xp∞Xn=01 + n(b + e1e2n + p2e1 + p1e2)s= c2−sgXp,n1(b + N)s−1 + cg! !sXp(1 −wb)∞Xn=01(n + wb)s ,(33)whereN = e1e2n + p2e1 + p1e2,andwb =be1e2+ p1e1+ p2e2.Consider the integer N = e1e2n + p2e1 + p1e2.As n ranges over 0 to ∞,and the pi over their domains, N will likewise run over this infinite range withthe exception of some integers < e1e2 at the beginning, specifically those integersthat equal p2e1 + p1e2 mod e1e2 for p2e1 + p1e2 > e1e2.

We denote these missingintegers by νi. Apart from these terms, the first sum in the second line of (33) willimmediately give a single Hurwitz ζ–function,ζ2(s, a | d) = c2−sg ζR(s −1, b) −Xi1(b + νi)s−1!+ cgs Xp(1 −wb)ζR(s, wb).

(34)(34) is a convenient form for numerical evaluation.It provides an explicitanalytical continuation of this integral Barnes ζ–function.For the derivative at s = 0 we find, after inserting the known values of theHurwitz ζ–function and its derivative,ζ′2(0, a | d) = c2g ζ′R(−1, b) +Xi(νi + b) ln(νi + b)!−c2g ζR(−1, b) −Xi(b + νi)!ln c+Xp(1 −wb)lnΓ(wb)/p(2π)−(1/2 −wb) ln(g/c). (35)Letting a tend to zero in (35) and comparing with the definition of ln ρ, (26),one finds thatln ρ2(d) = −c2g ζ′R(−1) +Xiνi ln νi!−c2g 112 +Xiνi!ln c−X′p(1 −w0)lnΓ(w0)/p(2π)−(12 −w0) ln(g/c)−12 ln(g/2πc)(36)8

where w0 = p1/e1 + p2/e2 and the dash means that the term p1 = p2 = 0 is to beomitted from the sum.Barnes gives a formula in terms of the multiple Γ-function,ζ′r(0, a | d) = lnΓr(a)ρr(d). (37)Formal expressions for the functional determinants are thuse−ζ′N (0) = e1/4g2 ρ22(d)Γ2(1)(38)ande−ζ′D(0) = e1/4g2ρ22(d)Γ2(d0)Γ2(d0 + 1).

(39)Our results, (34) and (35), (36), can be thought of as computationalformulae for these functions in terms of simpler ones.It is not necessary to rearrange the summation as in (33). We have done so inorder to extract the term ζR(s −1, b).

If the summation is left as in the first line of(33), it can immediately be turned into a sum of Hurwitz ζ–functions,ζ2(s, a | d) = cgs XpζR(s −1, wb) + (1 −wb)ζR(s, wb). (40)Then we have the alternative formζ′2(0, a | d) = ln(c/g)XpζR(−1, wb) + (1 −wb)ζR(0, wb)+Xpζ′R(−1, wb) + (1 −wb)ζ′R(0, wb)=112g6a2 −6a(d0 + 1) + (d0 + 1)2 + gln(c/g)+Xpζ′R(−1, wb) + (1 −wb) lnΓ(wb)/p(2π).

(41)In this way we do not need to find the missing integers (nor even the common factorc) but the price is the multiple evaluation of ζ′R(−1, wb) by a numerical procedure.There is no difficulty in this but (34) is faster and more accurate. Equation(41) constitutes a useful check.5.

The point groups.A limited test of our formulae is provided by the dihedral case, Γ = [q] in Coxeter’snotation [7,8]. (Sch¨onflies would write Cqv and it is Cq[Dq in Polya and Meyer9

[9,10]. Table 2 in [11] has a complete list of equivalents).

The degrees are d1 = q,d2 = 1, so c = 1, g = q and d0 = q.There are no missing integers νi and,furthermore, p2 = 0. The fundamental domain is the lune, or digon, (qq1).For q = 1 there is a single, equatorial reflection plane, the fundamental domainbeing a hemisphere, (111) (a spherical triangle with every angle equal to π).

Analternative notation for this domain is A1, [7]. In this extreme case, p1 is also zeroand the expressions rapidly collapse toln ρ2(1, 1) = −ζ′R(−1) −lnp(2π),ζ′2(0, 1 | 1, 1) = ζ′R(−1)andζ′(0, 2 | 1, 1) = ζ′R(−1) + lnp(2π).Thus, on the hemisphere, from (29) and (30),ζ′N(0) = 2ζ′R(−1) −lnp(2π) −14,ζ′D(0) = 2ζ′R(−1) + lnp(2π) −14,(42)which agree with the results exhibited by Weisberger [12].

Our value of ζN(0) = 1/6does not agree with [12].The sum of the Neumann and Dirichlet expressions should reduce to the full-sphere result derived by e.g. Horta¸csu, Rothe and Schroer [13] and later by Weis-berger [14].

We findζ′S2(0) = 4ζ′R(−1) −12 ≈−1.161684575agreeing with these earlier calculations.There are many discussions on spheresbounded equatorially by spheres.We give the explicit formulae for the next value of q, q = 2, corresponding toa quartersphere,ζ′N(0) = ζ′R(−1) −lnp(2π) −18,ζ′D(0) = ζ′R(−1) + lnp(2π) −18. (43)Adding these expressions gives half the full-sphere value.The results for higher values of q are shown in Fig.1, where we plot the effectiveaction W = −ζ′(0)/2.

It is shown in the appendix thatζ′N(0) −ζ′D(0) = −ln(2π)(44)for all [q], as born out by the numbers.For completeness we record the values of ζ(0) obtained from (16), relevant forthe conformal anomaly,ζN(0) = ζD(0) =112q (1 + q2). (45)10

We turn now to the extended dihedral group, [q, 2], of order 4q, obtained from[q] by adding a perpendicular reflection. It is the complete symmetry group of thedihedron.

(In [9,10] this group is Dqi (q even) and Dq[D2q (q odd). The Sch¨onfliesequivalent is Dqd.

)If q is odd, c = 1, d1 = q, d2 = 2, d0 = q + 1 and g = 2q, while, if q is even,c = 2, e1 = q/2 and e2 = 1. For odd q, the missing integers are 1, 3, .

. ., q −2.There are no missing integers if q is even.The fundamental domain is the spherical triangle (22q).When q = 1 thisdomain is the quartersphere lune and the results coincide with (43).

The groupisomorphisms are [1, 2] ∼= [2] ∼= [1]×[1] (or C2[D2 ∼= D1[D2 since C2 ∼= D1).Generally one has [q, 2] ∼= [q] × [1], in particular, [2, 2] ∼= [1]×[1]×[1] whichcorresponds to three perpendicular reflections with the eighthsphere, (222), as fun-damental domain.The hemisphere, quartersphere and eighthsphere are the intersections of S2with (R+×R2), (R+×R+×R) and (R+×R+×R+), respectively. The positive realaxis, R+, is the positive root space of the SU(2) algebra, A1 (cf [6]).

Fig.2 displaysvalues of W for bigger orders.The rotation part of [q, 2] is the complete symmetry group of the regular q-gon,{q}, and is the dihedral group in its guise as a group of rotations. Coxeter denotesit by [q, 2]+ and Polya and Meyer by Dq.

As stated, its structure is [q, 2]+ ∼= Dq.When q is odd there is the curious isomorphism [2q] ∼= [2, q].It is only a matter of substitution to work out the the values of (29) and (30)for the other reflection groups which are the complete symmetry groups of thespherical tessellations {3, 3}, {3, 4} and {3, 5}.We find −ζ′(0)/2 for (Dirichlet,Neumann)–conditions to be (0.45603, −0.34216) for Td = [3, 3], (0.2508, 0.001915)for Oh = [3, 4] and (−0.10538, 0.45014) for Ih = [3, 5].The fundamental domain of [p, q] is the spherical triangle (pqr). The rotationalpart of [p, q], i.e.

[p, q]+, is often denoted by (p, q, r).7. The Ces`aro-Fedorov formula.It is interesting to check the formula (16) by remembering that ζ(0) is a local objectrelated to the constant term in the short-time expansion of the heat-kernel.

Thegeneral formula for a two-dimensional domain, M, with boundary ∂M = S ∂Miisζ(0) =124πZMR dA + 112XiZ∂Miκ(l) dl +124πXαπ2 −α2α(46)where the α sum runs over all inward facing angles at the corners of ∂M.In the present case R = 2 and the extrinsic curvature, κ, vanishes since theboundaries of the fundamental domains are geodesic. Thereforeζ(0) = 1242g + p + q + r −1(47)11

where we have used the standard formula for the area of a spherical triangle. Thisagrees with (16) if the formula2d0(d0 −1) = g(p + q + r −3)(48)is taken into account.

In fact our derivation can be thought of as an analytical proofof this relation which is a slight generalisation of equation 4·51 in [7]. (Coxeter hasr = 2 and g = 2N1.

)Coxeter indicates a purely geometric proof and points out that (48) is equivalentto a formula discovered numerologically by Ces`aro [15] and is a special case of anearlier result of Fedorov [see 16]. (48) is virtually identical to the equation on p177of [15] with the correspondances X = d0, n = r, p = p and q = q.

An extensionto higher dimensions is possible using the generalisation of (46) that includes theresults of Fedosov on polyhedral domains, [17].7. Scaling and limits.The results given so far are for a unit sphere.

For radius R, simple scaling gives therelationζ′(0; R) = ζ′(0) + 2 ln R ζ(0)(49)where ζ′(0) = ζ′(0; 1) and ζ(0) = ζ(0; 1) = ζ(0; R).The effective action should incorporate an arbitrary scaling length, L, byWL = −12ζ′(0; R) + ln L ζ(0) = −12ζ′(0) + ln(L/R) ζ(0).The figures show just the first term.Consider the dihedral case [q] and let q and R tend to infinity in such a way thatthe equatorial width of the fundamental domain (qq1) remains fixed at β ≡πR/q.From (49) and (45), whence ζ(0) →q/12, we haveζ′(0; R) →limq→∞ζ′(0) + q6 lnβqπ. (50)The area of (qq1) is Aq = 2β2q/π and requiring the density, ζ′(0; R)/Aq, toremain finite as q →∞entails the leading behaviourζ′(0) →−q6 ln q + O(q).

(51)Numerically we findζ′(0) →−q6 ln q + 0.497509q ≈−q6 ln(q/19.79)(52)12

so that the density becomesζ′(0; R)Aq→π12β2 ln(6.299β). (53)Geometrically, it might be imagined that in the limit R = ∞, since the spherebecomes flat, the rescaled lune, (∞∞1), would be an infinite strip of width β.Defining the strip coordinates x = Rφ and y = R(π/2 −θ), the spherical Laplaciandoes become the usual Cartesian one as R →∞.

However, the influence of theinfinitely sharp corners at the poles persists, even though they are infinitely distant,producing an anomaly density of π/12β2. On the rectangular strip, infinite or not,the integrated anomaly equals 1/2 and so the density vanishes in the infinite case.8.

The three-sphere.The expressions for the three-sphere are (31) and (32). Then we require,ζ3(s, a | d) =Xm1(a + m1d1 + m2d2 + m3d3)s .We will not attempt to extract a single ζ–function as we did previously but will justreduce the sum to a finite one over Hurwitz ζ–functions in a not very symmetricalnor economic fashion.The residue classesm2 = d1n2 + p1,m1 = d2n1 + p2are introduced so that the denominator function reads (a + d1d2(n1 + n2) + p2d1 +p1d2 +m3d3).

The sums over n1 and n2 can be transformed by defining n = n1 +n2and doing the sum over n1 −n2 to yield the intermediate formζ3(s, a | d) =Xp1,p2∞Xn,m3=01 + n(a + d1d2n + p2d1 + p1d2 + m3d3)s .The further residue classesm3 = d1d2n3 + p3,n = d3n4 + p4are introduced and the sum and difference defined byn+ = n4 + n3,n−= n4 −n3.The denominator is independent of n−while the numerator equals 1 + d3(n+ +n−)/2 + p4. Since the range of n−is symmetrical about zero (from −n+ to n+ in13

steps of 2) the n−term gives nothing and there is a factor of (1 + n+) multiplyingthe rest. The sum may therefore be writtenζ3(s, a | d) =Xp,n(1 + n)(1 + d3n/2 + p4)(f + gn)swhere p = (p1, p2, p3, p4), f = a + d1d2p4 + p2d1 + p1d2 + p3d3, g = d1d2d3 and wehave set n = n+ for notational simplicity.The numerator is reorganised to(1 + n)(1 + d3n/2 + p4) = d32g2F + G(f + gn) + (f + gn)2whereF = (A −g + d1d2p4)(A −d1d2p4 −2d1d2),G = g + 2d1d2 −2Awith A being the combination A = a + d1p2 + d2p1 + d3p3.Thus, finally, we arrive at a finite sum of Hurwitz ζ–functions,ζ3(s, a | d) = d32g2Xp Fgs ζRs, fg+Ggs−1 ζRs −1, fg+1gs−2 ζRs −2, fg(54)which constitutes a possible, but inefficient, continuation of the Barnes ζ–function.9.

The honeycomb groups.The three-dimensional analogues of the polyhedral tessellations, {p, q}, of the two-sphere are the spherical honeycombs {p, q, r}, [7,8,18]. The reflection groups [p, q, r]are their complete symmetry groups, the fundamental domains being subspaces ofthe honeycomb cells.A numerical calculation using (54) and (32) produces thefollowing typical results for the Dirichlet effective actions.

For [3, 3, 3], W ≈44.4and for [3, 3, 4], W ≈−427.25.10. Conclusion.The results of this paper are strictly technical.

We have achieved our aim of pre-senting calculable formulae for the functional determinants of minimally coupledscalar fields on the fundamental domains of finite reflection groups. The problemhas devolved upon an evaluation of the derivative of the Barnes ζ–function.We could also extend our previous results on the vacuum energies [1] to minimalcoupling using the expressions for ζ(−n), (12), and ζ′(−n).

This straightforwardexercise will not be done here.The Ces`aro-Fedorov formula for the number of symmetry planes of a regularsolid proved in section 6 is one of a number of similar relations in higher dimensionsderivable from expressions for the coefficients in the short-time expansion of theheat-kernel. The details will be presented elsewhere.The conformal transformations taking a fundamental domain into the upperhalf–plane are known and so the results here described should also be obtainableusing standard conformal techniques.

This will be recounted at another time.14

Appendix.In this appendix we first work out an expression for the derivative of the ζ–function(8) at negative integers, ζ′(−n). For brevity we do not display the dependence ofthe Barnes ζ–function on the d.Differentiation of (10) first of all leads toζ′(−n) =nXr=0(−α2)r nr "2ζ′d(2r −2n, a) −ζd(2r −2n, a)nXk=n−r+11k#+(−1)nn+uXr=n+1α2r n!

(r −n −1)!r! R2r−2n + 12N2r−2nr−n−1Xk=n+11k!+(−1)n∞Xr=u+1+nα2r n!

(r −n −1)!r!ζd(2r −2n, a). (55)We substitute the integral form of the Barnes ζ–function into the last term andfind it as the s →−2n limit of 2n(−1)nn!

times2Z ∞0exp(−aτ) cosh ατ −n+uXr=0(ατ)2r(2r)! !τ s−1dτQi1 −exp(−diτ)(56)which equalsΓ(s)ζd(s, a −α) + ζd(s, a + α)−2u+nXr=0α2r(2r)!Γ(s + 2r)ζd(s + 2r, a).

(57)The pole cancellation gives the conditionζd(−2n, a −α) + ζd(−2n, a + α) −2ζd(−2n, a) == 2nXr=1α2r2n2rζd(2r −2n) + 2n+uXr=n+1α2r (2r −2n −1)!(2r)!N2r−2n. (58)Extracting the finite remainder yields, after using (58),1(2n)!ζ′d(−2n, a −α) + ζ′d(−2n, a + α) −2ζ′d(−2n, a)+2(2n)!nXr=1α2r2n2rψ(1 + 2n −2r) −ψ(1 + 2n)ζd(2r −2n, a) −ζ′d(2r −2n, a)15

−2n+uXr=n+1α2r (2r −2n −1)! (2r)!ψ(2r −2n) + ψ(1 + 2n)N2r−2n + R2r−2n.

(59)Multiplied by 2n(−1)nn!, (59) must be substituted into (55) to yield a calcu-lable formula for ζ′(−n). Doing so reveals that the remainder terms R2r−2n cancelbut, apart from this, there are no other simplifications apparent and we leave theanalysis at this point.We next derive the result (44) starting from (38) and (39) whencee−ζ′N (0)+ζ′D(0) = Γ2(d0)Γ2(d0 + 1)Γ2(1).

(60)It is necessary to use some properties of the multiple Γ–function.From (26) and (37) it is obvious thatlima→0 Γr(a) = 1a. (61)The other properties we need follow from the important recursion formula satisfiedby the Barnes ζ–function,ζr(s, a + di | d) −ζr(s, a | d) = −ζr−1(s, a | d′),(62)where d′ stands for the set of degrees d with the di element omitted.If this equation is differentiated, it quickly results that, [2],Γr(a)Γr(a + di) = Γr−1(a)ρr−1(d′).

(63)Setting a equal to zero in (63) and using (61) it follows thatΓr(di) = ρr−1(d′). (64)For the group [q], we recall that the degrees are d = (q, 1).

Then, choosingdi = d1 = q and setting a = 1, we have from (63)Γ2(1)Γ2(1 + q) = Γ1(1)ρ1(1)which is clearly independent of q since the quantities on the right-hand side arecalculated for the degree d′ = (1). Further, from (64), it is likewise clear that Γ2(q)is independent of q.

Therefore the quantity in (60),Γ2(q)Γ2(q + 1)Γ2(1)= ρ21(1)Γ1(1),16

is independent of q. The actual value is 2π, agreeing with the particular cases (42)and (43).Incidentally, from the general formulae (2), (3) and (62) it easily follows thatthe [q] conformal ζ–functions are related byζ(C)N(s) −ζ(C)D(s) = ζ1(2s, 1/2 | 1) = ζR(2s, 1/2)(65)so that, in particular,ζ(C)N′(0) −ζ(C)D′(0) = −ln(2)for all [q].17

References[1]Peter Chang and J.S.Dowker Nucl. Phys.

B395 (1993) 407. [2]E.W.Barnes Trans.Camb.Phil.Soc.

(1903) 376. [3]B.Allen, PhD Thesis, University of Cambridge, 1984.

[4]A.Chodos and E.Myers Can.J.Phys. 64 (1986) 633.

[5]A.Erdelyi,W.Magnus,F.Oberhettinger and F.G.Tricomi Higher Transcendentalfunctions McGraw-Hill, New York (1953). [6]P.B´erard & G.Besson Ann.Inst.Fourier 30, (1980) 237.

[7]H.S.M.Coxeter Regular Polytopes Methuen, London (1948). [8]H.S.M.Coxeter Regular Complex Polytopes 2nd.

Edn. Cambridge UniversityPress, Cambridge (1991).

[9]G.Polya and B.Meyer Comptes Rend. Acad.Sci.

(Paris) 228 (1949) 28. [10]B.Meyer Can.J.Math.

6 (1954) 135. [11]H.S.M.Coxeter and W.O.J.Moser Generators and relations for finite groupsSpringer-Verlag, Berlin (1957).

[12]W.I.Weisberger Comm. Math.

Phys. 112 (1987) 633.

[13]M.Hortacsu, K.D.Rothe and B.Schroer Phys. Rev.

D20 (1979) 3203. [14]W.I.Weisberger Nucl.

Phys. B284 (1987) 171.

[15]G.Cesaro Mineralogical Mag. 17 (1915) 173.

[16]E.S.Fedorov Mineralogical Mag. 18 (1919) 99.

[17]B.V.Fedosov Sov.Mat.Dokl. 4 (1963) 1092; ibid 5 (1964) 988.

[18]H.S.M.Coxeter and G.J.Whitrow Proc. Roy.

Soc. A200 (1950) 417.18

---

출처: arXiv:9306.154 • 원문 보기