Numerical evaluation of spherical GJMS determinants for even dimensions
aa r X i v : . [ h e p - t h ] O c t Numerical evaluation ofspherical GJMS determinantsfor even dimensions
J.S.Dowker Theory Group,School of Physics and Astronomy,The University of Manchester,Manchester, England
The functional determinants of the GJMS scalar operators, P k , oneven–dimensional spheres are computed via Barnes multiple gammafunctions relying on the numerical availability of the digamma func-tion. For the critical k = d/ ∼ − log det ) as k is varied in thereals. For odd dimensions these extrema occur at integer k .The multiplicative anomalies are given as odd polynomials in k andit is emphasised that that the Dirichlet–to–Robin factorisation, P l +1 , l ∈ Z , gives the same results as P k if k = l + 1 / ζ –function is rederived by an improved method. [email protected]; [email protected] . Introduction In a recent note I gave a simple quadrature for the (log of) the functionaldeterminant of the scalar GJMS operator on odd spheres. In this communication, Idiscuss the case of even spheres using a different method, which applies also to theodd case, so allowing a check.Critical operators present a slightly complicated analysis requiring the separatecomputation of Barnes’ ‘Stirling moduli’ and so I begin with the subcritical situationwhich corresponds to the restriction k < d/ k is the level of the GJMSoperator, P k . On the d –sphere this operator takes the product form, [1], P k ( d ) = k − Y j =0 (cid:0) B − α j (cid:1) , α j = j + 1 / , (1)where B ≡ p P + 1 / P = − ∆ + (cid:0) ( d − − (cid:1) / Y d ) on the sphere.There is an alternative, analytic form, P k ( d ) = Γ (cid:0) B + 1 / k (cid:1) Γ (cid:0) B + 1 / − k (cid:1) , (2)continuing k off the integers.The critical value of k is the one for which a zero mode first appears as k is increased, at whch point the operator becomes the analogue of the minimallycoupled scalar Laplacian.
2. The subcritical cases, k < d/ P k = log Γ d +1 ( d/ k ) Γ d +1 ( d/ k + 1)Γ d +1 ( d/ − k ) Γ d +1 ( d/ − k + 1) − M (cid:0) d, ( d − / , k ) (cid:1) . (3)The first part of this formula follows, after a little algebra, from the unques-tioning use of the product nature of P k . The second part corrects this expectation,and constitutes a multiplicative anomaly . There seems to be no way of determiningwhat this is except by direct calculation, as in [2] where an evaluation is detailed.I take the opportunity of correcting the expression displayed there.1 . The multiplicative anomaly The anomaly, M , can be thought of as composed of two parts, one comingfrom the factorisation of each bracket in (3) and the other from the product ofthese brackets. I find, M ( d, a, k ) = M ( d, a, k ) + M ( d, a, k ) , where, M ( d, a, k ) = − u X r =1 r (cid:18) k − X j =0 α rj (cid:19) H k ( r ) N r ( d, a ) , and M ( d, a, k ) = 12 k u X r =1 r u − r X t =1 t k − X i 2, both desired since reversing k inverts the operator and k = 1 / M ( d, k ) ≡ M ( d, k ) /k (1 − k ) M (4 , k ) = 14320 (28 k − M (6 , k ) = 1907200 (174 k − k + 955) M (8 , k ) = 11524096000 (4656 k − k + 304600 k − M (10 , k ) = 9236 k − k + 3236824 k − k + 9322110301771008000 . (5)For odd dimensions, the Neumann and Dirichlet anomalies are equal and op-posite so adding for the full sphere gives zero. I list a few cases of the M ( d, k ), M (3 , k ) = ± M (5 , k ) = ± k − M (7 , k ) = ± k − k + 28745) M (9 , k ) = ± k − k + 749695 k − . (6)I mention the curious fact that subtracting these Neumann and Dirichletanomalies produces the Dirac anomalies on spheres of one dimension lower, up tothe spin degeneracy factor. 4. The Dirichlet–to–Robin factorisation When k is a half–integer ( k = l + 1 / 2) the intertwinor form (2) gives thealternative factorisation P l +1 ( d ) = B l Y j =1 (cid:0) B − j (cid:1) = B l Y j =1 (cid:0) B + j (cid:1)(cid:0) B − j (cid:1) , (7)which is a well known Dirichlet–to–Robin boundary pseudo–operator, the simplestexample being P = B .It can be shown by calculation that log det P l +1 produces the same functionof k as log det P k if k = l + 1 / 2. This equality reinforces my continuation of k intothe reals. It will be considered further in a communication in preparation.3 . Computational method The method consists of integrating the multiple digamma function, ψ d , definedby, ψ d ( z ) = ∂∂z log Γ d ( z ) , because ψ d is expressible in terms of the ordinary digamma function which is avail-able numerically. This results in the combination appearing in (3),log Γ d ( z )Γ d ( z ) = Z z z dz ψ d ( z ) , (8)with ψ d ( z ) = ( − d − ( d − (cid:18) B ( d ) d − ( z ) ψ ( z ) + Q d ( z ) (cid:19) , (9)where the polynomial Q is given by, Q d ( z ) = − ( − d − d − X n =1 ( − n n B ( d − n ) d − n − ( d − z ) B ( n ) n ( z ) . The expression for ψ d , (9), follows, [2], on iteration of recursion relations given byBarnes, [6]. (See also Onodera, [7].) Again I present just one example, ψ ( z ) = − z ( z − z − ψ ( z ) + 22 z − z + 167 z − . It is then a simple matter to accurately evaluate the expression (3) since thereare no poles of the ψ –function in the integration intervals and the multiplicativeanomaly is explicit. I exhibit the results in the form of graphs which are more ex-pressive than just numbers. The plots show minus the logdet, which, for convenienceonly, I refer to as the effective action. I have treated k as a continuous parameter which I can sensibly do in viewof the polynomial nature of the multiplicative anomaly and the appearance of k injust the integration limits. See remarks in section 4. (Equivalently, the multipleΓ–functions in (3) are functions of their arguments.)Figures 1 to 3 plot the effective action for dimensions 2,4,6,8,10 and 12 againstthe scaled variable 2 k/d while Figure 4 shows that for d = 12 against a portion The same numerical method is adopted by Adamchik, [3], for the double Γ–function, G . Seealso Kamela and Burgess, [4], and Basar and Dunne, [5], for example. For ordinary scalar field theory there should be a factor of 1 / 4f the k range. The curves show a series of extrema of which the deepest one byfar is that close to the critical value k = d/ 2, where the effective action divergespositively. 5 . The critical case, k = d/ k = d/ P d = log ( d − d +1 ( d ) Γ d +1 ( d + 1)Γ d +1 (1) − M (cid:0) d, d/ (cid:1) = log Γ d +1 ( d ) Γ d +1 ( d + 1)Γ d +1 (1)Γ d +1 (1) + log Γ d +1 (1)+ log( d − − M (cid:0) d, d/ (cid:1) . (10)6he expression has been arranged so that the first term can be computed asbefore in terms of ψ , but at the expense of introducing the term log Γ d +1 (1) = log ρ d ,where ρ d is a d –ple Stirling modular form, [6].It is thus seen that, in this approach, the values of the ρ d are required and Itreat this as an independent calculation discussed in the next section.Using the values there, I give some numbers for critical effective actions ≡− logdets ≡ F ( d ), and a graph. I find, F (2) = − . , F (4) = − . , F (6) = − . F (8) = 10 . , F (10) = − . , F (12) = − . . The value of F (2) agrees with the standard result for the (minimal) Laplacian onthe two–sphere, i.e. ζ ′ R ( − − / 2, going back to 1979. 7. The Stirling moduli The way I have chosen to compute the moduli, ρ r , is via Barnes’ Binet–typeformula which I copy here, [6] p.411, log ρ r = − Z ∞ dzz (cid:20) − e − z ) r − r − X s =1 ( − s z s − r s ! B ( r ) s − (cid:18) − ( − r B ( r ) r r ! (cid:19) e − z (cid:21) , (11) This could be regarded as a Weierstrass regularised expression. There seems to be an error inthe formula in [6] which I have corrected. B ( r ) s are generalised Bernoulli numbers. This allows an adequate numeri-cal treatment after dividing the integration range into three – a small z polynomialpart, a large z , asymptotic part and an intermediate region where the exact inte-grand has to be used. The Bernoulli numbers can be calculated by recursion andstored for fast recall.I list a few values of the moduli ρ = 2 . , ρ = 3 . , ρ = 3 . ρ = 3 . , ρ = 3 . , ρ = 3 . ρ = 4 . , ρ = 4 . , ρ = 4 . , and plot a graph, Fig.6.This is a somewhat workaday technique. A more elegant, but particular, for-mula results from expressing the relevant Barnes ζ –function as a sum of Hurwitz ζ –functions and is outlined in the following section.8 . More on moduli I present some basic information and manipulation that warrants further ex-posure, although the results, if not the methods, are well known to workers in thefield. I use Barnes’ original notation, except for the Bernoulli polynomials.In the case that all the parameters are unity, Barnes, [8] p.431, showed that his ζ –function could be expressed as a sum of Hurwitz functions. Thus, [8], ζ d ( s, a ) = d X r =1 ( − d − r ( r − d − r )! B ( d ) d − r ( a ) ζ R ( s + 1 − r, a ) . (12)The definition of the multiple Γ is, ζ ′ d (0 , a ) = log Γ d ( a ) ρ d with the normalisation modulus, ρ d , given by,log ρ d = − lim a → (cid:0) ζ ′ d (0 , a ) + log a (cid:1) . Further, differentiating the recursion for Γ d , it follows that ρ d = Γ d +1 (1) so, ζ ′ d (0 , 1) = log ρ d − ρ d , (13)and therefore, d X r =1 ζ ′ r (0 , 1) = − log( ρ d /ρ ) , (14)which together with (12), constitutes a means of computing ρ d since ρ = 1 (fol-lowing from the ‘trivial’ ζ –function, ζ ( s, a ) = a − s ). Hence,log ρ d = d X r =0 A ( d ) r ζ ′ R ( − r ) , (15)in terms of the derivatives of the Riemann ζ –function, which are available to anyaccuracy in some computer languages. In contrast, the expression (11), although This follows essentially by expanding the degeneracy, a binomial coefficient. On p.433, Barnes,with some foresight, remarks ‘It is evident that such algebra is capable of almost indefinitedevelopment’. 9n integral, employs only standard functions and it can be adapted easily to thecase when the parameters are not all unity.Although the result is known, in various forms, I will derive the coefficients in(15), from (12), the coefficients in which, at a = 1, are Stirling numbers, but I donot use this. In fact there is no need to compute the sum (14) by brute force as Iwill now show.Moving to the required quantity, immediately from (12), ζ ′ r (0 , 1) = r X k =1 ( − d − k ( k − r − k )! B ( r ) r − k (1) ζ ′ R (1 − k )= r X k =1 ( − r − k ( k − r − k )! (cid:18) rk B ( r +1) r − k (1) + ( r − r − k ) k B ( r ) r − k − (1) (cid:19) ζ ′ R (1 − k )= r X k =1 ( − k k ! (cid:18) ( − r r ( r − k )! B ( r +1) r − k (1) − ( − r − ( r − r − − k )! B ( r ) r − k − (1) (cid:19) ζ ′ R (1 − k )(16)where the recursion, [9] p.186, B ( n +1) ν ( x ) = n − νn B ( n ) ν ( x ) + ( x − n ) νn B ( n ) ν − ( x ) , has been employed.Note now that the second term in brackets in (16) is the negative of the firstone, after setting r → r − 1. On trivial summation of (14) by cancellation, orotherwise, there results,log ρ d = d d X k =1 ( − d − k +1 k ! 1( d − k )! B ( d +1) d − k (1) ζ ′ R (1 − k ) . (17)I now say, for comparison, that Stirling numbers, s , can be introduced by therelation, s ( d, k ) = (cid:18) dk (cid:19) B ( d +1) d − k (1) , and (17) reads, log ρ d = 1( d − d X k =1 ( − d − k +1 s ( d, k ) ζ ′ R (1 − k ) . (18) At x = 1 this is the recursion for Stirling numbers. d (1) = 1( d − d X k =1 ( − d − k s ( d − , k − ζ ′ R (1 − k ) , which is the first line of (16) (really due to Barnes), is given by Quine and Choi,[12], for example. 9. Odd dimensions For odd dimensions, the same expression, (3), holds with zero multiplicativeanomaly and the numerical evaluation proceeds as in section 3. As an example,Figure 7 plots the interpolation provided by (3), for d = 9, of the values computedin [13] for integer k by a different process. It shows extrema at these integers. Thiscan be proved analytically. 10. Conclusion and remarks The functional determinants of the GJMS scalar operators, P k , on even–dimensional spheres have been computed via Barnes multiple gamma functions interms of the digamma function. For the critical k = d/ ∼ − log det P k ) as k is varied in the reals. Forodd dimensions these extrema occur at integer k . This is modified by the existenceof a multiplicative anomaly for even dimensions.I wish to draw attention to the fact that the continuations, in k , of log det P k ,where P k is given by (2), provided by the two (distinct) factorisations, (1) and (7),are identical.The values confirm an expectation that the determinant tends to unity as thedimension increases. For odd dimensions, the explicit integral given in [13] provesimmediately that this is so. Møller, [14], has shown, by a more complicated analysis,that it holds for odd and even dimensions for the simple Laplacian and the Diracoperator. These cases were discussed by B¨ar and Schopka, [15], who made thenumerical observation which can also be seen in earlier calculations, [16], [12].It is possible very easily to extend the calculations of the scalar GJMS operatorto a Dirac version where the B operator in (1) is replaced by the square root of theiterated (‘squared’) Dirac operator, and this will be detailed at another time. Prod-uct, higher derivative, higher spin propagation operators, and their determinantsand anomalies, have been recently investigated by Tseytlin, [17]. Useful relevantanalysis is also given by Aros and Diaz, [18]. References. 1. Branson,T.P. Trans.Am.Math.Soc. (1995) 3671.2. Dowker,J.S. J. Phys. A44 (2011) 115402.3. V.S.Adamchik, Contributions to the theory of the Barnes function , ArXiv:math. CA/0308086.4. Kamela,M. and Burgess,C.P. Can. J. Phys. (1999) 85.5. Basar,G and Dunne,G.V. J. Phys. A43 (2010) 072002.6. Barnes,E.W. Trans. Camb. Phil. Soc. (1903) 374.7. Onodera,K. Adv. in Math. (2010) 895.8. Barnes,E.W. Trans. Camb. Phil. Soc. (1903) 426.9. N¨orlund,N.E. Acta Mathematica (1922) 121.10. Kanemitsu,S., Kumagai,H. and Yoshimoto,M. The Ramanujan J. (2001)5.11. Vardi,I. SIAM J.Math.Anal. (1988) 493.12. J.R.Quine and J.Choi, Rocky Mountain J. Math. (1996) 719-729.13. Dowker,J.S. Numerical evaluation of spherical GJMS operators ArXiv:12309.2873.14. Møller,N.M. Math. Ann. (2009) 35.15. B¨ar,C. and Schopka,S. The Dirac determinant of spherical space forms, Geom.Anal. and Nonlinear PDEs (Springer, Berlin, 2003).16. Dowker,J.S. J. Math. Phys. (1994) 6076.17. Tseytlin,A.A. On Partition function and Weyl anomaly of conformal higherspin fields ArXiv:1309.0785.18. Aros,R. and Diaz,D.E.