Volume 2010, Article ID 941560,44pages doi:10.1155/2010/941560
Research Article
Renormalization, Isogenies, and Rational Symmetries of Differential Equations
A. Bostan,
1S. Boukraa,
2S. Hassani,
3J.-M. Maillard,
4J.-A. Weil,
5N. Zenine,
3and N. Abarenkova
61INRIA Paris-Rocquencourt, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France
2LPTHIRM and D´epartement d’A´eronautique, Universit´e de Blida, 09470 Blida, Algeria
3Centre de Recherche Nucl´eaire d’Alger, 2 Boulevard. Frantz Fanon, BP 399, 16000 Alger, Algeria
4LPTMC, UMR 7600 CNRS, Universit´e de Paris, Tour 24, 4`eme ´etage, case 121, 4 Place Jussieu, 75252 Paris Cedex 05, France
5XLIM, Universit´e de Limoges, 123 avenue Albert Thomas, 87060 Limoges Cedex, France
6St Petersburg Department of Steklov Institute of Mathematics, 27 Fontanka, 191023 St. Petersburg, Russia
Correspondence should be addressed to J.-M. Maillard,maillard@lptl.jussieu.fr Received 21 December 2009; Accepted 17 January 2010
Academic Editor: Richard Kerner
Copyrightq2010 A. Bostan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We give an example of infinite-order rational transformation that leaves a linear differential equation covariant. This example can be seen as a nontrivial but still simple illustration of an exact representation of the renormalization group.
1. Introduction
There is no need to underline the success of the renormalization group revisited by Wilson 1, 2 which is nowadays seen as a fundamental symmetry in lattice statistical mechanics or field theory. It contributed to promote 2d conformal field theories and/or scaling limits of second-order phase transition in lattice statistical mechanics.1 If one does not take into account most of the subtleties of the renormalization group, the simplest sketch of the renormalization group corresponds to Migdal-Kadanoffdecimation calculations, where the new coupling constants created at each step of thereal-spacedecimation calculations are forced2to stay in someslightly arbitraryfinite-dimensional parameter space. This drastic projection may be justified by the hope that the basin of attraction of the fixed points of the correspondingrenormalizationtransformation in the parameter space is “large enough.”
One heuristic example is always given because it is one of the very few examples of exact renormalization, the renormalization of the one-dimensional Ising model without
a magnetic field. It is a straightforward undergraduate exercise to show that performing various decimations summing over every two, three, orN spins, one gets exact generators of the renormalization group readingTN:t → tN, wheretiswith standard notationsthe high temperature variablettanhK. It is easy to see that these transformationsTN, depending on the integerN, commute together. Such an exact symmetry is associated with a covariance of the partition function per siteZt Ct·Zt2. In this particular case one recovers the very simpleexpression of the partition function per site, 2 coshK, as an infinite product of the action offor instanceT2on the cofactorCt. In this very simple case, this corresponds to the using of the identityvalid for|x|<1:
∞ n0
1x2n 1
1−x. 1.1
ForT3:t → t3one must use the identity ∞
n0
1x3nx2·3n ∞
n0
1−x3n1 1−x3n
1
1−x, 1.2
and forTN:t → tNa similar identity where the 3 in the exponents is changed intoN.
Another simple heuristic example is the one-dimensional Ising model with a magnetic field. Straightforward calculations enable to get an infinite number of exact generators of the corresponding renormalization group, represented as rational transformations3
TN:x, z−→TNx, z xN, zN, 1.3 where the first two transformations T2 and T3 read in terms of the twolow-temperature well-suited and fugacity-likevariablesxe4Kandze2H:
x2 xz1xz
x·1z2 , z2z·1xz xz , x3 x·
z2x2z1
z22zx
z2xzxzx2 , z3z·z2x2z1 z22zx .
1.4
One simply verifies that these rational transformations of twocomplexvariables commute.
This can be checked by formal calculations forTNandTMfor anyNandMless than 30, and one can easily verify a fundamental property expected for renormalization group generators:
TN·TMTM·TNTNM, 1.5 where the “dot” denotes the composition of two transformations. The infinite number of these rational transformations of twocomplexvariables1.3are thus a rational representation of the positive integers together with their product. Such rational transformations can be studied
“per se” as discrete dynamical systems, the iteration of any of these various exact generators corresponding to an orbit of the renormalization group.
Of course these two examples of exact representation of the renormalization group are extremely degenerate since they correspond to one-dimensional models.4Migdal-Kadanoff decimation will quite systematically yield rational5 transformations similar to 1.3 in two, or more, variables.6 Consequently, they are never except “academical” self-similar models exact representations of the renormalization group. The purpose of this paper is to provide simplebut nontrivialexamples of exact renormalization transformations that are not degenerate like the previous transformations on one-dimensional models.7In several papers3,4for Yang-Baxter integrable models with a canonical genus one parametrization 5, 6 elliptic functions of modulus k, we underlined that the exact generators of the renormalization group must necessarily identify with the various isogenies which amount to multiplying or dividingτ, the ratio of the two periods of the elliptic curves, by an integer.
The simplest example is the Landen transformation4which corresponds to multiplyingor dividing because of the modular group symmetryτ ↔1/τ, the ratio of the two periods is
k−→kL 2√ k
1k, τ←→2τ. 1.6
The other transformations8 correspond to τ ↔ N · τ, for various integers N. In the transcendental variableτ, it is clear that they satisfy relations like 1.5. However, in the natural variables of the model eK,tanhK, k sinh2K, not transcendental variables like τ, these transformations are algebraic transformations corresponding in fact to the fundamental modular curves. For instance, 1.6 corresponds to the genus zero fundamental modular curve
j2·j2− jj
·
j21487·jjj2
3·153·
16j2−4027jj16j2
−12·306· jj
8·3090,
1.7
or
59v3u3−12·56u2v2·uv 375uv·
16u216v2−4027vu
−64vu·
v21487vuu2
212·33·uv0,
1.8
which relates the two Hauptmodulsu123/jk,v123/jkL:
jk 256·
1−k2k43
k4·1−k22 , jkL 16·
114k2k43
1−k24·k2 . 1.9 One verifies easily that1.7is verified withjjkandjjkL.
The selected values ofk, the modulus of elliptic functions,k 0,1, are actually fixed points of the Landen transformations. The Kramers-Wannier dualityk ↔ 1/kmapsk 0 onto k ∞. For the Isingresp. Baxtermodel these selected values ofkcorrespond to the three selected subcases of the modelT ∞,T 0, and the critical temperatureT Tc, for which
the elliptic parametrization of the model degenerates into a rational parametrization4. We have the same property for all the other algebraic modular curves corresponding toτ ↔N·τ.
This is certainly the main property most physicists expect for an exact representation of a generator of the renormalization group, namely, that it maps a generic point of the parameter space onto the critical manifoldfixed points. Modular transformations are, in fact, the only transformations to be compatible with all the other symmetries of the Isingresp. Baxter model like, for instance, the gauge transformations, some extended sl2×sl2×sl2× sl2symmetry7, and so forth. It has also been underlined in3,4that seeing1.6as a transformation on complex variablesinstead of real variablesprovides two other complex fixed points which actually correspond to complex multiplication for the elliptic curve, and are, actually, fundamental new singularities9 discovered on the χ3 linear ODE 8–10. In general, this underlines the deep relation between the renormalization group and the theory of elliptic curves in a deep sense, namely, isogenies of elliptic curves, Hauptmoduls,10 modular curves and modular forms.
Note that an algebraic transformation like 1.6 or 1.8 cannot be obtained from any local Migdal-Kadanofftransformation which naturally yields rational transformations;
an exact renormalization group transformation like 1.6 can only be deduced from nonlocal decimations. The emergence of modular transformations as representations of exact generators of the renormalization group explains, in a quite subtle way, the difficult problem of how renormalization group transformations can be compatible with reversibility11 iteration forward and backwards. An algebraic modular transformation1.8corresponds to τ → 2τ and τ → τ/2 in the same time, as a consequence of the modular group symmetry τ ↔1/τ.
A simple rational parametrization12of the genus zero modular curve1.8reads:
u1728 z
z163, v1728 z2
z2563 u 212
z
. 1.10
Note that the previously mentioned reversibility is also associated with the fact that the modular curve1.8is invariant byu↔v, and, within the previous rational parametrization 1.10, with the fact that permutinguand vcorresponds13 to the Atkin-Lehner involution z↔212/z.
For many Yang-Baxter integrable models of lattice statistical mechanics the physical quantitiespartition function per site, correlation functions, etc.are solutions of selected14 linear differential equations. For instance, the partition function per site of the squareresp.
triangular, etc.Ising model is an integral of an elliptic integral of the third kind. It would be too complicated to show the precise covariance of these physical quantities with respect to algebraicmodular transformations like1.8. Instead, let us give, here, an illustration of the nontrivial action of the renormalization group on some elliptic function that actually occurs in the 2D Ising model: a weight-one modular form. This modular form actually, and remarkably, emerged11in a second-order linear differential operator factor denotedZ2 occurring8 forχ3, and that the reader can think as a physical quantity solution of a particular linear ODE replacing the too complicated integral of an elliptic integral of the third kind. Let us consider the second-order linear differential operatorDzdenotesd/dz:
αDz2
z256z1024
z·z16z64·Dz− 240
z·z162z64, 1.11
which has themodular formsolution
2F1 1 12, 5
12
,1; 1728 z z163
2·
z256 z16
−1/4
· 2F1
1 12, 5
12
,1; 1728 z2 z2563
.
1.12
Do note that the two pull-backs in the arguments of the same hypergeometric function are actually related by the modular curve relation1.8 see1.10. The covariance 1.12is thus the very expression of a modular form property with respect to a modular transformation τ ↔2τcorresponding to the modular transformation1.8.
The hypergeometric function at the rhs of1.12is solution of the second-order linear differential operator
βD2z z2416z16384
z256z64z·Dz− 60
z64z2562, 1.13 which is the transformed of operatorαby the Atkin-Lehner dualityz↔ 212/z, and, also, a conjugation ofα:
β
z16 z256
−1/4
·α·
z16 z256
1/4
. 1.14
Along this line we can also recall that themodular formfunction15
F j
j−1/12·2F1
1 12, 5
12
,1;123 j
, 1.15
verifies:
F
z163 z
2·z−1/12·F
z2563 z2
. 1.16
A relation like1.12is a straight generalization of the covariance we had in the one- dimensional model Zt Ct ·Zt2, which basically amounts to seeing the partition function per site as some “automorphic function” with respect to the renormalization group, with the simple renormalization group transformationt → t2being replaced by the algebraic modular transformation 1.8 corresponding to τ ↔ 2τ i.e., the Landen transformation 1.6.
We have here all the ingredients for seeing the identification of exact algebraic representations of the renormalization group with the modular curves structures we tried so many times to promote preaching in the desert in various papers 3, 4. However, even if there are no difficulties, just subtleties, these Ising-Baxter examples of exact algebraic
representations of the renormalization group already require some serious knowledge of the modular curves, modular forms, and Hauptmoduls in the theory of elliptic curves, mixed with the subtleties naturally associated with the various branches of such algebraic multivaluedtransformations.
The purpose of this paper is to present another elliptic hypergeometric function and other much simplerGauss hypergeometricsecond-order linear differential operators covariant by infinite-order rational transformations.
The replacement of algebraic (modular) transformations by simple rational transforma- tions will enable us to display a complete explicit description of an exact representation of the renormalization group that any graduate student can completely dominate.
2. Infinite Number of Rational Symmetries on a Gauss Hypergeometric ODE
Keeping in mind modular form expressions like 1.12, let us recall a particular Gauss hypergeometric function introduced by Vidunas in12
2F1
1 2,1
4
, 5 4
;z
1
4·z−1/4· z
0
t−3/41−t−1/2dt
1−z−1/2· 2F1
1 2,1
4
, 5 4
; −4z 1−z2
.
2.1
This hypergeometric function corresponds to the integral of a holomorphic form on a genus- one curvePy, t 0:
dt
y, with : y4−t3·1−t20. 2.2
Note that the function
Fz z1/4·2F1
1 2,1
4
, 5 4
;z
, 2.3
which is exactly an integral of an algebraic function, has an extremely simple covariance property with respect to the infinite-order rational transformationz → −4z/1−z2:
F
−4z 1−z2
−41/4· Fz. 2.4
The occurrence of this specific infinite-order transformation is reminiscent of Kummer’s quadratic relation
2F1a, b,1a−b;z 1−z−a· 2F1
a 2,1a
2 −b
,1a−b;− 4z 1−z2
, 2.5
but it is crucial to note that, relation2.4does not relate two different functions, but is an
“automorphy” relation on the same function.
It is clear from the previous paragraph that we want to see such functions as “ideal”
examples of physical functions covariant by an exact here, rational generator of the renormalization group. The function 2.3 is actually solution of the second-order linear differential operator:
Ω Dz21 4
3−5z
z·1−z·Dzω1·Dz, with
ω1Dz1 4
3−5z
z·1−z Dz1 4 ·dln
z31−z2
dz .
2.6
From the previous expression of ω1 involving a log derivative of a rational function it is obvious that this second-order linear differential operator has two solutions, the constant function and an integral of an algebraic function. Since these two solutions behave very simply under the infinite-order rational transformationz → −4z/1−z2, it is totally and utterly natural to see how the linear differential operatorΩ transforms under the rational change of variablez → Rz −4z/1−z2 which amounts to seeing how the two-order- one operatorsω1andDztransform. It is a straightforward calculation to see that introducing the cofactorCzwhich is the inverse of the derivative ofRz
Cz −1
4·1−z3
1z , 1
Cz dRz
dz , 2.7
Dzandω1, respectively, transform under the rational change of variablez → Rz −4z/1−
z2as
Dz−→Cz·Dz, ω1 −→ω1RCz2·ω1· 1
Cz, yielding : Ω−→Cz2·Ω.
2.8
Sincez → −4z/1−z2is of infinite-order, the second-order linear differential operator2.6 has an infinite number of rational symmetriesisogenies:
z−→ −4z
1−z2 −→16·1−z2·z
1z4 −→ −64·1−z21z4z
1−6zz24 −→ · · ·. 2.9 Once we have found a second-order linear differential operatorwritten in a unitary or monic form Ω, covariant by the infinite-order rational transformation z → −4z/1 − z2, it is natural to seek for higher-order linear differential operators also covariant byz →
−4z/1−z2. One easily verifies that the successive symmetric powers ofΩareof course also covariant. The symmetric square ofΩ,
D3z3 4
3−5z
1−zz ·D2z3 8
1−5z
1−zz2 ·Dz, 2.10
factorizes in simple order-one operators
Dz2 4
3−5z 1−zz
·
Dz1 4
3−5z 1−zz
·Dz, 2.11
and, more generally, the symmetricNth power16ofΩreads
DzN 4
3−5z z1−z
·
DzN−1 4
3−5z z1−z
· · ·
Dz1 4
3−5z z1−z
·Dz. 2.12
The covariance of such expressions is the straight consequence of the fact that the order-one factors
ωkDzk 4
3−5z
z·1−z, k0,1, . . . , N, 2.13 transform very simply underz → −4z/1−z2:
ωk−→ωkR Czk1·ωk·Cz−k. 2.14 More generally, let us consider a rational transformation z → Rz, the corresponding cofactorCz 1/Rz, and the order-one operatorω1 DzAz. We have the identity
Cz·Dz· 1
Cz
Dz−dlnCz
dz . 2.15
The change of variablez → Rzonω1reads
DzAz−→Cz·DzARz Cz·DzBz. 2.16 We want to impose that this rhs expression can be writtensee2.8as
Cz2·DzAz· 1
Cz, 2.17
which, because of2.15, occurs if
Bz Az−dlnCz
dz , 2.18
yielding a “Rota-Baxter-like”13,14functional equation onAzandRz dRz
dz 2
·ARz dRz
dz ·Az d2Rz
dz2 . 2.19
Remark 2.1. Coming back to the initial Gauss hypergeometric differential operator the covariance of Ω becomes a conjugation. Let us start with the Gauss hypergeometric differential operator for2.1:
H08z·1−z·D2z2·5−7z·Dz−1. 2.20
It is transformed byz → Rz −4z/1−z2into
H18z·1−z·Dz2−23z−5·Dz 4
1−z 1−z1/2·H0·1−z−1/2, 2.21
then byz → RRz R2z 16z1−z2/1z4into
H28z·1−z·Dz2−23z−1z5
z1 ·Dz16 z−1 z12
z1
√z−1
·H0·
z1
√z−1 −1
,
2.22
and more generally forz → RNRRR· · ·Rz· · ·
HNCN·H0·C−1N, where : CNz1/4·R−1/4N . 2.23
2.1. A Few Remarks on the “Rota-Baxter-Like” Functional Equation
The functional equation172.19 is the necessary and sufficient condition for Ω Dz Az·Dzto be covariant byz → Rz.
Using the chain rule formula of derivatives of composed functions:
dRRz
dz dRz
dz · dRz
dz Rz
, d2RRz
dz2 d2Rz
dz2 · dRz
dz Rz
dRz dz
2
·
d2Rz dz2 Rz
,
2.24
one can show that, for Az fixed, the “Rota-Baxter-like” functional equation 2.19 is invariant by the composition ofRzby itselfRz → RRz,RRRz, . . . .This result can be generalized to any composition of variousRz’s satisfying2.19. This is in agreement with the fact that2.19is the condition forΩ DzAz·Dzto be covariant byz → Rz it must be invariant by composition ofRz’sforAzfixed.
Note that we have not used here the fact that for globally nilpotent 11 oper- ators, Az and Bz are necessarily log derivatives of Nth roots of rational functions.
ForRz −4z/1−z2: Az 1
4 ·dlnaz
dz , Bz 1
4 ·dlnbz dz , az 1−z2·z3, bz z3· 1z4
1−z10.
2.25
The existence of the underlyingazin2.25consequence of a global nilpotence of the order- one differential operator, can however be seen in the following remark on the zeros of the lhs and rhs terms in the functional equation 2.19. When Rzis a rational function e.g.,
−4z/1−z2or any of its iteratesRnz, the lhs and rhs of2.19are rational expressions.
The zeros are roots of the numerators of these rational expressions. Because of 2.25 the functional equation2.19can be rewrittenafter dividing byRzas
dRz dz
·ARz Az d
dz
ln
dRz dz
1
4· d dz
ln
az·
dRz dz
4 . 2.26 One easily verifies, in our example, that the zeros of the rhs of2.26come from the zeros of ARz and not from the zeros ofRzin the lhs of2.26. The zeros of the log-derivative rhs of2.26correspond toaz·Rz4ρ, whereρis a constant to be found. Let us consider forRzthenth iterates of−4z/1−z2that we denoteRnz. A straightforward calculation shows that the zeros ofARnzoraRnz whereazdenotes the derivative ofaz namely,z−15z−3·z2actually correspond to the general closed formula:
55·az·
dRnz dz
4
−4·33·−4n0. 2.27
More precisely the zeros of 5·Rnz−3 verify2.27, or, in other words, the numerator of 5Rnz−3 divides the numerator of the lhs of2.27.
In another case forTz given by2.45, which also verifies 2.19 see below, the relation2.27is replaced by
55·az·
dTnz dz
4
−4·33·−7−24in 0. 2.28
More generally for a rational functionρx, obtained by an arbitrary composition of−4z/1− z2andTz, we would have
55·az·
dρz dz
4
−4·33·λn0. 2.29
whereλcorresponds to
ρx λ·z· · · , λ dρz dz
z0. 2.30
2.2. Symmetries ofΩ, Solutions to
the “Rota-Baxter-Like” Functional Equation
Let us now analyse all the symmetries of the linear differential operatorΩ DzAz·Dz by analyzing all the solutions of2.19 for a given Az. For simplicity we will restrict to Az 3−5z/z/1−z/4 which corresponds toRz −4z/z−12 and all its iterates 2.9. Let us first seek for othermore generalsolutions that are analytic atz0:
Rz a1·za2·z2a3·z3· · · . 2.31 It is a straightforward calculation to get, order by order from2.19, the successive coefficients an in2.31as polynomial expressionswith rational coefficientsof the first coefficienta1 with
a2−2
5 ·a1·a1−1, a3 1
75·a1·a1−1·7a1−17, a4− 2
4875·a1·a1−1·
41a21−232a1366 , . . . , an−n
5 ·a1·a1−1· Pna1 Pn−4,
2.32
wherePna1is a polynomial with integer coefficients of degreen−2. Since we have here a series depending on one parametera1we will denote itRa1z. This is a quite remarkable series depending on one parameter.18 One can easily verify that this series actually reduces as it should!to the successive iterates2.9of−4z/1−z2fora1 −4n. In other words this one-parameter family of “functions” actually reduces to rational functions for an infinite number of integer valuesa1 −4n.
Furthermore, one can also verify a quite essential property we expect for a representation of the renormalization group, namely, that twoRa1zfor different values of a1commute, the result corresponding to the product of these twoa1:
Ra1Rb1z Rb1Ra1z Ra1·b1z. 2.33 The neutral element must necessarily correspond to a1 1 which is actually the identity transformationR1z z. We have an “absorbing” element corresponding toa10, namely, R0z 0. Performing the inverse ofRa1z with respect to the composition of functions amounts to changinga1 into its inverse 1/a1. Let us explore some “reversibility” property of our exact representation of a renormalization group with the inverse of the rational transformations2.9. The inverse ofR−4z −4z/1−z2must correspond toa1−1/4:
R−1/4z −1 4·z−1
8z2− 5
64z3− 7
128z4− 21
512z5· · ·. 2.34 However, a straight calculation of the inverse ofR−4z −4z/1−z2gives a multivalued function, or if one prefers, two functions
S1−1/4z z−22√ 1−z
z −1
4 ·z− 1
8z2· · ·, S2−1/4z z−2−2√
1−z
z −4
z 21 4z1
8z2· · ·,
2.35
which are the two roots of the simple quadratic relationR−4z z:
z2−2· 1−2
z
·z10, 2.36
where it is clear that the product of these two functions is equal to 1. The radius of convergence ofS1−1/4zis 1.
Because of our choice to seek for functions analytical atz 0 our renormalization group representation “chooses” the unique root that is analytical atz 0, namely,S1−1/4z.
For the next iterate ofR−4z −4z/1−z2in2.9the inverse transformation corresponds to the roots of the polynomial equation of degree fourR16z z:
z4
4−16 z
·z3
632 z
·z2
4− 16 z
·z10, 2.37
which yields four roots, one of which is analytical atz 0 and corresponds toa1 1/−42 in our one-parameter family ofrenormalizationtransformations:
S11/16z 1 16z 3
128z2 53
4096z3 277
32768z4 3181
524288z5· · · , 2.38 itsmultiplicativeinverseS21/16z 1/S11/16z:
S21/16z 16
z −6− 17 16z− 67
128z2−1333
4096z3− 7445
32768z4· · ·, 2.39 and twoformalPuiseux seriesu±√
z:
S31/16z 1u1 2u23
8u31
4u4 27 128u5 5
32u6· · ·. 2.40 Many of these results are better understood when one keeps in mind that there is a special transformationJ:z↔1/zwhich is also a R-solution of 2.19and verifies many compatibility relations with these transformationsIddenotes the identity transformationR0z:
R−4·JR−4, S2−1/4·R−4J, R−4·S1−1/4S1−1/4·R−4Id, S11/16z S1−1/4·S1−1/4, S21/16z S1−1/4·S2−1/4,
J·S1−1/4S2−1/4, J·S2−1/4S1−1/4, . . . ,
2.41
where the dot corresponds, here, to the composition of functions. These symmetries of the linear differential operatorΩcorrespond to isogenies of the elliptic curve2.2.
It is clear that we have another one-parameter family corresponding toJ·Ra1with an expansion of the form
J·Ra1 b1 z −2
5·b1−1− 1
15·b21−1
b1 ·z− 2
975 ·b1−14b114b13 b21 ·z2
− 1
248625·b1−14b11
1268b12951b191
b13 ·z3
− 2
2071875 ·b1−14b11
3688b312766b12404b117
b14 ·z4· · ·.
2.42
Forb1 −1/4,b1 −1/42,b1 −1/43, this family reduces to themultiplicativeinverse of the successive rational functions displayed in2.9
−1
4·1−z2 z −→ 1
16· 1z4
1−z2·z −→ −1 64·
1−6zz24
1−z21z4·z −→ · · ·, 2.43
which can also be written as:
−1 4·
z 1
z
1
2, 1
16·
z 1 z
3
8 z
1−z2,
− 1 64·
z1
z
13 32−z
4 ·17−60z102z2−60z317z4 1−z21z4 , 1
256 ·
z 1 z
51
128 z
16·17−60z102z2−60z317z4
1−z21z4 16z·1−z21z4 z2−6z14 ,
− 1 1024·
z1
z
205 512− z
164 ·17−60z102z2−60z317z4 1−z21z4
−4z·1−z21z4
z2−6z14 −64z·1−z21z4
z2−6z14
120z−26z220z3z44 , . . . , 1
−4n · z 1
z
2 54n
4n−−1n
z
−4n−2 ·17−60z102z2−60z317z4 1−z21z4 z
−4n−6 ·1−z21z4 z2−6z14 z
−4n−8 ·1−z21z4
z2−6z14 120z−26z220z3z44 · · ·,
2.44
where we discover some “additive structure” of these successive rational functions.
In fact, due to the specificity of this elliptic curve occurrence of complex multipli- cation, we have another remarkable rational transformation solution of 2.19, preserving covariantlyΩ. Let us introduce the rational transformationidenotes√
−1:
Tz z·
z−12i 1−12i·z
4
, 2.45
we also have the remarkable covariance12:
2F1
1 2,1
4
, 5 4
;z
1−z/12i 1−12iz ·2F1
1 2,1
4
, 5 4
;Tz
, 2.46
which can be rewritten in a simpler way on2.3 see2.4.
It is a straightforward matter to see that Tz actually belongs to the Ra1z one- parameter family:
Tz Ra1z −724i·z· · · , a1−25·ρ, ρ 724i
25 , ρ1.
2.47
As far as the reduction of2.32to a rational function is concerned, it is straightforward to see that:
1−z2·1z4·Ra1z a1·z· · · − 2
175746796875·a1·a1−1·a14·a1−16·P8a1·z8· · ·
− 1
Nn ·a1·a1−1·a14·a1−16·Pna1·zn· · ·,
2.48
whereNnis a large integer growing withn, andPnis a polynomial with integer coefficients of degreen−4, or
1−12i·z4·Ra1z a1·z· · · − 4
1243125·a1·a1−1·a1724i·P6a1 iQ6a1·z6· · · 1
Nn·a1·a1−1·a1724i·Pna1 iQna1·zn· · ·,
2.49
wherePnandQnare two polynomials with integer coefficients of degree, respectively,n−3 andn−4.
Similar calculations can be performed forT∗zdefined by
T∗z z·
z−1−2i 1−2iz−1
4
, 2.50
for which we also have the covariance
2F1 1 2,1
4
, 5 4
;z
1−z/1−2i
1−1−2iz ·2F1 1 2,1
4
, 5 4
;T∗z
. 2.51
It is a simple calculation to check that any iterate ofTz resp. T∗zis actually a solution of2.19and corresponds toRa1zfor the infinite number of valuesa1 −7−24iN resp.−724iN. Furthermore, one verifies, as it shouldsee2.33, that the three rational functions R−4z,Tz, and T∗z commute. It is also a straightforward calculation to see that the rational function built from any composition ofR−4z,Tz, andT∗zis actually a solution of2.19. We thus have a triple infinity of values ofa1, namelya1 −4M·−7−24iN·
−724iP for any integerM,NandP, for whichRa1zreduces to rational functions. We are in fact describingsome subset of the isogenies of the elliptic curve2.2, and identifying these isogenies with a discrete subset of the renormalization group. Conversely, a functional equation like 2.19 can be seen as a way to extend the n-fold composition of a rational functionRz namelyRR· · ·Rz· · ·tonany complex number.
2.3. Revisiting the One-Parameter Family of Solutions of the
“Rota-Baxter-Like” Functional Equation
This extension can be revisited as follows. Keeping in mind the well-known example of the parametrization of the standard mapz → 4z·1−zwithzsin2θ, yieldingθ → 2θ, let us seek for atranscendentalparametrizationzPusuch that
R−4Pu P−4u or : R−4P·H−4·P−1, 2.52 whereHa1 denotes the scaling transformationz → a1·zhereH−4 : z → −4·zandP−1 denotes the inverse transformation of P for the composition. One can easily find such a transcendentalparametrization order by order
Pz z−2 5z2 7
75z3− 82
4875z4 1078 414375z5
− 452
1243125z6 57311
1212046875z7− 1023946
175746796875z8· · · ,
2.53
and similarly for its inversefor the compositiontransformation Qz P−1z z2
5z217
75z3 244
1625z4 45043 414375z5 2302
27625z6 128941
1939275z7 15365176
281194875z8· · ·.
2.54
This approach is reminiscent of the conjugation introduced in Siegel’s theorem15–17. It is a straightforward matter to seeorder by orderthat one actually has
Ra1Pu Pa1·u or : Ra1P·Ha1·P−1. 2.55
The structure of the one-parameter renormalization group and the extension of the composition ofntimes a rational functionRz namely,RR· · ·Rz· · ·tonany complex number become a straight consequence of this relation. Along this line one can define some
“infinitesimal composition” 0:
R1z P·H1·P−1z z·Fz · · ·, 2.56 where one can find, order by order, the “infinitesimal composition” functionFz:
Fz z−2 5z2− 2
15z3− 14
195z4− 154
3315z5− 22 663z6
− 418
16575z7− 9614
480675z8− 2622
160225z9· · ·.
2.57
It is straightforward to see, from 2.33, that the function Fz satisfies the following functional equations involving a rational functionRz in the one-parameter familyRa1z:
dRz
dz ·Fz FRz, dRnz
dz ·Fz F
Rnz
, where : Rnz RR· · ·Rz· · ·.
2.58
Fzcannot be a rational or algebraic function. Let us consider the fixed points ofRnz.
GenericallydRnz/dzis not equal to 0 or∞at any of these fixed points. Therefore one must haveFz 0 orFz ∞for the infinite set of these fixed points:Fzcannot be a rational or algebraic function, it is a transcendental function, and similarly for the parametrization functionPz. In fact, let us introduce the function
Gz 1−z·Fz, Gz z−7
5z2 4 15z3 4
65z4 28
1105z5 44
3315z6 44 5525z7 836
160225z8 1748
480675z9· · ·gn·zn· · · .
2.59
One actually finds that the successivegnsatisfies the very simplehypergeometric function relation:
gn1
gn 4n−9
4n1. 2.60
The function Gz is actually the hypergeometric function solution of the homogeneous operator
D2z1 4
13z−3
z·1−z·Dz3 4
6z2−3z1
1−z2·z2, 2.61
or of the inhomogeneous ODE
4z·1−z·dGz
dz 9z−3·Gz−z·1−z20. 2.62 One deduces the expression ofFzas a hypergeometric function
Fz z·1−z1/2· 2F1
1 4,1
2
, 5 4
;z
∂Ra1
∂a1
a11. 2.63
Finally we get the linear differential operator annihilatingFz
ΩF Dz21
4 · 5z−3
z1−z·Dz1
4 ·3−6z5z2 1−z2z2 Dz·
Dz−1
4· 3−5z z·1−z
, 2.64
which is, in fact, nothing butΩ∗the adjoint of linear differential operatorΩ see2.6. One easily checks19that the second-order differential equationΩFyz 0 transforms under the change of variablez → −4z/1−z2into the second-order differential equationΩRF yz 0 withΩRF Cz2·ωRF where the unitarymonicoperatorωFRis the conjugate ofΩF:
ωRF D2z−1
4 · 11z230z3
z·1−z1z·Dz1
4 ·312z50z212z33z4 z2·1−z21z2
1 Cz
·Dz·
Dz−1
4 · 3−5z z·1−z
·Cz
1 Cz
·ΩF ·Cz 1
Cz
·Ω∗·Cz
2.65
withCz 1/Rzand the “dot” denotes the composition of operators. Actually, the factors in the adjointΩ∗transform under the change of variablez → −4z/1−z2as follows20:
Dz−→Cz·Dz, ω1∗−→
ω∗1R
ω1∗·Cz, Ω∗−→ΩRF Cz·Ω∗·Cz 2.66
which is precisely the transformation we need to match with 2.58 and see the ODE Ω∗Fz 0 compatible with the change of variablez → −4z/1−z2:
Ω∗Fz 0−→Cz·Ω∗·CzFRz Cz·Ω∗·Cz
Rz·Fz
Cz·Ω∗Fz 0. 2.67
This is, in fact, a quite general result that will be seen to be valid in a more generalhigher genusframeworksee2.148,2.150in what follows.
Not surprisingly one can deduce from2.33 and the previous results, in particular 2.63, the following results forRa1z:
−4·∂Ra1
∂a1
a1−4FRz, −4n·∂Ra1
∂a1
a1−4n F
Rnz
, 2.68
whereRz −4z/1−z2andRnzdenotesRR· · ·RRz. Of course we have similar relation forTz,−4 being replaced by−7−24i. Therefore the partial derivative∂Ra1/∂a1that can be expressed in terms of hypergeometric functions for for a double infinity of values ofa1, namely,a1 −4M×−7−24iN.
One can, of course, check, order by order, that 2.58 is actually verified for any function in the one-parameter familyRa1z:
dRa1z
dz ·Fz FRa1z, 2.69
which corresponds to an infinitesimal version of2.33.
From2.56one simply deduces
z·dPz
dz FPz, 2.70
that we can check, order by order from2.53, the series expansion ofPz, and from2.57 the series expansion ofFz, but also
dQz
dz ·Fz Qz 2.71
that we can, check order by order, from2.54, the series expansion ofQz P−1zand from2.57. We now deduce that the log-derivative of the “well-suited change of variable”
Qzis nothing but themultiplicativeinverse of a hypergeometric functionFz:
dlnQz
dz 1
Fz, Qz λ·exp z
dz Fz
. 2.72
The functionQzis solution of the nonlinear differential equation
−4z2·1−z2·
Q·Q1·Q3 Q12
·Q2−2Q·
Q22 z·3−5z1−z·Q1·
Q·Q2−
Q12
5z2−6z3
·Q· Q12
0,
2.73
where the Qn’s denote the nth derivative of Qz. At first sight Qz would be a nonholonomic function, however, remarkably, it is a holonomic function solution of an order- five operator which factorizes as follows:
ΩQ
Dz 3−5z 1−z·z
·
Dz3
4 · 3−5z 1−z·z
·
×
Dz2
4 · 3−5z 1−z·z
·
Dz1
4 · 3−5z 1−z·z
·Dz,
2.74
yielding the exact expression ofQzin terms of hypergeometric functions:
Qz z·
2F1
1 2,1
4
, 5 4
;z 4
z 1−z·
2F1 1 4,3
4
, 5 4
;− z 1−z
4
, 2.75
that is, the fourth power of2.3, with the differential operator2.74being the symmetric fourth power ofΩ. From2.3we immediately get the covariance ofQz:
Q
− 4z 1−z2
−4·Qz, 2.76
and, more generally,QRa1 a1·Qz. SinceQzandFzare expressed in terms of the same hypergeometric function, the relation2.71must be an identity on that hypergeometric function. This is actually the case. This hypergeometric function verifies the inhomogeneous equation:
4·z·dHz
dz Hz−1−z−1/2 0, 2.77
where
Hz 2F1
1 2,1
4
, 5 4
;z
. 2.78
RecallingQPz z, one has the following functional relation onPz:
Pz·2F1
1 4,1
2
, 5 4
;Pz 4
z. 2.79
Noting thatQz41/4 Fz4 see2.3 can be expressed in term of an incomplete elliptic integral of the first kind of argument√
−1 z·2F1 1
4,1 2
, 5 4
;z4
EllipticF z,√
−1
, 2.80
one can find that2.79rewrites onPzas EllipticF
Pz1/4,√
−1
z1/4, 2.81
from which we deduce that the functionPzis nothing but a Jacobi elliptic function21 Pz
snz1/4,√
−14
. 2.82
InAppendix Bwe display a set of “Painlev´e-like” ODEs22verified byPz. From the simple nonlinear ODE on the Jacobi elliptic sinus, namely,S2·S3 0, and the exact expression ofPzin term of Jacobi elliptic sinus, one can deduce other nonlinear ODEs verified by the nonholonomic functionPz P1dPz/dz,P2d2Pz/dz2:
z3/2· P12
−1−P·P3/20, 2.83
P2− 3 4·
P12
P 3 4·P1
z 1 2 ·P3/2
z3/2 0. 2.84
2.4. Singularities of the Jacobi Elliptic FunctionPz
Most of the results of this section, and to some extent, of the next one, are straight consequences of the exact closed expression ofPzin terms of an elliptic function. Following the pedagogical approach of this paper we will rather follow a heuristic approach not taking into account the exact result2.82, to display simple methods and ideas that can be used beyond exact results on a specific example.
From a diff-Pad´e analysis of the series expansion ofPz, we got the sixty closest to z 0 singularities. In particular we got that Pz has a radius of convergence R 11.81704500807· · · corresponding to the followingclosest to z 0singularity z zs of Pz:
zs−11.817045008077115768316337283432582087420697· · · −4·2F1 1
4,1 2
, 5 4
; 1 4
− 1 16· π6
Γ3/48.
2.85
This singularity corresponds to a pole of order four:Pz z−zs−4. The functionPzhas many other singularities:
34·zs,161±240i·zs,−7±24i·zs,−119±120i·zs,· · · 54·zs,41±840i·zs,−527±336i·zs,−1519±720i·zs,· · · 74·zs,1241±2520i·zs,−567±1944i·zs,−3479±1320i·zs,· · ·
2.86
