We investigate the symmetry structure of the WDVV equations. We obtain an $r$-parameter group of symmetries, where $r = (n^2 + 7n + 2)/2 + \lfloor n/2 \rfloor$. Moreover it is proved that for $n=3$ and $n=4$ these comprise all symmetries. We determine a subgroup, which defines an $SL_2$-action on the space of solutions. For the special case $n=3$ this action is compared to the $SL_2$-symmetry of the Chazy equation. For $n=4$ and $n=5$ we construct new, Chazy-type, solutions

Martini, R.

Post, G.F.

English

Martini, Ruud

Post, Gerhard F.

University of Twente Research Information

Faculty of Mathematical SciencesUniversity of TwenteUniversity for Technical and Social SciencesP.O. Box 2177500 AE EnschedeThe NetherlandsPhone: +31-53-4893400Fax: +31-53-4893114Email: memo@math.utwente.nlMemorandum No. 1466Symmetries of the WDVV equations andChazy-type equationsR. Martini and G.F. PostOctober 1998ISSN 0169-2690SYMMETRIES OF THE WDVV EQUATIONS ANDCHAZY-TYPE EQUATIONSR. MARTINI AND G.F. POSTAbstract. We investigate the symmetry structure of the WDVV equa-tions. We obtain an r-parameter group of symmetries, where r =12 (n2 + 7n + 2) + bn2 c. Moreover it is proved that for n = 3 and n = 4these comprise all symmetries.We determine a subgroup, which defines an SL2-action on the spaceof solutions. For the special case n = 3 this action is compared to theSL2-symmetry of the Chazy equation. For n = 4 and n = 5 we constructnew, Chazy-type, solutions.MSC 1991 : 35Q99, 35N05, 17B66, 81T40.Keywords : WDVV equations, symmetries, Chazy equation.1. Introduction.In the physics literature on two-dimensional topological field theory, aremarkable and rather complicated system of partial differential equationsemerged [1, 2]. Roughly speaking the equations describe the conditionsfor a function F = F (t) of the variable t = (t1, . . . , tn), such that thethird-order derivatives define a set of structure constants of an associativealgebra. Commonly this system of nonlinear equations is known as theWitten–Dijkgraaf–H. Verlinde–E. Verlinde (WDVV) system.Recently the equations also appeared in four-dimensional supersymmetricgauge theory, see e.g. [3, 4, 5], suggesting an interesting relationship between4-dimensional N = 2 supersymmetric gauge theories and 2-dimensionaltopological field theories.Although extremely difficult to solve in general, these equations admitexact solutions and other features of integrability. It is therefore interestingto investigate its symmetry structure; this is the subject of the present paper.For all n we obtain an extensive symmetry group generalizing the resultsfor n = 3 and n = 4. The results for n = 3 and n = 4 were obtained in[6] by computer programs written in reduce [7], using methods which arefor example described in [8]; solving these overdetermined systems of partialdifferential equations is already difficult for n = 4.The symmetry group contains SL2(C). We discuss its intriguing actionespecially for the Chazy equation [9, 10] that is contained in the case n = 3.2. The WDVV equations.2.1. Definition. Following the conventions in Dubrovin [10], the WDVVequations are a system of partial differential equations for a function F (t) =Date: October 23, 1998.12 R. MARTINI AND G.F. POSTF (t1, . . . , tn) such that the third order derivatives,cαβγ(t) =∂3F (t)∂tα∂tβ∂tγobey the following conditions, cf. [10]1. (Normalization)c1αβ ={0 if α+ β 6= n+ 11 if α+ β = n+ 1(2.1)We introduce the metric ηαβ = ηαβ = c1αβ .2. (Associativity)The functionscγαβ(t) =n∑ε=1ηγεcεαβfor any t, must define an n-dimensional associative algebra with basise1, . . . , en and product given byeα · eβ =n∑γ=1cγαβeγThe normalization expresses the existence of a unity in the correspondingalgebra. Throughout we use subscripts instead of superscripts.This system of conditions, we call, following [10], the WDVV equations,[2, 1]. (We omit quasi-homogeneity). We assume the metric ηαβ to be instandard form, corresponding to an algebra with unity ∂t1 .The associativity condition may be written as an overdetermined systemof partial differential equations, namelyn∑λ=1∂3F (t)∂tα∂tβ∂tλ· ∂3F (t)∂tγ∂tδ∂tλ̄=n∑λ=1∂3F (t)∂tγ∂tβ∂tλ· ∂3F (t)∂tα∂tδ∂tλ̄(2.2)for any α, β, γ and δ. We denoted λ̄ = n+1−λ. From (2.1), the dependencyof F on t1 can be solved completely:F (t) =12t21tn +12t1n−1∑α=2tαtᾱ + f(t2, . . . , tn). (2.3)Substituting this expression for F in (2.2) we obtain the following systemfor f (subscripts denote partial derivatives)n−1∑λ=2(fαβλ · fγδλ̄ − fγβλ · fαδλ̄)+ηαβfγδn − ηγβfαδn + fαβnηγδ − fγβnηαδ = 0 (2.4)Hence the WDVV equations (2.1) and (2.2) are equivalent to (2.4).The number of (generating) equations in (2.4) grows rapidly with n. Thisnumber is(n3)for n ≤ 7.3. Description of the symmetriesWe split the discussion of symmetries in 4 parts.SYMMETRIES OF THE WDVV EQUATIONS AND CHAZY-TYPE EQUATIONS 33.1. Translations. The simplest symmetries of the WDVV equations arethe translations. Since the associativity equations (2.4) does not explicitlydepend on (t1, . . . , tn) we have the infinitesimal symmetryv = ∂ta (a = 1, . . . , n) (3.1)with corresponding symmetryta 7→ ta + ε (3.1’)(all non-occurring variables remain fixed).Similarly, we have the translation-type infinitesimal symmetriesv = P (t1, . . . , tn)∂F (3.2)where P is a polynomial of degree at most 2. Obviously, the correspondingsymmetry isF 7→ F + εP (t1, . . . , tn) (3.2’)Hence (3.1) and (3.2) yield n+(n+22)infinitesimal symmetries.3.2. Scalings. The scalings are most easily described in terms of f insteadof F . For instance the system (2.4) is independent of t1. Hence one canrescale t1. For f nothing changes, while F should be corrected to maintainthe normalization (2.1). So the first scaling is infinitesimallyv = t1∂t1 + t1Q∂F . (3.3)Here and hereafter,Q =12n∑s=1tsts̄ (3.4)The corresponding scaling ist1 7→ eεt1; f 7→ f (3.3’)Next we have for 2 ≤ a < n+12 the generatorv = ta∂ta − tā∂tā (3.5)with corresponding symmetryta 7→ eεta; tā 7→ e−εtā (3.5’)Finally we have the generatorsv =n−1∑i=1ti∂ti + 4tn∂tn (3.6)andv = tn∂tn + (t1Q− F )∂F (3.7)with corresponding scalingsti 7→ eεti (i = 1, . . . , n− 1); tn 7→ e4εtn (3.6’)andtn 7→ eεtn; f 7→ fe−ε (3.7’)4 R. MARTINI AND G.F. POSTUsing the equations (2.4), these last two scalings are easily checked. Totallywe find 3 + bn−22 c scaling symmetries.3.3. Linear-type transformations. The associativity condition is invari-ant under linear changes of the variables t1, . . . , tn. However ηij (defined asηij =∂3F (t)∂t1∂ti∂tj) changes. Hence the normalization spoils most of these sym-metries. Some, with a slight modification, remain; apart from the scalings,which were discussed above, there are two classes. The first class is easy tocheck:v = ta∂t1 + taQ∂F (a = 2, . . . , n) (3.8)with 1-parameter groupt1 7→ t1 + εta; f 7→ f (3.8’)The second class is not so obviousv = tn∂ta + tāQ∂F (a = 1, . . . , n− 1). (3.9)The corresponding transformation does not leave f invariant. In terms of Fit ista 7→ ta + εtn; F 7→ F + εtāQ+12ε2tnt2ā (3.9’)To check that (3.9’) is a symmetry, we use the infinitesimal criterion, seee.g. [8], section 2.3. One calculates the prolongation coefficients Φ(k,l,m) ofthe vector field (3.9)Φ(k,`,m) = δk,āη`m + δ`,āηkm + δm,āηk`−δk,nFa`m − δ`,nFakm − δm,nFak`and by direct calculation one checks thatn∑λ=1Φ(α,β,λ)Fγδλ̄ + Fα,β,λΦ(γ,δ,λ̄) − Φ(γ,β,λ)Fαδλ̄ + Fγ,β,λΦ(α,δ,λ̄) = 0 (3.10)holds for solutions F of (2.2). That the normalization is not spoilt, can bechecked directly from (3.9’).3.4. Quadratic symmetry. Calculations for n = 3 and n = 4 show thatthe WDVV equations admit one highly non-trivial symmetry. It can begeneralized to all n. The generator isv = tnn∑i=1ti∂ti + (12Q2 + 2tnF )∂F (3.11)The corresponding 1-parameter group isti 7→tiεtn + 1(i = 1, . . . n); F 7→ F(εtn + 1)2− 12εQ2(εtn + 1)3(3.12)The normalization is maintained. The corresponding transformation for fisf 7→ f(εtn + 1)2− 12ε(n−1∑i=212tit̄i)2(εtn + 1)3(3.13)SYMMETRIES OF THE WDVV EQUATIONS AND CHAZY-TYPE EQUATIONS 5To check that (3.12) is a symmetry, we again use the infinitesimal criterion.The prolongation coefficients of (3.11) areΦ(k,`,m) = −δk,nE(F`m)− δ`,nE(Fkm) + δm,nE(Fk`) (3.14)+ηk`tm + ηkmt` + η`mtk − tnFk`mHere we denoted the Euler vector field∑i ti∂ti by E. One can substitute(3.14) in (3.10). A straightforward calculation yields the result. Hence wehave proved3.5. Proposition.a. The WDVV equations (2.1) and (2.2) admit an r-parameter group ofsymmetries, where r = 12(n2 + 9n+ 2) + bn2 c.b. For n = 3 and n = 4 these are all (continuous) symmetries.Part b. was proved by explicit calculation, see [6].4. SL2-action and the Chazy equation4.1. Lie algebra structure. We have investigated the structure of the Liealgebra of infinitesimal symmetries described above. It turned out to be asemi-direct product. The elements of the form P∂F , corresponding to thesymmetries (3.2), form a commutative ideal. Dividing out this ideal leavesus essentially with vector fields of the formX =n∑i=1xi(t1, . . . , tn)∂ti .Here the coefficients xi have degree at most two (in case of (3.11)). Theelements for which all xi are linear in t1, . . . , tn form a subalgebra of g`n(C)by the isomorphism∑aijti∂tj 7→ (aij). In our case this subalgebra is iso-morphic to a set of lower-triangular matrices, with only non-zero elementsin the first column, the last row and on the diagonal. However, using thequadratic element (3.11) we obtain a subalgebra, isomorphic to s`2(C).4.2. We consider the three infinitesimal symmetries,Y = tnn∑i=1ti∂ti + (12Q2 + 2tnF )∂FH = −n∑i=1ti∂ti − tn∂tn − (t1Q+ 2F )∂FX = ∂tn +12t21∂FOne can check that[H,X] = 2X; [X,Y ] = H and [H,Y ] = −2YThese vector fields may be integrated to yield an SL2-action on the solutionspace. To describe this action, it is again convenient to shift from F to f :integrating X,Y and H above, we have(a 00 1a)· f = fa2(1 b0 1)· f = f (4.1)6 R. MARTINI AND G.F. POSTwhile the action of(1 0ε 1)is given by (3.13).Hence totally we obtain for A =(a bc d)∈ SL2(C)A · ti =tictn + d(i = 1, . . . , n− 1); A · tn =atn + bctn + d(4.2)A · f = f(ctn + d)2− 12c(n−1∑i=212 tit̄i)2(ctn + d)3(4.2a)4.3. Quasi-homogeneity. In Dubrovin [10] the WDVV equations aboveare supplemented with the requirement that any solution should be quasi-homogeneous. This means that there must exist constants d1, d2, . . . , dn anddF such thatn∑i=1diti∂F∂ti= dFF (4.3)Most symmetries above do not respect this quasi-homogeneity. However,we did not include (4.3) in our system, as we don’t mind that a symmetrychanges the quasi-homogeneity, i.e. the transformed solution is again quasi-homogeneous, but now with different di and dF . For the SL2-action aboveanother phenomenon occurs. For the special choice dn = 0, and hencedF = 2d1, see (2.3), quasi-homogeneity is preserved. This case is discussedfor n = 3 in Appendix C of [10]. Quasi-homogeneity requires d2 = 12 (if wenormalize d1 = 1), and we obtainF (t1, t2, t3) =12t21t3 +12t1t22 −116t42γ(t3).The associativity equation now turns intoγ′′′ − 6γγ′′ + 9γ′2 = 0 (4.4)which is the Chazy equation for γ, see [9]. The transformation (4.2a) boilsdown to the well-known SL2(C)-symmetry for γγ 7→ (ct3 + d)2γ + 2c(ct3 + d).Using the SL2(C)-action, one can find a 2-parameter family of rational so-lutions to the Chazy equation, starting with the seed solution γ = 1, see e.g.[11].5. Chazy-type equationsIt turns out that for higher n and special choices of the degrees d1, d2, . . . , dnthe system (2.4) reduces to a system of equations, resembling the Chazyequation. We investigated n = 4 and n = 5. The calculations for n = 4 arealready very complicated, while n = 5 is prohibitive.SYMMETRIES OF THE WDVV EQUATIONS AND CHAZY-TYPE EQUATIONS 75.1. Chazy-type equations for n = 4. We consider the case n = 4 withd1 = 1, d4 = 0 and hence dF = 2, and d2 > d3 > 0. We will denotex = t2, y = t3 and z = t4. Note that d2 + d3 = 1. Consequently, F ispolynomial in x and y, at most quartic in x. We putF (t) =12t21t4 + t1t2t3 −14x2y2γ(z) + g(x, y, z)where g does not contain the term with x2y2. Now look at (2.4) for α =β = 2 and γ = δ = 3. It reads2fxyz = fxxxfyyy − fxxyfxyy.Expressed in γ and g it yields−2xyγ′ + 2gxyz = gxxxgyyy − (−yγ + gxxy)(−xγ + gxyy)We compare the coefficients of xy left and right. To obtain γ′− 12γ2 6= 0 wehave only 2 possibilities:a.) d2 = d3 = 12 .b.) d2 = 23 ; d3 =13 .Let us discuss these 2 cases.Case a.) In this caseg(x, y, z) = g1(z)y4 + g2(z)xy3 + g3(z)x3y + g4x4Substitution of this g in (2.4) we obtain that g3 = Kg1, g4 = K4 g2 with K aconstant, andg′1 = 6g22 + γg1; (5.1)g′2 = 24Kg21 + γg2; (5.2)γ′ =12γ2 − 72Kg1g2 (5.3)Following [10], we denote ∇gi = g′i− γgi (i = 1, 2) and ∇γ = γ′− 12γ2 andΩ = −72Kg1g2. Then we are led to the equivalent system∇g1 = 6g22; ∇g2 = 24Kg21 ; ∇γ = Ω (5.4)Like Dubrovin, we put∇Ω = Ω′ − 2γΩand∇2Ω = (∇Ω)′ − 3γΩ.From (5.4) it now follows that∇2Ω + 12Ω2 = 0 (5.5)(The easiest way is to calculate first ∇(Ω3)). Expressed in γ, using Ω = ∇γ,we exactly obtain the Chazy equation (4.4).Conversely, suppose γ satisfies (4.4), fix a K and define g1 and g2 byΩ3 = −(72)3K3g31g32 and ∇Ω = −432Kg32 − 1728K2g21.Then one can check that g1 and g2 satisfy (5.1) en (5.2).8 R. MARTINI AND G.F. POSTCase b.) In case that d2 = 23 and d3 =13 we haveg(x, y, z) = g3(z)y6 + g2(z)xy4 + g1(z)x3Substition in (2.4) now gives that g2 = Kg31 and g3 = Kg1(g1− 12γg1), whereagain K is a constant. Moreover putting Ω = −72Kg41 we findg′′1 − 2γg′1 − γ′g1 = 0 and ∇γ = Ω (5.6)Eliminating g1 from the first equation (using g41 = − 172K∇γ) we obtain athird order equation for γ that can be expressed elegantly in Ω by(∇Ω)2 = 43Ω∇2Ω + 8Ω3 (5.7)5.2. Chazy equation for n = 5. The analysis for n = 5 is much moredifficult; while the system (2.4) is determined by 4 equations in case n =4, for n = 5 we have 10 equations, namely equations for fab5 for a, b ∈{2, 3, 4, 5}. We consider the case d2 = d3 = d4 = 12 , and again d1 = 1 andd5 = 0. Each of these 10 equations splits into several equations, once thepolynomial Ansatz is made; one is left with a system of approximately 100nonlinear ordinary differential equations for 15 functions in t5. We were notable to solve the system in general. However we found a solution similar tothe cases n = 3 and n = 4. We denote z = t5. For f we havef = −14(t2t4 +12t23)2γ(z) +(t42 − 2t2t33 − 2t2t34 + 3t23t24)g1(z)−12(t44 − 2t33t4 − 2t32t4 + 3t22t23)g2(z)and the remaining system of equations for g1, g2 and γ is exactly the system(5.1), (5.2) and (5.3) with K = −2.References[1] E. Witten, On the structure of the topological phase of two-dimensional gravity, Nucl.Phys. B 340, p. 281–332, (1990).[2] R. Dijkgraaf, H. Verlinde and E. Verlinde, Topological strings in d < 1, Nucl. Phys.B 352, p. 59–86, (1991).[3] G. Bonelli and M. Matone, Nonperturbative relations in N = 2 SUSY Yang-Mills andWDVV equation, Phys. Rev. Lett. 77, p. 4712 (1996).[4] A. Marshakov, A. Mironov and A. Morozov, WDVV-like equations in N = 2 SUSYYang-Mills theory, Phys. Lett. B 389, p. 43–52, (1996).[5] A. Marshakov and A. Mironov, More evidence for the WDVV equations in N = 2SUSY Yang-Mills theory, preprint hep-th/9701014, (1997).[6] M.L. Geurts, Symmetry calculations on the WDVV-system, Master thesis UniversityTwente, (1996).[7] A. Hearn, reduce user’s manual, version 3.6, Rand Corporation, (1995).[8] P.J. Olver, Applications of Lie groups to differential equations, Springer-Verlag, NewYork, (1986).[9] J. Chazy, Sur les équations différentielles dont l’intégrale générale possède un coupureessentielle mobile, C.R. Acad. Sc. Paris 150, p. 456–458, (1910).[10] B. Dubrovin, Geometry of 2D topological field theories, preprint SISSA-89/94/FM,(1994) and Springer lecture notes in mathematics 1620, p. 120–348, (1996).[11] L.A. Takhtajan, A simple example of modular-forms as tau-functions for integrableequations, Theo. Math. Phys. 93, p. 1308–1317, (1993).Faculty of Applied Mathematics, University Twente, P.O. Box 217, 7500AE Enschede, The Netherlands.

Symmetries of the WDVV equations and Chazy-type equations

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A simple example of modular-forms as tau-functions for integrable equations,T h e o .M a t h .P h y s .93,

Applications of Lie groups to dierential equations,

Geometry of 2D topological  theories, preprint SISSA-89/94/FM,

More evidence for the WDVV equations

Nonperturbative relations

On the structure of the topological phase of two-dimensional gravity,N u c l .

reduce user's manual, version 3.6, Rand Corporation,

Sur les  equations di erentielles dont l'int egrale g en erale poss ede un coupure essentielle mobile,C . R .A c a d .S c .P a r i s150,

Symmetry calculations on the WDVV-system,

WDVV-like equations

https://ris.utwente.nl/ws/files/5117422/1466.pdf

Symmetries of the WDVV equations and Chazy-type equations

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