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Differential Equations for Periods and Flat Coordinates

arXiv:hep-th/9108013v1 22 Aug 1991LBL-31104;UCB-PTH-91/39USC-91/022CALT-68-1738DOE RESEARCH ANDDEVELOPMENT REPORTDifferential Equations for Periods and Flat Coordinatesin Two Dimensionsional Topological Matter TheoriesW. Lerche⋆California Institute of Technology, Pasadena, CA 91125andD.-J.

SmitDepartment of Physics, University of California, Berkeley, CA 94720andN.P. WarnerPhysics Department, U.S.C., University Park, Los Angeles, CA 90089AbstractWe consider two dimensional topological Landau-Ginzburg models.

In order toobtain the free energy of these models, and to determine the K¨ahler potential forthe marginal perturbations, one needs to determine flat or ‘special’ coordinates thatcan be used to parametrize the perturbations of the superpotentials. This paper de-scribes the relationship between the natural Landau-Ginzburg parametrization andthese flat coordinates.

In particular we show how one can explicitly obtain the dif-ferential equations that relate the two. We discuss the problem for both Calabi-Yaumanifolds and for general topological matter models (with arbitary central charges)with relevant and marginal perturbations.

We also give a number of examples.⋆Address after Nov. 1, 1991: CERN, Geneva, Switzerland.July, 1991

−1 −1. IntroductionTopological Landau-Ginzburg theories†are not only of interest in their ownright, but they also determine the modular dependence of the Yukawa couplingsin string theories [2].

The correlation functions of such topological models [3] arecompletely determined by a prepotential (or free energy), F, and in particular thereis a set of ‘flat’ (or ‘special’) [4-7] coordinates, ti, i = 1, . .

., µ, in which the threepoint function can be written asCijk =∂3 F∂ti∂tj∂tk,(1.1)and for whichCijm Cklm = Cilm Ckjm. (1.2)These coordinates are referred to as flat since the two point function, ηij, is aninvertible, t-independent matrix, providing a natural, flat metric on the space ofchiral primary fields.

One set of flat coordinates is provided by taking the ti tobe the coupling constants of the chiral primary perturbations about the underlyingN = 2 superconformal field theory. In particular one has:Cijk(t) ≡φi φj φk exph XℓtℓZd2z φ(1,1)ℓ(z, z)i(1.3)where φ(1,1)ℓ≡G−−12eG−−12φℓ.

One could, in principle, consider perturbations by anychiral primary field; however, for several reasons it is natural, and perhaps necessary[1], to restrict ones attention to relevant and marginal perturbations.In string theory, only marginal perturbations are considered (relevant operatorswould generate space-time tachyons and thus are projected out). The three-point† By this we mean the topological, chiral primary sub-sector of two-dimensional N = 2 super-symmetric Landau-Ginzburg models, or equivalently, topologically twisted Landau-Ginzburgmodels [1].

−2 −functions, or structure constants, Cijk, determine the Yukawa couplings of the low-energy effective field theory. In addition to this, the K¨ahler potential, K, of theZamolodchikov metric is determined from the prepotential F. To obtain K onefirst passes to homogeneous coordinates zA, A = 0, .

. ., µ, such that ti = zi/z0;i = 1, .

. ., µ and views F(ti) as a function of the zA that is homogeneous of degree2.

That is, F(zA) ≡(z0)2F(zA/z0) = (z0)2F(ti). One then has [8][4]∗:K = −logi t∂F∂tj −i tj ∂F∂t.

(1.4)In this paper we will consider topological (or N = 2 supersymmetric) theoriesthat have a Landau-Ginzburg description [10]. One can obtain a set of coordinatesfor such a topological field theory simply by parametrizing the superpotential, W[3].

The problem is that these parameters are generally not the flat coordinates.One needs the flat coordinates to use (1.4). Moreover, for general coordinates thederivatives in (1.1) are covariant, making it difficult to determine F from Cijk.

How-ever, once given a parametrization of W in terms of flat coordinates ti, it is trivialto determine the structure constants Cijk(t) via simple polynomial multiplicationmodulo the vanishing relations:φi(t) φj(t) = Cijk(t) φk(t)mod∇W ≡0 ,(1.5)where φi(t) ≡−∂∂tiW(t) [3].Thus our purpose will be to determine how these flat coordinates can be relatedto general parametrizations of the Landau-Ginzburg potential.To date there have been several approaches to solving this problem. On Calabi-Yau manifolds the required coordinates can be related to the periods of the holo-morphic 3-form evaluated on an integral homology basis [5] [4-12].

One can some-times evaluate these periods explicitly as in [11]. One also knows that such periods∗It appears [9] that K is not uniquely defined given flat coordinates that obey only (1.1).

Webelieve that the ‘correct’ flat coordinates are those which obey (1.2) as well.

−3 −must satisfy a linear differential equation, and it turns out that there is an elemen-tary algorithm for determining this differential equation directly from the Landau-Ginzburg superpotential W. (A brief exposition of this has already been given in[13].) This method has been well known to mathematicians for many years (see, forexample, [14,15] [16]), but is apparently not well known in the physics community,and so we will give an exposition of the procedure, along with some examples, insection 2.In section 3 we will discuss the relationship between flat coordinates and thedifferential equations of section 2, and derive in detail the flat coordinate of a familyof K3 surfaces.

We will also discuss the role of the duality group of the Landau-Ginzburg potential. In section 4 we will describe how the Calabi-Yau techniquescan be generalized to general topological Landau-Ginzburg models.

This time oneconsiders periods of differential forms on the level curves of W, and then one showsthat by choosing the gauge carefully, one can solve the consistency conditions (1.1)and (1.2). The basic method is also known in the mathematics literature and is anapplication of the work of K. Saito.

A recent, rather brief exposition of this appearedin [6]. Our intention here is not only to simplify the exposition still further, butalso to show that if one restricts to relevant and marginal perturbations then thecalculations can be simplified.

Indeed (contrary to the expectations expressed in [6])it becomes relatively straightforward to solve topological models whose underlyingconformal theory has c > 3.2. Chiral Rings and Differential Equations for PeriodsIn this section we will, for simplicity, consider a d- dimensional (non-singular)hypersurface, V , defined by the vanishing of a homogenous polynomial, W, of degreeν in CP d+1 (the generalization to weighted projective spaces is elementary).

We willdenote the homogenous coordinates on CP d+1 by xA, A = 1, . .

., d + 2. We will alsoconsider W to be a function of the xA and of some (dimensionless) moduli µi.

−4 −If the first Chern class of V vanishes then there is a globally defined, holomorphicd- form, Ω, on V . This form can be represented [15] [17,18] byΩ=Zγ1W ω ;ω =d+2XA=1(−1)AxA dx1∧.

. .

∧ddxA∧. .

.∧dxd+2 ,(2.1)where γ is a small, one-dimensional curve winding around the hypersurface V . Moregenerally, the integralΩα =Zγpα(xA)W k+1 ω ,(2.2)where pα(xA) is a homogenous polynomial of degree kν, represents a (rational)differential d-form.

The form, Ωα, is an element of Lkq=0V(d−q,q). One finds [15][18]that Ωα represents a non-trivial cohomology element in Fk ≡Lkq=0 H(d−q,q)(V, IR) ifand only if pα is a non-trivial element of the local ring, R, of W. If we take the pα tobe a basis for R, then the corresponding forms, Ωα, are a basis for the cohomologyHd.

For the moment we will restrict our attention to these cohomologically non-trivial differential forms.The set of periods of a differential form, Ωα, is defined to be the integrals of Ωαover elements of a basis of the integral homology of V . This also has a convenientrepresentation:Π βα=ZΓβpα(xA)W k+1 ω ,(2.3)where Γβ is a representative of a homology basis in Hd+1(CP d+1 −V, ZZ).

The curveΓβ may be thought of as a tube over the corresponding cycle in Hd(V, ZZ).From now on, we will fix Γβ and consider the vector ̟α ≡Π βα . Considered asa function of the moduli µi of W, the vector ̟ satisfies a regular singular, matrix

−5 −differential equation ∂∂µi−Ai(µj)̟ = 0(2.4)for some matrices Ai. (The complete set of solutions to this differential equation isin fact all of the columns of the period matrix Π βα [15] [19].

)Our purpose in this section is to give an elementary procedure that generatesthe differential equation directly from W. The key ingredient is a technical resultestablished in [15]. That is, one considers a differential (d−1)-form of V defined byφ=Zγ1W ln XB

. .∧ddxB∧.

. .∧ddxC∧.

. .∧dxd+2o,(2.5)where the YB(xA) are homogenous of degree lν −(d + 1).

One finds thatdφ =Zγ1W l+1hl d+2XA=1YA∂W∂xA−W d+2XA=1∂YA∂xAiω . (2.6)Because this is an exact form, it provides us with a simple means of integrating byparts.

Equivalently, if pα(xA) in (2.2) has the form P YA ∂W∂xA then, modulo exactforms, we have from (2.6)Ωa ≡1kZγh 1W kXA∂YA∂xAiω . (2.7)One can iterate this procedure (if necessary) and so reduce the numerator until itlies in the local ring of W. Note that this procedure amounts to the most naiveform of partial integration.To derive the differential equation, simply differentiate under the integral toobtain∂̟α∂µi=ZΓh(∂µipα)W k+1 −(k + 1)pα(∂µiW)W k+2iω ,(2.8)and then partially integrate until all numerators have been reduced to elements

−6 −of the local ring of W. Expressing this reduced r.h.s. of (2.8) in terms of the Ωαimmediately yields (2.4).If one is interested in the dependence of Ωα on one particular modulus, µ0, thenone can reduce the first order system (2.4) to one linear, regular singular O.D.E.for ̟1 ≡RΩof order equal to or less than the dimension, µ, of the local ring of W.Note that the order is often much less then µ.

For example, if pα(xA) and W(xA; µ0)are invariant under some discrete symmetry, then the foregoing reduction procedurecan generate only those Ωβ for which pβ is also invariant. Hence the order of thedifferential equation cannot be greater than the number of such invariant pβ’s.We conclude this section by calculating a couple of examples.

First we considerthe cubic torus, that is, we take d = 1, (µ0 ≡α), and:W(xA) =13(x3 + y3 + z3) −α xyz . (2.9)Let ̟1 =RΓ1W ω and ̟2 =RΓxyzW 2 ω.

Then obviously∂∂α̟1 = ̟2 and∂∂α̟2 = 2Z x2y2z2W 3ω .One now uses the identity:(1 −α3) x2y2z2 = xz2nα2y∂zW + αz∂xW + x∂yWo,(2.10)and integrating by parts yields(1 −α3) ∂∂α̟2 = αZz3W 2 ω + 2α2Z xyzW 2 ω .Using z3 = z∂zW +αxyz in the first term and again integrating by parts gives then(1 −α3) ∂∂α̟2 = α̟1 + 3α2̟2 ,

−7 −and hence∂∂α̟1̟2= 01α(1−α3)3α2(1−α3)! ̟1̟2.Upon eliminating ̟2, this equation can be rewrittenh(1 −α3)∂2α −3α2∂α −ai̟1 = 0 .

(2.11)This example can easily be generalized to the series of potentials,W (N)(xA) =1NNXi=1xNi−αNYi=1xi ,N ≥3 ,(2.12)that describe N = 2 Landau-Ginzburg theories with central charge c = 3(N −2).Consider the following periods, which are associated with (N −l −1, l −1)-formson V (N),̟(N)l= (l −1)!ZΓQNi=1(xi)l−1(W (N))lω ,l = 1, . .

., N −1 .We find an equation in a “Drinfeld-Sokolov” form:∂∂α̟(N)=010· · ·00001· · ·00000· · ·00..................000· · ·01αb(N)1(1−αN)α2b(N)2(1−αN)α3b(N)3(1−αN)· · ·αN−2b(N)N−2(1−αN)αN−1b(N)N−1(1−αN)· ̟(N) ,

−8 −where the coefficients are recursively defined:b(N)l= l b(N−1)l+ b(N−1)l−1,for l = 1, . .

., N −2 ,with b(N)1= 1 and b(N)N−1 = 12N(N −1). The matrix equation yieldsh(1 −αN) ∂N−1∂αN−1 − N−1Xl=1b(N)lαl ∂l−1∂αl−1 i̟(N)1= 0 .

(2.13)Note that due to the high degree of symmetry of the perturbation, the order of thisequation is much less then the dimension of the corresponding local ring.Under the substitutions αN →z−1 and ̟1 →z1/N̟1, this equation transformsinto the following generalized hypergeometric differential equation [20] with regular,singular points at z = 0, 1, ∞:h z ∂∂zN−1−zz ∂∂z + 1Nz ∂∂z + 2N. .

.z ∂∂z + N −1Ni̟(N)1= 0 . (2.14)For N =5, this is identical to the equation that was discussed in [11]:hz3(1 −z) ∂4∂z4 + (6 −8z)z2 ∂3∂z3 + (7 −725 z)z ∂2∂z2 + (1 −245 z) ∂∂z −24625i̟(5)1= 0 .

(2.15)Equation (2.14) is solved [20] by̟(N)1=N−1FN−2h 1N , 2N , . .

., N −1N; 1, 1, . .

., 1; zi,whereAFBa1, a2, . .

., αA; b1, b2, . .

.bB; z≡QBk=1 Γ(bk)QAl=1 Γ(al)∞Xn=0znn!QAl=1 Γ(al + n)QBk=1 Γ(bk + n)(2.16)is the generalized hypergeometric function. In direct generalization of the results of

−9 −[11], a complete set of linear independent solutions to (2.14) for N > 3 is given byyk = z−k/N N−1FN−2h kN , kN , . .

., kN ;z}|{k + 1N, k + 2N, . .

., k + N −1N; z−1 i(2.17)(k = 1, . .

., N −1), where the overbrace indicates that the entry with value equalto one is to be omitted.The reduction method that we have described for obtaining the differentialequation (2.4) works far more generally than for the class of potentials W discussedabove. For example, the generalization to quasihomogenous spaces is straightfor-ward.

Moreover, one can apply these techniques to marginal deformations of moregeneral Landau-Ginzburg models. One does not have to restrict to Landau-Ginz-burg theories that have a sigma-model interpretation; one such generalization isobtained by formally combining theories with other ones so as to mimick the c = 3dsituation.

Then one applies the results derived above and makes the trivial observa-tion that so long as the marginal perturbations do not mix one component theorywith another, this tensoring of theories is irrelevant for the determination of thedifferential equation. Hence we need not restrict to theories with c = 3d.

As anillustration, considerW = x3y + y3 + z3 −α x3z ,(2.18)which describes a perturbation of an N =2 theory with c = 113 and µ = 14. We findh9(1 −α3) ∂2∂α2 −24α2 ∂∂α −4αi̟1 = 0 .

(2.19)In general, for theories that have effectively one modulus, the order of the differentialequation will be two if 3 ≤c < 6, three if 6 ≤c < 9, and so on.In the next section, we will describe how the linear equations (2.4) for the periodsare related to non-linear differential equations that determine the dependence ofthe flat coordinates on the moduli, µi. In section 4, we will further generalize themethod to arbitrary marginal and relevant perturbations of generic Landau-Ginz-burg potentials.

−10 −3. Non-Linear Equations, Duality and Monodromy GroupsTo obtain the flat or ‘special’ coordinates on the moduli space of a Calabi-Yaumanifold one expands the holomorphic (3, 0)-form in a basis of integral cohomology[5][11].

That is, one introduces a symplectically diagonal basis, {αa, βb}, of integralcohomology and writes the (3, 0)-form as:Ω= za αa + Gb βb. (3.1)(The periods za and Gb are integral linear combinations of the entries of an appro-priate row of the period matrix Πβα).

Since Ωis only defined up to multiplication byan arbitrary function f(za), the quantities za and Gb are only defined projectively.It was shown in [5][11][12] that the za define good projective coordinates on themoduli space, while the Gb satisfy Gb = ∂b(zaGa). It is thus the inhomogeneouscoordinates ζa = za/z0 that constitute the required flat, or special coordinates, ti.The crucial ingredient that leads to the flat coordinates is the choice of an integralcohomology basis.

This, in a very strong sense, means that we are choosing a locallyconstant frame for the cohomology fibration over the space of moduli.In the following, we will not restrict ourselves to 3- folds, but we will considerprojective coordinates ζa that are provided by the expansion of the the holomorphic(d, 0)-form on a general ‘Calabi-Yau’ manifold.In section 2 we saw how to derive a linear system of equations (2.4) that issatisfied by the periods, za. Obviously, if one multiplies Ωby f(za) then one willobtain a different set of equations, and thus (2.4) is not unique.

The appropriateinvariant equation is a non- linear system of equations for ζa that can be derivedfrom the linear system for za.For example, the linear, second order equationd2zdµ2 + p(µ) dzdµ + q(µ) z = 0(3.2)

−11 −gives rise to the non-linear, Schwarzian differential equation:ζ; µ= 2 I ,I ≡q −14p2 −12dpdµ(3.3)where ζ = z1/z2 and zi , i = 1, 2 are the two solutions of the linear equation (3.2).The Schwarzian differential operator on the left-hand-side of this equation is definedby:ζ; µ≡ζ′′′ζ′−32ζ′′ζ′2(3.4)and satisfies:y; x= −dydx2x; y(3.5)y; x=dzdx2hy; z−x; zi(3.6)y; x≡0if and only ify = ax + bcx + d ,(3.7)for some a, b, c, d ∈C and ad −bc ̸= 0.The quantity I in (3.3) is often referred to as the invariant of (3.2) since it isunchanged if one replaces (3.2) by the linear equation for f(µ)z(µ) (where f(µ) isan arbitrary function).Recall [21] that the ‘duality group’ ΓW of the superpotential W consists of thosetransformations of the moduli that are induced through quasihomogeneous changesof the variables, xA, that leave the form of the superpotential unchanged up to anoverall factor. That is, if W0(xB; µi) is a quasihomogeneous potential then one seeksquasi homogeneous changes of variable ˆxA whose Jacobian, det ∂ˆxA∂xB, is constant,or at worst a function, ∆(µi), of µi and for which:W0(ˆxA; µi) = h(µi)−1 W0(xA; ˆµi(µi)) ,(3.8)where ˆµi is some function of µi.

If one makes such a change of variables in theperiod integrals of Ωthen the result is changed by an overall factor of h(µi)∆(µi).

−12 −It follows that the linear system of differential equations must be covariant withrespect to the duality group of the superpotential. That is, if z(µi) is a solution,then so is ∆(µi)h(µi)z(ˆµi(µi)).

The corresponding system of non-linear equationsmust be invariant with respect to this duality group. This duality covariance andinvariance can be very instructive in understanding the properties of the linear andnon-linear equations.

Turning this around, it allows in principle to determine ΓWfrom the differential equations. In particular, given a second order equation (3.2),one can often read offthe solution of the associated Schwarzian equation in terms oftriangle functions, s(α, β, γ).

The parameters α, β and γ then determine the dualitygroup of the superpotential. For triangle functions, this group can be thought ofas being generated by reflections in the sides of hyperbolic triangles (with anglesπα, πβ and πγ) that cover the upper half-plane.It is thus of interest to understand the relationship between the foregoing dual-ity group ΓW, the monodromy group ΓM of the linear equations and the ‘modular’group Γ of the surface.

The integral homology basis undergoes an integral sym-plectic transformation when it is transported around singular points in the modulispace of the manifold. Consequently, the periods of the differential forms undergojust such symplectic transformations about these singular points.

This is directlyreflected in the monodromy around regular singular points of the solutions of thedifferential equations. The set of all such monodromies will generate a subgroup,ΓM, of the ‘modular group’, Γ.

The set of duality transformations ΓW of the su-perpotential maps the surface back to itself and will thus extend the group ΓM toan even larger subgroup of Γ. In some cases this extension is all of Γ, and then theduality group of W is ΓW = Γ/ΓM.We are uncertain as to the general validity of this conclusion, but a simpleillustration is provided by the cubic torus, (2.9).

The non-linear system associatedto (2.11) is given by the Schwarzianα; t= −12(8 + α3)(1 −α3)2 α(α′)2 ,(3.9)

−13 −which is solved by the triangle function α(t) = s(12, 13, 13; J(t)) [22][21]. The trans-formation properties of this function are well known; in particular, it is a modularform of Γ(3) ≡PSL(2, ZZ3), and this group is the monodromy group ΓM of thedifferential equation (2.11).

Both sides of (3.9) are invariant under Γ = PSL(2, ZZ),which is the full modular group of the torus. The quotient, the tetrahedral groupΓ/Γ(3), is precisely the duality group ΓW of the superpotential (2.9), [22][21].As a further example of the non-linear systems of equations that one can derivefor the flat coordinates, we will descibe in some detail the a very particular familyof K3 surfaces.

That is, we will consider the surface defined by the superpotential(2.12) for N =4. Equation (2.13) becomes:h(1 −α4) ∂3∂α3 −6α3 ∂2∂α2 −7α2 ∂∂α −αi̟(4)1= 0 ,(3.10)Before discussing the solution of this system, it is instructive to consider a generalthird order equation and see how one passes to the associated non-linear systemand obtains the invariants.

Our discussion will follow that of [23]. Consider thegeneric third order equation:̟′′′ + 3p(α) ̟′′ + 3q(α) ̟′ + r(α) ̟ = 0.

(3.11)One starts by partially removing the freedom to multiply a solution by an arbitraryfunction of α. This is done by requiring the vanishing of the coefficient of the secondderivative, and is accomplished by substituting ̟ ≡ye−Rp dα.The differentialequation then takes the form:y′′′ + 3Q(α) y′ + R(α)y = 0,(3.12)whereQ = q −p2 −p′R = r −3pq + 2p3 −p′′ .

(3.13)

−14 −Let y1, y2 and y3 be solutions of (3.12) and define s and t bys = y2y1,t = y3y1.Substituting y2 = sy1 and y3 = ty1 into (3.12), and using the fact that y1 is asolution, one obtains:3s′y′′1 + 3s′′y′1 + (3Qs′ + s′′′)y1 = 03t′y′′1 + 3t′′y′1 + (3Qt′ + t′′′)y1 = 0. (3.14)If one now differentiates these two equations again, and eliminates y′′′1 using (3.12)one obtains two more equations that are linear in y1, y′1 and y′′1.

These two equa-tions, along with (3.14), provide four linear equations for the three non-trivial,independent unknowns y1, y′1 and y′′1, and thus there are two indepedent 3 × 3 de-terminants that must vanish. The vanishing of these determinants gives two fourthorder, non-linear equations for s and t. Conversely, given a solution to these non-linear equations, one can eliminate y′′1 from the linear system described above toobtain a simple linear, first order equation for y1, whose solution is:y1 =s′′t′ −s′t′′−13.The other solutions are then obtained from y2 = sy1 and y3 = ty1.The actual non-linear system for s and t is fairly unedifying, but we will give ithere for the sake of completeness.

Define the following variables:u1 = s′′t′ −s′t′′u2 = s(3) t′ −s′t(3)u3 = s(4) t′ −s′t(4)v1 = s(3) t′′ −s′′t(3)v2 = s(4) t′′ −s′′t(4),(3.15)where s(i) ≡dis/dαi, and introduce the differential operators:D1(s, t; α) ≡u3 −2v1u1−43u2u12D2(s, t; α) ≡9v2u1−6u2(u3 + 4v1)u21+ 8u2u13. (3.16)

−15 −The non-linear system may then be written:D1(s, t; α) = 3Q(α) ≡I,D2(s, t; α) = 27(∂Q∂α −R) ≡J. (3.17)The operators D1 and D2 are invariant under fractional linear transformations:D1a2 + b2s + c2ta1 + b1s + c1t, a3 + b3s + c3ta1 + b1s + c1t; α= D1(s, t; α)D2a2 + b2s + c2ta1 + b1s + c1t, a3 + b3s + c3ta1 + b1s + c1t; α= D2(s, t; α) .

(3.18)The right-hand-sides of (3.17) define the quantities I and J, which are called theinvariants of the system.To solve (3.10) one needs to use some more of the theory of reduced differentialequations of the form (3.12)[23]⋆. The form of (3.12) can be preserved by a com-bined rescaling and reparametrization.

That is, one introduces a new parameter tand sets y = dtdα−1u. Under this transformation the resulting differential equationhas the form of (3.12), but with:α →t ,y →u ,Q →˜Q ≡ dtdα−2Q −23t; α R →˜R ≡ dtdα−3R −ddαt; α −3 d2tdα2 dtdα˜Q.It is interesting to note that Q transforms precisely like an energy momentum tensor,and that the combination W3 ≡R −32dQdα transforms homogeneously, i.e.

:W3 →˜W3 ≡˜R −32d ˜Qdt= dtdα−3W3 .It is precisely one of the classical W-generators [24]. One can fix the reparametriz-⋆A recent discussion of this subject may be found in [24].

−16 −ation invariance by requiring that ˜Q = 0, ort; α= 32 Q . (3.19)If one puts (3.10) in the form (3.12) one has:Q = α23(α4 + 11)(1 −α4)2R = α 11 + 36α4 + α8(1 −α4)3,From this one finds an extra bonus: W3 ≡0, or R = 32Q′.

This means that whenone passes to the equation for u(t), one obtains d3udt3 = 0, whose solutions are 1, tand t2. Therefore the solutions to (3.10) are:̟ = (1 −α4)−12 dtdα−1u(t) ;u(t) = 1 , t , t2 ,(3.20)where t(α) is the solution of (3.19):t; α= 32 Q ≡12α2 α4 + 11(1 −α4)2.

(3.21)Finally, changing variables z = α−4 in (3.21) one obtains:t; z= 121z2 + 381(z −1)2 −13321z(z −1).The solution of this equation is given by a triangle function, t(z) = s(0, 12, 14; z),which can, in turn, be re-expressed as the ratio of two solutions to the ordinaryhypergeometric equation with parameters α = 18, β = 38, and γ = 1. (The solutioncan, of course, also be expressed in terms of ratios of generalized hypergeometricfunctions (2.17).) We remark that the structure of the monodromy group of (3.10)is very similar to that of the quintic of [11], the difference being that all appearancesof 5 in the formulae of [11] must be replaced by 4.

This rule seems to hold for allN.

−17 −As we have seen, the modular dependence of the periods of this family of K3surfaces could have involved a non- trivial W3 invariant, but instead we found thatW3 vanished. In this sense, the structure is determined merely by the Virasoroalgebra, that is, by the Schwarzian differential equation (3.19).It would be very interesting to discover to what extent the higher dimensionalsurfaces defined by (2.12) might be similarly reduced, and to understand whetherthe appearance of such reduced W-algebras has any deeper meaning.

In particular,note that the solutions of (3.10) are algebraically related (inspection of (3.20) showsthat ̟1̟3 = ̟22), and as we have seen this is a consequence of the vanishing ofW3. It turns out that for the quintic, i.e.

for (2.15), one also has W3 ≡0 butW4 ̸= 0, and it is known that the Gb in (3.1) are homogenous functions of the za.Thus it appears that the vanishing of W-generators is closely connected to algebraicrelations between the solutions. We hope to discuss these issues elsewhere.4.

Flat Coordinates for Generic PerturbationsWe now wish to generalize the methods of section 2 to marginal and releventperturbations of arbitrary topological Landau-Ginzburg field theories⋆. The basicproblem is that general N =2 superconformal theories have no obvious analogue ofintegral cohomology.

As discussed in the previous section, it is this that leads oneto flat coordinates for ‘Calabi-Yau’ spaces.For general topological matter models, one can make a general ansatz for F, orfor W, in terms of the flat coordinates, and then evolve algebraic and differentialequations from consistency conditions of the topological matter models [3] [25] [26].In particular one requires that the Cijk be given by (1.1) and that they satisfy (1.2).However, solving the system (1.2) is extremely laborious, except in the simplestcases.⋆Our discussion will follow, and extend, that of [6]. Flat coordinates in generic topologicalmatter theories have also recently been discussed in [7].

−18 −Thus we like to obtain differential equations that determine the flat coordinatesmore directly from the superpotential. Let W0(xA) be a quasihomogenous superpo-tential and W(xA; si) be a parametrization of a general, versal deformation of W0 byelements φα of the chiral ring.

The problem is to determine the relationship betweenthe general coordinates si, and the flat coordinates ti. Once the parametrization ofW in terms of the ti is known, the free energy F and all correlation functions caneasily be computed.We will regard W(xA; si) as a quasihomogenous function of xA and si, and thusthe coupling constants si can be assigned dimensions.

(We will adopt the conventionthat both W0 and W have dimension equal to one.) Below we will actually consideronly marginal and relevant perturbations, whose corresponding coupling constantswill have vanishing or positive dimensions.

This will lead to the major simplificationthat all quantities will have polynomial dependence on the coupling constants withpositive dimension, and the only non-polynomial behavior will be via the marginalparameters.The coupling constant associated to the constant term in W(xA; si) (i.e., theunique coupling constant of dimension one) will play a distinguished, importantrole and it will be denoted by s1. The remaining coupling constants s2, s3, .

. .

willbe denoted generically by s′. We will takefW(xA, s′) = W(xA, si) −s1(4.1)as independent of s1.Let φα(xA; s′), α = 1, .

. ., µ, be any (polynomial) basis for the chiral ring andconsider integrals of the formu(λ)α= (−1)λ+1Γ(λ + 1)Zγφα(xA; s′)W λ+1dx1∧dx2.

. .∧dxn ,(4.2)

−19 −where the integral is taken over any compact homology cycle†γ in the set {x ∈Cn : W(x, s) ̸= 0}. The gamma function and the factor of (−1)λ+1 are introducedfor later convenience.

These integrals are related to the periods of differential formson the level surfaces of W, [19]. They satisfy some important recurrence relations[27][6]:∂ks1 u(λ)α= u(λ+k)α,k ∈ZZ(4.3)∂si u(λ)α=∞Xk=−1B(k) βi α(s′) u(λ−k)β(4.4)s1 u(λ)α= −∞Xk=0A(k−2) βα(s′) u(λ−k)β(4.5)These recurrence relations are derived by the same procedure as that employedin section 2.

Equation (4.3) is a trivial consequence of differentiation under theintegral. Equation (4.4) is also obtained by differentiating under the integral, but inthis second instance the numerator of the integrand is a polynomial (∂siW)φα(xA; s′)which might need to be reduced.

That is, by definition of the local ring of W thispolynomial may always be rewritten in the following form:(∂siW)φα(xA; s′) ≡Ciαβφβ(xA; s′) + q(0)Ai α (xA; s′) ∂W∂xA(xA; s′)(4.6)for some polynomial q(0)Ai α . One now integrates by parts∗to obtain∂si u(λ)α= Ciαβu(λ+1)β+ (−1)λ+1Γ(λ + 1)Z ( ∂∂xAq(0)Ai α )W λ+1dx1∧dx2.

. .∧dxn .† There are µ independent possible choices, but the choice is not important for the moment.

Thereader might find it helpful to consider the one-variable case, in which γ is some loop aroundsome subset of the zeros of W.∗That is, one uses the fact that 0 ≡R∂∂xAV AW λ+1dx1∧dx2. .

.∧dxn for any vector V A.

−20 −Once again one decomposes the numerator∂∂xAq(0)Ai α= B(0) βi α (s′) φβ(xA; s′) + q(1)Ai α (xA; s′) ∂W∂xA(xA; s′) ,and integrates by parts. In this manner one may recursively compute B(k) βi α(s′),k = −1, 0, 1, .

. ..

Note that after every integration by parts the polynomial degree(in xA) of the successive terms ( ∂∂xAq(k)Ai α ) decreases by one unit, and hence thisprocedure must terminate after a finite number of steps. Also note that Ciαβ ≡B(−1) βi α(s′) are essentially the structure constants of the local ring.Finally, equation (4.5) is obtained by taking s1 inside the integral, and rewritingit as s1 ≡W(xA, si) −fW(xA, s′).

The factor of W is cancelled immediately, whilefW(xA, s′)φα(xA; s′) is simplified by the identical, recursive reduction procedure de-scribed above.It is very convenient to make a “Fourier transformation” in the s1 variable.Specifically, it replaces f(s1) byR 0−∞es1/zf(s1) ds1. This has the effect of sending∂s1 →z−1 and s1 →−z2 ddz.

Let u(λ)α (z; s′) denote the transform of u(λ)α (s). Thenequations (4.4) and (4.5) may be rewritten as a linear system: ∂si −∞Xk=−1zkB(k) βi α(s′)!u(λ)β (z; s′) = 0(4.7) ∂z −∞Xk=−2zkA(k) βα(s′)!u(λ)β (z; s′) = 0(4.8)Observe that the u(λ)β (z; s′) are, by definition, covariant constant sections of a flatvector bundle whose connections are defined by B(k)iand A(k).To get more insight into why we make this construction, suppose that B(k)i≡0for k ≥1 and define Di = ∂si −B(0)i, Ciaβ = B(−1) βi α(s′).

Then the flatness of theconnection, or integrability of (4.7) impliesDi −z−1Ci , Dj −z−1Cj= 0 ,(4.9)

−21 −and separating out different orders in z, one gets the zero curvature equations‡Di , Dj≡D[i Cj] ≡Ci , Cj≡0 . (4.10)Hence the connection B(0)iis flat, the structure constants Ciaβ commute and arecovariantly constant.

If we now arrange that the basis {φα} of the chiral ring is, infact, given by {∂W∂si }, and letΓijk = B(0) kij,then one finds that Γijk = Γjik and equation (4.10) implies that Γ is the flatcoordinate connection we seek.We thus have solved the consistency conditions(1.1) and (1.2). The remainder of this section will essentially reduce the generalproblem to the foregoing simpler situation.Because the connection defined by A and B is flat, one already knows that onecan find a gauge transformation that will trivialize it.

More precisely, because theremight be non- trivial monodromy, one can find a matrix M such that♮A = (∂zM)M −1 + M ˜A(s′)zM −1Bi = (∂siM)M −1 . (4.11)Thus we can gauge away all of B and almost all of A.

The problem is that thematrix M will in general involve all powers in z and 1/z. Hence M will define abasis change involving u(λ+k)αfor all k ∈ZZ.

To control this, and indeed to preservequasihomogeneity, we want to restrict ourselves to changes of basis that are uppertriangular, that is, u(λ+k)αis only modified by addition of polynomials in s′ andu(λ+l)αfor l ≤k. This means that the change of basis must be analytic at z = 0.‡ Such equations have also been discussed in [5][7].♮These matrices are analytic in s′, but not in z, hence the form of the equation.

−22 −Now suppose that we can decompose M of equation (4.11) in the following manner:M = g0(z; s′) g∞(1z; s′) ,(4.12)where g0 is analytic at z = 0 and g∞is analytic at z = ∞. Now let A′ ≡g−10 Ag0 −g−10 (∂zg0) and B′i ≡g−10 Big0 −g−10 (∂zg0).

Then it is elementary to see thatA′ =h(∂zg∞)g−1∞+ g∞A0(s′)zg−1∞iB′i = (∂sig∞)g−1∞. (4.13)Moreover, by modifying g0 by multiplying by a suitable matrix, h(s′), one canfurther gauge away the z-independent term in B′.

Thus, provided that we can makethe split in (4.12) there is a z-analytic gauge choice that has A(k) ≡B(k)i≡0, k ≥0.The problem of finding the splitting (4.12) is called a Riemann-Hilbert problem,and is generically [28], but not always, solvable. Its solution is intimately connectedwith solving integrable models (see, for example, [29]).

We have, in fact, a vari-ational Riemann-Hilbert problem in that our matrices have parameters s′. Thismakes the problem much easier to address and it will be discussed further in theappendix.

In particular, we show in the appendix that g0 = I + O(s′), where I isthe identity matrix, and we will also show that g0 is analytic in s′ and preservesquasihomogeneity (i.e. the elements of the new basis have a well-defined scalingdimension).To get flat coordinates, we need to make the restriction to marginal or relevantperturbations (of dimension less than or equal to one), which means dim(si) ≥0.Let vα(z; s′) be the basis in which A(k) ≡B(k) ≡0 for k ≥0.

Observe that ifwe restrict to φα(xA; s′) of dimension strictly less than one, then the correspondingu(λ)α (s′) have dimensions strictly less than (P ωA)−λ (where ωA is the weight of xA).However, because the basis change has the form: g0 = 1+O(s′), is analytic in z ands′ and preserves quasihomogeneity, it follows that the vα of dimension strictly lessthan (P ωA)−λ are analytic, quasihomogenous combinations of s′ and the u(λ)α (s′).

−23 −In particular, such vα do not involve any u(λ+k)α(s′) for k ̸= 0). Furthermore, the vαof dimension equal to (P ωA)−λ must be analytic, quasihomogenous combinationsof s′, the u(λ)α , andu′0 ≡u(λ−1)0≡(−1)λΓ(λ)Z1W λdx1∧dx2.

. .∧dxn .

(4.14)One of the vα of dimension equal to (P ωA) −λ must be (a dimensionless mul-tiple of) u(λ−1)0= zu(λ)0 , while the rest of the vα must start with a u(λ)αterm forwhich φα(xA; s′) is a marginal operator. Let ˜s denote the (dimensionless) marginalparameters and let v′α and f(˜s) be such that {v′α, f(˜s)u′0} forms a set of linearlyindependent v(λ)αof dimension less than or equal to (P ωA) −λ.It follows from the foregoing that there is a quasihomogenous, analytic, invert-ible matrix ejα (s′) and a set of functions qj(˜s) such that∂∂sju′0 = eαjv′α −qj(˜s) u′0(4.15)and qj ≡0 if dim(sj) > 0.

Next observe that∂2∂si∂sju′0 =∂ieαjv′α + eαj(∂iv′α) −qj(∂iu′0) −(∂iqj)u′0=∂ieαj−qjeαiv′α + z−1eαjCiaβv′β +qiqj −∂iqju′0 .By linear independence of the v′α and u′0 we have ∂[i qj] ≡0 and hence qj ≡∂jq forsome function q(˜s).The foregoing combines to give us the following simple result. There is a ‘uni-versal’ function q(˜s) of all the marginal (dimension zero) parameters such that theintegralu0 = (−1)λΓ(λ)Z q(˜s)W λ dx1∧dx2.

. .∧dxn(4.16)

−24 −satisfies the following equation⋆∂2∂si∂sju0 = Cijα u(λ+1)α+ Γijk ∂∂sku0 . (4.17)Thus, the function q(˜s) is determined by requiring that (4.17) contains no termproportional to u0 itself.

The Cijα are just the structure constants of the chiralring and Γijk is the Gauss-Manin connection. Flat coordinates are determined bysimply requiring that Γ ≡0 on the r.h.s.

of (4.17).In practice one takes the si to be the flat coordinates ti, and considers a per-turbation of the formW(x; t) = W0(x) +Xµi(t) mi(x) ,(4.18)where mi(x) are monomials in the local ring (with degree less or equal than one),and the Landau-Ginzburg couplings, µi(t), are unknown functions to be determined.We note that it is elementary to explicitly write down the constraints implied by(4.17) since this only involves differentiating under the integral and integrating byparts, just like the reduction procedure described in section 2. The constraints takefirst the form of linear differential equations for the µi.

One determines q(˜t) interms of the µi(t) by requiring that the u0 piece in (4.17) vanishes. Substituting forq(˜t) then turns the linear system into the associated non-linear system (e.g., into aSchwarzian differential equation) that determines the µi(t).The function q(˜t) appears to be playing the rˆole of a conformal rescaling of thevielbein.

In particular we note that for the examples we computed, the functionq(˜t)−2 is precisely the conformal factor that takes the Grothendieck metric of [7] tothe flat metric.For conformal theories with c > 3, there are chiral primary fields of dimensionlarger than one. With the restriction that we have made on the perturbed superpo-tential, we cannot write these irrelevant chiral primaries as ∂W∂si for some si.

Thus⋆Remember that W is only perturbed by marginal and relevant operators, i.e., 0 ≤dim(sj) ≤1.

−25 −it might appear that these irrelevant chiral primaries play no role in determiningthe form of the equations that we derive from the procedure described above. Thisis not so.

It is important to remember to pass first to the basis for all the chiralprimaries in which one has B(k)i= 0 for k ≥0, and then one must use this basisin calculating the separate terms in expressions like those on the right-hand-side of(4.17).Finally we note that equations (4.17) and (4.10) can be recast in the familiarform˜Di ̟ = 0and ˜Di , ˜Dj= 0 ,(4.19)where ˜Di = ∂si + Γi + Ci∂1 and ̟i = ∂iu0. The first equation is a generalization ofthe matrix differential equation (2.4) we discussed in section 2.5.

Examples RevisitedIt is instructive to reconsider first the torus example (2.9) of section 2, but nowwith an additional, relevant perturbation:W =13(x3 + y3 + z3) −α(t) xyz −s β1(t)xy −12s2β2(t)z −16s3β3(t) . (5.1)Here, t is a dimensionless, flat coordinate (the modular parameter of a torus), and sis a parameter of dimension 1/3.

The dependence of the Landau-Ginzburg couplingconstants on the relevant perturbation parameter s is already fixed by its dimension,so we will have to determine only the dependence on the modular parameter. Thetwo-parameter perturbation is certainly not the most general one (which was con-sidered previously in [25]), but the extension is obvious.

The specific perturbationwe chose is however the most general one consistent with the ZZ3 × ZZ3 symmetrygenerated by (x, y, z, s) →(ωx, ω2y, z, s) and (x, y, z, s) →(ωx, ωy, ωz, ωs).Let u0 ≡(−1)λΓ(λ)R q(t)W λdx1∧dx2. .

.∧dxn. We want to solve for α(t) and βi(t)by requiring the connection Γ in (4.17) to be flat; this corresponds to the vanishing

−26 −of terms proportional to u(λ)αin this equation. In particular, we obtain∂2u0∂t2=q′′qu0 + (−1)λ+1 Γ(λ + 1)Zdx1∧dx2.

. .∧dxn1W λ+1 ×n h2q′q + α′′α′iα′q xyz + sh2q′q + β′′1β′1iβ′1q xy+ 12s2h2q′q + β′′2β′2iβ′2q z + 16s3h2q′q + β′′3β′3iβ′3qo+(−1)λ+2 Γ(λ + 2)Zdx1∧dx2.

. .∧dxn1W λ+2 ×n(α′)2q x2y2z2 + 2sα′β′1q x2y2z + s2(β′1)2q x2y2 + s2α′β′2q xyz2 + s3[β′1β′2+ 13α′β′3]q xyz + 14s4(β′2)2q z2 + 13s4β′1β′3q xy + 16s5β′2β′3q z + 136s6(β′3)2qo.

(5.2)We could obtain equations for q, α, βi by considering all the different powers of s inthis equation, but it is easier to just concentrate on the s = 0 pieces. We can thususe (2.10) and subsequently z3s=0= (z∂zW + αxyz)s=0to integrateR x2y2z2W λ+2by parts to reduce its degree.

The vanishing of the connection Γ corresponds to thevanishing of the terms proportional to1W λ+1, i.e. (−1)λ+1Γ(λ + 1)Zdx1∧dx2.

. .∧dxn1W λ+1h2q′q + α′′α′ +3α2α′(1 −α3)iα′q xyz ≡0 .This determines q(t)q(t) =1 −α(t)3α′(t)1/2.

(5.3)Integrating (5.2) (with s = 0) by parts, substituting (5.3) and requiring the van-ishing of all terms that are proportional to u0, we then indeed obtain directly theSchwarzian differential equation (3.9) for α(t),α; t= −12(8 + α3)(1 −α3)2α(α′)2 ,which is associated to the linear equation (2.11). However, by using the methodsderived in the foregoing section, we can now also solve for the couplings βi(t) of the

−27 −relevant perturbations. To obtain β1(t), it is easiest to consider the s = 0 piece of∂2u0∂s∂t; by integrating by parts, we find the condition1W λ+1h2q′q + β′1β1+2α2α′(1 −α3)iβ1q xy ≡0 ,and this gives β1(t) = A(α(t)′)1/2(1−α(t)3)1/6 (where A is an integration constant).Similarly, from the s independent piece of ∂2u0∂s2 we obtain1W λ+1hβ2 +β21α(1 −α3)iq z ≡0 ,which yields β2(t) = −A2α(t)α(t)′(1 −α(t)3)−2/3.

Finally, for β3 we consider thepiece linear in s of ∂2u0∂s2 , and use the identityx2y2 =1(1 −α3)nαxz∂xW + x2∂yW + α2xy∂zW+ 2sαβ1 xyz + sβ1x∂xW + s2β21 + 12s2α2β2xyoPartial integration givesZ1W λ+1sqhβ3 +β31(1 −α3)i≡0and thus determines β3(t). These results coincide with the expressions derived in[25].

One can also check that for the choice of α, βi give above, the connection Γ iscompletely flattened.To illustrate that our method may be used for theories with arbitrary centralcharges, reconsider the potential (2.18) with additional, relevant perturbations:W = x3y + y3 + z3 −α(t) x3z −s1β1(t) z −s2β2(t) x2 ,We find for the1W λ+1 piece in the si = 0 part of ∂2u0∂t21W λ+1h2q′q + α′′α′ −83α2α′(1 + α3)iα′q x3z ,

−28 −and this determinesq(t) = α′(t)−1/21 + α(t)34/9 .The u0 piece in (4.17) then vanishes if α satisfies the differential equationα; t=409(1 + α3)2α(α′)2 .This is precisely the Schwarzian form of the linear equation (2.19). Moreover, β1may be obtained from the si = 0 piece in∂2u0∂t∂s1: β1(t) = α′(t)1/2[1 + α(t)3]−1/3.Similarly, we find β2(t) = α′(t)1/2[1 + α(t)3]−1/9.Hence, we can compute thefollowing term of the free energy:F(si, t) =12s1s22α′(t)1/2[1 + α(t)3]1/9 + .

. .

.It is clear that we could compute the other terms of F in a similar way.AcknowledgementsThe work of W.L. was supported by DOE contract DE- AC0381ER40050 andthat of D.-J.S.

by NSF grant PHY 90-21139. N.W.

is supported in part by fundsprovided by the DOE under grant No. DE-FG03-84ER40168 and also by a fellow-ship from the Alfred P. Sloan foundation.D.-J.S.

thanks O. Alvarez and A. Giveon for discussions.N.W. would liketo thank D. Morrison and S. Katz for telling him about the reduction proceduredescribed in section 2 of this paper.

He would also like to thank K. Uhlenbeck forvery helpful discussions on the Riemann-Hilbert problem. D.-J.S.

and N.W. aregrateful to the MSRI in Berkeley and the organizers of the Workshop on MirrorSymmetry for creating a stimulating environment in which some of the ideas in thispaper were developed.

W.L. and N.W.

would also like to thank the Aspen Centerfor Physics for providing a excellent environment that was very conducive to thefurther progress of this research.

−29 −While this manuscript was in preparation we received the preprint [30] in whichsimilar results to those of section 2 are derived.APPENDIXOur purpose here is to show why one can find a matrix g0(z, s′) that is analyticin z and s′ in the neighbourhood of z = 0,s′ = 0 and which will gauge thepotentials A(k) and B(k) to zero for k ≥0. As was shown in section 3, it suffices toshow that the matrix M(z; s′) defined in (4.11) can be factorized as in (4.12).

TheBirkhoffdecomposition theorem [28] [31] implies that any matrix M(z; s′) can bedecomposed according to:M = g0(z; s′) Λ(z; s′) g∞(1z; s′) ,(A1)where g0 is analytic at z = 0, g∞is analytic at z = ∞and Λ(z; s′) is a diagonalmatrix whose entries are integral powers⋆of z. We need to show that Λ = I,where I is the identity matrix.

More simply, it suffices to show that all the integralpowers of z in Λ are, in fact, zero and hence Λ is z independent and can thus beabsorbed into g0 or g∞. Since integers can only be continuous functions of s′ bybeing constant, we can establish the desired result in a region about s′ = 0 bysimply showing that it is true at s′ = 0.Let φα(x) = φα(x, s′ = 0), and recall that W0(x) ≡fW(x, s′ = 0).

By assump-tion W0(x) and φα(x) are quasihomogeneous of weight 1 and of some weight λα⋆These integers are related to the Chern class of the relevant vector bundle over S2.

−30 −respectively. It follows that W0 ≡PA ωAxA ∂W0∂xA and henceW0(x) φα(x) ≡XAωAxA ∂W0∂xA φα(x)=XA ∂∂xAωAxAW0φα−ωAW0φα −ωAxAW0∂φα∂xA=XA∂∂xAωAxAW0φα−h XAωA+ λαiW0φα.Therefore W0(x) φα(x) is a total derivative at s′ = 0.

It follows that A(k)(s′) ≡0for k ≥0. Now takeg0(z, s′) = I +Xj∞Xk=0zks′jB(k)j (s′ = 0) + O((s′)2)where I is the identity matrix.

This yields the desired gauge for all values of z butwith s′ = 0. As described above, the Birkhofftheorem then guarantees that it canbe done in a region about s′ = 0.

Also note that g0(z, s′ = 0) = I.To see that the solution can be made quasihomogeneously, consider the differ-ential equation that needs to be solved:g−10 Ag0 −g−10 ∂zg0 = z−2P−2(s′) + z−1P−1(s′)g−10 Bjg0 −g−10 ∂sjg0 = z−2Qj(s′)where P−1, P−2 and Qj are unknowns. This means that we must solve:∂sjg0 = g0hg−10 Bjg0i+(A2)where []+ means: take only the non-negative powers of z in a power seriesexpansion about z = 0.

The fact that the system (A2) is integrable follows fromthe general observations above. (It could probably be proved more directly.) If g0 isknown to n-th order in s′, then (A2) determines the (n+1)-st order term.

Thus one

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