wdvv and dzm

22 September 1997

PHYSICS LETTERS A

ELSEZVIER Physics Letters A 234 (1997) 171-180

WDVV and DZM Robert Carroll ’

Mathematics Department, University of Illinois, Urbana, IL 61801, USA

Received 23 April 1997; revised manuscript received 10 July 1997; accepted for publication 11 July 1997 Communicated by C.R. Doering

Abstract

We show how the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations and the Darboux-Zakharov-Manakov (DZM) system can be characterized via a background family of functions. Connections to the Hirota bilinear identity are indicated. @ 1997 Elsevier Science B.V.

1. Introduction

The background literature for WDVV in terms

of topological field theory (TFT) goes back to Refs. [ 1,2] for example and an extensive develop-

ment appears in Ref. [ 31, connecting the equations to Frobenius manifolds and Egorov geometry. A recent paper [4] develops this point of view on Riemann surfaces and other recent work in Refs. [5,6] con-

nects matters to N = 2 SUSY Yang-Mills (YM) or

Seiberg-Witten (SW) theory. On the other hand the DZM system goes back to Refs. [7,8] for example

and more recently there have been extensive devel- opments in Ref. [9] (cf. also Refs. [lo-131). We will exhibit here some of the connections between

WDVV and DZM in a somewhat different abstract manner which reveals the purely algebraic character of certain features. The origin of DZM via the Hirota bilinear identity is spelled out following Bogdanov and Konopelchenko and this indirectly connects WDVV to Hirota. For simplicity we do not give a

survey of background ideas on Egorov geometry from

’ E-mail: [email protected].

Refs. [3,14,4] but will mention some points of con- tact as we go along (cf. Refs. [ 15-171 for an extended treatment of these matters in a broader context).

2. DZM

To see how the DZM theory can arise independently in perhaps maximum generality we follow Ref. [9]

(cf. also Refs. [lo-12,9,13,18-201).

The generality here is to suggest a basic role of Hirota in DZM and hence in WDVV. Thus first a

background situation here goes back to Refs. [ 18,191 where the Hirota bilinear identity was derived from the D-bar framework. This connection involves alge-

braic techniques from Sato theory on one side and analytic techniques from D-bar on the other. Connec- tions based on Hirota as in Refs. [ 18,191, or more

generally in Ref. [ 91, form a bridge or marriage between the two types of technique and touch upon the intrinsic meaning of the whole business. Now for the background derivation of Refs. [ 18,191 we consider the (matrix) formula

03759601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PII SO375-9601(97)00588-4

172 R. Carroll/Physics Letters A 234 (1997) 171-180

&w, A, A>

= JJ cCl(x,A’,~‘)Ro(A’,X’,A,X) dh' AdA',

a

&,3(x’, A, X)

=-- JJ Ro( A, ii, A’, i’)tj(x’, A’, A’) dA’ A dA’.

R

(2.1)

Multiply by 4(x’, A, A) on the right in the first equation and by +(x, A, A) on the left in the second to obtain

JJ @(x,A,i)&.x’,A,ii)] dAAd;i

R

~,~?(x,A,A)$(x’,h,;i) dA=O, (2.2)

which is the Hirota bilinear identity when 30 N C is a small circle around 00.

We go next to Refs. [ 11,9,13,20] (a complete discussion of this with details and derivations will appear in Ref. [ 151). Here we go directly to Ref. [ 91 and take (i) $(A,p,g) = g-‘(p)x(A,p,g)g(A), with ?1=(A-~~)-‘(~(A,~~)~rlasA--,~u)andgN exp( c Kixi) where Ki( A) are commuting meromor- phic matrix functions. It is assumed now that there is some region G c C where R( A, p) = 0 in G with respect to A and ,u (we will take this to mean that R = 0

whenever A or p are in G). Also G contains all ze- ros and poles of the g(A) and G contains a neighbor- hood of 00. For Ki( A) a polynomial in A this involves only {co} c G whereas for g = (A - a) -’ it requires {a, oo} c G. Some examples are also used where g N exp( c xiA_‘) with G a unit disc. For now we think of G as some region containing co and 7 = ( A - ,u) -’ which leads to (gt N g( A, x) and g2 N g( A, x’) )

gAA( A, ,x) = 2tiS( A - ,u)

+ J d*v x(v+)g,(V)R(~,A)gl(A)-‘3

C

&~*(A,/_L) = 277i&A - ,u)

- J d*v g2(A)R(A,y)g&T1A*(r+). (2.3) C

One can also take

R(x, A’,A) = g(x, A’)Ro(A’,A)g-‘(x, A)

and in Eq. (2.3) we should think of R as an Ro term and write (2.3) as

3~ [g&.@(A,pu,gr)g;‘(A)] = 27Zi&A - ,u)

x gO)RoW)g;'(Ah (2.4)

Then a(~, A) = gt (v) Ro(v, A)g,‘( A) plays the role of R in Ref. [ 181 and I$. (2.4) becomes, for gt analytic,

&+(A,pu,gr) =2+g,‘(~)&A- p)gt(A)

+ J d2v +(v,cL,gt)Ro(v,A) (2.5) (actually a N 81 here). One can also stipulate an equation (ii) JiR(A,pu,X) = Ki(A)R - RKi(p). Some calculations now give (iii) x*(,u, A, g) = --A( A, ,u, g) and, generally and aside from the formulas (2.3), we know that the functions A, x*, and the gi are analytic in C/G so by Cauchy’s theorem

(A, CL E G)

0 = - J

x(y,A,g,)g,(v)gz’(v)x*(v,CL) dv

= I x(y, A)gd4g,‘(~>xbu, v> dv. (2.6)

Note here that from Eq. (2.3) one knows A( A, ,u) - (A-,u)-’ forA+pandX*(A,p) N (A-_cL)-’ as well. Then from EQ. (2.6) and a residue calculation one obtains for gt = gz another proof of (iii), requiring only that the gi be analytic in C/G. Finally we can write (2.6) in terms of $ via (i), namely

O= J +(V,A,gl)@(pu,v,g;!) dv, (2.7)

aG

which is a more familiar form of the Hirota bilinear identity (but now generalized considerably).

One can derive the DZM equations immediately from the Hirota bilinear identity (2.6) as in Ref. [ 91.

R. Carroll/Physics Letters A 234 (1997) 171-180

Thus write g(Y) = exp[Ki(Y)xi] with Ki = Ai(V - Ai)-’ SO that

gt(v)gz’(v) = exp[Ai(v - Ai)-‘(xi - x:)]

(gl N g(v,A,xi),g2 N g(Y3Ai7xj)).

Look at Eq. (2.6) and differentiate in xi (with xi

fixed); then let XI -+ Xi to obtain

o= S[

aix(v,~,gl)lX(h,v,gl) dG

+x(v,p,gd--&W.gd dv. I 1 Computing residues yields (g N gt )

(2.8)

-aiA(A,~,g) + ~(Ai,p,g)Ai,~(A,Ai,g)

+ --+(A& - x(A,P,g)* =O. I

’ (2.9)

Using the relation (i)

$(A,P*g) = g-‘(P)X(A&g)g(A)

this is immediately seen to be equivalent to

J#(A+Pu,g) =rCI(Ai,~u,g)Ai~(A,Ai,g). (2.10)

To derive the DZM system take for G a set of three

identical unit discs Di with centers at A = 0. The func-

tions Ki( A) have the form Ki( A) = Ai/A for A E Di and Ki( A) = 0 for A $.! Di (the Ai are commuting ma-

trices and each Ki is defined in its own copy of C in

Refs. [ 12,9] - alternatively take, e.g., three distinct discs in C and translate variables accordingly). Eval-

uating Eq. (2.10) for independent variables A, I_L E {Oi, O,i, Ok} one obtains

ai@(A,P) =clr(Oi,~)Ai+(A,Oi),

ailfi(A,O,i) =+(Oi,Oj)AiJI(A,Oi),

Ji$(Oi,P) =cl,(Oi,~)Ai+(Oi,Oj),

J&(O.i~Ok) =+(Oj,Oi)A&(Oi,Ot). (2.11)

Now one can integrate equations containing A, p over aG with weight functions p(A), p( ,u) with the result (no sum over repeated indices)

di@ = fifi* difj = fljifi, 8ifj = fiflij*

dipjk = PjiPik,

where

173

(2.12)

0 = s ~~cL)$(A,cL)P(A) dA dl.L>

pij = (Aj)“2+(Oj,0i)Af’2,

fi = ,$I2 s

cCr(A,Oi)p(A) dA>

fi = /P(p)q(Oi,p)AJJ2 dpu. (2.13)

The system of Eqs. (2.12) implies that

diajfk = [(aj?i>fLT’laiJ”k + [(4~j’j>.T,r’lajfkT

&aj@ = [ (dj~~)~,~l]&@ + [ (aJj)~j~i]dj@,

(2.14)

aiajfk = (&fk>(ft”ajfi> f (ajfk)(fI”aifi)3

aidj@ =a,@f,-‘(ajfi) +aj@f,y*(&fj). (2.15)

The first system in Eq. (2.14) is the matrix DZM equa-

tion with the first system in (2.15) as its dual partner.

At this stage the development is purely abstract; no reference to Egorov geometry or TFT is involved. In this note, regardless of origin, we will refer to

aifj = Pjifi (i #j) 3

@jk = PjiPik (i $ .i $ k) T

Pij = Pjit (2.16)

as a (reduced) DZM system (see Remark 3.1 for the last condition). In addition one will want the condition

a&j=afj=o (a=x&) (2.17)

discussed below (cf. Remark 3.1 in particular). One also recalls that

akPij=PikPkj (i#j $k),

a&j F c?iPij + ajpji

+ ~PinrPmj=” (i#j) (2.18)

m#.j

are referred to as Lame equations. They correspond to vanishing conditions Rij,ik = 0 and Rij,ij = 0 re-

spectively for the curvature tensor of the associated


Egorov metric (see Section 3). Compatibility conditions for Eqs. (2.15) give the equations for rotation

coefficients Pij as in Eq. (2.12). One has the free-

dom to choose the weight function i5 keeping the rotation coefficients invariant, and this is described by

the Combescure symmetry transformation

(f;)-‘a,J = ~L:‘ai~j.

Similarly the dual DZM system admits

(2.19)

The function @ is considered as a wave function for two linear problems (with different potentials) corresponding to the DZM and dual DZM systems. A general Combescure transformation changes solutions for both the original system and its dual (i.e. both p and j5 change). We note also that, according to Refs. [ 2 1,221, the theory of Combescure transformations coincides with the theory of integrable diagonal systems of hydrodynamic type.

It is natural now to ask whether some general WDVV equations (see below) arise directly from DZM as formulated here, without explicit reference to Egorov geometry, etc. We emphasize however that the role of Egorov geometry and its many important connections to integrable systems, TFI, etc. is fundamental here (cf. Section 3) ; we are in fact using the

Egorov geometry to isolate some algebraic features,

after which the geometry disappears (cf. in Refs. [ 61 for example where the need for a general context is indicated). With this in mind consider a scalar situation

where fj N +j and assume flij = pji in Eq. (2.12) as indicated in Eq. (2.16). Then aifj = pjifi and ajfi = piifj which implies Jif: = ajff. This cor-

responds to the existence of a function G such that f,’ = &G = Gi. Then look at (2.15) where (no sums)

fkaiajfk = fkfiaifkajfi + fkfjajfkaifj

f;L f;

GkiGij Gkj Gji * fkaiajfk = 4Gi + -.

4Gj (2.20)

Also from 2fkajfk = Gkj one gets 2aifkdjfk +

2fkdiajfk = Gkji. Similarly 2fidjfi = Gij implies

2akfidjfi + 2fiakajfi = Gijk and Zfjdifj = Gji im-

plies Gjik = 2dk fjaifj + 2fjakaifj. Hence, using (2.20), we get

2ftGijk = 4f$%fkaj fk + 4fi3aidj fk

= GikGjk + fz GkiGij ~ GkjGji 7 -

I I Gj *

This implies

2Gijk GikGjk GkiGij Gkj Gji - -

=-+ Gi Gk + Gj

(2.21)

(2.22)

(such a formula also appears in Ref. [ 171 with ori-

gins in the Egorov geometry; cf. also Remark 3.5). Similarly, ff = a$ and

a@ = fifi + fif$‘jai@ = 4 (GiGij + GiG,) ,

(2.23)

while from Eqs. (2.14) and (2.15) one has

djdi@ = fiajfi + fjaifj,

= f$j fi + fjiai fj. (2.24)

Consequently

fifjWj@ = i( fifj + fj_?i)Gijv

$ifjaidj@ = i( fifj + fjfi)Gij.

Further

(2.25)

aifj ajfi Pij = Pji = fi = fi

=+ fifjpij = ;Gij, fifjpij = &Gij. (2.26)

Note dS0 from (2.12) that a$jk = Pj~Pik, akpij =

PikPkj, and aj&k = pij@jk, while from (2.26) we have

(akfi)fjpij + fi(akfj)&j + fifjakpij = IGijk.

(2.27)

This leads to

4Gijk = fkfjPki&j + fkfipijpjk + fifjpikpkjt

= fkfj&Pkj + fkfiajflik •I- fifjakhj. (2.28)

Such relations seem interesting in themselves. Now one can reverse the arguments and, starting

with G satisfying (2.22)) define

pij = i[Gij/(GiGj)“*] =pji.

R. Carroll/Physics Letters A 234 (1997) 171-180 115

Then immediately

akpij = Gijk

Gii 2(GiG,i)‘J2 - 4(GiGj)3f2

[GikGj + GiGjk]

GikGjk GkiGij Gkj Gji --- Gk + Gi + Gj 1

G, - [ Gik I 21 4(GiGj)‘f2 Gi

= 4Gk~$!!j ,12 = PikPkj. (2.29)

Also for fi = fi one has

ajfi =

Gij 2( Gi) ‘I2

= L2(GiGj)'12Pij =flijfj, 2G!‘*

(2.30)

which is the reduced DZM system (2.16). This shows

that (2.21) characterizes reduced DZM and we have

Theorem 2.1. Referring to Eq. (2.16) as the (reduced) DZM system we stipulate the N indices. Then

a solution of DZM yields a function G satisfying

Eqs. (2.22), (2.26), (2.27), and (2.28) for example.

Conversely given G satisfying (2.22) one can define Pij = i [ Gij/( GiGj)‘i2] such that (2.29) and (2.30)

hold for fi = (G,j) ‘/*, which corresponds to reduced

DZM.

Remark 2.2. One can also develop an analogy of

Eq. (2.21) in a matrix situation (i.e. fj, pijv etc. are matrices). However this requires commutativity as-

sumptions fifj = fjfi for example, along with Gij = G,ii and, e.g., f[‘Gij = Gij f;'. We do not pursue this here.

In order to go from Eq. (2.22) to (reduced) DZM to WDVV in a purely algebraic manner one takes a basis of solutions f/i, for Jifj = pjifi. Here we assume N variables xi which will be denoted by ui in con- formity with standard notation involving WDVV and

Egorov geometry. Classical theory cited in Ref. [4] for example yields N( N - 1) /2 functions flij (i # j) depending on N( N - 1) /2 arbitrary functions of one variable, along with N functions fi depending on N arbitrary functions fi(u’) = fi(O,. . . ,O,u’,O,. . . ,O) determining initial values. Then one chooses a basis

of solutions fipy 1 < p Q N, associated, e.g., to N

successive choices of $ accompanied by N - 1 ze- ros, and we can write (iv) fjp = JjG’ = GT. One should examine this a little more extensively. Thus assume the pij are given and consider the equations for

fi in Eqs. (2.16) and (2.17) (cf. Section 3 and (vi)

therein for more details). For N = 2 there is one pt2

and two equations &f2 = /3]2fl and &fl = /?12f2, leading to (v> 4W:! - (&logPd4h - P:,.h = 0, ala,fl-(allogP]2)a2fl --j?i2fl =Oforwhicha basis of solutions fl = fll, f: = f12, fi = f2], and

fg = f22 can be envisioned (roughly speaking these are hyperbolic equations for which two arbitrary functions should appear in the integration). For N = 3 we

have pt2. p13, and p23 with

&f2 =P21f1, 6f3 = P31fl, 82f3 =P23f2,

a3f2 = P23f37 a3fI = P13f3, a2f1 = p12f2,

(2.31)

leading to equations similar in form to (v) plus others

of third order for example. However it is better to simply think of Eq. (2.3 1) as a system of N = 3

ordinary differential equations of first order. To see this we use the stipulation dfj = 0 = (c &) fj from

(2.17) so that for N = 3 we can write, e.g.

4f2 = P21f19 alf3 = P3]f19

aIfl = -a2fl - a3fl = -b2f2 - h3f3,

or equivalently

(2.32)

3, (;i) =(‘%$7r3) (F). (2.33)

For such systems there is a well known integration

theory leading to a basis fjp as indicated earlier.

Remark 2.3. More generally (cf. Section 3) we will want a basis of solutions of aifj = pji fi satisfying afj = Zfj for a “spectral parameter” z and the con- struction is essentially the same. Thus for N = 3 for

example, instead of Eqs. (2.32) and (2.33) one has

alf1 = Zfl -a2f1 -a3fl = Zfl -P12f2-p13f3 lead-

ing to (2.33) with a z in the ( 1, 1) position in the matrix . In this situation we can provide functions Gp ( z )

(- GP(Z,tk>>, 1 6 p f N, satisfying Eq. (2.22) with

176 R. Carroll/Physics Lelters A 234 (1997) 171-180

afjp = (C&)(G~)‘/*= ~X(G;)-*~*G;~ k

=z(G,P)‘/*+~G;~=~zG;.

k

(2.34)

For z = 0 this reduces to xk G$ = 0.

Remark 2.4. One (degenerate) road to WDVV goes as follows. Set

(2.35)

The WDVV equations can be written as

~‘sCrjkCt?ms = ~rsCi-j!Ckntsv

vij = Clij = vji = const, (2.36)

for which some discussion is given below. Then putting Eq. (2.35) in Eq. (2.36) we obtain (cf. (iv) )

(2.37)

Changing indices a + n and b + p one has

x (GkGe)“* [ ” P

- ( GfG;)‘/2] (2.38)

and one needs conditions which guarantee (2.38). As- sume there is a functional relation GP = GP (6) for some function G (with no a priori restrictions on G) . Then d,Gk = a&$ and this yields (we write G for G for simplicity)

SK_ &Gkd,G &G dGGec3,,G G;

G:, - acG'dpG = a,c = aGGeapG = G’,’ (2.39)

which gives Eq. (2.38). However we will see in Re- mark 3.3 that the stipulation GP = GP(G) leads to a degenerate geometrical situation.

3. WDW

As background for WDVV we refer first to Ref. [ 141 where one considers a 2D TFI with N primary fields & and double correlation functions (4i4j) = qij = vji (cf. also Refs. [ 23-271 for detailed information on some low dimensional situations). The triple COITdatOrS Cijk = (+i$j+k) determine the structure of the operator algebra of the model via

@i ' +j = C;$f’k, Ci = qk”‘Cij”t, (Vii) = (vij)-‘9

(3.1)

along with (. . . $i4j . . .) = Ci(. . . & . . .). This is a

commutative algebra with unity 41 where ct;i = yij with cij = 6;. The Symmetry Of cijk is equivalent to (ab,c) = ( a, c an such algebras (a, b,c E A) are b ) d called Frobenius algebras (FA) . It is of concern here to consider algebras A(t), (t = t’, . . . , tN) with cijk = Cijk ( t) and vii = const (we assume here that A ( t) has no nilpotents; the decomposable case). Then in the TIT situation One can write cijk = &aj&F(t) for a function F called the primary free energy. The conditions of associativity give rise to the WDVV equations

a3F a3F

VP’ atpatfiaty ataat”atu = rl w d3F a3F

atpatpata at?atAatV *

(3.2)

Connections to the Egorov geometry arise via the Darboux-Egorov (DE) integrable system (cf. (2.16))

akYij(U) =Yik(U)Ykj(u) (i #j $k, no Sum),

N

@ij= cak yij=O,

( ) 1 (3.3)

with yij ( U) = rji (u) for i $ j (evidently yij m pij for Eqs. (2.16) and (2.17)). The ui are new coordinates ui = u’(t) defined via

N atk sum sum c;(t) = c -7;,

m=, sum at’ at] (3.4)

and the functions yij (u) are expressed via the com- ponents of the metric Qij where

atk at”! &j(U) = ~krn~~ = &?iiaij,


(3.5)

Next one recalls that a diagonal metric ds* = C;” gii( u) (dU’)* determines curvilinear orthogonal coordinates in a Euclidean space if and only if its curvature vanishes. This is called an Egorov metric if the rotation coefficients yij = (lYjdm/ Jgm) for (i # j) satisfy yij ( U) = yji ( U) . Equivalently a potential V = V(U) exists such that gii(U) = &V(U) (i = 1 . . 7 N) . Vanishing of the curvature of the Egorov metric is equivalent to the integrable system (3.3) where (3.3) corresponds to the compatibility conditions of the system (vi) 6’j#i = yij$j (i $ j, no sum), &,bj = z$j, which is related to the N-wave interaction system. Note here that compatibility requires

aaj*i = (aYij)$j + Yija*j = (aYij>$j + ZYij

= aja& = Zaj$i = ZYij+j, (3.6)

so &‘yij = 0. The Egorov zero curvature metric ds* is called a invariant if agii( u) = 0 for i = 1,. . . , N and then it can be specified uniquely by its rotation coefficients (no arbitrary constants) via solution of the system (vi) for z = 0 (in particular a+j = 0 as in (2.17)). The same is true for the corresponding flat coordinates t’. Thus consider the system (vii) aj$i = yij+j with a@i = 0 for some solution yij = yji of (vi). Then via yij = aj&/& and 6’gii = 0 it follows that @; = 6 is a solution of (vii). Conversely any solution (CI of (vii) determines a a invariant Egorov metric with the same rotation coefficients via gii = (+ii )*. Let @it (u) , . . , $iN be a basis in the space of solutions of (vii) (cf. Section 2); then one can show that the scalar product vij = C;” @mi (u) #mj (u) is nonde- generate and does not depend on u (cf. Ref. [ 151 for details). The flat coordinates t’ are then determined by quadratures from the system

i$tj = (jlil lj$ Z &$I!, I+: = qjk$ik, (3.7)

and one notes that tl = q,it’ = V is the potential of this metric. Indeed one obtains tj via (3.7) (i.e. &tj = i,bil $b{ = &vjk#ik ) and in particular

&tl = aiT jfj = &~,j~jk8+b&

= &$il = gii = (/li: z&V fv t] = V (3.8)

Note here that from aj#i = yij+j and yij = yji one has (?j@i/$j) = (a&j/@i) which implies aj+T = dilli,?;

hence in particular this holds for (Gj = $jt . It follows that ds2 is in fact a a invariant Egorov metric of zero curvature. Finally one proves as in Ref. [ 141 that any solution of the WDVV equations (in the decomposable case) is determined by a solution Yij = yji of the integrable system (3.3) and by N arbitrary constants via gii = ($ii)*, vij = Cf”$~~i&ln,j~ (3.7), and the formula

N ‘$mi$nrj$mk Cijk(t) =c ccI .

I ml (3.9)

One also has the orthogonality conditions

atj 2

= gii !&k.i

ad atj - giiT@ = a,ir]jp (i=l,...,N), (3.10)

and the transformations

Note that since &? = b’iV one can distinguish between the $ip via different functions VP as in Theorem 2.1. Then for (clip N (&VP) ‘/* one can define the ,$ as

5: = C $fjp$ji = C (ajVpajVi) “* , (3.12)

which will agree with the definition via ,$’ = ~(&d/at’)+j,Qj~ (cf. Eq. (3.10)). Note also

(3.13)

(3.14)

This material is all gathered together in Ref. [ 151 where many details are spelled out.

Remark 3.1. In connection with the Lame equations (2.18) let US note the following. From pij = aj fi/ fj = i [Gij/( GiGj)“*] we have

a&j = 1 PikPkj = a&j + ajflji + c &iPmj m m#.j

= 4( GiGj)"* m l c(Y) =o. (3.15)


Consequently we have

Theorem 3.2. The condition Apia = 0 of Eq. (2.18) or (3.15) corresponds to B2 = 0 where B is the matrix B = (pij). Then fitting our functions GP or G to the geometric environment involves a stipulation

(3.16)

together with Fq. (2.22).

Remark 3.3. Let us reconsider the resolution [;p = Gy with GP = GP(6) of Remark 2.3 with regard to the geometrical meaning connected with yij N Pij, +j N fj, and UT = dP’/6Vj N (+mj/+ml) from (3.14). One obtains then from Rq. (2.39)

(3.17)

SO Eq. (3.14) involves

f nk -N fn1u;l fpk _ fplu:: -=-- f ne f”l$ fP( m

UI 4 - u; G sun aup j-=~.yTacy”---$&-.

UP” ue uk ue atk (3.18)

Hence the Jacobian of the transformation ti -+ d, namely J = (d (uj) /a (t’) ) will be “totally” degenerate. Consequently GP = GP(G) is not a resolution with significant meaning and we will now use the un- derlying geometry to obtain WDVV, while eventually phrasing matters in terms of Gp (but GP $ GP( G) ).

To produce a mechanism leading from DZM to WDVV via the GP we first recall a few more formulas from Refs. [ 15,3,14]. First note that (3.4) corresponds to

ad aup k aup -- ati atj = Cij,t,*

since

c aup aup

z-y_

ati at3

(3.19)

(3.20)

The WDVV equations (2.36) can be also expressed as

k nt CijCek = c;jcy;, (3.21)

and this (plus &c;i = a$$) follows as compatibil-

ity conditions for (viii) a& = z c +k. Here tj, as

given in Eq. (3.11) via (ix) ti = C(&4j/&i)#j#j,, will not do since Eq. (3.11) corresponds to z = 0 and we must go to the z dependent $j (note however that z = 0 is needed to produce a uniquely defined zero curvature Egorov metric). A direct calculation is con- ceivable but the argument below will suffice to char- acterize WDVV via the Gy .

First we note that it is really quite wise to follow Refs. [ 3,141 in writing ta and ui, with summation on repeated Greek indices understood. This will enable us to make sense of formulas involving derivatives of ti N ta or @j with respect to t@ and ui together. In this context then let us isolate some key features of the WDVV-DZM correspondence from Refs. [ 3,141. To begin we elaborate on the introduction of the spectral parameter z in (v) and in the equations

a& = zc;p(t)5y (aIt = ~6~). (3.22)

In an ad hoc spirit an elementary calculation shows that compatibility of Eq. (3.22) is equivalent to WDVV in the form (3.21) along with (x) a& = a,&. Simi- larly compatibility of (vi) is equivalent to the DE system (3.3) (cf. (3.6) ) . Hence let us go to the (Gjp ( z ) N fjp(z) 3 fjp(Z,tk) of Remark 2.3 with q(z) E g(Z,rk) (1 6 j,p < N). FromRefs. [3,14] we let M be a space of parameters P and consider a Frobe- nius algebra (FA) deformation ta -+ fa( t, z) (cf. Refs. [3,14] for the language of FA and Frobenius manifolds (FM) ). For 0, the Levi-Civita connection of the z = 0 metric one writes

WZ)U=~,U+ZWU (V,(Z) -a/al,),

and then 27 is flat, uniformly in z, if and only if WDVV holds with ccrar = &a&F. Here the c& are based on the z = 0 metric and WDVV as in Eq. (3.21) for example expresses an associativity condition for the corresponding FA. Further any [a satisfying (3.22) is a gradient ta = &i for some function i and one obtains a fundamental system of solutions of (3.22) via tt = a&. Actually the i are specified via

a, asi = z&a< i, (3.23)


and one will have a coordinate family &. ( 1 < y < iv’)

with Si = a&,, where & = tY + zuy + 0( z*). The uy

are uniquely determined up to a transformation uy -+ uy + TitP for T a constant matrix. Further there is a

formula,

$-I al(t(n),z)

@il = $i(U,Z), (3.24)

which establishes a l-l correspondence between solu-

tions of (vi) and (3.22). Here d is a scaling parameter

arising in the consideration of self-similar solutions of WDVV and we take it to be 2 here in order to elimi- nate the z factor in (3.24) (cf. Refs. [3,14,16,17] for

a more comprehensive treatment of WDVV) . With this background we pick now G;(z) N

f;,, (z ) N I,!& (z ) as in Remark 2.3, so (vi) is sat- isfied, along with Eq. (3.3). Note that this implies

Jpij = 0 SO (3.15) holds true automatically. From (3.24) we then obtain la(u, z) and tp via

~$4 = aifa = $!‘il $F = $i/ilr]a”k@ik, (3.25)

which reduces to Eq. (3.7) for z, = 0. This leads to (3.22) whose compatibility implies WDVV. Hence

Theorem 3.4. Let G$’ N f$ N I,$$, be functions of z

as in Remark 2.3, satisfying (2.22) and (2.34). Then

one obtains (vi), and (3.3) holds for yij N pij, which

implies that (3.15 ) follows automatically. Further one obtains WDVV with Cijk as in (2.35) by means of the

map $jk 4 (5 of (3.25). This means that such a q characterizes WDVV as well as reduced DZM with

(2.17).

Remark 3.5. For convenience we list together the stipulations on the GP. Thus generically

(A) C Gjk = 2zGj, k

GikGkj GkiGij + Gkj Gji (B) 2Gijk=T+T 7. (3.26)

, J

Then it follows that

(C) Cl (GintGmj)/Gml = 0, “I

and to see this we note from (3.26) that

2zGij = C Gijk k

GikGkj =2zC,+iCT.

k

(3.27)

In terms of the metric gii N & N ff, - Gf the conditions (A) and (B) for G’ can be written in the

form

2didkgjj =

aigkkajgkk + akgiidjgii + dkgjjaigj,j

gkk gii gjj (3.28)

where Gji = aigjj = G,!i = Jjgii, etc.

Acknowledgement

The author would like to thank Y. Nutku for stim- ulating conversations on WDVV and related topics.

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wdvv and dzm

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