A scheme for solving quasiclassical string equations is developped to prove
that genus-zero Whitham hierarchies describe the deformations of planar domains
determined by rational conformal maps. This property is applied in normal
matrix models to show that deformations of simply-connected supports of
eigenvalues under changes of coupling constants are governed by genus-zero
Whitham hierarchies.Comment: 12 pages, 3 figure

Alonso, Luis Martinez

Medina, Elena

Physics Letters B

English

arXiv

©2006 Elsevier B.V.
Partially supported by MEC project FIS2005-00319A scheme for solving quasiclassical-string equations is developed to prove that genus-zero  hitham hierarchies describe the deformations of planar domains determined by rational conformal-maps. This property is applied in normal matrix models to show that deformations of simply-connected supports of eigenvalues under changes of coupling constants are governed by genus-zero Whitham hierarchies.Ministerio de Educación y Ciencia, SpainDepto. de Física TeóricaFac. de Ciencias FísicasTRUEpu

Martínez Alonso, Luis

Medina Reus, Elena

Docta Complutense

Physics Letters B 641 (2006) 466–473www.elsevier.com/locate/physletbGenus-zero Whitham hierarchies in conformal-map dynamics ✩Luis Martínez-Alonso a,∗, Elena Medina ba Departamento de Física Teórica II, Universidad Complutense, E-28040 Madrid, Spainb Departamento de Matemáticas, Universidad de Cádiz, E-11510 Puerto Real, Cádiz, SpainReceived 17 February 2006; accepted 24 August 2006Available online 25 September 2006Editor: L. Alvarez-GauméAbstractA scheme for solving quasiclassical string equations is developed to prove that genus-zero Whitham hierarchies describe the deformations ofplanar domains determined by rational conformal maps. This property is applied in normal matrix models to show that deformations of simply-connected supports of eigenvalues under changes of coupling constants are governed by genus-zero Whitham hierarchies.© 2006 Elsevier B.V. All rights reserved.MSC: 58B20Keywords: Whitham hierarchy; Conformal maps; Normal matrix models1. IntroductionConformal mapping methods have been effectively applied in the analysis of interfacial free-boundary problems involving planardomains [1]. They have provided many exact solutions [2–4] which stimulated the research on possible underlying integrablestructures. Thus, Wiegmann and Zabrodin discovered [5,6] that deformations of simply-connected domains with respect to changesof their exterior harmonic moments, treated as independent variables, are described by the dispersionless Toda hierarchy. They alsoformulated an algebro-geometric analysis [7,8] of the deformations of multiply-connected domains in terms of Whitham equationsfor Abelian differentials.Recent research has shown [9,10] that many exact solutions of Laplacian growth models correspond to a special type of domainscalled algebraic or quadrature domains. In the simply-connected case the complement of a quadrature domain D is the image ofthe exterior of the unit disk under a conformal map given by a rational function(1)z(w) = r w +N0∑n=0u0,nwn+k∑s=1Ns∑n=1us,n(w − as)n ,where the coefficient r is a positive number and the k poles as = 0 lie inside the unit circle. In this work we prove that thedeformations of rational conformal maps under changes of the parameters (r, u0,n, us,m, as) such that z(ā−1s ) are kept constant, turnout to be described by a solution of the genus-zero Whitham hierarchy W(n) with n = 2k +2 punctures [11]. It should be noted thataccording to a recent general result by Takasaki [12] the Whitham hierarchy W(n) is the quasiclassical limit of the n-componentKP hierarchy.✩ Partially supported by MEC project FIS2005-00319.* Corresponding author.E-mail address: luism@fis.ucm.es (L. Martínez-Alonso).0370-2693/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2006.08.082L. Martínez-Alonso, E. Medina / Physics Letters B 641 (2006) 466–473 467Our analysis is based on solving a system of quasiclassical string equations which leads to the characterization of the conformalmap (1) as a function z(w, t) of the Whitham times t . For k = 0 our result agrees with [5] (see also [13]) since W(2) is thedispersionless Toda hierarchy. However, for k  1 the analysis of [5] does not apply because the Schwarz function of the boundaryof D has poles outside D, consequently there are infinite exterior harmonic moments of D different from zero and, since z(w)depends on a finite number of parameters only, these harmonic moments are not independent variables.Deformations of quadrature domains naturally arise in the analysis of partition functions of N × N normal matrix models [14](2)ZN =∫e1h̄W(M,M†) dM dM†,with quasiharmonic potentials W(z, z̄) := −zz̄ + V (z) + V (z) of the form(3)V (z) =N0+1∑n=1znt0,n +k∑s=1(−ts,0 log(z − βs) +Ns−1∑n=1ts,n(z − βs)n).Integrating over eigenvalues and ignoring normalization factors, the partition function reduces to(4)ZN =∫ ∏i>j|zi − zj |2e 1h̄∑j W(zj ,z̄j )∏jd2zj .In the large N limit (N → ∞, h̄N fixed) the eigenvalues densely occupy a bounded quadrature domain D in the complex plane(the support of eigenvalues). As a consequence of our analysis we prove that for simply-connected supports of eigenvalues thecorresponding rational conformal map z = z(w, t) as a function of the coupling constants t of the partition function represents asolution of the Whitham hierarchy W(2k + 2).2. String equations in Whitham hierarchiesThe elements of the phase space for a genus-zero Whitham hierarchy W(M + 1) are characterized by M + 1 punctures qα (α =0, . . . ,M), where q0 := ∞, of the extended complex p-plane and an associated set of local coordinates of the form(5)z0 = p +∞∑n=1c0,npn, zi = dip − qi +∞∑n=0di,n(p − qi)n, i = 1, . . . ,M.In what follows Greek and Latin suffixes will label indices of the sets {0, . . . ,M} and {1, . . . ,M}, respectively. We will henceforthsuppose that there exist positively oriented closed curves Γμ in the complex planes of the variables zμ such that each function zμ(p)determines a conformal map of the right-exterior of a circle γμ := z−1μ (Γμ) on the exterior of Γμ (we will assume that the circle γ0encircles all the γi ) (see Fig. 1).The flows of the Whitham hierarchy can be formulated as the following infinite system of quasiclassical Lax equations(6)∂zα∂tμ,n= {Ωμn, zα},associated to the series of time parameters {t0,n: n  1; ti,n: i = 1, . . . ,M,n  0}. Here the Poisson bracket is defined as {F,G} :=∂pF∂xG − ∂xF∂pG, x := t01 and the Hamiltonian functions are(7)Ωμn :=(znμ)(μ,+), (n  1), Ωi0 := − log(p − qi),Fig. 1. Right-exteriors of γμ and Γμ .468 L. Martínez-Alonso, E. Medina / Physics Letters B 641 (2006) 466–473where (·)(i,+) and (·)(0,+) stand for the projectors on the subspaces generated by {(p −qi)−n}∞n=1 and {pn}∞n=0 in the correspondingspaces of Laurent series. These hierarchies [12] are the dispersionless limits of the multi-component KP hierarchies. In particular,for M = 0 and M = 1 they represent the dispersionless versions of the KP and Toda hierarchies, respectively.In our analysis we will use an extended Lax formalism with Orlov functions(8)mα(zα, t) =∞∑n=1ntα,nzn−1α +tα,0zα+∑n2vαnznα,which verify the same Lax equations (6) as the variables zα , and such that {zα,mα} = 1. The parameter t0,0 in (8) is defined by(9)t0,0 := −M∑i=1ti,0.The Whitham hierarchy can be formulated as the following system of equations(10)dzα ∧ dmα = dω, ∀α,where ω is the one-form(11)ω :=∑μ,nΩμn dtμ,n.To see how to get from the system (10) to the Whitham hierarchy, note that by identifying the coefficients of dp∧dtμn and dx ∧dtμnin (10) we obtain(12)∂zα∂p∂mα∂tμn− ∂mα∂p∂zα∂tμn= ∂Ωμn∂p,∂zα∂x∂mα∂tμn− ∂mα∂x∂zα∂tμn= ∂Ωμn∂xand, in particular, since Ω01 = p, for (μ,n) = (0,1), the system (12) implies {zα,mα} = 1. Thus, using this fact and solving (12)for ∂tμnzα and ∂tμnmα , the Lax equations for (zα,mα) follow.A natural form of characterizing solutions of Whitham hierarchies is provided by systems of string equations(13){Pi(zi,mi) = P0(z0,m0),Qi(zi,mi) = Q0(z0,m0), i = 1,2, . . . ,M,where {Pα,Qα}Mα=0 satisfy {Pα(p,x),Qα(p,x)} = 1. Given a solution (zα(p, t),mα(p, t)) of a system (13), if we denotePα(p, t) := Pα(zα(p, t),mα(p, t)), Qα(p, t) := Qα(zα(p, t),mα(p, t)),it is clear that(14)dPα ∧ dQα = dPβ ∧ dQβ, ∀α,β.On the other hand {Pα(p,x),Qα(p,x)} = {zα,mα} = 1, so that solutions of a system of string equations verify(15)dPα ∧ dQα = dzβ ∧ dmβ, ∀α,β.Theorem. Let (zα(p, t),mα(p, t)) be a solution of (13) which admits expansions of the form (5), (8), (9) and such that the co-efficients of the two-forms (15) are meromorphic functions of the complex variable p with finite poles at {q1, . . . , qM} only. Then(zα(p, t),mα(p, t)) is a solution of the Whitham hierarchy.Proof. In view of the hypothesis of the theorem the coefficients of the two-forms (15) with respect to the basis{dp ∧ dtα,n, dtα,n ∧ dtβ,m}are determined by their principal parts at qμ (μ = 0, . . . ,M), so that by taking (15) into account we may writedzα ∧ dmα =M∑μ=0(dzμ ∧ dmμ)(μ,+), ∀α.Moreover the terms in these decompositions can be found by using the expansions (8) of the functions mμ as followsdzμ ∧ dmμ = dzμ ∧( ∞∑n=1nzn−1μ dtμ,n +dtμ,0zμ+∑n2dvμnznμ)= d( ∞∑n=1znμ dtμ,n + log zμ dtμ,0 −∑n21n − 1dvμnzn−1μ),L. Martínez-Alonso, E. Medina / Physics Letters B 641 (2006) 466–473 469so that(dzμ ∧ dmμ)(μ,+) = d( ∞∑n=1(znμ)(μ,+) dtμ,n − (1 − δμ0) log(p − qμ)dtμ,0)= d(∑nΩμn dtμ,n).Thus we finddzα ∧ dmα = dω = d(∑μ,nΩμn dtμ,n), ∀αand, consequently, this proves that the functions (zα(p, t),mα(p, t)) determine a solution of the Whitham hierarchy. 3. Integrable dynamics of quadrature domainsLet us consider a rational conformal map z(w) of the form (1) with k poles as inside the unit circle and define(16)z̃(w) := z(w̄−1) = rw+N0∑n=0ū0,nwn +k∑s=1Ns∑n=1ūs,n wn(1 − wās)n .In order to establish the connection between the deformations of the conformal map z(w) and the genus-zero Whitham hierarchieswe introduce the change of variablep = Rw := rw + u0,0,where r and u0,0 are the first coefficients of z(w) in (1). As a function of the new variable p the conformal map is normalized atinfinityz(p) = p +O(1/p), p → ∞.Moreover, if we defineq0 := ∞, qs := Rbs, qs+k := Ras, q2k+1 := u0,0,it is clear that z(p) and z̃(p) become rational functions of p with poles at (q0, qs, q2k+1) and (q0, qs+k, q2k+1) respectively.We are going to prove that deformations of z(w) with respect to the coefficientsu := (r, u0,n, us,m, as),such that βs = z(w)|bs are kept constant, are described by the Whitham hierarchy W(2k + 2). To this end we introduce Whithamvariables (zα,mα) on the 2k + 2 punctures qα of the p-plane(17)⎧⎪⎪⎪⎨⎪⎪⎪⎩z0 = z, m0 = z̃ (near q0 = ∞),zs = 1z−βs , ms = −(z − βs)2z̃ (near qs),zs+k = 1z̃−β̄s , ms+k = (z̃ − β̄s)2 z (near qs+k),z2k+1 = z̃, m2k+1 = −z (near q2k+1).It is clear that these variables are rational functions of p with possible poles at the punctures qα only. Moreover, they satisfy thesystem of string equations(18){ 1zs+ βs = ms+kz2s+k = −m2k+1 = z0,−msz2s = 1zs+k + β̄s = z2k+1 = m0.Obviously the functions zα are of the form (5). On the other hand due to (1) and (16) it follows that the functions mα defined in(17) verify expansions of the form (8)m0 = z̃ =N0+1∑n=1nt0,nzn−10 +t0,0z0+ · · · , w → ∞,ms = −z−2s z̃ =Ns−1∑n=1nts,nzn−1s +ts,0zs+ · · · , w → bs,ms+k = z−2s+kz =Ns−1∑nts+k,nzn−1s+k +ts+k,0zs+k+ · · · , w → as,n=1470 L. Martínez-Alonso, E. Medina / Physics Letters B 641 (2006) 466–473(19)m2k+1 = −z =N0+1∑n=1nt2k+1,nzn−12k+1 +t2k+1,0z2k+1+ · · · , w → 0,where the time parameters t := (tα,n) are rational functions in u(20)tαn = Qα,n(u).The functions Qα,n(u) satisfy certain constraints which can be characterized by considering the map(21)C :f → Cf, Cf (w) := f (w̄−1).Observe that in terms of the variable p(22)Cf (p) = f (Ip),where Ip = r2/(p − u0,0) + u0,0 is the inversion with respect to the circle |p − u0,0|2 = r2. From (17) it is clear that(23)z2k+1 = Cz0, m2k+1 = −Cm0, zs+k = Czs, ms+k = −Cms,which implies(24)Q2k+1,n(u) = −Q0,n(u), Qs+k,n = −Qs,n(u).Furthermore, we can prove that(25)∑αQα,0(u) = 0.Indeed from (17) we deducem0 dz0 = ms dzs = z̃ dz, m2k+1 dz2k+1 = ms+k dzs+k = −z dz̃.Hence2πi∑αQα,0 =∑α∮Γαmα dzα =∑α∮γαz̃ ∂pz dp = 0,where we have taken into account that z̃ ∂pz is a rational function of p with poles at the punctures qα only, and the fact that∑αγα ∼ 0 in C \ {q1, . . . , q2k+1}.Notice that due to (24) the constraint (25) can be rewritten as(26)ImQ0,0 +∑sImQs,0(u) = 0.Under appropriate conditions one can determine u as a function of (t, βs, β̄s). To this end we consider the system(27){tα,n = Qα,n(u),βs = z(bs,u),where the time parameters(28)t := (tα,n), α = 0, . . . ,2k + 1; n = 0, . . . , Ñα; Ñ0 = Ñ2k+1 = N0 + 1, Ñs = Ñs+k = Ns − 1,are assumed to satisfy(29)t2k+1,n = −t̄0,n, ts+k,n = −t̄s,n,∑αtα,0 = 0.Firstly, we observe that u constitutes a set of(30)2(N0 +∑sNs + k)+ 3,real variables given by r and the real and imaginary parts of (u0,n, us,m, ai). On the other hand, in view of (24) and (25) we mayignore the equations corresponding to α = s + k, 2k + 1 in (27). In this way the system (27) reduces to 2(N0 + 2 + ∑s Ns + k)real equations, but due to (26) one of them is a consequence of the others. Therefore, we are lead to a system of equal number ofequations and unknowns which under appropriate conditions will determine u, and consequently (zα,mα) as functions of (t, βs).L. Martínez-Alonso, E. Medina / Physics Letters B 641 (2006) 466–473 471In this way we have determined a rational solution (zα(p),mα(p)) of the system of string equations (18) which depends on(t, β1, . . . , βk) and satisfies the asymptotic conditions (5) and (8). Therefore, from the above theorem we conclude that this solutionevolves with respect to t according to the Whitham hierarchy W(2k + 2).It is interesting to notice the following identity involving the times tα0 and the area T of the domain D. Let Γ be the positivelyoriented boundary of D and take small positively oriented closed curves Γ ′s around the points βs of the z-plane, then we have that(31)t0,0 = 12πi∮Γ0z̃ dz =∑s12πi∮Γ ′sz̃ dz + 12πi∮Γz̃ dz =∑s12πi∮Γsz−2s z̃ dzs +12πi∮Γz̄ dz = −∑sts,0 + 1πT .4. Exact solutionsThe above analysis can be used to generate solutions of W(2k + 2) of the form (1). Let us consider the case in which only simplepoles arisez = rw + u0 +k∑s=1vsw − as , z̃ =rw+ ū0 +k∑s=1v̄sw1 − āsw .By identifying the coefficients of z00 and z−10 in the expansion of m0 = z̃ as z0 = z → ∞ (w → ∞), we get(32)ū0 −k∑s=1v̄sās= t0,1, r(r −k∑s=1v̄sā2s)= t0,0,and identifying the coefficient of z−1s in the expansion of ms = −(z − βs)2z̃ as zs → ∞ (w → bs ) yields(33)v̄sā2s(r −k∑s′=1vs′ ā2s(1 − as′ ās)2)= ts,0.Finally, the equations z(w)|w=bs = βs read(34)rās+ u0 +k∑s′=1vs′ ās1 − āsas′ = βs.The system (32)–(34) determines (r, u0, vs, as) in terms of (t0,0, t0,1, ts,0, βs).ExamplesA solution of W(4) is obtained by deforming the conformal map (aircraft wind)z = rw + u0 + vw − a , z̃ =rw+ ū0 + v̄w1 − āw .The solution of the corresponding equations (32)–(34) is given by:a = r2 − At0,0r(β̄ − t0,1)(1 − A), v = (r2 − t0,0)(r2 − t0,0A)2r3(β̄ − t0,1)2(1 − A)2,u0 = r4 − r2t0,0A + r2t0,1(β̄ − t0,1)(1 − A) + |t0,0|2A − r2t0,0r2(β̄ − t0,1)(1 − A),withr2 = 2|t0,0|2A2|β̄ − t0,1|2(2A3 − 3A2 + A) + (t0,0 + t1,0)(A − 1) + (t0,0 + t0,0)A,where A = |a|2 is implicitly defined by2|β̄ − t0,1|4A5 +[2(t0,0 + t1,0) − 2(t0,0 + t0,0) − 5|β̄ − t0,1|2]|β̄ − t0,1|2A4+ 4[(t0,0 + t0,0) − (t0,0 + t1,0) + |β̄ − t0,1|2]|β̄ − t0,1|2A3+ [(t0,0 + t1,0)2 − (t0,0 + t0,0)2 + 4|t0,0|2 − 2(t0,0 + t0,0)|β̄ − t0,1|2 + 2(t0,0 + t1,0)|β̄ − t0,1|2 − |β̄ − t0,1|4]A2− 2(t0,0 + t1,0)2A + (t0,0 + t1,0)2 = 0.472 L. Martínez-Alonso, E. Medina / Physics Letters B 641 (2006) 466–473Fig. 2. Solution corresponding to k = 1. Fig. 3. Solution corresponding to k = 2.It can be proved that a reduction of the system (32)–(34) is obtained by setting(35)z = rw + u +l∑s=1(vsw − as +v̄sw − ās), z̃ = rw+ u +l∑s=1(v̄sw1 − āsw +vsw1 − asw),and by assumingu, t0,0, t0,1 ∈ R, ts+l,0 = t̄s,0, βs+l = β̄s .In the simplest case l = 1 the reduced system of implicit equations readsu − v1a1− v̄1ā1= t0,1, r(r − v1a12− v̄1ā21)= t0,0, v̄1(rā21− v̄1(1 − ā21)2− v1(1 − a1ā1)2)= t1,0,(36)rā1+ u + v1ā11 − a1ā1 +v̄1ā11 − ā21= β1,and determines a solution of W(6).Figs. 2 and 3 exhibit deformations of domains with respect to changes of the area T such that the Whitham times ts,0 are keptconstant. Observe that the evolution of the boundary develops cusp-like singularities [15].5. Whitham times and coupling constants of normal matrix modelsLet us assume that the support of eigenvalues of the normal matrix model (2) is a simply-connected domain D. For example thisis the case if k = 1 [8]. Then by applying the saddle point method to the large N limit of (4) it follows that(37)z̄ = V ′(z) + 12πi∫ ∫Ddz′ ∧ dz̄′z′ − z , z ∈ D.Let us consider now the rational conformal map z(w) associated to D, from (37) it follows that the function z̃(w) can be extendedas a meromorphic function of z outside D by(38)z̃(z) = V ′(z) + 12πi∫ ∫Ddz′ ∧ dz̄′z′ − z .In fact z̃(z) represents the Schwarz function of the boundary Γ of D. By using (3) and (17) one can rewrite (38) as(39)z̃(z) =N0+1∑n=1nzn−10 t0,n −k∑s=1(Ns−1∑n=1nzn+1s ts,n + zsts,0)+ 12πi∫ ∫Ddz′ ∧ dz̄′z′ − z .By comparing this identity with our definition (19) of the Whitham times and by taking into account (31), we conclude that theWhitham times coincide with the coupling constants of the normal matrix model.Finally, we notice that by using the strategy deployed in the proof of the Theorem of Section 2 and by taking (39) into account,it follows that the deformations with respect to the parameters βs and β̄s are described by the flows(40)∂zα∂βs= ts,0 ∂zα∂ts,1+Ns−2∑n=1nts,n∂zα∂ts,n+1+ (Ns − 1)ts,Ns−1{(zNss)(s,+), zα},L. Martínez-Alonso, E. Medina / Physics Letters B 641 (2006) 466–473 473and(41)∂zα∂β̄s= ts+k,0 ∂zα∂ts+k,1+Ns−2∑n=1nts+k,n∂zα∂ts+k,n+1+ (Ns − 1)ts+k,Ns−1{(zNss+k)(s+k,+), zα}.References[1] M.Z. Bazant, D. Crowdy, Conformal mapping methods for interfacial dynamics, in: S. 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Genus-zero Whitham hierarchies in conformal-map dynamics

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Genus-zero Whitham hierarchies in conformal-map dynamics

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