Table Of Content

Topologi ally Linked Polymers are Anyon Systems ∗ 1,2, Fran o Ferrari 1 Institute of Physi s, University of Sz ze in, ul. Wielkopolska 15, 70-451 Sz ze in, Poland 2 Max Plan k Institute for Polymer Resear h, 10 A kermannweg, 55128 Mainz, Germany 4 0 We onsider the statisti al me hani s of a system of topologi ally linked polymers, su h as for 0 instan e a dense solution of polymer rings. If the possible topologi al states of the system are 2 distinguished usNing the Gauss linking number as a to2pNologi al invariant, the partition fun tion of n anensemNbleof losedpolymers oin ides with the pointfun tionofa(cid:28)eldtheory ontaining a set of omplex repli a (cid:28)elds and Abelian Chern-Simons (cid:28)elds. Thanks to this mapping to a (cid:28)eld theories, some quantitative predi tions on the behavior of topologi ally entangled polymers J have been obtained by exploiting perturbative te hniques. In order to go beyond perturbation 7 theory,a onne tion betweenpolymers andanyonsis established here. Itis shown in this waythat the topologi al for es whi h maintain two polymers in a given topologi al on(cid:28)guration have both ] h attra tive andrepulsive omponents. Whentheseopposite omponentsrea h a sort ofequilibrium, c the system (cid:28)nds itself in a self-dual point similar to that whi h, in the Landau-Ginzburg model e for super ondu tors, orresponds to the transition from type I to type II super ondu tivity. The m s4ig−npi(cid:28)la atn e of self-dualityin polymer physi sisillustrated onsidering theexampleoftheso- alled - on(cid:28)gurations, whi h are ofinterest in thebio hemistryof DNApro esses like repli ation, t trans riptionandre ombination. The aseofstati vortexsolutionsoftheEuler-Lagrangeequations a is dis ussed. t s . t ξµ = ( ,t) µ = 1,2,3 = (x1,x2) a In this letter we onsider the statisti al me hani s of tor r , where , r and m x3 = t a system of topologi ally linked polymers[1℄, su h as for plays the role of time. We need also to intro- - instan e a dense solution of polymer rings ( atenanes). du ethetwo-dimensional ompletely antisymmetri ten- d ǫij i = 1,2 n If the possible topologi al states of the system are dis- ǫsoµνrρ µ,,ν,ρ = 1,a2n,d3its three dimensional ounterpart o tinguished using the Gauss linking number as a topolog- , . These tensoǫ1r2s =ar1e uniqǫu1e2l3y=de1- c i al invariant,it hasbeen shownin [2℄ that the partition (cid:28)ned by the following onventions: and . [ N fun tion o2fNan ensemble of losed polymers oin ides We use middle latin letters as two-dimensional spa e in- 1 with the point fun tion of a (cid:28)eld theory ontaining di es and middle greek letters as three dimensional in- N v a set of omplex repli a (cid:28)elds and Abelian Chern- di es. Sum over repeated spa e indi es will be every- Γ 2 a Simons (C-S) (cid:28)elds. Thanks to this mapping to (cid:28)eld where understood. The traje tories of the polymers, 0 a=1,...,N theories, some quantitative predi tions on the behavior , will be treated as ontinuous urves. Usu- 1 1 of topologi ally entangled polymers have been obtained ally,thesetσrajeΓ tor=ies(are(pσar)a,mt e(tσriz))ed0bymσeansLofthLeir a a a a a a a a a 0 by exploiting perturbative methods, see Ref. [3℄ and ref- ar -length : { r | ≤ ≤ }, 4 eren es therein. If onewishes to go beyond perturbation being the total length of the traje tory. To make a on- 0 theory, however, one is fa ed with the problem of (cid:28)eld ne tion with anyons, however,it is onvenientto use the / t t t theorieswhi h ontainmulti- omponents alar(cid:28)eldsand time as a parameter. Of ourse, the oordinate is a m gauge (cid:28)elds intera ting together. Clearly, it is not easy not a good parameter if weΓdo not introdu e a suitable a to study theories of this kind even in the simplest ase of se tioning for the urves . To this purpose, let us - N = 2 t d . The idea whi h we wish to pursue here in or- noti e that, with respe t to the −dire tion, ea h urve Γ n der to ir umvent su h di(cid:30) ulties is to establish a link dtaa(hσaa)s=ma0xima and minima de(cid:28)nedsby the ondition: o between polymers and anyons [4℄. Anyon (cid:28)eld theories dσa . The number of minima, a, is equal to the c : have been in fa t intensively investigated. It is known numberoΓfmaxima. Letτuspi kupagivenτpointofmini- v for instan e that their lassi al equations of motion ad- mum on a and all it a,1. Starting from a,1 and going i X mitvortexsolutions[5℄. Inaspe ialregionofthespa eof alongthe urveafter hoosingarbitrarilyadire tion,one r parameters, alledtheBogoml'nyiself-dualpoint[6℄, the willen ouτntersu essivelyapointoτfmaximum,thatwill a attra tive and repulsive for es between vorti es vanish. be alled a,2, apointofminimum a,3 andsoon. Inthis 2s τ ,...,τ This point orresponds in the Landau-Ginzburg model way we obtain aset of a points a,1 a,2sa. Now it Γ s Γ ,...,Γ forsuper ondu torsto the transitionfromtypeItotype is possible to split a into a open paths a,1 a,sa II super ondu tivity. whi h onne tpairwise ontiguouspointsofmaximaand The (cid:28)rst obsta le in this program is that anyons are minima: 2+1 livingin dimensions,while polymer traje toriesare τ t τ a,I a,I+1 intrinse ally de(cid:28)ned in three dimensions. For this rea- ≤ ≤ Γ = ( (t),t) (τ )= (τ ) saosnt,imone.e Ao o rodridniaRntge3lym, uastpobientsionfgtlehde othurteea-nddimreengsairodneadl a,Ia  ra,Ia (cid:12)(cid:12)(cid:12) rraa,,IIaa(τaa,,IIaa+)1=ra,rIaa,−Ia1+(τ1a,Iaa,)Ia+1  (cid:12) Eu lidean spa e will be identi(cid:28)ed by a three ve -  (cid:12) (1) 2 I I 1,...,2s τ where a is a y li index su h that a ∈ { a} at the points of maxima and minima a,Ia in order to I +2s =I (τ ) Γ and a a a. Thepointsra,Ia a,IA are onsidered re onstru t the losed urve a. as (cid:28)xed points. This se tioning pro edure is expli itly The polymer a tion A an be divided into two ontri- s=3 illustrated in the ase by Fig.; 1. By onstru tion, butions: t = + free top A A A (3) τ a,6 τa,2 The free part Afree is: τ Γa,1 Γ Γa,5 Afree =Xa=21IX2as=a1Zτaτ,aI,aIa+1dt(−1)Ia−1gaIa(cid:12)(cid:12)drad,Ita(t)(cid:12)(cid:12) a,4 Γ a,2 ( 1)Ia−1 (cid:12)(cid:12) (cid:12)(cid:12)(4) a,3 Γ The fa tors − appearing in Eq. (4) are neg essary ττa,5 a,4 Γa,6 rtoelamtaedketAoftrheee lpeonsgittihvseLdae,(cid:28)Ianitoef.tTheheoppaenram heatienrssΓaaI,Iaaaares a,3 follows. Letusdis retizeΓa,Ia onsideringitasarandom n τ hainswith segments. IfEq.(4)wouldbethea tionof a,1 a set of free two-dimensional hains, in the limit of large n n/g valuesof andkeeping(cid:28)xedtheratio a,Ia,onewould 2s−platΓa s=3 obtain the well known result [7℄: La,Ia ∼ n/ga,Ia. Here FIG. 1: Se tioning pro edure for a with . things arealittle bit more ompli ated, be auseone has to take into a ount the third dimension, whi h in turn there is a one-to-one orresponden e between the points t [τa,I,τa,I+1] is also the variable whi h parametrizes the urves. As a of the −axis in the interval and the points Γa,I onsequen e, the above relation governing the length of of the traje tories . As a onsequen e, it is possible to parametrizethe traje toriesΓa,Ia by meansof the the traje tories is repla ed by: t n variable . L2a,Ia ∼|τa,Ia+1−τa,Ia|2+ g |τa,Ia+1−τa,Ia| (5) We are now ready to onstru t the partition fun tion a,Ia of a system of topologi allyentangled polymers. For the sakeof larity,itis onvenienttousesomeofthenomen- The se ond term in Eq. (3) is: leature of knot theory. In parti ular, from thΓe mathe- 2s1 τ1,I+1 dξµ (t) a 1,I mati al point of view, ea h losed traje tory an be =ıλ dt B ( (t),t) top µ 1,I A dt r regarded as a knot, while an ensemble of two or more I=1Zτ1,I X traje tories de(cid:28)nes a link. Moreover, with some slight 2s abuse of language, a knot or a link with points of maxima and minima will be alled a 2s−plat. The on- κ 2s2 dξ2µ,J(t) +ı dt C ( (t),t) µ 2,J ept of plats arises quite naturally in the pro esses of 2π dt r (6) J=1 repli ation,re ombinationand trans ription of DNA [8℄. X To start with, let us writ2eSthe partition Sfun= tionn fosr a Bµ = ( ,B0) Cµ = ( ,C0) linkwiththetopologyofa −plat,where a=1 a. In the above equation B and BC B sa aretwoAbelian C-S ve tor(cid:28)elds. We haveput 3 ≡ 0 The integers are now kept (cid:28)xed, so that thPe link is C C Γa 2sa and 3 ≡ 0 in order to stress the role of time of the omposed by knots whi h are −plats. In order κ λ third spatial oordinate. and are real parameters. to avoid a proliferation of indi es, we will restri t our- κ oin ides with the C-S oupling onstant (see below). selves to a system of two linked polymers, thus setting N = 2 The spurious gauge degrees of freedom have been elimi- . A tually, this is not a big limitation, be ause nated using the gauge ondition (Coulomb gauge): the two-polymer problem ontains all the ingredients of N ı =√ 1 the −polymer problem. Putting − , the desired ∇ =∇ =0 ·B ·C (7) polymer partition fun tion is given by: S CS (λ)= B C e−ıSCS (t)e−A so that the gauge (cid:28)xed C-S a tion looks as follows: Z Z D µD ν Zboundary Dra,Ia (2) S = κ d3ξ B ǫij∂ C +C ǫij∂ B onditions CS 0 i j 0 i j 4π (8) Γ Z The boundary onditions on the traje tories a,Ia are (cid:2) (cid:3) thoseofEq.(1). Theseboundary onditionsarerequired The C-S (cid:28)elds a t asauxiliary(cid:28)elds whi h impose topo- Γ a Γ Γ by the fa t that the open paths a,Ia, with (cid:28)xed and logi al onstraints on the losed traje tories 1 and 2. I = 1,...,2s a a , must join together in a suitable way To show this, one has to integrate out these (cid:28)elds from 3 thepartitionfun tion(2)andthentoperformaninverse dis uss the steri intera tions, in part be ause we prefer (λ) Fourier transformationof Z : to on entrate on the pure topologi al intera tions, in 2π dλ part be ause the in lusion of these repulsive for es om- (m)= eimλ (λ) pli ates enormously the task of (cid:28)nding non-perturbative Z 2π Z (9) Z0 analyti solutionsofthetwo-polymerproblemunder on- sideration. Thus,wewillassumethatinoursystemsteri The details of this al ulation have been already ex- intera tions are negligible. This approximation is valid plainedin Ref.[3℄andwill notberepeatedhere. There- (m) for instan e in polymer solutions whi h are very dilute sult is the new partition fun tion Z , whi h des ribes or in whi h the temperature is near the so- alled theta- the (cid:29)u tuations of two polymer rings onstrainedby the point. The hypothesis of very low monomer densities is ondition: naturalin thepresent ontext,be ausewearedis ussing m=χ(Γ1,Γ2) the statisti al properties of two isolated polymer rings, (10) whi h are supposed to be very long and thin. χ(Γ ,Γ ) where 1 2 is the Gauss linking number and the To establish the onne tion with anyons, we need to m topologi al number must be an integer. A few om- passinthepartitionfun t(ito)n(2)fromthestatisti alsum ments are in order at this point. First of all, in [3℄ the overthetraje toriesra,Ia toapathintegralover(cid:28)elds. topologi al onstraint (10) has been derived from the C- Th(λe)details of the derivation of the partition fun tion Z in terms of (cid:28)elds in the general ase and in luding S path integral using the Lorentz gauge, while here the also the steri intera tions will be presented elsewhere. Coulomb gauge (7) has been hosen. Formally, things works exa tly in the same way in both gauges. How- Here we will4restri t ourselves to links wshi h=hsave=th1e 1 2 stru ture of −plats, putting hen eforth . ever, one should keep in mind that the (cid:28)nal expression χ(Γ ,Γ ) of the Gauss linking number 1 2 obtained starting This ase is parti ularly interesting for biologi al appli- from the Lorentz gauge di(cid:27)ers from the expression that ations, sin e most of the l4inks produ ed by in vitro en- oneobtainsstartingfromtheCoulombgauge. Of ourse, zymologyexperimentsare −plats4[8℄.pMlaotrseover,(cid:16)small(cid:17) the two expressions are equivalent due to gauge invari- links up to seven rossinngs are all − . r an e,but thewayinwhi hthewindingofonetraje tory Introdu ing a set of omplex repli a (cid:28)elds: around the other is ounted, is realized in a very di(cid:27)er- Ψ~ = (ψ1 ,...,ψnr ) ent way. Another important point is that our approa h Ψ~∗a,Ia = (ψa1∗,Ia,...,ψan,rI∗a) (11) allowsamorere(cid:28)ned lassi(cid:28) ationoflinks thanthebare a,Ia a,Ia a,Ia Gauss linking number. Indeed, two links whi h have the same Gauss linking number an be further distinguished and a onvenient notation for ovariantderivatives: by their plat-stru ture, i. e. by the number of maxima κ κ ( λ, )=∇ ıλ ( , )=∇ ı and minima of the knots whi h are omposing the links. D ± B ± B D ±2π C ± 2πC (12) Finally, it is not di(cid:30) ult to add also the repulsive steri intera tionsbetweenthemonomersinthepartitionfun - 4 the −platpartitionfun tion anbeexpressedasfollows: tion (2). It is interesting to noti e that, sin e we have x t hosen the oordinate 3 ≡ in orderto parametrizethe 2 2 urves, the usual δ−fun tioVn(p,ott)e=ntivalδ(wh)δi (ht)dves ribes Z4−plat(λ)=nli→m0 D(fields) O1,I O2,Je−S4−plat thesefor estakestheform: r 0 r , 0being Z IY=1 JY=1 a temperature dependent onstant. Due to the presen e (13) δ 4−plat ofthe −fun tionin time, thesteri intera tionsbe ome The a tion S ontains the free (cid:28)eld a tion and the instantaneous in our ase. In the following we will not topologi al terms: τ1,2 1 2 1 2 S4−plat = dt dx Ψ~∗1,1∂tΨ~1,1+ 4g D(−λ,B)Ψ~1,1 +Ψ~∗1,2∂tΨ~1,2+ 4g D(λ,B)Ψ~1,2 + τ1Z,1 Z (cid:26) 11 (cid:12)(cid:12) (cid:12)(cid:12) 12 (cid:12)(cid:12) (cid:12)(cid:12) (cid:27) τ2,2 (cid:12) (cid:12) (cid:12) (cid:12) 1 κ 2 1 κ 2 + dt d Ψ~∗ ∂ Ψ~ + ( , )Ψ~ +Ψ~∗ ∂ Ψ~ + ( , )Ψ~ x 2,1 t 2,1 4g D −2π C 2,1 2,2 t 2,2 4g D 2π C 2,2 (14) τ2Z,1 Z (cid:26) 21 (cid:12)(cid:12) (cid:12)(cid:12) 22 (cid:12)(cid:12) (cid:12)(cid:12) (cid:27) (cid:12) (cid:12) (cid:12) (cid:12) d dx dx ∂ = ∂ x ≡ 1 2 is the in(cid:28)nitesimal volume element in the two-dimensional spa e and t ∂t. Let us note that in 4 n r thepartitionfun tion(13)wehavealreadyintegratedout propagate anyoni parti les with spin and the statis- B ,C 0 0 the time omponents of the Chern-Simons (cid:28)elds , ti sofboson. Onedi(cid:27)eren efromanyoni (cid:28)eldtheoriesis whi h play the role of Lagrange multipliers and impose duetothefa tthat,inthe aseofrealparti les,one on- the following onstraints: siders the evolution of the whole system from an initial T T 1 2 2 2 time anda(cid:28)naltime . Here,instead,timeintervals =2 Ψ~2,1 Ψ~2,2 θ(τ2,2 t)θ(t τ2,1) measure the extensions of the traje tories Γa,Ia in the B − − − (15) x (cid:18)(cid:12) (cid:12) (cid:12) (cid:12) (cid:19) 3 dire tion. For this reason, in the partition fun tion 4π (cid:12) (cid:12)2 (cid:12) (cid:12)2 = (cid:12)Ψ~ (cid:12) (cid:12)Ψ~ (cid:12) θ(τ t)θ(t τ ) (13) part of the system evolves within the time inter- C κ 1,1 − 1,2 1,2− − 1,1 (16) [τ1,1,τ1,2] (cid:18)(cid:12) (cid:12) (cid:12) (cid:12) (cid:19) v[τal ,τ ] and another part within the time interval (cid:12) (cid:12) (cid:12) (cid:12) 2,1 2,2 . This inhomogeneity in the time intervals and (cid:12) (cid:12) (cid:12) (cid:12) In the above equation B and C are the magneti (cid:28)elds the use of the Coulomb gauge, whi h makes the intera - asso iatedtotheve torpotentialsBandCrespe tively: tions instantaneous, is the ause of the presen e of the θ = ∂ B ∂ B =ǫij∂ B Heaviside −fun tions in Eqs. (15)(cid:21)(16). In this sense, 1 2 2 1 i j B − (17) = ∂ C ∂ C =ǫij∂ C the omplete analogywith anyons an be re overedonly 1 2 2 1 i j C − (18) in the limit θ(t) θ(t)=0 t<0 and is the Heaviside theta fun tion if τ1,1 =τ2,1 T1 τ1,2 =τ2,2 =T2 θ(t)=1 t 0 (fields) ≡ (21) and if ≥ . The symbol D in Eq. (13) is a shorthand for the (cid:28)eld integration measure: Throughout the rest of this letter we will assume that 2sa (fields)= Ψ~∗ Ψ~ theabove onditionissatis(cid:28)ed. Thθisistoavoidte hni al D D a,IaD a,Ia (19) ompli ations in dealing with the −fun tions. Z IYa=1 The onne tiontotheanyonproblemsuggesttoapply Finally, the operatorsOa,Ia are given by: tothea tionofEq.(14)theBogomol'nyitransformations [6℄. Using these transformation in a suitable way, it is Oa,Ia = ψa1,Ia(ra,Ia(τa,Ia+1),τa,Ia+1) possible to rewrite S4−plat in the following form: ψ1∗ ( (τ ),τ ) a,Ia ra,Ia a,Ia a,Ia (20) S4−plat =F4−plat+I4plat (22) Eqs. (13)(cid:21)(20) establish the desired onne tion between polymers and anyons. In fa t, the a tion (14), together Negle ting surfa e terms and introdu ing the notation D± = D1 ıD2 Di i = 1,2 i with the onstraints (15)(cid:21)(16), oin ides formally with ± , where , , denotes the −th the a tion of2(asm+ultsi-)lay=er4edanyon system. TΨ~h∗e nu,Ψ~mber spa e omponents of the ovariant derivatives (12), one of layers is 1 2 , while the (cid:28)elds a,Ia a,Ia has that: T2 2 2 F4−plat =ZT1 dtZ dx"Xa=1IXa=1Ψ~∗a,Ia∂tΨ~a,Ia+ 1 2 1 2 1 κ 2 1 κ 2 + D+( λ, )Ψ~1,1 + D+(λ, )Ψ~1,2 + D−( , )Ψ~2,1 + D−( , )Ψ~2,2 4g − B 4g B 4g −2π C 4g 2π C (23) 11 (cid:12) (cid:12) 12 (cid:12) (cid:12) 21 (cid:12) (cid:12) 22 (cid:12) (cid:12) (cid:21) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) and I4−plat = λ2 T2dt dx |Ψ~g11,11|2 − |Ψ~g11,22|2 × Ψ~2,1 2− Ψ~2,2 2 − |Ψ~g21,21|2 − |Ψ~g22,22|2 × Ψ~1,1 2− Ψ~1,2 2 ZT1 Z (cid:20)(cid:18) (cid:19) (cid:18)(cid:12) (cid:12) (cid:12) (cid:12) (cid:19) (cid:18) (cid:19) (cid:18)(cid:12) (cid:12) (cid:12) (cid:12) (cid:19)(cid:21) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)(24) 4−plat It is easy to re ognize from Eq. (23) that F on- instead Coulomb-like potentials, whi h attra t or repel Γ sists in the sum ofΨ~the self-dual a tions of four di(cid:27)erent themonomersbelongingtodi(cid:27)erenttraje tories a,Ia. It families of anyons a,Ia. These families are oupled to- isdi(cid:30) ulttoguesswhatisthein(cid:29)uen eoftheseCoulomb gether by the onstraints (15)(cid:21)(16). The se ond ontri- for es on the statisti al behavior of the polymers. The 4,plat 4−plat λ bution to S , the intera tion term I , ontains problem is that their strength is proportional to (see 5 λ Eq. (24)) and is not a real oupling onstant with a the self-dual regime. This does not mean however that physi al meaning. At the end, we are interested to elim- atta tions and repulsions between monomers disappear, λ inate and to express the polymer partition fun tion in be auseCoulomb-likefor esstillhideintheself-dualpart m 4−plat terms ofthe topologi alnumber , aswedid in Eq. (9). of the a tion F . m is in fa t the true physi al parameter whi h des ribes Wewouldlikenowtoexploitfurthertheanalogyofthe the topologi al state of the system. What is possible to polymerproblemwithanyons(cid:28)eldtheoriesandsear hfor on lude from Eqs. (22)(cid:21)(24) is that topologi al for es possiblenon-trivial lassi alsolutionswhi hminimizethe 4−plat have attra tive and repulsive omponents, whi h inter- polymer free energy S . Of ourse, one should keep fere with the steri intera tions. This result on(cid:28)rms at in mind that, even in the ase of the simplest model of anon-perturbativelevelapreviousresult obtained using anyons, vortex solutions outside the self-dual point have perturbative methods [3℄. Also in Ref. [9℄ it has been been found only by means of numeri al methods. Also shown in the limit in whi h (cid:29)u tuations in the parti le at the self-dual point, there is no lear and detailed un- densities arerelatively small that the C-S (cid:28)elds generate derstanding of the dynami s of C-S vorti es [10℄. Here e(cid:27)e tive Coulomb intera tions in anyon models. the situation is further ompli ated by the presen e of Remarkably, our approa h reveals that the two- repli a (cid:28)elds and multiple families of anyons whi h are polymersystemunder onsiderationhasaself-dualpoint. mixed togetherdue to the onstraints(15)-(16). In view g Thispointisrea hedwhentheparameters aIa areequal: of these limitations, it seems reasonable to restri t our- g11 =g12 =g21 =g22 =g (25) stieolvness, wtohit hhessaetlifs-fdyuathliety orengdiimtieon(s25∂)tψaaωn,dIat=o s0tafotir esvoelruy- a=1,2 I =1,2 ω =1,...,n r value of , and . As a onse- If the above relations are satis(cid:28)ed, it is easy to see that 4−plat = 0 quen e, from now on we will onsider the pure self-dual I , so that the only surviving terms in the a - 4−plat a tion: tion S are those oming from the self-dual ontri- butions in F4−plat. We note that the existen e of the S4−plat = self-dual point (25) is not submitted to the assumption (T2−T1) d D ( λ, )Ψ~ 2+ D (λ, )Ψ~ 2+ of Eq. (21). Indeed, let us suppose that the points of 4g x + − B 1,1 + B 1,2 maxima and minima τa,Ia of the traje tories Γa,Ia are Zκ (cid:20)(cid:12)(cid:12) 2 κ(cid:12)(cid:12) (cid:12)(cid:12) 2 (cid:12)(cid:12) not aligned as spe i(cid:28)ed by Eq. (21). The only di(cid:27)eren e D−(−2π,C)Ψ~(cid:12)2,1 + D−(2π(cid:12),C)Ψ~(cid:12)2,2 (cid:12)(27) in the expression of the intera tion term I4−plat is that (cid:12) (cid:12) (cid:12) (cid:12) (cid:21) T1 T2 (cid:12) (cid:12) (cid:12) (cid:12) now the values of and , whi h give the boundaries (cid:12) (cid:12) (cid:12) (cid:12) Of ourse, stati solutions do not (cid:28)t very well with the of the integration over time, should be repla ed by the physi alboundary onditionswhi hshouldbeimposedto following ones: T1 the lassi al(cid:28)eldsatthepointsofmaximaandminima T T =max[τ ,τ ] T =min[τ ,τ ] 2 1 1,1 2,1 2 1,2 2,2 and . Forthisreason,wearesupposinghereimpli itly (26) T T 2 1 thatpolymersareverylongandthattheinterval − The hoi e(26)ofintegrationlimitsinI4−plat isdi tated islarge. Inthiswaytherelevant ontributionstothefree θ 4−plat t by the presen e of the −fun tions of Heaviside in the energyS omefrominstants whi harefarenough t=T t=T 1 2 onstraints (15)(cid:21)(16). Physi ally, Eq. (26) is motivated from the extrema lo ated at and . At these by the fa t that, in the Coulomb gauge, all intera tions intermediate points the traje tories (cid:29)u tuate in su h a t be omeinstantaneous. Therefore,ifmaximaandminima way that, at ea h (cid:28)xed instant , they visit more or less arenotaligned,itiseasytoseethatthetraje toriesmay with the same frequen y the same lo ations of the two onlyintera tinthetimeinterval[T1,T2],whereT1andT2 dimensional spa e spanned by the ve tors ra,Ia(t). are given by Eq. (26). Substituting the new integration From the a tion (27) one (cid:28)nds the following Euler- limits (26) in Eq. (24), it is lear that. if Eq. (25) is Lagrangeequations of motion: satis(cid:28)ed, the e(cid:27)e tive Coulomb intera tions in I4−plat D ( λ, )ψω =D (λ, )ψω =0 4−plats + − B 1,1 + B 1,2 (28) vanish identi ally, so that the a tion S oin ides κ κ on e again with the pure self-dual part F4−plat. Thus, D− −2π,C ψ2ω,1 =D− 2π,C ψ2ω.2 =0 (29) to have self-duality we only need that the e(cid:27)e ts of the (cid:16) (cid:17) (cid:16) (cid:17) steri intera tions are negligible, su h as for instan e in To these equations one should also add the onstraints the ase of very low monomer on entration or at the (15)(cid:21)(16) whi h determine the transverse omponents of , theta-point. the (cid:28)elds B C and the Coulomb gauge (cid:28)xing (7). In terTphreetesdelfa-dsutahleityreq ounirdeimtioennt(t2h5a)tmthaeytbreajpe htyosrii easllΓyai,nIa- Etoqtsa.l,(2a8f)t(cid:21)er(2s9e)p,awreatoinbgtaitnhe12re×alnranedquiamtiaognins,awryhip har tsomin- mustbehomogeneous,inthesensethattheyshouldhave pletely spe ify the lassi al (cid:28)eld on(cid:28)gurations. It seems equalpersisten elengths. Inthis ase,theattra tiveand natural to try repli a symmetri solutions of the kind: 4−plat repulsivefor esdes ribedbythetermI ounterbal- ψω =ψ ψω∗ =ψ∗ an ethemselvesinasortofequilibriumwhi hestablishes a,Ia a,Ia a,Ia a,Ia (30) 6 a = 1,2 I = 1,2 ω = 1,...,n t=T a r 2 where , and . Moreover, arealignedattheheight , self-dualityo urswhen Γ we swit h to polar oordinates in order to express the all open urves a,Ia have the same length. Parti ular omplex (cid:28)elds: polymer on(cid:28)gurations of this kind may exist in nature, ψa,Ia =eıφa,Iaρ1a/,I2a (31) efovreri,nwsteanh aeveafsteeerntthheatprtoh eersesqoufirDemNeAntrsepfolir tahteiopnr.esHeonw e- ofaself-dualpointaremu hmoregeneral: Weonlyneed The onsisten y of Eqs. (28)(cid:21)(29) requires that: that the e(cid:27)e ts of the steri intera tions are negligible, c φ = φ ρ = a su h as for instan e in the ase of very low monomer a,1 a,2 a,1 − ρ (32) a,2 on entration or at the theta-point. c c Theexisten eoftheself-dualpoint(25)showsthatthe 1 2 Here and areρarbitrary onstantsdi(cid:27)erentfromzero. physi s of intera ting polymers is mu h ri her than pre- Alsothedensities a,Ia mustnotvanish. Thereareother viouslyexpe ted. Forinstan e, the lassi alequationsof waystomakeEqs.(28)(cid:21)(29) onsistent,buttheyleadei- motionadmitnon-trivialsolutions,whi haretheanalogs ther to unphysi alorto trivial solutions. After imposing of vortex-like solutions in anyon models. By analyti al the onsisten y onditions (32) in Eqs. (28)(cid:21)(29), there methods it is just possible to investigate stati (cid:28)eld on- remainstillfourindependentrelations,whi h, anbeused (cid:28)gurations. These su(cid:27)er the limitations mentioned after to (cid:28)x the omponents of the ve tor (cid:28)elds B C: Eq. (27) and annot be de(cid:28)ned if the requirements of λB = ∂ φ + 1ǫ ∂jlnρ Eqs. (21) are not ful(cid:28)lled. In fa t, if maximta and min- i i 1,1 ij 1,1 2 (33) ima are lo alized at di(cid:27)erent heights on the −axis, the κ C = ∂ φ 1ǫ ∂jlnρ a tion S4−plat be omes expli itlyθdependent on time due i i 2,1 ij 2,1 2π − 2 (34) tothepresen eoftheHeaviside −fun tionsinthe(cid:28)elds B and C. This time dependen e is very mild. Its only It is easy to prove that the Coulomb gauge requirement e(cid:27)e t is that the boundaries of integration over time in φ1,1 φ2,1 4−plat (7) is satis(cid:28)ed only if the phases and are on- the various terms omposing the a tion S do not stant. Finally, insertingEqs.(33)(cid:21)(34)inthe onstraints oin ide. Yet, this is su(cid:30) ient to spoil the possibility of (15)(cid:21)(16), it is possible to determine the remaining de- having purely stati (cid:28)eld on(cid:28)gurations. For these rea- ρ ρ 1,1 2,1 grees of freedom and : sons, any on lusion on the meaning of the stati solu- c tions in the ase of polymers should be taken with some 2 ∆lnρ = 4λn ρ 1,1 r 2,1 are. Having in mind these aveats, it is interesting to ρ − (35) (cid:18) 2,1 (cid:19) note that, in the solutions whi h we have studied here, it seems to prevail a repulsive for e between ouples of Γ ∆lnρ2,1 = 4λnr(cid:18)ρ1,1− ρc11,1(cid:19) (36) tisrawjeh attorEieqs. (3a2,I)asbueglgoensgtsin,gbet oauthseetshaemdeepnosiltyimeseρr.a,1Tahries onstrainedtobeinverselyproportionaltotheir ounter- ρ a,2 A further study of the above two equations requires nu- parts at ea h point x. meri al methods. It is worthing to point out that, despite their lose Con luding, we have su eded in establishing a on- resemblan es,the me hanisms used to establish the self- ne tionbetweentopologi allylinkedpolymersandanyon dual regime in our polymer model and in usual anyon 4 (cid:28)eld theories. We have limited ourselvesto −plats, but (cid:28)eldtheoriesaredi(cid:27)erent. Inanyon(cid:28)eldtheories,infa t, it is possible to generalize our results to the ase of any theexisten eoftheself-dualpointrequiresthatthee(cid:27)e - 2s −plat. Inourapproa h,linkswhi hbelongtodi(cid:27)erent tive Coulomb intera tions, whi h arise after performing 2s −plat on(cid:28)gurations are distinguished. This allows a the Bogomol'nyi transformations, are an eled against better lassi(cid:28) ation of the topologi al states of polymers true Coulomb intera tions of harged parti les [11℄. In than the bare Gauss linking number. In view of possi- thepolymer model, instead, thee(cid:27)e tiveCoulombinter- 4−plat ble appli ations to knot theory, it would be ni e to ex- a tions, represented by the term I , ounterbalan e tendalsotonon-abelianC-S(cid:28)eldtheoriesourte hniques themselves and the presen e of other intera tions is not based on the Coulomb gauge and on the se tioning pro- needed. Thisself-balan ingtransitiontoself-dualitymay edure of loops des ribed above. o uralsoinotherphysi alproblemswheremulti-layered The analogy between polymers and anyons has been systems of non- harged parti les are relevant. Another exploited to investigate the statisti al me hani s of appli ationofourapproa harepolymerbrushes. Inthat 4 −plats, a parti ular lass of links, whi h is relevant in ase,C-S (cid:28)eldtheories in the Coulombgaugeareable to biologi al appli ations. It turns out that, as anyon par- take into a ount the e(cid:27)e ts related to the winding of 4 ti les in multi-layered systems, also −plats have a self- neighboring polymer traje tories. dual point. Within the hypothesis of Eq. (21), whi h Part of this work has been arried out during a visit τ demands that the points of minima a,2Ia+1 are aligned to the Max Plan k Institute for Polymer Resear h. This t = T 1 at the same height , while the points of maxima visit has been funded by a grant from the German A a- 7 demi Ex hangeServi e(DAAD), whi hisgratefullya - [4℄ F. Wil zek, Phys. Rev. Lett. 49, 957 (1982); F. Wil zek knowledged. The author wishes also to thank the Max and A. Zee, ibid.51 (1983), 2250. [5℄ R. Ja kiw and S.-Y. Pi, Phys. Rev. D42 (1990), 3500. Plan k Institute for Polymer Resear h for the ni e hos- [6℄ E. Bogomol'nyi, Sov. J. Nu l. Phys. 24 (1976), 449. pitality. He is also indebted to V. A. Rostiashvili and [7℄ H. Kleinert, Path Integrals in Quantum Me hani s, T. Vilgis, whi h parti ipated in the early stages of this Statisti s, Polymer Physi s, and Finan ial Markets, work,fortheirinvaluablehelpandtheir onstanten our- (World S ienti(cid:28) Publishing, 3nd Ed., Singapore, 2003). agement. [8℄ S. A. Wasserman and N. R. Cozzarelli, S ien e 232 (1986), 952; D. W. Sumners, Noti es of the Am. Math. So . 42 (5) (1995), 528. [9℄ Y.-H. Chen, F. Wil zek and E. Witten, Int. Jour. Mod. ∗ Phys. B 3 (7) (1989), 1001. Ele troni address: ferrariuniv.sz ze in.pl [10℄ G. Dunne, Aspe t of Chern-Simons Theories, published [1℄ For a review on topologi ally linked polymers see for in- in Topologi al Aspe ts of Low Dimensional Systems, A. stan e: Comtet et al. (Eds), SpringerVerlag 2000. A. L. Kholodenko and T. A. Vilgis, Phys. Rep. 298 [11℄ Z. F. Ezawa and A. Iwazaki, Phys. Rev. B47 (1993), (1998), 251. 7295; S. M. Girvin and A. H. Ma Donald, Perspe tives [2℄ F. Ferrari, H. Kleinert and I. Lazzizzera, Int. J. Mod. in Quantum Hall E(cid:27)e ts, S. D. Sarma and A. Pin zuk Phys.B14 (2000) 3881 [arXiv: ond-mat/0005300℄. (Eds), Chap. 5, Wiley,New York,1997. [3℄ F. Ferrari, AnnalenPhys. 11 (2002) 255.