]>NUPHB114673114673S0550-3213(19)30159-210.1016/j.nuclphysb.2019.114673Quantum Field Theory and Statistical SystemsFig. 1A plat obtained by joining together the ends of a set of braided strings both at the top and at the bottom. The ends of the strings that are nearest to each other are connected together in pairs with the help of arcs. In the figure only the arcs are visible, while the braided stings are inside the dashed rectangular area.Fig. 1Fig. 2Representation of a trefoil knot in terms of a two-dimensional diagram. The τi's, i = 1,…,4, denote the heights of the points of minima and maxima.Fig. 2Fig. 3The figure shows one of the crossings which are present in the diagram of the trefoil knot of Fig. 2.Fig. 3Fig. 4A link formed by two polymer rings Γ1 and Γ2.Fig. 4Fig. 5Sectioning procedure for a 2s-plat Γa with s = 3 into a set of directed paths Γa,Ia (see text for details).Fig. 5Fig. 6Feynman diagram representation of the interactions in Eqs. (59) and (65).Fig. 6Fig. 7Example of a process in which the three body interactions of topological origin do not vanish in the zero replicas limit na,Ia→0 appearing in Eq. (63). The process describes the interaction of the trajectories Γa,Ia, a = 1,2,3, Ia = 1,2 forming a 6-plat in which three loops Γ1,Γ2,Γ3 are linked together.Fig. 7Fig. 8Example of configuration of a 4-plat.Fig. 8Knots, links, anyons and statistical mechanics of entangled polymer ringsFrancoFerrariafranco@feynman.fiz.univ.szczecin.plJarosławPaturejab⁎MarcinPia̧tekacpiatek@fermi.fiz.univ.szczecin.plYaniZhaodaFaculty of Mathematics and Physics, University of Szczecin, Wielkopolska 15, 70–451 Szczecin, PolandFaculty of Mathematics and PhysicsUniversity of SzczecinWielkopolska 15Szczecin70–451PolandFaculty of Mathematics and Physics, University of Szczecin, Wielkopolska 15, 70–451 Szczecin, PolandbLeibniz-Institut für Polymerforschung Dresden e.V., 01069 Dresden, GermanyLeibniz-Institut für Polymerforschung Dresden e.V.Dresden01069GermanyLeibniz-Institut für Polymerforschung Dresden e.V., 01069 Dresden, GermanycBogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, RussiaBogoliubov Laboratory of Theoretical PhysicsJoint Institute for Nuclear ResearchDubna141980RussiaBogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, RussiadMax Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, GermanyMax Planck Institute for Polymer ResearchAckermannweg 10Mainz55128GermanyMax Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany⁎Corresponding author.Editor: Hubert SaleurAbstractThe field theory approach to the statistical mechanics of a system of N polymer rings linked together is extended to the case of links whose paths in space are characterized by a fixed number 2s of maxima and minima. Such kind of links are called 2s-plats and appear for instance in the DNA of living organisms or in the wordlines of quasiparticles associated with vortices nucleated in a quasi-two-dimensional superfluid. The path integral theory describing the statistical mechanics of polymers subjected to topological constraints is mapped here into a field theory of quasiparticles (anyons). In the particular case of s=2, it is shown that this field theory admits vortex solutions with special self-dual points in which the interactions between the vortices vanish identically. The topological states of the link are distinguished using two topological invariants, namely the Gauss linking number and the so-called bridge number which is related to s. The Gauss linking number is a topological invariant that is relatively weak in distinguishing the different topological configurations of a general link. The addition of topological constraints based on the bridge number allows to get a glimpse into the non-abelian world of quasiparticles, which is relevant for important applications like topological quantum computing and high-TC superconductivity. At the end an useful connection with the cosh-Gordon equation is shown in the case s=2.1IntroductionKnots and links are a fascinating subject and are researched in connection with several concrete applications both in physics and biology [1–32]. A beautiful review from a theoretical physicist point of view about knot theory and polymers can be found in Ref. [33], Chapter 16. In this paper we study the statistical mechanics of a system of an arbitrary number of entangled polymer rings. Mathematically, two or more entangled polymers form what is called a link. Single polymer rings form instead knots. We will restrict ourselves to links in the configurations of 2s-plats. Roughly speaking, 2s-plats are knots or links obtained by braiding together a set of 2s strings and connecting their ends pairwise [34]. More precisely, a 2s-plat is obtained from a 2s-braid β=B2s by closing it with 2s simple arcs [34,35]. The way in which a general braid can be closed is not unique. In the case of plats, the 2s strings forming the braid are closed by arcs in the way illustrated in Fig. 1. Clearly, after the plat-type closure shown in Fig. 1, it turns out that 2s-plats consist of closed paths characterized by s maxima and s minima. The result of the operation of plat closure of Fig. 1 is a knot (or link) diagram on the two dimensional plane, with a system of overpasses and underpasses simulating the three dimensional structure of the original knot (or link), see Figs. 2 and 3.11Technically, an overpass is defined as a subarc in the diagram of a knot whose path is not interrupted at least at one crossing. Going along the subarc in both directions allowed, we will encounter sooner or later a crossing in which the path is interrupted. A maximal overpass is the longest overpass that it is possible to obtain without breaking the lines of the subarc. For instance, the trefoil knot of Fig. 2 contains two maximal overpasses. Overpasses and underpasses meet at points that are called crossings.Besides plats, the other mostly used convention for braid closure is called strand closure [36]. While in a plat closure the number of strings is always even (2s), in the case of strand closures this number can be both even or odd. Plats are very general constructions in knot theory. Indeed, it has been shown that any unoriented knot or link in S3 can be realized as a 2s-plat [35,37]. This is due to a theorem of Alexander [38] stating that all knots and links admit a representation as closed braids [38,39]. It turns out that the number 2s of strings in a braid that should be used in order to represent a given knot or link as a plat is bound from below. For instance, an unknotted ring, also called the unknot and denoted according to the Rolfsen table 01, is the only knot type that can be constructed from the closure of two strings. The trefoil knot 31 requires instead a minimal number of strings equal to four. In a general link composed by N unknots, it is easy to realize that the smallest value of s is reached when the unknotted rings have only a maximum and one minimum, i.e. smin=2N. The least number of strings corresponding to a plat closure representation of a knot or link is equal to twice the so-called bridge number, a topological invariant that denotes the minimal number of maximal overpasses necessary to represent a given knot [40].A physical realization of 2s-plats in the case of polymers could be a set of ring-shaped polymers in which some of the monomers are grafted to two membranes or surfaces located at two different heights. If the chains are rigid enough, the formation of turn points, a fact that could change the number of maxima and minima of the plat obtained in this way, can be eliminated. In nature 2s-plats occur for example in the DNA of living organisms [13,29,30,41]. Indeed, it is believed that most knots and links formed by DNA are in the class of 4-plats [13]. This fact may depend on the rigidity of DNA, that prevents the formation of further bendings necessary to build six or higher order plats in the short strands of DNA arising after the action of the so-called topoisomerases, the enzymes discussed in [13]. The relevance of plats in biology and biochemistry have inspired the research of Ref. [42], in which 4-plats have been studied with the methods of statistical mechanics and field theory. In particular, in [42] it has been established an analogy between polymeric 4-plats and anyons, showing in this way the tight relations between two-component systems of quasiparticles and the theory of polymer knots and links. After the publication of [42], interesting applications of two-component anyon systems in topological quantum computing have been proposed [43–45]. These applications are corroborated by the results of experiments concerning the detection of anyons obeying a nonabelian statistics, see for example [46]. While these results have appeared in 2005 and are still under debate [45,47], other systems in which non-abelian anyon statistics could be present have been discussed [48,49]. A more recent extensive report on topological quantum computing with non-abelian anyons may be found in [50]. Physical systems that have been proposed in order to implement topological quantum computations are quasiparticles in the form of vortices nucleated in a quasi two-dimensional superfluid. Concrete experimental realizations of such systems are mentioned in [50].Motivated by these advances, we study here the general case of 2s-plats in which N polymer rings are entangled together to form a link. Let us recall that two links are topologically equivalent if they can be transformed one into the other by means of continuous deformations. Here the topology of the link will be determined using two topological invariants, namely the Gauss linking number (GLN) and the bridge number discussed above. The simultaneous application of the Gauss linking number and of the bridge number goes beyond the limitations of the abelian anyon models that are obtained using only the GLN and allows to get a glimpse into the non-abelian world that is relevant in topological quantum computing and high-TC superconductivity.The first result in this work is to derive a field theoretical model describing the statistical mechanics of a link composed by N rings concatenated together to form the configuration of a 2s-plat. The points of maxima and minima of the plat are kept fixed. To mimic the situation in which these points correspond to polymer bonds that are grafted on membranes, the heights of these points in the z-direction are taken to be arbitrary. Links of this type have already been discussed in Ref. [51], where they have been called deformed plats to distinguish them from the mathematical plats, in which maxima and minima are distributed respectively at the two heights zmax and zmin. Let us notice that the analogy between polymers and quasiparticles requires that the maxima are all at the height zmax and the minima at the height zmin. The configuration of a 2s-plat allows to split the link into 2s-directed paths. In principle the partition function of 2s-directed chains could be formulated using the path integral formalism and mapped into a field theory using standard techniques explained for instance in [33]. This strategy is however complicated by the topological constraints based on the Gauss linking number. The approach of Ref. [52], that uses a set of BF-fields to impose conditions on the Gauss linking numbers of N topologically entangled rings, cannot be applied to the present system. The reason is that such approach is valid for a general link and does not take into account the further constraints that are necessary to keep the desired 2s-plat configuration. The problem of implementing the GLN constraints has been solved in the case of 4-plats in Ref. [42]. Here the results of [42] are generalized to the more complicated situation in which there are 2s-paths, each of them having the possibility of winding up around another. The topological field theories implementing the conditions on the GLN are quantized in the Coulomb gauge. This gauge has the advantage of making the connection between polymers and quasiparticles particularly evident. The details of our method, that could not be provided in a short letter like [42], are explained in details.After the passage to the second quantized fields is realized, a model describing a gas of quasiparticles is obtained. All the nonlocalities and strong nonlinearities due to the topological constraints that characterize the original first quantized theory of polymers disappear in the field theoretical formulation. The polymer paths become wordlines describing the motion of the quasiparticles, while the densities of monomers may be regarded as quasiparticle densities of a multi-layered anyon gas. The evolution of the positions of the quasiparticles is followed from the initial time τ0=zmin to the final time τ1=zmax, where the time t flows along the z-axis. As already mentioned, systems of this type, in which the worldlines of quasiparticles are braided together, are relevant in quantum computing, as they have been proposed as logic gates in prototypes of topological quantum computers.In the context of quantum computing, a remarkable feature of the field theoretical model derived here starting from the partition function of polymer links in the configuration of a 2s-plat is that it admits self-dual points. The action of the quasiparticles can be in fact minimized by self-dual solutions of the classical equations of motion. From the polymer point of view, this self-duality has a simple physical explanation. Due to the topological constraints, the lines of knots and links can attract or repel themselves. For instance, unknotted rings are known to repel also in the absence of excluded volume interactions. On the contrary, when the rings are linked together, their lines get closer, a fact that has been experimentally verified by measuring the average distance between the centers of mass of two DNA rings linked together and can be predicted using field theoretical methods [53]. The more complex is the topological configuration of the link, the smaller is the distance [53]. In the particular case of a 4s-plat, it was already shown in [42] that, after a Bogomol'nyi transformation, it is possible to single out in the two-body forces of entropic origin that are related to the topological constraints contributions that, apart from proportionality constants, are exactly of the form of the excluded volume forces, but can be both attractive and repulsive. For certain values of the parameters of the model, namely the Kuhn lengths and the total lengths of the 2s open chains composing the plat, it turns out that the attractive and repulsive components of these forces disappear giving rise to the self-dual point. With respect to [42] we prove here always in the case of a 4-plat that the vortex solutions may be explicitly constructed after solving a cosh-Gordon equation. Let us finally recall that polymer 2s-plats can be realized in the laboratory with present technologies [54]. Our results show that at least in 4-plats there are self-dual conformations that are particularly stable. The effects of the presence of these conformations could be experimentally measurable.Apart from the existence of self-dual solutions, the field theoretical model developed in this work has also phenomenological consequences that are relevant for the statistical mechanics of polymers. First of all, its Lagrangian contains a local, analytic and nonperturbative expression of the interaction terms which describe the topological forces acting on the monomers. These forces, which appear due to the constraints that limit the topological configuration of the 2s-plat, have two-body and three-body components. The two-body interactions have already been studied with the help of the method of the effective potential in [55]. It has been found there that the monomers of two heavily entangled polymer rings attract themselves due to the topological constraints counterfeiting the excluded volume interactions typical of polymers in a good solution. What is unespected is the presence of three-body interactions in a polymer system subjected to topological constraints imposed with the help of the Gauss linking number. The appearance of three-body forces of topological origin is surprising because the Gauss linking number is able to take into account only the topological relations between pairs of knotted polymer rings. For this reason, one could expect that this type of constraints should be rather associated with interactions between pairs of monomers belonging to two different chains. Indeed, the explicit expression of the Gauss linking number can be interpreted as a (nonlocal) two-body potential related to forces acting on the bonds located on two different polymers. Three-body forces have been proved to vanish in the case of links with two polymers only, see Ref. [55]. However, we show here that there are processes in which three-body forces are relevant if the number of loops involved in the link is equal to three or higher.This paper is organized as follows. First, we split the lines of the N polymer rings forming a 2s-plat into a set of 2s open chains. The splitting procedure and the definition of a suitable “time” variable that parametrizes the 2s chains is carefully described in Section 2. In Section 3 it is shown how it is possible to implement and simplify in the partition function of the 2s-plat the constraints that fix the possible topological configurations in which the system of polymer rings linked together can be found. The constraints are imposed using the Gauss linking number. The treatment follows the method already established in Ref. [56], but its generalization to the case in which the link is splitted into a set of 2s open chains parametrized by the special “time” coordinate instead of the usual arc-lengths is new. To eliminate the nonlinearities and nonlocalities introduced by the topological constraints, which necessarily have memory since they must remember the global geometry of the ring in space, we use a set of abelian BF-fields. These fields generate electromagnetic type interactions acting on the monomers and create in this way the necessary “reaction” forces that forbid the system to escape the topological constraints. Next, the BF-field theory is quantized in the non-covariant Coulomb gauge. This gauge leads to several simplifications and is very convenient in order to establish the analogy with anyon systems. How the “covariance” of the theory is recovered is shown in Appendix C in the particular case of a 4-plat. This example is very helpful to interpret the meaning of the Gauss linking number in the Coulomb gauge, which is otherwise apparently more related to the winding number of open polymers than to the Gauss linking number itself. In Section 4 the passage from first quantized polymer chains to second quantized fields is performed. The case of general interactions between the monomers is considered. After the second quantization procedure and the introduction of replica complex scalar fields, the densities of monomers of the original polymer rings can be regarded as the densities of a system of multilayered gas of quasiparticles. The topological BF-fields are eliminated by integrating them out from the partition function. In this way quartic and sestic interactions terms appear in the action, corresponding to two and three body interactions respectively. In Section 5 some phenomenological consequences on the statistical mechanics of the 2s-plat coming from the field theoretical model obtained in Section 4 are presented. In Section 6 we limit ourselves to 4-plats, switching off the non-topological interactions. In this particular case, studied in Ref. [42], it is known that the Hamiltonian of the 4-plat is minimized by self-dual solutions. Here the classical equations of motion are reduced to a cosh-Gordon equation. It is shown how the explicit expression of the classical configurations minimizing the Hamiltonian of the 4-plat can be constructed out of the solution of this cosh-Gordon equation. Finally, our conclusions are drawn in Section 7.2Polymers as 2s-platsLet's consider N closed loops Γ1,…,ΓN of lengths L1,…,LN respectively in a three-dimensional space with coordinates (r,z). The vector r=(x,y) spans the two dimensional space R2. The N loops will be labeled using as indices the first letters of the latin alphabet: a,b,c,…=1,…,N. We will assume and ensure by means of suitable constraints that Γ1,…,ΓN form a 2s-plat. The heights of the points belonging to a 2s-plat will be measured here using the z coordinate. As it will be shown in Subsection 3.1, the choice of a special direction is not decreasing the degree of generality. Similar setups have been already studied in the literature, see for instance [51,57]. The trefoil diagram in Fig. 2 provides an example of a knot in the 4-plat configuration characterized by two points of minima and two maxima. Another example of 4-plats, this time a link composed by two concatenated rings, is given in Fig. 4.In the following we will deal with the deformed 2s-plats studied in Refs. [51,57], in which the maxima and minima are at different heights. s is kept constant as a requirement and the locations of the points of maxima and minima are fixed, i.e. they are not allowed to fluctuate. Summing over all possible values of s, i.e. over all integers s≥smin and integrating over all allowed positions of the maxima and minima for each value of s, the partition function of N polymer rings without further constraints apart from being linked together should be recovered. To perform such sum over s and the integration over the locations of the maxima and minima is however very complicated and it is not necessary for the aims of the present work. To establish the desired analogy between systems of linked polymer rings and quasiparticles, in fact, we need actually not only that the maxima and minima are fixed points, but also that the maxima and minima are respectively at the same heights zmax and zmin.Let us denote with the symbols τa,Ia, Ia=0,…,2sa−1, the heights of the maxima and minima of each loop Γa, for a=1,…,N. Of course, it should be that(1)∑a=1Nsa=s. We choose τa,0 to be the height of the point of absolute minimum of the loop Γa. Starting from this point, we select the orientation of Γa in such a way that, proceeding along this loop according to that orientation, we will encounter in the order a point of maximum at the height τa,1, the next point of minimum at the height τa,2 and so on. We denote with 2sa the total number of maxima and minima of the loop Γa. The heights of these points will be: τa,0,τa,1,τa,2,…,τa,2sa−1. Clearly, the point with height τa,2sa−1 is a point of maximum. To simplify the notations, it is convenient to add the height(2)τa,2sa≡τa,0. The introduction of two symbols τa,0 and τa,2sa for the height of the same point, that of the absolute minimum of the loop Γa, will be useful in the future in order to write formulas in a more compact form. In the following, the loops Γ1,…,ΓN will be decomposed into a set of directed paths Γa,Ia, a=1,…,N and Ia=1,…,2sa, whose ends are made to coincide in such a way that they form the topological configuration of two linked rings. Due to the analogy of these paths with the trajectories of two-dimensional quasiparticles, they will be called hereafter “trajectories”. An example of such trajectories when s=3 and N=1 is presented in Fig. 5. In the general case, the set of points belonging to Γa,Ia can be described by the formula:(3)Γa,Ia={ra,Ia(ta,Ia)|a=1,…,N;Ia=1,…,2sa{τa,Ia−1≤ta,Ia≤τa,IaIa oddτa,Ia−1≥ta,Ia≥τa,IaIa even} where the additional boundary conditions:(4)ra,Ia(τa,Ia)=ra,Ia+1(τa,Ia)Ia=1,…,2sa−1(5)ra,1(τa,0)=ra,2sa(τa,0) are understood. These conditions are needed in order to connect together the trajectories Γa,Ia so that the loop Γa is reconstructed. The variables ta,Ia defined in Eq. (3) are very convenient when considering curvilinear integrals around a loop Γa that is split into many trajectories Γa,Ia. The technical details of how these variables have been introduced and an example of how they work in curvilinear integrals are presented in Appendix A.3Fixing the topological properties of a 2s-plat: the case of the Gauss linking numberIn the case of a 2s-plat composed by N loops Γ1,…,ΓN, it is possible to specify the winding number between any two trajectories Γa,Ia and Γb,Ib composing the plat. These winding numbers cannot change due to the thermal fluctuations, because the end points (r(τa,Ia−1),τa,Ia−1) and (r(τa,Ia),τa,Ia) of each trajectory Γa,Ia must be fixed in our construction. This fact can be used to constrain the 2s-plat to stay in very complex topological configurations. In the following, however, we will not adopt this strategy. The topological configurations of the system will rather be imposed by applying the Gauss linking number.3.1The standard approach of imposing the constraints with the Gauss linking numberThe Gauss linking number is a link invariant expressing the topological states of two closed trajectories linked together. Due to the fact that it can only be applied to pairs of loops, here we restrict ourselves for simplicity to the case of a 2s-plat composed by only two loops Γ1 and Γ2. Note that each of these two loops is a plat too having sa points of maxima and sa points of minima with a=1,2. For consistency, it should be that s=s1+s2. The Gaussian linking number is defined as follows(6)χ(Γ1,Γ2)=14πϵμνρ∮Γ1dx˜1μ(d1)∮Γ2dx˜2ν(d2)(x˜1(d1)−x˜2(d2))ρ|x˜1(d1)−x˜2(d2)|3 where the x˜aμ(da)'s and the arc-lengths da's, a=1,2 have been defined in Appendix A, after Eq. (133). The Gauss linking number has the advantage that it is easy to be implemented in a field theory because it is related to an abelian BF-model [56]. The price of this simplicity is that the set of transformations that do not change the value of the GLN is larger than that of the continuous deformations and contains also transformations that break the lines of the polymers. As a consequence, many inequivalent topological configurations characterized by the same value of the Gauss linking number are allowed. For example, the unlink and the Whitehead link are clearly topologically inequivalent, but they are equivalent according to the Gauss linking number, because this topological invariant is equal to zero in both cases. For this reason, the GLN is a weak topological invariant. However, it is easy to realize that the additional requirement that the conformations of a link can change only within the class of 2s-plats with a fixed number 2s of maxima and minima removes in part the limitations of the GLN. If we start from an unlink in the form of a 4-plat, for instance, it is not possible to obtain a Whitehead link acting on the unlink with transformations that keep fixed both the GLN and the number of maxima and minima. Indeed, the minimum allowed number of maxima and minima of the Whitehead link is six, so that this link cannot be reduced to a 4-plat. Vice-versa, it will not be possible to obtain a 4-plat unlink acting on a 6-plat Whitehead link. For the sake of generality, in the present work we will allow for arbitrary values of s. The treatment of the constraints in the proposed field theoretical model can be made mathematically rigorous by imposing additionally that s coincides with the least possible value smin for a link of a given type, i.e. with its bridge number. As already explained before, in a link composed by N unknots smin=2N. If we would like to select a link consisting of N trefoil knots, for example, it turns out that smin=4N. Of course, this way of specifying the topology of the knots composing the link is very rough. Only the condition smin=2N determines uniquely a set of N unknotted rings. In all the other case different mixtures of knots of different topological types are allowed.Having in mind the analogy with systems of quasiparticles that will be established here, the 2s chains of the open polymers could also be viewed as the trajectories of 2s particles moving on a two-dimensional space, while the time t flows along the z-direction. This is very important for the realization of the mapping between polymers and quasi-particles that is one of the main results of the present paper. We would like to stress that the choice of a particular direction in space does not spoil the generality of our treatment. Indeed, the only effect of a rotation could be that the number of maxima and minima of the plat could change to a new value s′ such that s′≠s. This fact does not represent a problem, because our calculations are valid for any value of s. On the other side, the value of the Gauss linking number does not depend on the way in which the system is rotated. Following the original implementation of the field theoretical formulation of the statistical mechanics of polymer links in which the topological constraints are imposed with the help of the GLN, see Ref. [56], this link invariant can be associated to a BF-model. In the presence of a preferred direction non-covariant gauge fixings are the most convenient. In the case of static knots — i.e. knots that not subjected to thermal fluctuations —, the light-cone gauge has been applied, see for instance Refs. [51] and [57]. In this work we prefer to use the Coulomb gauge, whose consistency in the frame of the so-called Chern-Simons field theories has been rigorously tested [58]. As it will be shown in Appendix C, the GLN is unaffected by our gauge choice.Coming back to Eq. (6), the trajectories of the two loops Γ1 and Γ2 will be topologically constrained by the GLN condition(7)m12=χ(Γ1,Γ2) m12 being a given integer and χ(Γ1,Γ2) is defined in (6). The constraint (7) is imposed by inserting the Dirac delta function δ(m12−χ(Γ1,Γ2)) in the partition function of the 2s-plat, where the statistical sum over all conformations of Γ1 and Γ2 is performed. Of course, the analytical treatment of such a delta function in a path integral is difficult. Some simplification is obtained by passing to the Fourier representation(8)δ(m12−χ(Γ1,Γ2))=∫−∞+∞dλ122πe−iλ12(m12−χ(Γ1,Γ2)). Even in the Fourier representation, the difficulty of having to deal with the Gauss linking number in the exponent appearing in the right hand side of Eq. (8) remains. Formally, this link invariant introduces a term that resembles the potential of a two-body interaction which is both nonlocal and nonpolynomial. For this reason, the treatment of the Gauss linking number in any microscopical model of topologically entangled polymers is complicated. The best strategy to deal with this problem consists in rewriting the delta function δ(m12−χ(Γ1,Γ2)) as a correlation function of the holonomies of a local field theory, namely the so-called abelian BF-model [56,59,60](9)δ(m12−χ(Γ1,Γ2))=∫−∞+∞dλ12e−iλ12m12ZBF(λ12) where(10)ZBF(λ12)=∫DBμ12(x)DCμ12(x)e−iSBF[B,C]×e−ic˜12∮Γ1dx˜1μ(d1)Bμ12(x˜1(d1))e−id˜∮Γ2dx˜2μ(d2)Cμ12(x˜2(d2)). In the above equation we have put x≡(x,t) to be dummy integration variables spanning the whole three-dimensional space R3. Moreover, SBF[B,C] denotes the action of the abelian BF-model(11)SBF[B,C]=κ4π∫d3xBμ12(x)∂νCρ12(x)ϵμνρ. Above ϵμνρ, μ,ν,ρ=1,2,3, is the completely antisymmetric ϵ-tensor density defined by the condition ϵ123=1. κ is the coupling constant of the BF-model. Finally, the constants c˜12 and d˜ are given by:(12)c˜12=λ12d˜=κ8π2. While there is some freedom in choosing c˜12 and d˜, one unavoidable requirement in order that Eq. (9) will be satisfied is that one of these parameters should be linearly dependent on κ. In this way, it is easy to check that κ may be completely eliminated from Eq. (10) by performing a rescaling of one of the two fields Bμ12 and Cμ12. This is an expected result, because κ does not appear in the left hand side of Eq. (9), so that it cannot be a new parameter of the theory. By introducing the currents:(13)ζ12μ(x)=c˜12∮Γ1dx˜1μ(d1)δ(3)(x−x˜1(d1))ξ12μ(x)=d˜∮Γ2dx˜2μ(d2)δ(3)(x−x˜2(d2)) ZBF(λ12) may be rewritten in the more compact way:(14)ZBF(λ12)=∫DBμ12(x)DCμ12(x)e−iSBF[B,C]e−i∫d3x[ζμ12(x)Bμ12(x)+ξ12μ(x)Cμ12(x)]. With Eq. (14) the goal of transforming the nonlinear and nonlocal interaction appearing in the right hand side of Eq. (8) is achieved. The right hand side of Eq. (14) represents in fact a local field theory, the BF-model, interacting with the trajectories Γ1 and Γ2. Of course, the price paid for that simplification is the introduction of the fields Bμ12 and Cμ12.3.2How to impose constraints on a link composed by plats using the Gauss linking numberIn all the above discussion, the two trajectories Γ1 and Γ2 have been parametrized with the help of the arc-lengths d1 and d2. However, in the present case the loops Γ1,…,ΓN are realized as a set of open paths Γa,Ia connected together by the conditions (4)–(5). The trajectories Γa,Ia's are directed paths ra,Ia(ta,Ia)=(xa,Ia1(ta,Ia),xa,Ia2(ta,Ia)) parametrized by the variables ta,Ia. This difference of parametrization introduces several important changes. Apart from the fact that we have to deal with many trajectories, also one degree of freedom, represented by the third coordinate xa3(sa), disappears due to the change (135). As a consequence, the method illustrated in the previous Subsection in order to express the Gauss linking number as an amplitude of the BF-model, in particular Eq. (9), should be changed appropriately. Thus, we rewrite the partition function ZBF(λ12) of Eq. (10) using the variables ta,Ia to parametrize the trajectories Γa,Ia. The way in which the curvilinear integrals along the loops Γ1 and Γ2 appearing in Eq. (10) should be replaced by integrals over the trajectories Γa,Ia is shown in Eqs. (133) and (134). As a result, we arrive at the following expression of the partition function ZBF(λ12):(15)ZBF(λ12)=∫DBμ12(x)DCμ12(x)e−SBF[B,C]e−i∫d3x[ζ12(x,t)⋅B12(x,t)+ζ123(x,t)B312(x,t)]×e−i∫d3x[ξ12(x,t)⋅C12(x,t)+ξ312(x,t)C312(x,t)] where SBF[B,C] coincides with the action (11) and(16)ζ12(x,t)=c˜12∑I1=12s1∫τ1,I1−1τ1,I1dt1,I1r˙1,I1(t1,I1)δ(2)(x−r1,I1(t1,I1))δ(t−t1,I1)(17)ξ12(x,t)=d˜∑I2=12s2∫τ2,I2−1τ2,I2dt2,I2r˙2,I2(t2,I2)δ(2)(x−r2,I2(t2,I2))δ(t−t2,I2)(18)ζ123(x,t)=c˜12∑I1=12s1∫τ1,I1−1τ1,I1dt1,I1δ(2)(x−r1,I1(t1,I1))δ(t−t1,I1)(19)ξ123(x,t)=d˜∑I2=12s2∫τ2,I2−1τ2,I2dt2,I2δ(2)(x−r2,I2(t2,I2))δ(t−t2,I2).3.3The Coulomb gaugeNow we use the Fourier representation of the topological constraints of Eq. (9), but with the partition function ZBF(λ12) written in the form of Eq. (15). In this way the path integral over all conformations of the 2s-plat can be split into path integrals over all conformations of the trajectories Γa,Ia. The latter can be regarded as the trajectories of a two-dimensional system of 2s particles interacting with abelian BF fields. In order to establish an explicit analogy between polymers and two-dimensional particles evolving in time, it is convenient to choose a non-covariant gauge like the Coulomb gauge. Similar approaches like that proposed here can be found in [51,57]. Interestingly, in [57] Chern-Simons field theories quantized in noncovariant gauges have also been applied to express the knot and link invariants of 2s-plats, called in [57] Morse knots. In Refs. [51] and [57] knots and links are however static, they do not fluctuate, and the calculations have been performed in noncovariant gauges different from the Coulomb gauge.To begin with, we impose the Coulomb gauge condition on the B and C fields(20)∂iBi12=∂iCi12=0 where i=1,2 labels the first two components of the vector potentials Bμ12=(B12,B312) and Cμ12=(C12,C312). After the gauge choice (20), the action of the BF model (11) becomes(21)SBF,CG[B,C]=κ4π∫d3x[B312ϵij∂iCj12+C312ϵij∂iBj12] with ϵij=ϵij3 being the two-dimensional completely antisymmetric tensor. The gauge fixing term vanishes in the pure Coulomb gauge where the conditions (20) are strictly satisfied. Also the Faddeev-Popov term, which in principle should be present in Eq. (21), may be neglected because the ghosts decouple from all other fields.The requirement of transversality of (20) in the “spatial” directions x1,x2 implies that the components Bi12 and Ci12 of the BF fields may be expressed in terms of two scalar fields b12 and c12 via the Hodge decomposition:(22)Bi12=ϵij∂jb12Ci12=ϵij∂jc12. After performing the above substitutions of fields in the BF action of Eq. (21), we obtain(23)SBF,CG[B,C]=κ4π∫d3x[B312Δc12+C312Δb12].Now we compute the propagator of the BF fields(24)Gμν(x,t;y,t′)=〈Bμ12(x,t),Cν12(y,t′)〉. Only the following components of the propagator are different from zero:(25)G3i(x,t;y,t′)=δ(t−t′)2κϵij∂yjlog|x−y|2(26)Gi3(x,t;y,t′)=−G3i(x,t;y,t′). The path integration over the scalar fields b12 and c12 in the partition function ZBF(λ) is gaussian and can be performed analytically eliminating completely the gauge fields. A natural question that arise at this point is the interpretation of the topological constraint (7) in the Coulomb gauge. As a matter of fact, the BF propagator in the Coulomb gauge breaks explicitly the invariance of the BF model under general three-dimensional transformation. It seems thus hard to recover the form (6) of the Gauss linking number in this gauge. Of course, an equivalent constraint should be obtained in the Coulomb gauge due to gauge invariance. In Appendix C it will be shown by a direct calculation in the case of a 4-plat that this is actually true. The computation of the expression of the equivalent of the Gauss linking number in the Coulomb gauge for a general 2s-plat is however technically complicated and will not be performed here.4The partition function of a plat4.1Directed polymers with topological constraintsIn order to write the partition function of a 2s-plat, we follow the strategy explained in the previous Section of dividing each trajectory Γa, a=1,…,N, into 2sa open paths Γa,Ia, Ia=1,…,2sa. The statistical sum Zpol({m}) of the system is performed over all conformations ra,Ia(ta,Ia) of the trajectories Γa,Ia using path integral methods, i.e.:(27)Zpol({m})=∫boundaryconditions[∏a=1N∏Ia=12saDra,Ia(ta,Ia)]e−(Sfree+SEV)∏a=1N−1∏b=a+1Nδ(mab−χ(Γa,Γb)). In the above equation the boundary conditions on the trajectories ra,Ia(ta,Ia) enforce the constraints (4) and (5). The free part of the action Sfree is given by(28)Sfree=∑a=1N∑Ia=12sa∫τa,Ia−1τa,Iadta,Ia(−1)Ia−1ga,Ia|dra,Ia(ta,Ia)dta,Ia|2. The parameters ga,Ia>0 are proportional to the inverse of the Kuhn lengths of the trajectories Γa,Ia. They are also related to the total lengths of the trajectories Γa,Ia according to the formula provided in Appendix B. Let us note that Sfree is a positive definite functional thanks to the factors (−1)Ia−1, which compensate the fact that the increment dta,Ia is negative when Ia is even. The contribution SEV to the total action takes into account the interactions between the monomers which arise because we treat the trajectories Γa,Ia as directed paths moving in a random media. The mechanism through which these interactions appear after the integration over the non-white random noises is explained in Ref. [61]. Explicitly, SEV is given by(29)SEV=12∑a=1N∑b=1N∑Ia=12sa∑Ib=12sb∫τa,Ia−1τa,Iadta,Ia∫τb,Ib−1τb,Ibdtb,Ib(−1)Ia+Ib−2Ma,Ia;b,IbV(ra,Ia(ta,Ia)−rb,Ib(tb,Ib))δ(ta,Ia−tb,Ib) where(30)Ma,Ia;b,Ib={0if a=b and Ia=Ib1otherwise Due to the matrix Ma,Ia;b,Ib the interactions between a trajectory with itself are forbidden. We note that the presence of the delta functions δ(ta,Ia−tb,Ib) is necessary to express the fact that the trajectories Γa,Ia and Γb,Ib for Ia≠Ib may interact only if both ta,Ia and tb,Ib belong to the common interval [τa,Ia−1,τa,Ia]∩[τb,Ib−1,τb,Ib]. The potential V(r) can be any two-body potential. If the random noises are gaussianly distributed as in Ref. [61], then(31)V(r)=V0δ(r) V0 being a positive constant. Again, the factors (−1)Ia+Ib−2 appearing in SEV are necessary in order to compensate the fact that the increments dta,Ia and dtb,Ib are negative for even values of Ia and Ib respectively. Finally, the Dirac delta functions inserted in the right hand side of Eq. (27) impose the topological constraints on each pair of trajectories (Γa,Γb), a=1,…,N−1, b=a+1,…,N.4.2Passage to Field Theory I: the topological statesAccording to Eq. (9), the physically relevant contributions coming from the topological conditions mab=χ(Γa,Γb), a=1,…,N−1, b=a+1,…,N, are encoded in the Fourier transform Zpol({λ}) of the original probability function Zpol({m}). Notice that Zpol({λ}) is obtained from Zpol({m}) by the relation(32)Zpol({m})=∏a=1N−1∏b=a+1N∫−∞+∞dλabe−iλabmabZpol({λ}). It is easy to realize that(33)Zpol({λ})=∫[∏a=1N−1∏b=a+1NDBμabDCμab]e−iSBF∫boundaryconditions[∏a=1N∏Ia=12saDra,Ia(ta,Ia)]e−(Sfree+SEV+Stop) where(34)SBF=∑a=1N−1∑b=a+1Nκ4π∫d3xBμab(x)∂νCρab(x)ϵμνρ and(35)Stop=i∑a=1N−1∑b=a+1Nλab∑Ia=12sa∫τa,Ia−1τa,Iadta,Ia[r˙a,Ia(ta,Ia)⋅Bab(ra,Ia(ta,Ia),ta,Ia)+B3ab(ra,Ia(ta,Ia),ta,Ia)]+iκ8π2∑a=1N−1∑b=a+1N∑Ib=12sb∫τb,Ib−1τb,Ibdtb,Ib[r˙b,Ib(tb,Ib)⋅Cab(rb,Ib(tb,Ib),tb,Ib)+C3ab(rb,Ib(tb,Ib),tb,Ib)]. After going back to the parametrization of the loops Γa with the help of the arc-lengths using Eqs. (133) and (134) and integrating out the BF fields, it is possible to recover in the expression of Zpol({λ}) the factors ∏a=1N−1∏b=a+1Ne+iλabχ(Γa,Γb) that originate from the Fourier representation of the Dirac delta functions ∏a=1N−1∏b=a+1Nδ(mab−χ(Γa,Γb)). The integration over the BF fields in Zpol({λ}) can be performed applying the formula:(36)∫∏a=1N−1∏b=a+1NDBμab(x)DCμab(x)e−i(SBF+Stop)=∏a=1N−1∏b=a+1Ne+iλabχ(Γa,Γb). Let us note that in the above equation the gauge fields have been quantized using the covariant Lorentz gauge.4.3Passage to Field Theory II: the non-topological interactionsAnalogously to what has been done in the case of the topological interactions, also the interaction terms in SEV can be made linear and local with the help of auxiliary fields. The strategy to achieve this goal is a straightforward generalization of that followed by de Gennes and co-workers in Refs. [62].For our purposes, it will be convenient to introduce the set of real scalar fields φa,Ia, a=1,…,N and Ia=1,…,2sa. The action of these fields is(37)Sφ[J]=Sφ[0]+i∫d3xφa,Ia(x)Ja,Ia(x) where (here we use the convention that repeated upper and lower indices are summed):(38)Sφ[0]=∫d3xd3y[φa,Ia(x)φb,Ib(y)V˜−1(x−y)(M−1)a,Ia;b,Ib](39)V˜−1(x−y)=V−1(x−y)δ(x3−y3) and(40)∫d2yV(x−y)V−1(y−z)=δ(x−z). In other words, V−1(x−y) is the operator that inverts the potential V(r) appearing in SEV. The currents Ja,Ia(x) are defined as follows(41)Ja,Ia(x)=∫τa,Ia−1τa,Iadta,Iaδ(2)(x−ra,Ia(ta,Ia))δ(x3−ta,Ia)(−1)Ia−1. M−1 is the inverse of the matrix (we consider a,Ia and b,Ib as composite indexes denoting respectively the rows and columns) defined in Eq. (30).Supposing that M is a n×n-dimensional matrix, it is easy to find its inverse, which is given by:(42)M−1=(n−2n−1−1n−1…−1n−1−1n−1n−2n−1…−1n−1⋮⋮⋱⋮−1n−1−1n−1…n−2n−1) In words, M−1 is the matrix whose diagonal elements are n−2n−1, while all the other elements are −1n−1. Let us note that in the present case n=N(s1+s2+…+sN). It is possible to show that, apart from an irrelevant constant(43)∫∏a=1N∏Ia=12saDφa,Iae−Sφ[J]=e−SEV where SEV is written in the form of Eq. (29).4.4Passage to Field Theory III: second quantizationPutting all together, the probability function Zpol({λ}) of Eq. (32) may be expressed in terms of the auxiliary fields Bμab(x), Cμab(x) and φa,Ia(x) as follows(44)Zpol({λ})=∫D(fields)e−iSBFe−Sφ[0]∏a=1N∏Ia=12sa∫Dra,Ia(ta,Ia)e−Spart(ra,Ia) where each of the actions Spart(ra,Ia), a=1,…,N and Ia=1,…,2sa, formally coincides with the action of a particle immersed in the external potential φa,Ia(ta,Ia) and in an external magnetic field that consists in a linear combination of the fields Bμab and Cμab:(45)Spart(ra,Ia)=∫τa,Ia−1τa,Iadta,Ia[(−1)Ia−1ga,Iar˙a,Ia2(ta,Ia)+iφa,Ia(ra,Ia(ta,Ia),ta,Ia)(−1)Ia−1+ir˙a,Ia(ta,Ia)⋅Aa(ra,Ia(ta,Ia),ta,Ia)+iA3a(ra,Ia(ta,Ia),ta,Ia)]. In Eq. (45) we have put(46)Aμ1(r,t)=∑b=2Nλ1bBμ1b(r,t)(47)Aμa(r,t)=∑b=a+1NλabBμab(r,t)+κ8π2∑c=1a−1Cμca(r,t)a=2,…,N−1(48)AμN(r,t)=κ8π2∑c=1N−1CμcN(r,t) and(49)D(fields)=[∏a=1N−1∏b=a+1N∫DBμabCμab][∏a=1N∏Ia=12sa∫Dφa,Ia]. Let us note that with Eq. (44) we have succeeded to rewrite the probability function Zpol({λ}) in such a way that the trajectories ra,Ia(ta,Ia) do not interact directly with each other. They interact only indirectly via the fields φa,Ia and Aμa.The problem of passing to second quantized path integral in the case of a particle with partition function:(50)Zparta,Ia=∫Dra,Ia(ta,Ia)e−Spart(ra,Ia) is very well known in polymer physics [53,56,59,63]. After introducing na,Ia-multiplets of complex replica fields:(51)Ψ→(x,t)=(ψa,Ia1(x,t),…,ψa,Iana,Ia(x,t))(52)Ψ→⁎(x,t)=(ψa,Ia1⁎(x,t),…,ψa,Ia⁎na,Ia(x,t)) we obtain(53)Zparta,Ia=limna,Ia→0∫DΨ→a,IaDΨ→a,Ia⁎ψa,Ia1⁎(ra,Ia(τa,Ia),τa,Ia)ψa,Ia1(ra,Ia(τa,Ia−1),τa,Ia−1)e−Spart(Ψ→a,Ia⁎,Ψ→a,Ia) where(54)Spart(Ψ→a,Ia⁎,Ψ→a,Ia)=∫τa,Ia−1τa,Iadta,Ia∫d2x[Ψ→a,Ia⁎∂∂tΨ→a,Ia+14ga,Ia|(∇−i(−1)Ia−1Aa)Ψ→a,Ia|2+i|Ψ→a,Ia|2(Aa3+φa,Ia(−1)Ia−1)]. In writing Eq. (54) and in all the formulas below we follow the convention that, whenever products of Ψ→a,Ia⁎ with Ψ→a,Ia appear, also the scalar product over the replica multiplets is implicitly understood.Eventually, the probability function Zpol({λ}) of Eq. (44) becomes(55)Zpol({λ})=∫D(fields)e−iSBFe−Sφ[0]∏a=1N∏Ia=12saZparta,Ia with Zparta,Ia given by Eq. (53). From the actions Spart(Ψ→a,Ia⁎,Ψ→a,Ia) shown in Eq. (54), we see that the topological forces are tightly related to the non-topological forces mediated by the potential V(x−y). This can be realized from the fact that the fields φa,Ia and the third component of the vector fields A3a are coupled in the same way with the matter fields Ψ→a,Ia and Ψ→a,Ia⁎. This interplay between topological and non-topological interactions remains explicit after the integration over the auxiliary φa,Ia. After performing these integrations, we arrive at the final expression of Zpol({λ}):(56)Zpol({λ})=[∏c=1N−1∏d=c+1N∫DBμcdDCμcd][∏a=1N∏Ia=12salimna,Ia→0∫DΨ→a,Ia⁎DΨ→a,Iaψa,Ia1⁎(ra,Ia(τa,Ia),τa,Ia)ψa,Ia1(ra,Ia(τa,Ia−1),τa,Ia−1)]e−iSBFe−Smatter where SBF has been already defined in Eq. (33) and(57)Smatter=Smatter1+Smatter2 with(58)Smatter1=∑a=1N∑Ia=12sa∫τa,Ia−1τa,Iadta,Ia∫d2x[Ψ→a,Ia⁎(∂∂t+iA3a)Ψ→a,Ia+14ga,Ia|(∇−i(−1)Ia−1Aa)Ψ→a,Ia|2] and(59)Smatter2=∑a,b=1N∑Ia=12sa∑Ib=12sb∫τa,Ia−1τa,Iadta,Ia∫d2xd2y×M4a,Ia;b,Ib|Ψ→a,Ia(x,t)|2V(x−y)|Ψ→b,Ib(y,t)|2. Looking at Eqs. (56)–(59), we see that the original polymer partition function (33) has been transformed into a field theory of two-dimensional quasiparticles. The action Smatter1 in Eq. (58) is formally equivalent to the action of a multicomponent system of anyons subjected to the interactions described by the action Smatter2 in Eq. (59). Similar systems have been discussed in connection with the fractional quantum Hall effect and high-TC superconductivity [64]. The only differences in our case are the boundaries of the integrations over the time, which in this work depend on the heights of the points of maxima and minima of the two trajectories Γ1,…,ΓN. Moreover, here the quasiparticles are bosons of spin na,Ia, a=1,…,N and Ia=1,…,2sa considered in the limit na,Ia→0.At this point, we quantize the BF fields using the Coulomb gauge and perform the integration over the third components B3ab and C3ab. The generalization of Eq. (23) to the case of N loops Γ1,…,ΓN is straightforward. The BF action SBF becomes in the Coulomb gauge:(60)SBF=∑a=1N−1∑b=a+1Nκ4π∫d2xdt[B3abΔcab+C3abΔbab] where bab and cab are scalar fields related to the Hodge decomposition (22). The third components of the BF fields play the role of Lagrange multipliers. They can be easily integrated out in the probability function Zpol({λ}) of Eq. (56). As a result of this operation, the following constraints are imposed:(61)κ4πΔcab+λab∑Ia=12sa|Ψ→a,Ia|2θ(τa,Ia−t)θ(t−τa,Ia−1)=0{a=1,…,N−1b=2,…,N(62)Δbab+12π∑Ib=12sb|Ψ→b,Ib|2θ(τb,Ib−t)θ(t−τb,Ib−1)=0{b=2,…,Na=1,…,b−1 The final form of the probability function Zpol({λ}) in the Coulomb gauge is(63)Zpol({λ})=[∏c=1N−1∏d=c+1N∫DBcdDCcd]×[∏a=1N∏Ia=12salimna,Ia→0∫DΨ→⁎a,IaDΨ→a,Iaψa,Ia1⁎(ra,Ia(τa,Ia),τa,Ia)ψa,Ia1(ra,Ia(τa,Ia),τa,Ia)]×e−Smatter,CG where(64)Smatter,CG=Smatter,CG1+Smatter2. Here Smatter2 is the same of Eq. (59) while(65)Smatter,CG1=∑a=1N∑Ia=12sa∫τa,Ia−1τa,Iadta,Ia∫d2x14ga,Ia[|∇Ψ→a,Ia|2+i(−1)Ia−1Aa⋅Ja+|Ψ→a,Ia|2(Aa)2]. In the above equation the Ja's are the currents(66)Ja=Ψ→a,Ia∇Ψ→a,Ia⁎−Ψ→a,Ia⁎∇Ψ→a,Ia. The BF-fields cease to be independent degrees of freedom because, thanks to the constraints (61)–(62), they can be expressed as functions of the matter fields Ψ→a,Ia⁎, Ψ→a,Ia. As a matter of fact, these constraints can be solved analytically with respect to the remnants bab, cab of the original gauge fields. Remembering that Biab=ϵij∂jbab and Ciab=ϵij∂jcab, we write down directly the components of the fields Bab and Cab:(67)Ciab(x,t)=−2λabκ∫d2y∑Ia=12sa|Ψ→a,Ia(y,t)|2ϵij(x−y)j|x−y|2θ(τa,Ia−t)θ(t−τa,Ia−1)=0,a=1,…,N−1,b=2,…,N,(68)Biab(x,t)=−∫d2y14π2ϵij(x−y)j|x−y|2∑Ib=12sb|Ψ→b,Ib(y,t)|2θ(τb,Ib−t)θ(t−τb,Ib−1)=0,b=2,…,N,a=1,…,b−1. The above expressions of the BF-field should be inserted in Eqs. (46)–(48) which define the fields Aa appearing in the action (58). Let us note that the fields Aa written in terms of the solutions (67)–(68) do not contain the parameter κ as expected. Putting all together, it is possible to conclude that the total energy density of the system of plats contains quartic and sextic interactions in the matter fields Ψ→a,Ia⁎, Ψ→a,Ia. This conclusion is in agreement with previous calculations performed in [55], where it has been shown that the topological constraints generate quartic and sextic corrections due to the presence of the topological constraints. The difference is that in [55] the approximate method of the effective potential has been used, while the present calculations are exact.5A statistical model of a 2s-plat composed by N-linked polymersUsing Feynman diagrams, the nontopological quartic interactions in Eq. (59) may be represented by the four-vertex in Fig. 6-(a). The quartic interactions of topological origin described by the contributions to Smatter,CG1 of Eq. (65) in which the fields Aa are coupled to the currents Ja, correspond to the four-vertex of Fig. 6-(b). The sextic interactions, also of topological origin, consisting in the terms in Smatter,CG1 proportional to (Aa)2, are displayed in Fig. 6-(c). Let us note that in both the four-vertex and the six-vertex of Figs. 6-(b) and 6-(c) the external legs depart from a solid circle. This circle symbolizes the fact that these vertices contain non-perturbative contributions coming from the path integral summation over the field Bμab and Cμab. The strengths g4 and g6 of the quartic and sextic interactions of topological origin are respectively proportional to:(69)g4∼λab8π2g6∼λabλac16π4 As it is clear from Eq. (8), the λab's are Fourier coefficients varying in the interval (−∞,+∞). For this reason, g4 and g6 cannot be considered as real coupling constant. However, the parameters λab may be interpreted as chemical potentials that specify how easy is the linking of two trajectories Γa and Γb. To small values of λab correspond big values of the linking number mab and viceversa.An important feature of the model described in Eqs. (63) and (64) is that the interactions of topological origin have sextic interactions, in which the monomers of three different loops are involved. The appearance of three-body forces was up to now not supposed to be possible in the case of topological constraints imposed using the Gauss linking number. As a matter of fact, this link invariant controls only the linking between pairs of polymer rings. In the case N=2, in which we have just two loops, these three-body interactions are suppressed as showed in Ref. [55], because they vanish when the limit in which the numbers of replicas na,Ia approach zero is performed in the probability function of Eq. (63). However, not all diagrams with three-body interactions disappear when N>2. An example of nontrivial contribution in which interactions of three monomers are taking place is shown in Fig. 7.Another characteristic of the model describing the statistical mechanics of 2s-plats introduced here is the existence of vortex solutions of the equations that minimize the energy of the static field configurations. An example of such solutions will be presented in the next Section in the case N=2.6Self-dual solutions of the two-polymer problemIn this Section we restrict ourselves for simplicity to 4-plats. Moreover, the non-topological interactions contained in Smatter2 will be ignored. We will also suppose that the replica numbers are independent of Ia, i.e.:(70)Ψ→(x,t)=(ψa,Ia1(x,t),…,ψa,Iana(x,t))a=1,2 and Ia=1,2(71)Ψ→⁎(x,t)=(ψa,Ia1⁎(x,t),…,ψ⁎naa,Ia(x,t))a=1,2 and Ia=1,2. In Eqs. (51) and (52) each pair of complex fields Ψ→a,Ia⁎,Ψ→a,Ia had a separate replica index na,Ia, but it is easy to check that Ia-independent replica indexes are possible too without jeopardizing the passage to field theory and in particular the calculations made in Section 4. The partition function of a 4-plat formed by two linked polymers is obtained by putting N=2 and s1=s2=1 in the general partition function of a 2s-plat given in Eq. (63). Accordingly, the action Smatter,CG in Eq. (64) in this particular case becomes(72)Smatter,CG=∫τ1,0τ1,1dt∫d2x{Ψ→1,1⁎[∂∂t−14g1,1D2(−λ12,B12)]Ψ→1,1+Ψ→1,2⁎[∂∂t−14g1,2D2(λ12,B12)]Ψ→1,2}+∫τ2,0τ2,1dt∫d2xΨ→2,1⁎{[∂∂t−14g2,1D2(−κ8π2,C12)]Ψ→2,1+Ψ→2,2⁎[∂∂t−14g2,2D2(κ8π2,C12)]Ψ→2,2}. In the above equation D denotes the covariant derivatives, which are of two types depending if they are defined with respect to the field B12 or to the field C12:(73)D(±λ12,B12)=∇±iλ12B12D(±κ8π2,C12)=∇±iκ8π2C12. As mentioned at the end of the previous Section, the fields B12 and C12 are not independent degrees of freedom, because they are fully determined by the constraints (61)–(62). In the present case N=2, s1=s2=2, the required conditions are:(74)ϵij∂iBj12=−12π(|Ψ→21|2+|Ψ→22|2)θ(τ2,1−t)θ(t−τ2,0)(75)ϵij∂iCj12=−4πλ12κ(|Ψ→11|2+|Ψ→12|2)θ(τ1,1−t)θ(t−τ1,0). We will consider now the static field configurations that minimize the action Smatter,CG of Eq. (72). From Ref. [42] it is known that this action admits self-dual solutions in the case in which the parameters ga,Ia, a=1,2 and Ia=1,2 are all equal. To this purpose, for any constant γ and gauge field a we define the new covariant derivatives D±(γ,a):(76)D±(γ,a)=D1(γ,a)±iD2(γ,a) where D1 and D2 denote the first and second components of the covariant derivative D. In terms of the D±'s, the self-duality equations may be expressed as follows:(77)D+(−λ12,B12)ψ1,1n1=0(78)D+(λ12,B12)ψ1,2n1=0(79)D−(−κ8π2,C12)ψ2,1n2=0(80)D−(κ8π2,C12)ψ2,2n2=0. We notice in the constraints (74) and (75) the cumbersome presence of the Heaviside θ-functions. They are required in order to take into account the fact that the heights of the points belonging to the trajectories Γa,Ia are only partially overlapping. As a consequence, to avoid complications, we will assume that τ1,0=τ2,0=τ0 and τ1,1=τ2,1=τ1, i.e. all trajectories will start and end at the same height. In this way the Heaviside θ-functions are no longer needed. Moreover, we will restrict ourselves to replica symmetric solutions by putting:(81)ψ1,I11=⋯=ψ1,I1n1=ψ1,I1forI1=1,2ψ2,I21=⋯=ψ2,I2n2=ψ2,I2forI2=1,2. After these simplifications, the self-duality conditions (77)–(80) and the constraints (74) and (75) become:(82)[∂1−iλ12B112+i(∂2−iλ12B212)]ψ1,1=0(83)[∂1+iλ12B112+i(∂2+iλ12B212)]ψ1,2=0(84)[∂1−iκ8π2C112−i(∂2−iκ8π2C212)]ψ2,1=0(85)[∂1+iκ8π2C112−i(∂2+iκ8π2C212)]ψ2,2=0 and(86)ϵij∂iBj12=−12πn2(|ψ2,1|2+|ψ2,2|2)(87)ϵij∂iCj12=−4n1πλ12κ(|ψ1,1|2+|ψ1,2|2). At this point we pass to polar coordinates by performing the transformations:(88)ψa,Ia=eiωa,Iaρa,Ia1/2. After the above change of variables in Eqs. (82)–(87) and separating the real and imaginary parts, we obtain:(89)∂1ω1,1−λ12B112+12∂2logρ1,1=0(90)−∂2ω1,1+λ12B212+12∂1logρ1,1=0(91)∂1ω1,2+λ12B112+12∂2logρ1,2=0(92)−∂2ω1,2−λ12B212+12∂1logρ1,2=0(93)∂1ω2,1−κ8π2C112−12∂2logρ2,1=0(94)∂2ω2,1−κ8π2C212+12∂1logρ2,1=0(95)∂1ω2,2+κ8π2C112−12∂2logρ2,2=0(96)∂2ω2,2+κ8π2C212+12∂1logρ2,2=0(97)ϵij∂iBj=−12πn2(ρ2,1+ρ2,2)(98)ϵij∂iCj=−4n1πλ12κ(ρ1,1+ρ1,2). To solve equations (89)–(96) with respect to the unknowns ωa,Ia and ρa,Ia, we proceed as follows. First of all, we isolate from Eq. (89) and Eq. (91) the same quantity λ12B112. By requiring that the expressions of λ12B112 provided by Eqs. (89) and (91) are equal, we obtain the consistency condition(99)∂1ω1,1+12∂2logρ1,1=−∂1ω1,2−12∂2logρ1,2 A possible solution of Eq. (99) is(100)ω1,1=−ω1,2andρ1,1=A1ρ1,2 where A1 is at most a function of x1. As well, we require that the two different expressions of the quantity λ12B212 obtained from Eqs. (90) and (92) are equal. On this way one obtains a condition analogous to (99), which may be solved by applying the ansatz (100) and additionally requiring that A1 is a constant. In a similar way, it is possible to extract from equations (93)–(96) the conditions:(101)ω2,1=−ω2,2andρ2,1=A2ρ2,2 with A2 being a constant.Thanks to Eqs. (100) and (101), the number of unknowns to be computed is reduced. For instance, if we choose as independent degrees of freedom ω1,1,ω2,1,ρ1,1 and ρ2,1, the remaining classical field configurations ω1,2,ω2,2,ρ1,2 and ρ2,2 can be derived using such equations. As a consequence, the system of equations (89)–(98) reduces to:(102)λ12B112=∂1ω1,1+12∂2logρ1,1(103)λ12B212=∂2ω1,1−12∂1logρ1,1(104)κ8π2C112=∂1ω2,1−12∂2logρ2,1(105)κ8π2C212=∂2ω2,1+12∂1logρ2,1(106)∂1B212−∂2B112=−12πn2(ρ2,1+A2ρ2,1)(107)∂1C212−∂2C112=−4n1πλ12κ(ρ1,1+A1ρ1,1) where we have used the fact that ϵij∂iBj=∂1B212−∂2B112 and ϵij∂iCj=∂1C212−∂2C112. Eqs. (102)–(107) contain the unknowns ω1,1,ω2,1,ρ1,1 and ρ2,1 that will be determined below.By subtracting term by term the two equations resulting from the derivation of Eqs. (102) and (103) with respect to the variables x2 and x1 respectively, we obtain as an upshot the relation:(108)λ12(∂1B212−∂2B112)=∂1∂2ω1,1−∂2∂1ω1,1−12Δlogρ1,1 with Δ=∂12+∂22 being the two-dimensional Laplacian.Assuming that ω1,1 is a regular function satisfying the relation(109)∂1∂2ω1,1−∂2∂1ω1,1=0 Eq. (108) becomes:(110)λ12(∂1B212−∂2B112)=−12Δlogρ1,1. An analogous identity can be derived starting from Eqs. (104) and (105):(111)κ4π2(∂1C212−∂2C112)=Δlogρ2,1. The compatibility of (110) and (111) with the constraints (106) and (107) respectively leads to the following conditions between ρ1,1 and ρ2,1:(112)Δlogρ1,1=λ12n2π(A2ρ2,1+ρ2,1)(113)Δlogρ2,1=−λ12n1π(ρ1,1+A1ρ1,1). The fact that ρ1,1 and ρ2,1 appear in a symmetric way in Eqs. (112) and (113), suggests the following ansatz:(114)ρ2,1=A3ρ1,1 with A3 being a constant. It is easy to check that with this ansatz Eqs. (112) and (113) remain compatible provided:(115)A2A3=n1n2andA3A1=n1n2. We choose A1 to be the independent constant, while A2 and A3 are constrained by Eq. (115) to be dependent on A1:(116)A2=(n1n2)2A1A3=n1n2A1. We are now left only with the task of computing the explicit expression of ρ1,1. This may be obtained by solving the equation:(117)Δlogρ1,1=λ12n1π(A1ρ1,1+ρ1,1) The other quantities ρ2,1, ρ1,2 and ρ2,2 can be derived using the relations (114), (100) and (101) respectively. Eq. (117) may be cast in a more familiar form by putting: η=ln(ρ1,1A1). After this substitution, Eq. (117) becomes the Euclidean cosh–Gordon equation with respect to η:(118)Δη=2λ12n1πA1coshη Next, it is possible to determine the magnetic fields B12 and C12 from Eqs. (106) and (107). In the Coulomb gauge, in fact, the two-dimensional vector potentials B12 and C12 can be represented using two scalar fields b12 and c12 as follows (see also Eq. (22)):(119)B12=(−∂2b12,∂1b12)C12=(−∂2c12,∂1c12) Performing the above substitutions in Eqs. (106) and (107), it turns out that b12 and c12 satisfy the relations:(120)Δb12=−n12π(ρ1,1+A1ρ1,1)(121)Δc12=−4n1πλ12κ(ρ1,1+A1ρ1,1) The solution of Eqs. (120) and (121) can be easily derived with the help of the method of the Green functions once the expression of ρ1,1 is known. Finally, the phases ω1,1, ω1,2, ω2,1 and ω2,2 are computed using Eqs. (102)–(105). In fact, remembering that we assumed that ω1,1=−ω1,2 and ω2,1=−ω2,2 in (100) and (101) respectively, we have only to determine ω1,1 and ω2,1. By deriving Eq. (102) with respect to x1 and Eq. (103) with respect to x2, we obtain:(122)λ12∂1B112=∂12ω1,1+12∂1∂2logρ1,1λ12∂2B212=∂22ω1,1−12∂2∂1logρ1,1 On the other side, by adding term by term the above two equations and using the fact that in the Coulomb gauge the magnetic field B12 is completely transverse, it is possible to show that:(123)Δω1,1=0 Proceeding in a similar way with Eq. (104) and (105) it is possible to derive also the relation satisfied by ω2,1:(124)Δω2,1=07ConclusionsIn this work a 2s-plat composed by N polymers forming a nontrivial link has been considered. In a 2s-plats the number s of maxima and minima is fixed. With respect to the links of polymer rings discussed for instance in Ref. [52], that are not subjected to this constraint, the set of possible conformations of the links under investigation is limited. The differences that arise in the entropy and free energy of a 2s-plat in comparison with general links are important both for biological and technological applications. Apart from the novelty of the subject, the representation as 2s-plats of systems of knotted rings with non-trivial topological properties presents several advantages. One of them is the previously discussed possibility of using the bridge number, a topological invariant that improves the treatment of the topological constraints based only on the GLN of Refs. [56] and [52]. The more refined topological invariants that have been proposed up to now, for instance in [63] and [65], lead to field theoretical models that are far more complicated than the relatively simple model derived in this work. The feature of 2s-plats that is more relevant for our purposes is that 2s-plats may be decomposed into a set of 2s open and monotonic curves. Such curves can be interpreted as the conformations of 2s open polymer chains directed along an arbitrary direction. We have assumed here that this direction coincides with the z-axis. In Subsection 3.1 we have shown that this choice does not decrease the generality of our results.The nontrivial interactions and the topological constraints make the energy density of the system complicated and nonlocal, but we have seen that it can be simplified with the introduction of auxiliary fields. The final model which we obtain is a standard field theory involving a set of complex scalar fields with sextic interactions at most. This model allows some phenomenological predictions that were a priori not obvious and that will be summarized below.1.In the general case of a 2s-plat the two-body interactions between the monomers, expressed in Eq. (29) by a potential V(r2−r1), can be screened or enhanced by the interactions in Eq. (35) that arise due to the presence of the topological constraints. This interference between non-topological and topological interactions is visible for example in Eq. (54). In fact, it is easy to realize that in the action Spart(Ψ→a,Ia⁎,Ψ→a,Ia) the terms containing the third components of the BF-fields can be absorbed after a shift of the fields φa,Ia. This hints to a strong interplay between topological and non-topological interactions, since the former are mediated by the BF-fields, while the latter are propagated by the fields φa,Ia. Let us note that the effects of the forces of topological origin may result both in a reciprocal attraction or repulsion between the monomers. On the contrary, the short range two-body potential (31), which applies to the situation in which the polymers are immersed in a solution, can only be attractive if V0<0 or repulsive if V0>0.2.The field theoretical model of polymeric 2s-plats defined by Eqs. (63)–(68) shows that three-body forces become relevant in a system of N polymers linked together in which the topological constraints are imposed by means of the Gauss linking number. These three-body forces have been represented in the form of a Feynman diagram in Fig. 6-(c) and are described in Section 5. An example of process in which there are interactions between three monomers at once has been shown in Fig. 7. The existence of three-body interactions acting on the monomers was not predicted by previous calculations. This is probably because only the case N=2 has been mainly treated so far. When N=2, it turns out that the contribution of sextic interactions terms in the action of Eq. (65), which are responsible for the presence of the three-body forces, vanishes in the zero replica limit [55]. Besides, the appearance of three-body forces is not trivial and not easy to be predicted, because the Gauss linking number involves only interactions between pairs of monomers. Let us notice that the strengths of the two- and three-body forces are respectively proportional to λab and λab2. The λab's are Fourier coefficients, so that they are not fixed, but can take any value from −∞ to +∞. This complicates the task of evaluating the strengths of these interactions in different regimes.3.By using the splitting procedure presented in Section 2 and thanks to the introduction of auxiliary fields, the problem of the statistical mechanics of a 2s-plat has been mapped into the dynamics of a system in which quasiparticles of different kinds are mixed together. In Ref. [42] it has been shown that systems of this type admit vortex solutions. Out of the self-duality regime, vortex magnetic lines associated with quasi-particles of different kind can repel or attract themselves. After a particular choice of the parameters of the theory, in which the coefficients ga,Ia, a=1,2 and Ia=1,2 are all equal, a self-dual point is reached in which attractive and repulsive forces balance themselves and disappear. A similar phenomenon, but in a different model, has been recently found in Ref. [66]. In this work, the self-dual vortex conformations have been computed exactly and explicitly up to the solution of a cosh-Gordon equation.The topological properties of the link formed by the 2s-plat have been described here by using the Gauss linking invariant, which is related to the abelian BF-model of Eq. (34). When the BF-model is quantized in the Coulomb gauge, the topological constraints requiring that the Gauss linking numbers between pairs of rings are constant are apparently lost, being replaced by constraints on the winding numbers of the 2s open chains composing the plat. In Appendix C it has been shown in the case s=4 that these constraints on the winding numbers of the chains are compatible with the original topological constraints on the Gauss linking numbers.While abelian anyon field theories like those of Eq. (34) may be significant in quantum computing [67], it is rather nonabelian statistics that plays the main role in this kind of applications. Despite its limitations, our model is able to capture also part of the non-abelian features of the system. The reason is that the constraints on the Gauss linking numbers are applied together with the constraint on the 2s-plat configuration, which cannot be destroyed because the 2s points of maxima and minima are kept fixed. As mentioned in the Introduction, if we start from an unlink consisting of a 4-plat, the system will never be able to attain the configuration of a Whitehead link and vice-versa, a 6-plat Whitehead link cannot turn into a 4-plat. By choosing s=smin, where 2smin is the least possible number of maxima and minima necessary to represent a given link, another topological invariant is added to our treatment besides the GLN, namely the bridge number. This combination of two invariants is much more powerful than the GLN alone. Of course, it should be kept in mind that it is impossible to constrain the topology of a knot or link with the help of a finite set of topological invariants. While the combined constraint provided by the GLN and the bridge number is much more powerful than the GLN alone, still the paths of the polymers are allowed to cross themselves during thermal fluctuations, implying that the freedom of changing topology remains. Moreover, in a forthcoming publication we will show how the present formalism can be applied to include in our approach not only the bridge number, but also much stronger constraints than the Gauss linking number. This possibility has been mentioned at the beginning of Section 3. This will pave the way to the treatment of polymer knots or links constructed from tangles. It is worthing to stress at this point that so far there is no satisfactory analytical model for the statistical mechanics of knots, at least comparable with that of links derived in [56]. The problem in the case of knots is that knot invariants are too complicated to be implemented in a field theory. For this reason, the derivation of a field theory describing the fluctuations of single knots in a solution, even with the restriction of fixing the 2s points of maxima and minima due to the plat configuration, would be an important progress in understanding the behavior of knotted polymer rings. Besides, these results could be relevant in biochemistry because nontrivial knot and link configurations appearing as a major pattern in DNA rings are mostly in the form of tangles [13].AcknowledgementsF. Ferrari would like to thank E. Szuszkiewicz for pointing out Ref. [71] and inspiring the present work. He is also grateful to J. Wang for bringing to the attention the following papers that could be relevant for further developments toward the inclusion of non-abelian topological invariants: [68–70]. J. Paturej would like to acknowledge the support from the Polish Ministry of Science and Higher Education (Iuventus Plus: IP2015 059074). F. Ferrari and J. Paturej wish to thank heartily also M. Pyrka, V. G. Rostiashvili and T.A. Vilgis for fruitful discussions. The simulations reported in this work were performed in part using the HPC cluster HAL9000 of the Computing Centre of the Faculty of Mathematics and Physics at the University of Szczecin. This work results within the collaboration of the COST Action CA17139.Appendix AParametrization of the paths Γa,Ia'sIn this Appendix we consider the set of directed paths Γa,Ia, a=1,…,N and Ia=1,…,2sa resulting after the decomposition of the loops Γ1,…,ΓN in Section 2. In the general case, the set of points belonging to Γa,Ia can be described by the formula:(125)Γa,Ia={ra,Ia(za,Ia)|a=1,…,N;Ia=1,…,2sa{τa,Ia−1≤za,Ia≤τa,IaIa oddτa,Ia≤za,Ia≤τa,Ia−1Ia even} where the additional boundary conditions (4) and (5) are understood. They are necessary for granting the continuity of the loops Γa. For convenience, we report these conditions below:(126)ra,Ia(τa,Ia)=ra,Ia+1(τa,Ia)Ia=1,…,2sa−1(127)ra,1(τa,0)=ra,2sa(τa,0) In Eq. (125) the two-dimensional vector ra,Ia(za,Ia) represents the projection of the trajectory Γa,Ia onto the plane x,y perpendicular to the z-axis. Let us note that we are using the same indexes Ia to label the trajectories Γa,Ia and the points τa,Ia. However, in the first case Ia=1,…,2sa, while in the second case we have chosen Ia=0,…,2sa−1. The range of the indices Ia in the variables za,Ia's and of the ta,Ia's that will be introduced later in this Appendix (see also Eq. (3) in Section 2) is the same as that of the indices labeling the trajectories Γa,Ia's, i.e. Ia=1,…,2sa.The disadvantage of the variables za,Ia's is that by definition they are always growing. In this way, the fact that the whole loop Γa is continuous and has a given orientation is not taken into account. Better variables, respecting both the continuity and orientation of the trajectories Γa,Ia, are the ta,Ia's, which are defined as follows:(128)ta,Ia=za,Iawhen Ia is odd(129)ta,Ia=−(za,Ia−τa,Ia)+τa,Ia−1when Ia is even. Assuming for instance that Ia is odd, then for any two consecutive trajectories Γa,Ia and Γa,Ia+1 the range of the variables ta,Ia and ta,Ia+1 is given by:(130)τa,Ia−1≤ta,Ia≤τa,IaIa odd1≤Ia≤2sa−1 Instead, if Ia is even:(131)τa,Ia−1≥ta,Ia≥τa,IaIa even2≤Ia≤2sa. Let us recall that by our conventions the trajectories labeled by odd Ia's are oriented from a point of minimum to a point of maximum, while trajectories with even values of Ia go from a point of maximum to a point of minimum. Accordingly, the new variables ta,Ia have been chosen in such a way that they increase from the minimum to the maximum when Ia is odd, while they decrease from the point of maximum to that of minimum when Ia is even. Finally, we provide the definition of the curves Γa,Ia parametrized with the help of the new variables ta,Ia's:(132)Γa,Ia={ra,Ia(ta,Ia)|a=1,…,N;Ia=1,…,2sa{τa,Ia−1≤ta,Ia≤τa,IaIa oddτa,Ia−1≥ta,Ia≥τa,IaIa even} This is exactly Eq. (3). Of course, the boundary conditions (126) and (127), or equivalently (4) and (5), are always understood.The variables ta,Ia arise in a natural way when a curvilinear integral around the loop Γa is split into many trajectories Γa,Ia. In fact, let's consider for example integrals of the kind(133)I=∮Γadx˜aμ(da)Aμ(x˜a(da)) where the symbol x˜aμ(da)=(r˜a(da),x˜a3(da)) denotes the points of the trajectory Γa parametrized in terms of the arc-length da, 0≤da≤La. Aμ(x˜a(da)) is an abelian gauge field on R3. It is easy to show that, after splitting the loop Γa into the trajectories Γa,Ia, on each of these trajectories it is possible to change the arc-length da with the parameters ta,Ia. If one does that, the curvilinear integral I of Eq. (133) becomes parametrized by the variables ta,Ia and may be expressed as follows(134)I=∑Ia=12sa∫τa,Ia−1τa,Ia[dra,Ia(ta,Ia)dta,Ia⋅A(ra,Ia(ta,Ia),ta,Ia)+A3(ra,Ia(ta,Ia),ta,Ia)] where(135)ta,Ia=x˜a3(da)ra,Ia(ta,Ia)=ra,Ia(x˜a3(da))=r˜a(da). Of course, the above equation is valid only if da is restricted on the trajectory Γa,Ia, i.e., δa,Ia−1≤da≤δa,Ia. The δa,Ia's denote the values of the arc-length at the points of maxima and minima of the 2sa-plat Γa. Clearly, x˜a3(δa,Ia)=τa,Ia.Appendix BThe length L of a directed polymer as a function of the heightIn this Appendix we consider the partition function(136)Z=∫Dr(z)e−S where S is the action of the free open polymer, whose path Γ is parametrized by means of the height z defined in some interval [τ0,τ1]:(137)S=g∫τ0τ1dz|drdz|2 We want now to determine how the total length of the curve Γ depends on the constant parameter g. To understand what we mean by that, let us consider the standard case of an ideal chain whose path is parametrized with the help of the arc-length σ. We denote with a the average statistical length (Kuhn length) of the N segments composing the polymer. In the limit of large N and small a such that the product Na is constant, the total length L of the polymer satisfies the relation(138)L=Na We wish to obtain a similar identity connecting L with N and g in the present situation, which is somewhat different. To this purpose, we first dicretize the interval of integration [τ0,τ1] splitting it into N small segments of length:(139)Δz=τ1−τ0N As a consequence, we may approximate the action as follows:(140)S∼g∑w=1N|ΔrwΔz|2Δz where the symbol Δrw means(141)Δrw=rw+1−rw and(142)rw=r(τ0+wΔz) The discretized partition function becomes thus the partition function of a random chain composed by N segments:(143)Zdisc=∫∏w=1Ndrwe−∑w=1Ng|Δrw|2Δz Using simple trigonometric arguments it is easy to realize that the length of each segment is:(144)ΔL=|Δrw|2+(Δz)2 This is of course an average length, dictated by the fact that, from Eq. (143), the values of |Δrw| should be gaussianly distributed around the point:(145)|Δrw|2=Δzg In the limit Δz→0, the distribution of length of Δrw becomes the Dirac δ-function:(146)limΔz→012gΔze−g|Δrw|2/Δz∼δ(|Δrw|−Δzg) If N is large enough, we can therefore conclude that the total length of the chain Γ is:(147)L∼NΔL=NΔzg+(Δz)2 Since NΔz=τ1−τ0, we get:(148)L2=|τ1−τ0|2+N(τ1−τ0)g In the limit N→∞, while keeping the ratio Ng finite, Eq. (148) becomes the desired relation between the length of Γ and g which replaces Eq. (138).Appendix CThe expression of the Gauss linking invariant in the Coulomb gaugeTo fix the ideas, we will study here the particular case of a 4-plat. In the partition function (33) we isolate only the terms in which the BF fields appear, because the other contributions are not connected to topological constraints and thus are not relevant. As a consequence, we have just to compute the following partition function:(149)ZBF,CG(λ)=∫DBμDCμe−iSBF,CG−Stop where the BF action in the Coulomb gauge SBF,CG has been already defined in Eq. (21) and Stop has been given in Eq. (35). In the case of a 4-plat, Stop becomes:(150)Stop=iλ∫τ1,0τ1,1dt[dx1,1μ(t)dtBμ(r1,1(t),t)−dx1,2μ(t)dtBμ(r1,2(t),t)]+iκ8π2∫τ2,0τ2,1dt[dx2,1μ(t)dtCμ(r2,1(t),t)−dx2,2μ(t)dtCμ(r2,2(t),t)] where we recall that xa,Iμ(t)=(ra,I(t),t), a=1,2, I=1,2. For simplicity of the notation, in this Appendix we use λ instead of λ12. Using the Chern-Simons propagator of Eqs. (25)-(26), it is easy to evaluate the path integral over the gauge fields in Eq. (149). The result, after two simple Gaussian integrations, is:(151)ZBF,CG(λ)=exp{iλ2π∑I,J=12(−1)I+J−2ϵij∫τ0τ1d(x1,Ii(t)−x2,Ji(t))(x1,Ij(t)−x2,Jj(t))|r1,I(t)−r2,J(t)|2} In the above equation we have put for simplicity:(152)τ0=max[τ1,0,τ2,0]τ1=min[τ1,1,τ2,1] For instance, if the polymer configurations are as in Fig. 8, we have that τ0=τ1,0 and τ1=τ2,1. Moreover, we remember that in our notation ra,I(t)=(xa,I1(t),xa,I2(t)). Apparently, the elements of the loops Γ1 and Γ2 which lie below τ0 and above τ1 do not take the part in the topological interactions. Thus is due to the presence of the Dirac δ-function δ(t−t′) in the components of the Chern-Simons propagator (25)-(26). However, we will see later that also the contributions of these missing parts are present in the expression of ZBF,CG(λ). In order to proceed, we notice that the exponent of the right hand side of Eq. (151) consists in a sum of integrals over the time t of the kind:(153)D1,I;2,J(τ1)−D1,I;2,J(τ0)=ϵij∫τ0τ1d(x1,Ii(t)−x2,Ji(t))(x1,Ij(t)−x2,Jj(t))|r1,I(t)−r2,J(t)|2 The above integrals can be computed exactly. It is in fact well known that the function D1,I;2,J(t) is the winding angle of the vector r1,I(t)−r2,J(t) at time t:(154)D1,I;2,J(t)=arctan(x1,I1(t)−x2,J1(t)x1,I2(t)−x2,J2(t)) Thus, the quantity D1,I;2,J(τ1)−D1,I;2,J(τ0) is a difference of winding angles which measures how many times the trajectory Γ1,I turns around the trajectory Γ2,J in the slice of time τ0≤t≤τ1. At this point, without any loss of generality, we suppose that the configurations of the curves Γ1 and Γ2 are such that the maxima and minima τa,I are ordered as follows:(155)τ2,0<τ1,0<τ2,1<τ1,1 As example of loop configurations that respect this ordering is given in Fig. 8. As a consequence, we have:(156)τ0=τ1,0andτ1=τ2,1 Now we notice that the logarithm of the gauge partition function ZBF,CG(λ) in Eq. (151) contains a sum of differences of the winding angles defined in Eq. (154):(157)2πlogZBF,CG(λ)iλ=[D1,1;2,1(τ2,1)−D1,1;2,1(τ1,0)+D1,2;2,2(τ2,1)−D1,1;2,2(τ2,1)+D1,2;2,1(τ1,0)−D1,2;2,1(τ2,1)+D1,1;2,2(τ1,0)−D1,2;2,2(τ1,0)] Further, assuming that the curves Γ1 and Γ2 are oriented as in Fig. 8. If we start from the minimum point at τ0=τ1,0, we can isolate in the right hand side of Eq. (157) the following four contributions:1.In the time slice τ1,0≤t≤τ2,1 the angle which measures the winding of the trajectory Γ1,1 around the trajectory Γ2,1 is given by the difference D1,1;2,1(τ2,1)−D1,1;2,1(τ1,0).2.In the region τ2,1≤t≤τ1,1 only the loop Γ1 continues to evolve, going first upwards with the trajectory Γ1,1 and then downwards with Γ1,2. After this evolution, the winding angle between the two loops Γ1 and Γ2 has changed by the quantity D1,2;2,2(τ2,1)−D1,1;2,2(τ2,1).3.Next, in the region τ2,1≥t≥τ1,0, the winding angle which measures how many times the trajectory Γ1,2 winds up around Γ2,2 is given by the difference D1,2;2,1(τ1,0)−D1,2;2,1(τ2,1).4.Finally, in the region τ1,0≥t≥τ2,0 only the second loop Γ2 continues to evolve, going first downwards with the curve Γ2,2 and then upwards with Γ2,1. The net effect of this evolution is that the winding angle between Γ1 and Γ2 changes by the quantity D1,1;2,2(τ1,0)−D1,2;2,2(τ1,0). It is thus clear that the right hand side of Eq. (157), apart from a proportionality factor iλ, counts how many times the loop Γ1 winds around the second loop Γ2. If we wish to identify the quantity in the right hand side of Eq. (157) with the Gauss linking number χ(Γ1,Γ2), we should check for consistency that it takes only integer values as the Gauss linking number does. Indeed, it is easy to see that, modulo 2π, the following identities are holding:(158)D1,1;2,1(τ2,1)=D1,1;2,2(τ2,1)D1,1;2,2(τ1,0)=D1,2;2,2(τ1,0)D1,2;2,2(τ2,1)=D1,2;2,1(τ2,1)D1,1;2,1(τ1,0)=D1,2;2,1(τ1,0) For example, the first of the above equalities states that the angle formed by the vector r1,1−r2,1 connecting the trajectories Γ1,1 and Γ2,1 at the height τ2,1 is equal to the angle formed by the vector r1,1−r2,2 connecting the trajectories Γ1,1 and Γ2,2 at the same height. The reason of this identity is trivial: At that height, the trajectories Γ2,1 and Γ2,2 are connected together at the same point. Applying the above relations to Eq. (157), one may prove that:(159)2πlogZBF,CG(λ)iλ=0mod2π As a consequence, we can write:(160)ZBF,CG(λ)=eiλχ(Γ1,Γ2) where χ(Γ1,Γ2) is the Gauss linking number. 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