Self–dual solutions of a field theory model of two linked rings

In this work the connection established in [7,8] between a model of two linked polymers rings with fixed Gaussian linking number forming a 4-plat and the statistical mechanics of non-relativistic anyon particles is explored. The excluded volume interactions have been switched off and only the interactions of entropic origin arising from the topological constraints are considered. An interpretation from the polymer point of view of the field equations that minimize the energy of the model in the limit in which one of the spatial dimensions of the 4-plat becomes very large is provided. It is shown that the self-dual contributions are responsible for the long-range interactions that are necessary for preserving the global topological properties of the system during the thermal fluctuations. The non self-dual part is also related to the topological constraints, and takes into account the local interactions acting on the monomers in order to prevent the breaking of the polymer lines. It turns out that the energy landscape of the two linked rings is quite complex. Assuming as a rough approximation that the monomer densities of half of the 4-plat are constant, at least two points of energy minimum are found. Classes of non-trivial self-dual solutions of the self-dual field equations are derived. One of these classes is characterized by densities of monomers that are the squared modulus of holo-morphic functions. The second class is obtained under some assumptions that allow to reduce the self-dual equations to an analog of the Gouy-Chapman equation for the charge distribution of ions in a double layer capacitor. In the present case, the spatial distribution of the electric potential of the ions is replaced by the spatial distribution of the fictitious magnetic fields associated with the presence of the topological constraints. In the limit in which two of the spatial dimensions are large in comparison with the third one, we provide exact formulas for the conformations of the monomer densities of the 4-plat by using the elliptic, hyperbolic and trigonometric solutions of the sinh-Gordon


Introduction
In the 70s of the twentieth century, R.G. de Gennes discovered a relationship between the statistical mechanics of long polymer molecules and magnetic systems described by multicomponent complex field theories with O(N ) symmetry [1,2].This discovery gave rise to a new research direction and resulted in the Nobel Prize in Physics in 1991.The idea of using field theory techniques to study properties of polymers was developed by many authors, like for instance [3][4][5].A goal that was set was to construct a theoretical framework to investigate topologically linked polymers [6].This research program has been implemented in the particular case of polymer rings linked together to form a 2s−plat in Refs.[7,8].2s−plats are links whose paths in space are characterized by a fixed number 2s of maxima and minima along a given direction called here the height, for instance the z−axis.A physical realization of a 2s−plat could be a set of polymers rings or knots attached to two membranes consisting of two parallel planes in the xy axis.It was developed in [8] a path integral approach to the statistical mechanics of such a system composed by an arbitrary number of polymer rings linked together.The topological states of the link have been distinguished using a topological invariant known as the Gauss linking number.The topology of the knots forming the links has been left unspecified.It was shown in [7,8] that the derived path integral formulation can be mapped into a field theory of quasiparticles known as anyons.The height z on the polymer side can be interpreted as time in the anyon counterpart.The monomer densities of the N loops become densities of a mixture of N types of anyons.When s = 2, it was demonstrated in [7] that this field theory admits self-dual solutions.The integrability of these self-dual configurations have been explored in [8], where a relation with the cosh-Gordon equation has been established.
In this paper, we return to this subject providing a deeper insight into the meaning of self-duality in the physics of polymers subjected to topological constraints and deriving solutions that minimize the energy of a system of two linked polymer loops.More in details, using the Bogomol'nyi transformation, it is shown that the energy describing two polymer loops in a 4-plat conformation can be splitted in the limit of long polymers into a self-dual term and a term describing short-range interactions.Since the interactions have been switched off for simplicity, i.e., the system is in a solution at the theta point, the interactions mentioned here are of purely topological origin.They are necessary in order to enforce the topological constraints.In the limit in which all monomers are equal, i.e., in the case, where the 4-plat becomes homopolymeric only the self-dual component survives.The reason for which a system of homopolymers becomes self-dual is explained in Section 2. Next, some exact solutions that minimize the energy of the system at the self-dual point are derived.We show that the 4-plat has a complex energy landscape.In the case in which the loops are diblock copolymers, we show with a simple approximation (one of the two loops is supposed to have a constant monomer distribution) that there are at least two points of energy minimum.A class of solutions minimizing the energy in the homopolymeric case consists of field conformations that are holomorphic functions.Finally, we explore the case in which one of the dimensions, x or y, becomes negligible.This can happen for instance in a confined system in which the y−dimension is constrained to be small, though it should be big enough to contain the link formed by the two loops, which is intrinsecally three-dimensional.In this situation, not only the solutions of the cosh-Gordon equation mentioned above become relevant.There are also solutions of the sinh-Gordon and Liouville equations.In the present paper we consider one-dimensional integrable examples of the first two of these equations.In this way, starting from certain real (not complex) solutions of the sinh-/cosh-Gordon equation it has been possible to derive exact formulae for (i) field configurations that describe conformations of the 4-plats; (ii) observables such as monomer densities.To achieve this goal the so-called elliptic, hyperbolic and trigonometric solutions are employed.These are translationally invariant solutions of the Euclidean sinh-/cosh-Gordon equation that depend on only one variable.These types of solutions have been used for instance in the construction of classical string solutions in AdS3 and dS3 [9].Using elliptic, hyperbolic and trigonometric solutions, we find exact formulae for the monomer densities of the 4-plat that minimize the energy at the self-dual point.This is the main result of this work.
This paper is organized as follows.In Section 2, we formulate the problem and then, following [7,8], we re-derive the partition function for a system composed of two linked loops in a topological state described by a fixed Gauss linking number.In the field-theoretical formulation, the polymer partition function can be understood as a correlation function of a mixture of four types of anyons.Referring to the methods coming from the physics of anyons, we derive the self-duality conditions and find their solutions minimizing the energy mentioned above.Then we show that the self-duality equations reduce to the two-dimensional Euclidean sinh-Gordon or cosh-Gordon equations, depending on the sign of the integration constant.We show that there is one more possibility, i.e., when the integration constant equals zero, we obtain the Liouville equation. 1In Section 3, we calculate the translationally invariant solutions of the Euclidean sinh-Gordon and cosh-Gordon equations and the polymer densities expressed by them.Here we make an extensive use of the analytical methods developed in [9].In Section 4, we present our conclusions.

Solvable example of topological entanglement
We consider in this paper links formed in space by two concatenated polymer rings with the additional property that the paths of the rings have a fixed number of maxima and minima with respect to a particular direction, let's say the direction of the z-axis; z will measure the "height".In the case in which the link has a total number of s minima and s maxima, the system is called a 2s-plat.In the following, we will limit ourselves to the class of 4-plats which is particularly interesting for biological applications [10].Since s = 2, each ring has only one point of maximum and only one point of minimum.Let's Γ 1 denote the path of the first ring and Γ 2 the path of the second one.Here, m a and M a , a = 1, 2 will be respectively the points of minimum and maximum of Γ a (see Fig. 1).Each loop Γ a will be further decomposed into two monotonic curves Γ u a and Γ d a .The loops Γ u a run upwards in the z-direction and Γ d a run downwards according to the orientation of the loops given in Fig. 1.
In order to distinguish the different topologies of the link the Gauss linking number χ(Γ 1 , Γ 2 ) will be used, though the treatment can be generalized to the more powerful Vassiliev invariants.As shown in Ref. [11], in the case of the decomposition of two loops into monotonic curves Γ u a , Γ d a , a = 1, 2 along a preferred direction z, χ(Γ 1 , Γ 2 ) can be written as follows 2 where W ΓΓ ′ (z 0 , z 1 ) is the winding number of two monotonic curves Γ, Γ ′ between the two heights z 0 2 See also [8].
and z 1 : Since z is a privileged direction, we distinguish the two-dimensional spatial components of the coordinates x i , i = 1, 2 and the z component x 0 .In general, vectors will be denoted with the symbol (v, v 0 ), where v = (v 1 , v 2 ) is the projection of the vector in the xy-plane.Note that the two curves Γ, Γ ′ in Eq. (2.2) do not need to be defined in the same interval of heights.For this reason, the integration limits in the definition (2.2) of the winding number are in the interval [z 0 , z 1 ] in which both curves have points at the same height.For instance, in the situation of Fig. 1 in which The partition function Z(µ) of the system composed by the two linked loops Γ 1 , Γ 2 may be written as follows: The coordinates x a (m a ) and x a (M a ) denoting respectively the locations of the points of maximal and minimal height of Γ a are fixed.Moreover, the topological constraint with µ being a constant is imposed in Eq. (2.3) using a Dirac delta function.Finally, A pol is the term associated to chain connectivity: For simplicity, no interactions have been added, though the treatment can be easily extended to include the excluded volume potential.In the above formulae x u,d a (z) are curves describing the paths of Γ u,d a .The quantities g a,u 's and g a,d 's are constants related to the Kuhn length and characterizing the flexibility of the chains Γ u a and Γ d a .At this point, the Fourier transform can be applied to represent the Dirac delta function δ (χ(Γ 1 , Γ 2 ) − µ) in the following form: (2.6) This allows to rewrite the partition function Z(µ) in the simpler form: where The Fourier transformation from Z(µ) to Z(λ) is an anologue of the passage from the microcanonical ensemble to the canonical ensemble, but the role of the Hamiltonian H is repleced here by the Gauss linking number χ(Γ 1 , Γ 2 ) and the Boltzmann factor β = (kT ) −1 is repleced by iλ.
As shown in Ref. [8], the exponential e −iλχ(Γ 1 ,Γ 2 ) , which contains a very complicated dependence on the conformations x u,d a (z), may be simplified rewriting it as the partitian function of an abelian BF-model: where Here and below it is assumed that repeated upper and lower indices that label spatial coordinates are summed.The sources of the magnetic field (B a , B 0 a ), a = 1, 2 appearing in Eq. (2.9) are imaginary currents flowing inside the loops Γ 1 and Γ 2 : (2.12) Let's notice that in Eq. (2.9) the fields B a (x, t) satisfy the Coulomb gauge condition: The Coulomb gauge arises naturally when the curves Γ 1 , Γ 2 are parametrized using the z coordinate and divided into monotonic curves Γ u,d a , a = 1, 2. Indeed, it is easy to show that So that the longitudinal component of the currents in the xy-plane is vanishing.Thus, the longitudinal component of the magnetic fields B a (x, t) have no sorces and can be put to zero by imposing the condition (2.13).
In order to prove Eq. (2.9), one needs to integrate out the fields B a (x, t), B 0 a (x, t) on the right hand side of that equation.This amounts to a Gaussian integration that may be easily performed using the propagator: Applying the identity (2.9) it is possible to convert the partition function Z(λ) to the following form: where Dx u a (z) e −S u a , (2.17) and (2.20) The 2 × 2 matrix C ab is given by Let us note S u a and S d a are formally equal to the actions of two particles immersed in the magnetic fields generated by the vector potentials B 1 , B 2 and interacting with the external potentials B 0 1 , B 0 2 .Accordingly, Z u a (λ) may be interpreted as the transition amplitudes of particles x u a (z) to pass from an |.An analogous interpretation can be given to Z d a (λ).This analogy with quantum mechanics allows to pass from paths to fields using the procedure of second quantisation. Putting it is possible to show that the one-particle transition amplitudes Z u,d a (λ) satisfy the pseudo-Schrödinger equations: and being the partition functions of the complex scalar fields, and (2.27) The integrations over t are made over the intervals [m a , M a ] ∋ t.
Further processing of the expression (2.16) into a more useful form is a bit difficult because of the factors (Z u a ) −1 and (Z d a ) −1 in (2.24) and (2.25).However, it can be dealt with using the so-called replica method.Indeed, by introducing n replica fields: the partition function in Eq. (2.16) may be rewritten as a product of Gaussian integrals, In this form the z-components B 0 a of the vector potentials play the role of Lagrange multipliers.They may be easily integrated out producing the constraints: where The Heaviside theta functions are necessary in order to take into account the fact that the loops Γ 1 and Γ 2 are defined in the different ranges of heights [m 1 , M 1 ] and [m 2 , M 2 ].Putting into (2.30) and (2.31) the Gauss law constraint, i.e., one gets relations typical for electrostatics: problem, but may also be interpreted as the correlation function of a mixture of four types of anyon particles with densities |Ψ u a | 2 and Ψ d a 2 , a = 1, 2 and the action: (2.36) For simplicity, from now on we assume that m 1 = m 2 = 0 and M 1 = M 2 = T .This implies that the Heaviside theta functions in the constraints (2.30) and (2.31) satidfied by the vector potentials B a are no longer necessary.
The analogy with anyons suggests to investigate the action (2.36) with the methods of self-dual systems.Following [12,13] and [7], we introduce to this purpose the covariant derivatives: C ab B b,j , (2.37) where j = 1, 2 labels the spatial coordinates and a = 1, 2 labels the contributions coming from loops Γ 1 and Γ 2 .One gets the following identities: where where

.44)
I sd is the self-dual part of the action, (2.45) and I C accounts for the Coulomb-like interactions, . (2.46) In writing the action in Eqs.(2.43)-(2.46)terms that are total derivatives have been neglected.It turns out that the term I T is negligibly small when the height τ in which the paths of both loops are defined become large.To show this, it is sufficient to perform in Eqs.(2.44), (2.45) and (2.46) the change of variable τ σ = t.After doing this the self-dual contribution I sd and the Coulomb interaction terms I C pick up a factor τ , but not I T .From now on we will working in the limit of large τ , in which (2.47) Apart from the fact that the variable t describes the height of the polymer loops and not time, the above action is formally equal to that of a systems of anyons.In other words, as it is reasonable to expect, in the limit τ → ∞ the monomer distribution does not changes very much at different heights, so that it becomes possible to talk about static solutions similarly to the case of anyons with the difference that here static meanas absence of changes along the z-axis.
Proceeding analogously as in the case of anyons, on the basis of Eq. (2.47) we define the density of energy per unit of height z, . (2.48) An interesting case is when the monomer densities of Γ u a or Γ u a can be considered as constant.For instance, assuming that where E sd (z) is the energy density of the self-dual part, (2.50) The energy in Eq. (2.49) is minimized by the self-duality conditions: which are satisfied.There are two distinct minima corresponding to the following cases: (2.53) Most interesting is probably the homopolymer case in which all legs Γ u,d 1 and Γ u,d 2 are homogeneous, so that 1 Remarkably, if all parameters g a,u and g a,d are equal, then the Coulomb-like short range interactions disappear and the system becomes self-dual, i.e., E(z) = E sd (z).The vanishing of the short-range interactions reminds the case of solutions at high monomer concentration and good solvents, in which the interactions act on each monomer symmetrically from any direction, so that their total effect is negligible.The situation is similar here.The term I C of Eq. (2.46) accounts for the short range interactions and they vanish in the limit τ → ∞ and when the loops are homogeneous, see condition (2.54).As already mentioned, in the large τ limit the monomer distribution is not depending on the z direction implying that the short range forces due to the topological constraints acting on a monomer from above are counter balanced by the forces acting from below.In the xy directions all legs Γ u a and Γ d a are equal under the conditions (2.54).It is thus likely that the short range interactions of topological origin become isotropic as in the case of polymer solutions at high monomer concentrations.Of course, what is not cancelled are the long-range interactions because they are necessary to keep the topology of the link.These long term interactions are taken into account by the self-dual contributions in Eq. (2.45).
In the remaining part of this work some solutions of the self-duality equations (2.51) will be derived.First, we consider the self-duality equations for the fields Ψ u a , i.e., We attempt the ansatz i.e., Ψ u a is a holomorphic function of the complex variable w = x 1 + ix 2 .With this setting it turns out that ∂ 1 Ψ u a + i∂ 2 Ψ u a = 0.In this way Eq.(2.55) simplifies to (2.57) The fields may be derived by solving the Gauss constraints (2.32), The derivation of the expression of the fields Ψ d b is straightforward.
A connection between the polymer problem treated here and nonlinear models will be established in the following.Since we are now considering the energy E(z) coming from the partition function Z PF associated to the Green function (2.29), the appropriate expression of the energy is obtained in the limit n → 1 in which the number of replicas is equal to one.In Eq. (2.59) the subpartition functions Z u a , Z d a are given by Eqs.(2.26) and (2.27).Putting all together in the field equations (2.51) and the Gauss constraints (2.32), we obtain the equations for , B a,i , i.e.: C ab B b,2 ψ u a = 0, (2.60) C ab B b,2 = 0, (2.66) (2.68) By requiring that the expressions of B b,1 calculated using Eqs.(2.65) and (2.67) are the same we obtain the consistency conditions: (2.69) These conditions are satisfied if we require that (2.70) The A a are real constants that may be positive or negative.
Subtracting the first of the above equations from the second one, we get 1 and substituting (2.73) in Eq. ( 2.68) we obtain the final result: where C −1 ab is the inverse of the matrix C ab , i.e., Making explicit the dependence on the physical parameters, one gets: (2.76) The constants A 1 , A 2 may take any real value.Due to the fact that the two loops Γ 1 and Γ 2 have the same properties and lengths by assumption, it is natural to search for solutions such that (2.77) In this case, what remains to be solved is the following equation for ρ u 1 , i.e.,

.78)
Taking A 1 > 0 in (2.78), i.e., A 1 = |A 1 | and putting: This is the cosh-Gordon equation.Thus, Eq. ( 2.78) can be reduced to the sinh-Gordon and cosh-Gordon equations, respectively, depending on the sign of the parameter A 1 .There is one more possibility, i.e., taking A 1 = 0 in Eq. (2.78) and substituting ρ u 1 = e ϕ .This leads to the Liouville equation, ∆ϕ = λ π e ϕ .In this situation, however, the calculation of the density ρ u 1 is more sophisticated (some singularities appear) but rather feasible (cf.[14]).We leave this case as an open problem for further research. 4 Translationally invariant solutions

Elliptic solutions of sinh-Gordon and cosh-Gordon equations
In this subsection we construct solutions to Eqs. (2.80) and (2.81), which depend on only one variable, cf.[9]. 5 To begin with, let us write Eqs.(2.80) and (2.81) in the unified form and t = +1 for the cosh-Gordon equation, and t = −1 for the sinh-Gordon equation.Here we are searching for solutions of Eq. (3.82) in the form i.e., the solutions translationally invariant in the x 2 direction.(For translationally invariant solutions in the x 1 direction, there is exactly the same analysis and results as will be discussed below.)In this case, Eq. (3.82) reduces to the ordinary differential equation (ODE), which can be integrated to 1 2 The quantity E in Eq. (3.85) is an arbitrary constant.Substituting, Finally, after one more substitution, Eq. (3.87) takes the standard Weierstrass form: with constants g 2 , g 3 fixed as Our goal is to find real solutions of the Weierstrass Eq. (3.89) defined in the real domain.Indeed, it is noticeable that Eq. (3.82) is real.Recall, η was introduced in (2.79) as a logarithm of positive function ρ 11 .Hence, η and then y in Eq. ( 3.89) must be real-valued.Before we proceed to the construction of real solutions of Eq. (3.89), let us recall some properties of the Weierstrass elliptic function ℘(z; g 2 , g 3 ). 7The latter is a doubly periodic complex function of one complex variable z, satisfying the complex domain Weierstrass ODE, dy dz (3.91) The periods of ℘(z; g 2 , g 3 ) are related with the three roots e 1 , e 2 , e 3 of the cubic polynomial Q(y) = 4y 3 − g 2 y − g 3 .Let us introduce the discriminant ∆ ≡ g 3 2 − 27g 2 3 .One can distinguish three cases.
1.If ∆ > 0 then the cubic polynomial Q(y) will have three real roots.For e 1 > e 2 > e 3 the function ℘ has one real period 2ω 1 and one imaginary period 2ω 2 which are related to the roots: Here, K(k) is the complete elliptic integral of the first kind and 2. If ∆ < 0 then the cubic polynomial Q(y) will have one real root and two complex, which are conjugate to each other.Let e 2 be the real root and e 1,3 = a ± ib with b > 0.Then, the function ℘ has one real period and one complex, which is not purely imaginary.In this case it is convenient to consider as fundamental periods the complex one and its conjugate: The real period is just the sum of the two fundamental periods 2ω 1 and 2ω 2 .
3. If ∆ = 0 then at least two of the roots are equal and the Weierstrass function ℘ takes a special form, which is not doubly periodic, but trigonometric.
In cases 1. and 2. the Weierstrass function ℘ obeys the following half-period relations where It is known how to uncover real solutions to Eq. (3.91).As spoted in [9], Eq. (3.91) can be understood as the conservation of energy for the one-dimensional classical mechanical system, namely, for a point particle with vanishing energy moving under the influence of the cubic potential V ℘ (y) = −4y 3 + g 2 y + g 3 = −Q(y).
• If ∆ > 0 then one can expect to have two real solutions of Eq. (3.91), one being unbounded with y > e 1 and a bounded one with e 3 < y < e 2 (see Fig. 2).
• If ∆ < 0 then one can expect to have only one real solution of Eq. (3.91), which is unbounded with y > e 2 .
• The unbounded real solutions are given by ℘(x), where x ∈ R. Indeed, the Weierstrass function ℘ has a second order pole at z = 0 and it is real on the real axis.So, to determine the unbounded real solution it is enough to restrict z to the real axis.
• The bounded real solutions are determined by ℘(x + ω 2 ), where x ∈ R. Precisely, ℘(x + ω 2 ) is a real periodic solution that oscillates between e 2 and e 3 , with period equal to 2ω 1 .The proof of this statement is as follows (cf.[9]).First, it is clear that the argument of the Weierstrass function can be shifted by an arbitrary constant and ℘ still solves Eq. (3.91).Second, using the periodic properties of ℘ and the fact that ω 2 is purely imaginary in the case ∆ > 0, one gets the relation: Finally, the above claim stems from the fact that the Weierstrass function obeys the half-period identities ℘(ω 1 + ω 2 ) = e 2 and ℘(ω 2 ) = e 3 .
To conclude, when ∆ > 0 then Eq. (3.91) has two real solutions: ) corresponding to the unbounded and bounded solutions respectively, whereas for ∆ < 0 there is only one real solution given by (3.97).At this point, we are ready to calculate: -the real solutions of Eq. (3.89); -the translationally invariant solutions of the sinh-/cosh-Gordon equation (3.82); -analytic expressions for ρ u 1 and Eq. (3.89) is then solved by The second solution is valid only when there are three real roots.The coefficients g 2 and g 3 are given by (3.90) and related cubic polynomial is It is easy to obtain all three roots of Q(q), they are Recall, we set e 1 > e 2 > e 3 when all roots are real.If only one root is real this will be e 2 , and e 1 will be the complex root with a positive imaginary part.Taking into account (3.102) we have the following orderings of the roots q 1,2,3 (E): 8 i. if t = +1 then q 2 > q 1 > q 3 and e 1 = q 2 , e 2 = q 1 , e 3 = q 3 ; ii.t = −1, E > m 2 then q 1 > q 2 > q 3 and e 1 = q 1 , e 2 = q 2 , e 3 = q 3 ; iii. t = −1, E < −m 2 then q 2 > q 3 > q 1 and e 1 = q 2 , e 2 = q 3 , e 3 = q 1 ; iv.t = −1, |E| < m 2 then e 2 = q 1 (real), e 1 = q 2 and e 3 = q 3 (complex).Going forward let us repeat it again, the unbounded solution ranges from e 1 to infinity when ∆ > 0 and from e 2 to infinity when ∆ < 0. The bounded solution ranges from e 3 to e 2 .Then, using these informations (see Tab.1) and (3.88), namely, (3.86), one can write down for instance the explicit form of the sought (bounded) solutions: and its ranges for all values of ν 1 written in the  [0, +∞) [0, +∞) -Table 3: The ranges of ρ u 1 and ρ u 2 .

Coinciding roots and hyperbolic or trigonometric solutions
Elliptic solutions to the Eq.(3.84) with t = −1 have an interesting limit, where they are no longer given by elliptic functions.More precisely, the Weierstrass ℘-function degenerates to hyperbolic or trigonometric functions when two roots coincide.These solutions have been studied in [9].We will use these results in the present subsection.To begin with let us quote the result regarding the function ℘(z, g 2 , g 3 ) reported in [9], i.e.: parameters The ranges of ρ d 1 and ρ d 2 .

.110)
Looking at Fig. 3 and Fig. 4, and the orderings of the roots (points i.-iv.), one may see two possibilities for degeneracy, i.e., in cases with t = −1, namely, for E = −m 2 and E = m 2 .Below, we calculate solutions corresponding to these situations.
In this case the roots are e 1 = e 2 = e 0 = 1 12 m 2 and e 3 = − 1 6 m 2 .The real half-period ω 1 diverges and the imaginary half-period becomes ω 2 = iπ/m.Applying (3.109) to the solution (3.99) one gets • the densities: • the densities: In such a situation the roots are e 1 = 1 6 m 2 and e 2 = e 3 = −e 0 = − 1 12 m 2 .The imaginary half-period ω 2 diverges and the real half-period is ω 1 = π/m.This time, applying the formula (3.110) to Eq. (3.99) one gets • the densities: As a final remark in this subsection let us note that among the solutions listed above, one can find soliton-like structures such as (3.112) and (3.116).They can be seen as "superpositions" of soliton-like profiles.9

Concluding remarks
In [7] and [8] a connection has been established between polymers and the statistical mechanics of nonrelativistic anyon particles.While this connection is intriguing, many of its aspects remained unclear.
First of all, the equations of motion of anyon particles admit self-dual static solutions that minimize their energy.So far, it was unclear what is the meaning of self-duality from the polymer point of view.Second, explicit self-dual configurations minimizing the energy (2.48) were missing.That energy resembles that of the Abelian Higgs model in the limit in which the density of half of the fields is constant, see Eq. (2.49), but is more complicated in the general case.Within this work, we have considered the splitting established in [7] of the polymer action of a 4-plat composed by two linked rings into a self-dual part I sd and a non-self-dual contribution I C .It has been noticed that this splitting corresponds to two different kinds of the interactions of entropic origin that are present in a polymer system that is subjected to topological constraints.The term I C is responsible for Coulomb-like interactions.The potentials in I C describe local interactions that can be both attractive or repulsive.The repulsive component is necessary to prevent that the polymer lines cross themselves, as this would break the topology of the link.On the other side, the more the two rings are linked together, the more their centers of mass are getting closer.This phenomenon is well known [15,16] and explains why topological constraints are responsible also for attractive forces between the monomers.Of course, topological properties are global in nature, so that local interactions are not enough to preserve them.The self-dual term I sd accounts for the long range interactions that are necessary to globally enforce the topological relations.Indeed, the Chern-Simons vector potentials B a , a = 1, 2, that mediate such interactions, appear only inside I sd .
The self-dual regime occurs when the rings are homopolimeric, see Eq. (2.54).We have shown using a scaling argument that in the limit τ → ∞ static self-dual solutions of the model described by the action (2.36) or equivalently (2.43) make sense also in the polymer case and not only in its anyon particles interpretation.In this limit the height of the 4-plat becomes infinite and the monomer density becomes independent of z.
Next, the energy landscape of the 4-plat has been explored.It turns out that it is quite complex as it is characterized by at least two points of minimum in the simplified situation in which the monomer density of the legs of the rings point downwards (see Fig. 1) is assumed to be constant.
As an application of the established polymer-anyon model, classes of explicit solutions of its selfdual equations have been derived.First, we show that the densities of monomers |ψ u a | 2 and |ψ d a | 2 for the components Γ u,d a of the loops Γ a , a = 1, 2, may be described by holomorphic functions provided As a further step, we prove that, under some assumptions, the self-dual equations (2.51) may be identified with the sinh-Gordon or cosh-Gordon equations for the monomer densities (2.80)-(2.81).To some extent, it is possible to conclude that Eq. (2.80) is the analog of the Gouy-Chapman equation [19][20][21], but instead of the spatial distribution of the electric potential of charged particles, our equation describes the spatial distribution of the fictitious magnetic fields associated with the presence of the topological constraints.In the limit in which the two spatial dimensions are large in comparison with the third one, we provide exact formulas fo the conformations of the monomer densities by using the elliptic, hyperbolic and trigonometric solutions of the sinh-Gordon and cosh-Gordon equations which have been used for instance in the construction of classical string solutions in AdS3 and dS3 [9].