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Inspired by recent advances in the chromosome capture techniques, a method is proposed to study the structural organization of systems of polymers rings with topological constraints.To this purpose, the system is divided into compartments and a simple condition is provided in order to determine if two compartments are in contact or not. Next, a set of contact matrices $\bar T_{ab}$ is defined that count how many times during a simulation a compartment $a$ was found in contact with a non-contiguous compartment $b$ in conformations with a given energy or temperature. Similar strategies based on correlation maps have been applied to the study of knotted polymers in the recent past. The advantage of the present approach is that is coupled with the Wang-Landau algorithm. Once the density of states is computed, it is possible to generate the contact matrices at any temperature. This gives an immediate overview over the changes of phases that polymer systems undergo. The information on the structure of knotted polymers and links stored in the contact matrices is the result of averaging hundred of billions of conformations and visualized by means of colormaps. The obtained color patterns allow to identify the main properties of the structure of the system under investigation at any temperature. The method is applied to detect the structural rearrangements following the phase transitions of a knotted polymer ring and a circular polycatenane composed by four rings in a solution. It is shown that the colormaps have a finite number of patterns that can be clearly associated with the different phases of these systems. Colormaps also bring new knowledge, for instance predicting the average number of tails appearing in the conformations of the considered polymers at a given temperature., Comment: 27 pages, 11 figures, plain LaTex + RevTex4-1

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. ... ., Comment: 27 pages, 4 figures. arXiv admin note: text overlap with arXiv:1605.03920 by other authors

In this paper, we recall Richardson's solution of the reduced BCS model, its relationship with the Gaudin model, and the known implementation of these models in conformal field theory. The CFT techniques applied here are based on the use of the free field realization, or more precisely, on the calculation of saddle-point values of Coulomb gas integrals representing certain (perturbed) WZW conformal blocks. We identify the saddle-point limit as the classical limit of conformal blocks. We show that this observation implies a new method for calculating classical conformal blocks and can be further used in the study of quantum integrable models., Comment: 8 pages, based on a talk given at the XII. International Symposium on Quantum Theory and Symmetries, July 24-28 , 2023, Prague, Czech Republic; to be published in Journal of Physics: Conference Series

The thermal properties of coarse grained knotted polymers containing two kinds of monomers $A$ and $B$ fluctuating in a solution are investigated on a simple cubic lattice using the Wang-Landau MC algorithm. These knots have a more complex phase diagram than knots formed by homopolymers, including the possible presence of metastable states. Two different setups are considered: i) charged block copolymers in a ion solution and ii) neutral copolymers with the $A$ monomers above and the $B$ monomers below the theta point. A precise interpretation of the peaks observed in the plots of the specific heat capacity is provided. In view of possible applications in medicine and the construction of intelligent materials, it is also shown that the behavior of copolymer rings can be tuned by changing both their monomer configuration and topology. We find that the most stable compact states are formed by charged copolymers in which very short segments with $A$ monomers are alternated by short segments with $B$ monomers. In such knots the transition from the compact to the expanded state is very fast, leading to a narrow and high peak in the specific heat capacity which appears at very high temperatures. The effects of topology allow to tune the radius of gyration of the knotted polymer ring and to increase or decrease the temperatures at which the observed phase transitions or rearrangements of the system occur. While we observe a general fading out of the influence of topology in longer polymers, our simulations have captured a few exceptions to this rule., Comment: 35 pages, 19 figures

The so-called 2d/4d correspondences connect two-dimensional conformal field theory (2d CFT), N=2 supersymmetric gauge theories and quantum integrable systems. The latter in the simplest case of the SU(2) gauge group are nothing but the quantum-mechanical systems. In the present article we summarize our recent results and list open problems concerning an application of the aforementioned dualities in the studies of spectral problems for some Schrodinger operators with Mathieu-type periodic, periodic PT-symmetric and (Heun's) elliptic potentials., Comment: 12 pages, 1 fig., based on a talk at ISQS 25, 6-10 June, 2017, Prague, Czech Republic; to be published in Journal of Physics: Conference Series

It is known that the classical limit of the second order BPZ null vector decoupling equation for the simplest two 5-point degenerate spherical conformal blocks yields: (i) the normal form of the Heun equation with the complex accessory parameter determined by the 4-point classical block on the sphere, and (ii) a pair of the Floquet type linearly independent solutions. A key point in a derivation of the above result is the classical asymptotic of the 5-point degenerate blocks in which the so-called heavy and light contributions decouple. In the present work the semi-classical heavy-light factorization of the 5-point degenerate conformal blocks is studied. In particular, a mechanism responsible for the decoupling of the heavy and light contributions is identified. Moreover, it is shown that the factorization property yields a practical method of computation of the Floquet type Heun's solutions. Finally, it should be stressed that tools analyzed in this work have a broad spectrum of applications, in particular, in the studies of spectral problems with the Heun class of potentials, sphere-torus correspondence in 2d CFT, the KdV theory, the connection problem for the Heun equation and black hole physics. These applications are main motivations for the present work., Comment: 28 pages, revised and extended version

The Schroedinger eigenvalue problems for the Whittaker-Hill potential $Q_{2}(x)=\tfrac{1}{2} h^2\cos4x+4h\mu\cos2x$ and the periodic complex potential $Q_{1}(x)=\tfrac{1}{4}h^2{\rm e}^{-4ix}+2h^2\cos2x$ are studied using their realizations in two-dimensional conformal field theory (2dCFT). It is shown that for the weak coupling (small) $h\in\mathbb{R}$ and non-integer Floquet parameter $\nu\notin\mathbb{Z}$ spectra of hamiltonians $H_{i}\!=\!-{\rm d}^2/{\rm d}x^2 + Q_{i}(x)$, $i=1,2$ and corresponding two linearly independent eigenfunctions are given by the classical limit of the "single flavor" and "two flavors" ($N_f=1,2$) irregular conformal blocks. It is known that complex non-hermitian hamiltonians which are PT-symmetric (= invariant under simultaneous parity P and time reversal T transformations) can have real eigenvalues. The hamiltonian $H_{1}$ is PT-symmetric for $h,x\in\mathbb{R}$. It is found that $H_{1}$ has a real spectrum in the weak coupling region for $\nu\in\mathbb{R}\setminus\mathbb{Z}$. This fact in an elementary way follows from a definition of the $N_f=1$ classical irregular block. Thus, $H_{1}$ can serve as yet another new model for testing postulates of PT-symmetric quantum mechanics., Comment: 23 pages, 1 figure

The Nekrasov-Shatashvili limit of the N=2 SU(2) pure gauge (Omega-deformed) super Yang-Mills theory encodes the information about the spectrum of the Mathieu operator. On the other hand, the Mathieu equation emerges entirely within the frame of two-dimensional conformal field theory (2d CFT) as the classical limit of the null vector decoupling equation for some degenerate irregular block. Therefore, it seems to be possible to investigate the spectrum of the Mathieu operator employing the techniques of 2d CFT. To exploit this strategy, a full correspondence between the Mathieu equation and its realization within 2d CFT has to be established. In our previous paper [1], we have found that the expression of the Mathieu eigenvalue given in terms of the classical irregular block exactly coincides with the well known weak coupling expansion of this eigenvalue in the case in which the auxiliary parameter is the noninteger Floquet exponent. In the present work we verify that the formula for the corresponding eigenfunction obtained from the irregular block reproduces the so-called Mathieu exponent from which the noninteger order elliptic cosine and sine functions may be constructed. The derivation of the Mathieu equation within the formalism of 2d CFT is based on conjectures concerning the asymptotic behaviour of irregular blocks in the classical limit. A proof of these hypotheses is sketched. Finally, we speculate on how it could be possible to use the methods of 2d CFT in order to get from the irregular block the eigenvalues of the Mathieu operator in other regions of the coupling constant., Comment: 41 pages, 1 figure

The subject of this work is a three-dimensional topological field theory with a non-semisimple group of gauge symmetry with observables consisting in the holonomies of connections around three closed loops. The connections are a linear combination of gauge potentials with coefficients containing a set of one-dimensional scalar fields. It is checked that these observables are both metric independent and gauge invariant. The gauge invariance is achieved by requiring non-trivial gauge transformations in the scalar field sector. This topological field theory is solvable and has only a relevant amplitude which has been computed exactly. From this amplitude it is possible to isolate a topological invariant which is Milnor's triple linking invariant. The topological invariant obtained in this way is in the form of a sum of multiple contour integrals. The contours coincide with the trajectories of the three loops mentioned before. The introduction of the one-dimensional scalar field is necessary in order to reproduce correctly the particular path ordering of the integration over the contours which is present in the triple Milnor linking coefficient. This is the first example of a local topological gauge field theory that is solvable and can be associated to a topological invariant of the complexity of the triple Milnor linking coefficient., Comment: 9 pages, no figures, LaTeX + RevTeX 4

The field theory approach to the statistical mechanics of a system of N polymer rings linked together is generalized to the case of links that have a fixed number $2s$ of maxima and minima. Such kind of links are called plats and appear for instance in the DNA of living organisms. The topological states of the link are distinguished using the Gauss linking number. This is a relatively weak link invariant in the case of a general link, but its efficiency improves when $2s-$plats are considered. It is proved that, if we restrict ourselves to $2s-$plat conformations, the field theoretical model established here is able to take into account also the interactions of topological origin involving three chains simultaneously. It is shown that these three-body interactions have nonvanishing contributions when three or more rings are entangled together, enhancing for instance the attractive forces between monomers. The model can be used to study the statistical mechanics of polymers in confined geometries, for instance when $2s$ extrema of a few polymer rings are attached to membranes. Its partition function is mapped here into that of a multi-layer electron gas. Such quasi-particle systems are studied in connection with several interesting applications, including high-$T_c$ superconductivity and topological quantum computing. At the end an useful connection with the cosh-Gordon equation is shown., Comment: 35 pages, 8 figures. arXiv admin note: substantial text overlap with arXiv:1304.5980

Combining the semiclassical/Nekrasov-Shatashvili limit of the AGT conjecture and the Bethe/gauge correspondence results in a triple correspondence which identifies classical conformal blocks with twisted superpotentials and then with Yang-Yang functions. In this paper the triple correspondence is studied in the simplest, yet not completely understood case of pure SU(2) super-Yang-Mills gauge theory. A missing element of that correspondence is identified with the classical irregular block. Explicit tests provide a convincing evidence that such a function exists. In particular, it has been shown that the classical irregular block can be recovered from classical blocks on the torus and sphere in suitably defined decoupling limits of classical external conformal weights. These limits are "classical analogues" of known decoupling limits for corresponding quantum blocks. An exact correspondence between the classical irregular block and the SU(2) gauge theory twisted superpotential has been obtained as a result of another consistency check. The latter determines the spectrum of the 2-particle periodic Toda (sin-Gordon) Hamiltonian in accord with the Bethe/gauge correspondence. An analogue of this statement is found entirely within 2d CFT. Namely, considering the classical limit of the null vector decoupling equation for the degenerate irregular block a celebrated Mathieu's equation is obtained with an eigenvalue determined by the classical irregular block. As it has been checked this result reproduces a well known weak coupling expansion of Mathieu's eigenvalue. Finally, yet another new formulae for Mathieu's eigenvalue relating the latter to a solution of certain Bethe-like equation are found., Comment: 47 pages, 3 figures

In this work the correspondence between the semiclassical limit of the DOZZ quantum Liouville theory on the torus and the Nekrasov-Shatashvili limit of the N=2* (Omega-deformed) U(2) super-Yang-Mills theory is used to propose new formulae for the accessory parameter of the Lame equation. This quantity is in particular crucial for solving the problem of uniformization of the one-punctured torus. The computation of the accessory parameters for torus and sphere is an open longstanding problem which can however be solved if one succeeds to derive an expression for the so-called classical Liouville action. The method of calculation of the latter has been proposed some time ago by Zamolodchikov brothers. Studying the semiclassical limit of the four-point function of the quantum Liouville theory on the sphere they have derived the classical action for the Riemann sphere with four punctures. In the present work Zamolodchikovs idea is exploited in the case of the Liouville field theory on the torus. It is found that the Lame accessory parameter is determined by the classical Liouville action on the one-punctured torus or more concretely by the torus classical block evaluated on the saddle point intermediate classical weight. Secondly, as an implication of the aforementioned correspondence it is obtained that the torus accessory parameter is related to the sum of all rescaled column lengths of the so-called "critical" Young diagrams extremizing the instanton "free energy" for the N=2* gauge theory. Finally, it is pointed out that thanks to the known relation the sum over the "critical" column lengths can be expressed in terms of a contour integral in which the integrand is built out of certain special functions., Comment: 41 pages, published version

Anyon systems are studied in connection with several interesting applications including high $T_C$ superconductivity and topological quantum computing. In this work we show that these systems can be realized starting from directed polymers braided together to form a nontrivial link configuration belonging to the topological class of plats. The statistical sum of a such plat is related here to the partition function of a two-component anyon gas. The constraints that preserve the topological configuration of the plat are imposed on the polymer trajectories using the so-called Gauss linking number, a topological invariant that has already been well studied in polymer physics. Due to these constraints, short-range forces act on the monomers or, equivalently, on the anyon quasiparticles in a way that closely resembles the appearance of reaction forces in the constrained systems of classical mechanics. If the polymers are homogeneous, the anyon system reaches a self-dual point, in which these forces vanish exactly. A class of self-dual solutions that minimize the energy of the anyons is derived. The two anyon gas discussed here obeys an abelian statistics, while for quantum computing it is known that nonabelian anyons are necessary. However, this is a limitation due to the use of the Gauss linking invariant to impose the topological constraints, which is a poor topological invariant and is thus unable to capture the nonabelian characteristics of the braided polymer chains. A more refined treatment of the topological constraints would require more sophisticated topological invariants, but so far their application to the statistical mechanics of linked polymers is an open problem., Comment: title changed, several references added, typos corrected, text improved. 42 pages, 5 figures

In this work we study the Nekrasov--Shatashvili limit of the Nekrasov instanton partition function of Yang--Mills field theories with ${\cal N}=2$ supersymmetry and gauge group SU(N). The theories are coupled with fundamental matter. A path integral expression of the full instanton partition function is derived. It is checked that in the Nekrasov--Shatashvili (thermodynamic) limit the action of the field theory obtained in this way reproduces exactly the equation of motion used in the saddle-point calculations., Comment: 9 pages, no figures, LaTeX + amssymb. arXiv admin note: text overlap with arXiv:1202.5135

In this work we study the Nekrasov-Shatashvili limit of the Nekrasov instanton partition function of Yang-Mills field theories with N=2 supersymmetry and gauge group SU(n). The theories are coupled with fundamental matter. The equation that determines the density of eigenvalues at the leading order in the saddle-point approximation is exactly solved. The dominating contribution to the instanton free energy is computed. The requirement that this energy is finite imposes quantization conditions on the parameters of the theory that are in agreement with analogous conditions that have been derived in previous works. Using methods borrowed from the theory of matrix models, a field theoretical expression of the full instanton partition function is derived. It is checked that in the Nekrasov-Shatashvili (thermodynamic) limit the action of the field theory obtained in this way reproduces exactly the equation of motion used in the saddle-point calculations., Comment: 16 pages, LaTeX + RevTex 4, no figures, final version accepted by Phys. Lett. B

The correspondence between the semiclassical limit of the DOZZ quantum Liouville theory and the Nekrasov-Shatashvili limit of the N = 2 (Omega-deformed) U(2) super-Yang-Mills theories is used to calculate the unknown accessory parameter of the Fuchsian uniformization of the 4-punctured sphere. The computation is based on the saddle point method. This allows to find an analytic expression for the N_f = 4, U(2) instanton twisted superpotential and, in turn, to sum up the 4-point classical block. It is well known that the critical value of the Liouville action functional is the generating function of the accessory parameters. This statement and the factorization property of the 4-point action allow to express the unknown accessory parameter as the derivative of the 4-point classical block with respect to the modular parameter of the 4-punctured sphere. It has been found that this accessory parameter is related to the sum of all rescaled column lengths of the so-called 'critical' Young diagram extremizing the instanton 'free energy'. It is shown that the sum over the 'critical' column lengths can be rewritten in terms of a contour integral in which the integrand is built out of certain special functions closely related to the ordinary Gamma function., Comment: 26 pages, published version

The so-called Poghossian identities connecting the toric and spherical blocks, the AGT relation on the torus and the Nekrasov-Shatashvili formula for the elliptic Calogero-Moser Yang's (eCMY) functional are used to derive certain expressions for the classical 4-point block on the sphere. The main motivation for this line of research is the longstanding open problem of uniformization of the 4-punctured Riemann sphere, where the 4-point classical block plays a crucial role. It is found that the obtained representation for certain 4-point classical blocks implies the relation between the accessory parameter of the Fuchsian uniformization of the 4-punctured sphere and the eCMY functional. Additionally, a relation between the 4-point classical block and the $N_f=4$, ${\sf SU(2)}$ twisted superpotential is found and further used to re-derive the instanton sector of the Seiberg-Witten prepotential of the $N_f=4$, ${\sf SU(2)}$ supersymmetric gauge theory from the classical block., Comment: 25 pages, no figures, latex+JHEP3, published version

Zamolodchikov's recursion relations are used to analyze the existence and approximations to the classical conformal block in the case of four parabolic weights. Strong numerical evidence is found that the saddle point momenta arising in the classical limit of the DOZZ quantum Liouville theory are simply related to the geodesic length functions of the hyperbolic geometry on the 4-punctured Riemann sphere. Such relation provides new powerful methods for both numerical and analytical calculations of these functions. The consistency conditions for the factorization of the 4-point classical Liouville action in different channels are numerically verified. The factorization yields efficient numerical methods to calculate the 4-point classical action and, by the Polyakov conjecture, the accessory parameters of the Fuchsian uniformization of the 4-punctured sphere., Comment: 26 pages, 9 eps figures. Version 2: references added

Using the techniques developed by Ponsot and Teschner we derive the formulae for analytic continuation of the general 4-point conformal block., Comment: 17 pages, LaTeX