Weak-strong duality of the non-commutative Landau problem induced by a two-vortex permutation, and conformal bridge transformation

A correspondence is established between the dynamics of the two-vortex system and the non-commutative Landau problem (NCLP) in its sub- (non-chiral), super- (chiral) and critical phases. As a result, a trivial permutation symmetry of the point vortices induces a weak-strong coupling duality in the NCLP. We show that quantum two-vortex systems with non-zero total vorticity can be generated by applying conformal bridge transformation to a two-dimensional quantum free particle or to a quantum vortex-antivortex system of zero total vorticity. The sub- and super-critical phases of the quantum NCLP are generated in a similar way from the 2D quantum free particle in a commutative or non-commutative plane. The composition of the inverse and direct transformations of the conformal bridge also makes it possible to link the non-chiral and chiral phases in each of these two systems.


Introduction
Correspondences and connections between different systems, models and theories play an important role in physics.An example of this kind is the AdS/CFT correspondence or the Gauge/Gravity duality [1]- [9].On a simpler level an interesting example of the hidden correspondences between some integrable systems is provided by the Newton-Hooke duality, based on conformal mapping [10]- [12], and its generalizations in the form of the coupling constant metamorphosis [13,14].In the same vein, the properties of many important physical systems can be derived or explained by their hidden connection with the simplest system of a free particle.One of the striking examples of this is the construction of (multi)soliton solutions of the classical KdV equation and equations of its hierarchy based on the Schrödinger problem for a one-dimensional free particle.The construction, related to the inverse scattering method, employs the Darboux covariance of the corresponding Lax pair formulation for this integrable system [15,16].In a similar way, the Bäcklund transformations make it possible to connect various integrable systems and generate more complex solutions for them starting from simpler ones, in particular, from trivial, identically equal to zero solutions of the same or related systems [17].
Some time ago, the construction of the conformal bridge transformation (CBT) made it possible to connect various harmonically confined quantum mechanical systems with a free (or asymptotically free) particle in spaces of various dimensions and geometric backgrounds, including those of a magnetic monopole and a cosmic string [18]- [21], see also refs.[22,23].In this way, explicit and hidden symmetries as well as super-symmetries of various systems can be derived from a free particle.Also, their eigenstates, coherent and squeezed states can be obtained from certain states of the free particle quantum system.The CBT construction, based on a non-unitary non-local similarity transformation, is analogous to the Weierstrass transformation [18].At the same time, it turns out to be closely related to the PT -symmetry [24] through the underlying Dyson map [25], which transmutes the topological nature of the sl(2, R) conformal symmetry generators [26,27].The non-local CBT generator has the nature of the eighth-order root of the identity operator in the quantum phase space and also turns out to be related to the unitary Bargmann-Segal transformation, which connects the Hilbert space in the coordinate representation with the Fock-Bargmann space of the holomorphic representation of Heisenberg algebra [18].
The usual Landau problem, also covered by the CBT [18,28], underlies the quantum Hall effect.The properties of the fractional quantum Hall effect can be explained in terms of fractional statistics of the corresponding quasiparticle (anyon) excitations [29,30].Anyons can be realized in the form of point vortices through the mechanism of statistics transmutation by using the Chern-Simons gauge theory [31,32].Point vortices also appear in many non-linear field systems and phenomena, including superconductivity, superfluidity, and Bose-Einstein condensate physics [33]- [37].Recently, the Landau problem has attracted attention in the study of the non-relativistic conformally invariant Schwartzian mechanical system associated with the low energy limit of the Sachdev-Ye-Kitaev model [38]- [41].A generalization of the usual Landau problem to the case of noncommutative quantum mechanics has been actively studied in the context of physics associated with noncommutative geometry [42]- [56].
The present work is devoted to establishing a correspondence between the classical and quantum dynamics of the simplest system of two point vortices and the non-commutative Landau problem (NCLP) in its sub-(non-chiral), super-(chiral), and critical phases.This will be achieved by introducing an imaginary mirror particle into the latter system.The established correspondence will allow us to reveal a kind of the weak-strong coupling duality in the NCLP induced by a trivial permutation symmetry of vortices.By an appropriate generalization of the CBT construction we relate the chiral and non-chiral quantum phases of both systems with a free particle in commutative and non-commutative plane.Using the same transformation, the quantum dynamics of the two-vortex systems with non-zero total vorticity will be related to a vortex-antivortex system of zero total vorticity.
The paper is organized as follows.In Section 2 we briefly review the symplectic structure and integrals of motion for the general case of a system of N point vortices.In Section 3 we focus on the case of superintegrable two-vortex systems.First we consider the twovortex systems of non-zero total vorticity emphasizing the important difference of their classical and quantum dynamics in the cases of the vorticity strengths of the opposite and equal signs.Then we consider the vortex-antivortex system of zero total vorticity with its emergent (1+1)D "isospace" Lorentz symmetry.Section 4 is devoted to the NCLP in sub-(non-chiral), super-(chiral) and critical phases and to establishing its correspondence with the two-vortex systems by introducing an imaginary mirror particle into it.We show there that through the identified correspondence, a trivial permutational symmetry of vortices induces a weak-strong coupling duality in the NCLP.In Section 5 we study the conformal bridge transformation for the NCLP and two-vortex systems in chiral and non-chiral phases.To do this, we consider the integral of motion, which is a certain linear combination of generators of angular and time translations in the NCLP.This integral, and its analog in the two-vortex system, along with the integral of angular momentum, will allow us to find two different canonical sets of variables that have dual properties with respect to them in the non-chiral and chiral phases.Section 6 is devoted to the summary, discussion and outlook.In Appendix, we discuss the basics of the simplest conformal bridge transformation, a generalization of which is used for the constructions in Section 5. There, we also provide an explanation for the difference between non-chiral and chiral phases for the systems under consideration in the light of the outer Z 2 automorphism of the conformal sl(2, R) algebra, which has the nature of a PT -inversion.

N-vortex system and its symmetries
A system of N ≥ 2 point vortices on a plane can be described by Lagrangian [57,58] where ǫ ij is an antisymmetric tensor, ǫ 12 = 1, x a are coordinates of a vortex with index a and strength γ a , x a = x b for a = b ⇒ r 2 ab = ( x a − x b ) 2 > 0. We do not distinguish between superscripts and subscripts of a two-dimensional space and imply summation over repeated indices.The vortex coordinates and their strengths γ a are assumed to be dimensionless.
It was already shown by Kirchhoff [59] that the Euler-Lagrange equations of motion admit a Hamiltonian description given by Symplectic structure with Dirac-Poisson brackets (DPBs) of such a form can also be obtained from Lagrangian of Landau problem by setting the mass parameter to zero [60].System (2.1) is characterized by the Noetherian integrals of motion associated with translations, {P i , x j a } = −δ ij , and rotations, {M, x a i } = ǫ ij x j a .They generate a centrally extended Lie algebra e Γ (2) of the two-dimensional Euclidean group, with total vorticity Γ playing a role of a central charge, and Casimir element These intergrals together with Hamiltonian generate the e Γ (2)⊕u(1) algebra, which reduces to e(2) ⊕ u(1) when Γ = 0. System (2.3) is maximally superintegrable in the case of N = 2 vortices, completely integrable for N = 3 (unlike the three-body problem of gravitating mass points [59]), and is not integrable, chaotic for N ≥ 4 [57,58].In general case of N vortices, equations of motion (2.2) are also invariant under rescaling x i a → e α x i a , t → e 2α t.At the same time, Lagrangian is quasi-invariant, L → L + d dt (Ct), C = α a<b γ a γ b , but the action S = Ldt is rescaled, S → e 2α S, and this symmetry of equations of motion is not Noetherian, see also [61].

Two-vortex system
In the case of the two-vortex superintegrable system, Hamiltonian (2.3) reduces to a function of the Casimir element (2.6) of the e Γ (2) symmetry, From now on, we will consider the two-vortex system.

Nonzero total vorticity
Let us first consider the case of nonzero total vorticity Γ = 0.It is convenient to describe the system by the vector integral P and the relative coordinate vector r, where ̺ is a reduced vorticity.Equations of motion Ṗi = 0 , ṙi = −ωǫ ij r j , ω = ΓR −2 , R 2 = r 2 , lead to a circular motion dynamics, For vorticities of the opposite sign, κ < 0, the vortices occupy the closest positions on the circles with a common "center of vorticity" X 0 , while for κ > 0 they are always in the opposite positions on their circular orbits of radii Identical vortices with γ 1 = γ 2 move on the same circle of radius R/2, see Fig. 1.Angular momentum (2.4) can be expressed in terms of the integrals P 2 i and R 2 , It can take arbitrary values, M Γ ∈ R, if κ < 0, but only nonzero values of the sign −ε Γ = −sgn Γ, in the case of κ > 0. For κ < 0, the angular momentum takes, in particular, zero value, Introduce now generators A ± and B ± of the two copies of classical Heisenberg algebra, where Z = e −iωt z, z = r 1 +ir 2 , and Z are dynamical, explicitly depending on time, integrals of motion, d dt Z = ∂Z/∂t + {Z, H Γ } = 0. Quadratic integrals constructed from them, generate the algebra sl(2, R) Hamiltonian and angular momentum are represented in terms of the compact generators J 0 and L 0 not depending explicitly on time, and the integral J 0 is related, in turn, to the Casimir element (2.6) of the e Γ (2) symmetry, Hamiltonian together with the dynamical integrals J ± generate a peculiar nonlinear deformation of the conformal sl(2, R) algebra, Dropping the exponential time-dependent factors in A ± , and changing the notation A ± → a ± and B ± → b ± = B ± in (3.7) and (3.8), we do not modify the algebraic relations (3.9), (3.10), and (3.12).Then, the canonical quantization of the system entails the commutation relations (3.14)In the case of vorticities of opposite sign, κ = γ 1 γ 2 < 0, the energy eigenvalues E na are bounded from below, while the angular momentum eigenvalues take integer values M na,n b ∈ Z.For κ > 0, the angular momentum takes nonzero integer values of the sign of (−Γ), −ε Γ M na,n b ∈ Z >0 , whereas energy eigenvalues are bounded from above.Thus, the two-vortex system with κ < 0 is non-chiral, while the dynamics of the sytem with κ > 0 is chiral in the sense of eigenvalues of the angular momentum operator.

Zero total vorticity
The case of zero total vorticity can be interpreted as the vortex-antivortex system.It can be obtained from the κ < 0, Γ = 0 case by taking the limit Γ → 0. For this, we denote γ 1 = γ, γ 2 = −γ + Γ, and find that the variables Π i and χ i given by form the canonical set of the variables, {χ i , Π j } = δ ij , {χ i , χ j } = {Π i , Π j } = 0. Hamiltonian (3.1) and angular momentum (3.6) reduce here to The integrals Π i , M 0 and H 0 generate the e(2) ⊕ u(1) algebra with C 0 = Π 2 i > 0 to be the Casimir of the e(2) subalgebra.
The equations of motion are Πi = 0, χi = γ 2 Π i / Π 2 , and the evolution of the system is see Fig. 2. The distance between vortices is constant, | r| = | Π/γ| = R, while the speed of the vortex and antivortex is inverse to it, The rectilinear orbits of the vortex-antivortex system can be obtained by appropriately applying the limit Γ → 0 to the circular trajectories of the system with κ < 0 and nonzero total vorticity shown on panel (a) of Fig. 1.For instance, by setting t 0 = 0, γ 1 = γ, γ 2 = −γ + Γ, P 1 = µΓR, P 2 = νΓR − γ, µ, ν ∈ R, and taking Γ → 0, we reduce Eqs.(3.4), (3.5) to Eq. (3.18) with χ together with the integral generate the conformal sl(2, R) algebra, whose Casimir is related to the angular momentum (3.16 With the change H 0 → H 0 , the first and third relations in (3.21) are transformed into By dropping in (3.19) the terms that explicitly depend on time, i.e. changing there χ i 0 → χ i , the algebraic relations (3.21), (3.22) are not modified.
Note that in the vortex-antivortex system (Γ = 0), the first order in time derivative term in Lagrangian (2.1) with N = 2 can be presented as 1  2 They are generated by the dynamical integral D, while the canonical variables Π i and χ i undergo the rescaling transformations The linear in time t term in integral D is associated with quasi-invariance of the Lagrangian under transformation (3.23), L ′ − L = d dt (−αγ 2 t).So, the vortex-antivortex system is characterized by the hidden "isospace" (1+1)D Lorentz symmetry.With the "isospace" Minkowski metric η ab , the angular momentum integral (3.16) takes the form M 0 = 1 2 γη ab δ ij x i a x j b , and the vortexantivortex pair is described, in particular, by the so(1, 1) ⊕ so(2) symmetry generated by D and M 0 .These two generators, along with H 0 and K will play a key role in the CBT.
At the quantum level, one can work in representation with diagonal operators χj .The plane wave functions ψ Π (ξ) = 1 2π exp(iχ j Π j ), Π 2 i > 0, are the normalized for delta function eigenstates of Πj = −i ∂/∂χ j , which are eigenstates of Ĥ0 of energy The angular momentum operator takes integer eigenvalues n ∈ Z.

Non-commutative Landau problem
Let us compare now the two-vortex system with the non-commutative Landau problem (NCLP).We take Lagrangian for the NCLP in symmetric gauge in the form1 where X i are the coordinates of the particle of mass m and charge e = 1 in magnetic field B, while θ is a non-commutativity parameter.The speed of light c = 1.In the case β := Bθ = 1, system (4.1) is described by the Hamiltonian structure [47,52] We will see that the cases β < 1 and β > 1 correspond to non-chiral and chiral phases of the two-vortex system in spite of some differences in classical and quantum properties of the two systems.At the critical value β = 1, which separates the non-chiral and chiral phases, DPBs (4.2) blow up.The Dirac-Bergmann analysis [62] applied to the system (4.1)shows that at β = 1, the corresponding matrix of brackets of the constraints degenerates, dimension of the reduced phase space decreases from four to two, and the system is described by the coordinates X i with DPBs {X i , X j } = θǫ ij .Hamiltonian turns into zero, H = 0 [46,49,52].The spatial translation generators are generates rotations of X i , and takes values on a half-line.So the critical case β = 1, which reveals a chiral nature, is essentially different from the cases with β = 1 because of a trivial dynamics, Ẋi = 0.The angular momentum coincides here, up to a sign, with Hamiltonian of a 1D oscillator, which at the quantum level takes half-integer values M n = −ε θ (n + 1  2 ), ε θ = sgn θ, n = 0, 1, . ... The critical case β = 1 of the NCLP corresponds to the two-vortex system with one of the parameters γ a set equal to zero, i.e., to a single-vortex system.Indeed, putting there γ 2 = 0, we obtain the second-class, φ i 1 = p i 1 − 1 2 γ 1 ǫ ij x j 1 ≈ 0, and the first-class, φ i 2 = p i 2 ≈ 0, constraints with respect to the symplectic structure σ = dp i a ∧dx i a in the initial phase space.Constraints φ i 2 ≈ 0 mean that the variables x i 2 are of a pure gauge nature, and one can forget about the degrees of freedom corresponding to index a = 2. Reduction to the subspace of the constraints φ i 1 ≈ 0 results in the two-dimensional symplectic manifold described by coordinates x i 1 with the DPBs {x i 1 , x j 1 } = γ −1 1 ǫ ij and zero Hamiltonian.Identifying θ = γ −1 1 and X i = x i 1 , we conclude that the critical case of the NCLP is equivalent to the reduced, single-vortex system with a trivial dynamics and a chiral nature of the angular momentum.In what follows we assume that β = 1.

Non-critical phases
The vector integral generates translations of X i .Solutions to Eqs. (4.4) are Like in the two-vortex system, the angular momentum integral together with P i generate the centrally extended Lie algebra of the two-dimensional Euclidean group of the form (2.5) with −B playing the role of the central charge.The Hamiltonian is presented in terms of its Casimir element, cf.Eq. (2.6), Let us define now the coordinate vector of an "imaginary mirror particle", which is characterized by three important properties : it has zero DPBs with X i , it is translated by P i , and the DPBs betwen its components do not depend on B, In terms of X i and Y i the Hamiltonian and angular momentum are expressed as and we also have Up to a total time derivative, Lagrangian (4.1) takes the form similar to (2.1) at N = 2, (4.15) The four-dimensional phase space can be described, up to a canonical transformation, in terms of any pair of the vector variables X i , P i , Y i and P i .The Hamiltonian and angular momentum take the simplest form in terms of the pairs (P i , P i ) and (X i , Y i ).
Comparing the NCLP with the two-vortex system, one can establish, up to a permutation 1 ↔ 2 of the values of index a, the following correspondence between them : The difference of the two systems is reflected in a linear dependence of Because of this difference the signed frequency of rotational motion in the NCLP is given in terms of the parameters θ, B and m, while in the two-vortex system the signed frequency is energy-dependent.According to (4.17), the subcritical phase β < 1 corresponds to the case of vorticities of the opposite sign, γ 1 γ 2 < 0. In particular, the vortex-antivortex system, Γ = 0, is similar to a free particle in non-commutative plane, B = 0.The supercritical phase with β > 1 corresponds to the chiral case of vorticities of the same sign, γ 1 γ 2 > 0, see Eqs (4.8), (4.12).
The case of the ordinary Landau problem, θ = 0, B = 0, also can be covered by correspondence relations (4.16), (4.17).For this, note that in the limit θ → 0, we get In the two-vortex system we change the variables,

and redefine Hamiltonian by shifting and rescaling it, H
).According to (4.16), (4.17), the ordinary Landau problem corresponds to the limit γ 2 → ∞ in the two-vortex system.This yields In correspondence with Eqs.(3.4), (3.5), x i (t) describes rotational motion with angular velocity ω = Γ/R 2 along the circle of radius R = ( π 2 /Γ 2 ) 1/2 centered at X i 0 = − 1 Γ ǫ ij P j .Taking into account correspondence (4.16), (4.17), we find the analogs of the variables a ± and b ± of the two-vortex system, where ε a = sgn (B(β − 1)), ε b = sgn B. The analogs of quadratic quantities (3.8) given in terms of the variables a ± and b ± yield us the sl(2, R) ⊕ sl(2, R) ∼ = AdS 3 algebra of the non-commutative Landau problem, and here By analogy with the two-vortex system, the quantum states |n a , n b , n a , n b = 0, 1, . .., are eigenstates of the operators â+ â− and b+ b− with eigenvalues n a and n b .They are eigenstates of the energy ).Thus, in the supercriticial (chiral) phase β > 1, the angular momentum takes nonzero integer values of the sign of magnetic field B, sgn B • M na,n b ∈ Z >0 , while in the subcritical phase β < 1 it takes integer values of both signs, including zero value, M na,n b ∈ Z, in correspondence with the analogous picture in the two-vortex system.
The case of a free particle in non-commutative plane, B = 0, θ = 0, is similar to the vortex-antivortex system with zero total vorticity Γ = 0.For B = 0, we have the vector P i = P i ∼ Π i , and the vector coordinate see Eq. (3.15), which form a canonical set of variables, With the Hamiltonian and angular momentum the system looks like a free scalar particle in two-dimensional Euclidean space.However, it is necessary to bare in mind that the coordinates X i of a free particle in non-commutative plane, {X i , X j } = θǫ ij , are given by X i = X i − 1 2 θǫ ij P j .They form a 2D vector with respect to the angular momentum M 0 = ǫ ij X i P j + 1 2 θP2 i , and covariantly transform under the exotic Galilean boosts, where G i is a dynamical integral.Unlike X i , the 2D vector X i transforms non-covariantly under the exotic Galilean boosts, {X i , G j } = −δ ij t − 1 2 mθǫ ij , and plays a role analogous to the Newton-Wigner coordinates for a Dirac particle 2 .The coordinates Y i of imaginary particle also transform non-covariantly under the exotic Galilean boosts, {Y i , As X i (t) = X i 0 + 1 m P i t, one can construct the quadratic dynamical integrals which together with the Hamiltonian H 0 generate the conformal sl(2, R) algebra of the form (3.21).The phase space function D = X and H 0 generate the same sl(2, R) algebra.Combining the coordinates X i and Y i into the "isospace" vector X i c = (X i , Y i ), we find that in index c = 1, 2 it behaves like a (1+1)dimensional Lorentz vector with respect to the global transformations generated by D, cf.(3.23), At the quantum level, the non-covariant with respect to the exotic Galilean boosts vector X i allows us to work in representation diagonal in Xi , in which Pj = −i∂/∂X j , Xj = X j + i 1 2 θ ǫ jk ∂/∂X k , and Ŷj = X j − i 1 2 θ ǫ jk ∂/∂X k .Similarly to the vortex-antivortex system, the wave function ψ P (X ) = 1 2π exp(iX j P j ) is the eigenfunction of the Hamiltonian Ĥ0 of eigenvalue E P = 1 2m P 2 i , and M0 takes integer eigenvalues n ∈ Z.

Vortex permutation and weak-strong duality of the NCLP
In the two-vortex system, the transformation I : is the obvious symmetry of the nature of inversion, I 2 = id.It acts as a rotation by π on the relative coordinate r i = x i 1 − x i 2 , I : r i → −r i , but does not change the energy, the momentum vector and the angular momentum integrals.In the NCLP, the correspondence (4.16), (4.17) leads to the analog of the permutation of vortices, The Hamiltonian H is invariant under transformation (4.26)only when β = 0 or β = 2.
The first case corresponds either to a free particle in non-commutative plane (B = 0, θ = 0), which is analogous to the vortex-antivortex system with Γ = 0, or to the ordinary Landau problem (θ = 0, B = 0).The second case with θ = 2/B is similar to the system of identical vortices with γ 1 = γ 2 .For β = 0, 2, the transformation relates the modes of a weak and strong coupling (in the sense of energy levels spacing ∆E = |ω * |) of the NCLP with respect to the critical value β = 1 within the same sub-or super-critical phase, I : As I 2 = id, transformation (4.26) can be treated as a kind of a weak-strong coupling duality.

Conformal bridge
In this section we show how the quantum NCLP can be generated from a free particle system in a commutative or non-commutative plane by means of the conformal bridge transformation [18,26,27].In the sub-(β < 1) and super-(β > 1) critical phases the symplectic structure of the system is distinct.Though the components of non-commutative coordinate X i and momentum P i form the 2D vectors with respect to the angular momentum M, all their brackets among themselves have different signs in these two phases while the signs of the magnetic field B and the non-commutativity parameter θ are maintained fixed.Moreover, their brackets blow up at β = 1, and the system decreases the dimension of the phase space from four to two.Coherently with these changes, in the supercritical phase the system acquires chirality: the angular momentum integral M takes values of only one sign.As a consequence, generation of the two phases of the NCLP from a free particle is carried out in distinct ways.The reason is that in supercritical case the chiral nature of the angular momentum is incompatible with vector properties of Darboux coordinates.
The correspondence between the NCLP and the two-vortex system allows then to generate the latter system from the free particle in a plane separately in the cases of vorticities γ a of the opposite or equal signs.
In connection with appearance of chirality, we consider one more integral M := 1 2B (P 2 i + (1 − β)P 2 i ), which is a linear combination of the angular momentum and the Hamiltonian, We also introduce the two-component object Pi obtained from P i by a spatial reflection, One finds then that the integral P i is transformed as a vector by both integrals M and M, while P i and Pi are transformed as 2D vectors only by M and M, respectively: (5.4)

Canonical variables for two phases of the NCLP
Consider now the subcritical phase β < 1 of the NCLP.One can define canonical (Darboux) variables (q i , p i ), {q i , q j } = {p i , p j } = 0, {q i , p j } = δ ij , by means of relations The non-commutative vector variables are expressed then in a form The canonical variables (5.5) are defined so that P i | B=0 = P i | B=0 = p i , and Suchwise the q i reduces at B = 0 to the commutative coordinate X i of the free particle in non-commutative plane, see Eqs. (4.20), (4.21).On the other hand, one has Bǫ ij q j , that corresponds to the case of an ordinary Landau problem in commutative plane.Finally, the case B = θ = 0 corresponds to a free particle in commutative plane: Integrals M and M, and Hamiltonian (4.3) take the form where H osc is a Hamiltonian of a planar isotropic harmonic oscillator.One finds Define the "linearly polarized" creation-annihilation operators, From now on we identify the parameter ε with the sign of magnetic field, ε = sgn B .We have then two pairs of circularly polarized operators, b± ε and b± −ε , for magnetic field of a fixed sign.The quantum analogs of the angular momentum, integral M, and Hamiltonian take the form where Nε = b+ In this alternative way we reproduce the results of Sec.4.1 for the subcritical phase of the NCLP.
For the supercritical (chiral) phase of the NCLP, β > 1, we define the canonical variables ( q i , p i ), σ = d p i ∧ d q i , by relations similar to (5.5), (5.14) These canonical variables are obtained from those for the subcritical phase by changing Relations inverse to (5.14) are According to Eqs. (5.3) and (5.14), the canonical variables ( q i , p i ) are not the 2D vectors with respect to the angular momentum M, that we emphasize by supplying their characters with a tilde.They transform, however, like 2D vectors under the action of the integral M. In terms of ( q i , p i ), we have ) wherefrom one finds Analogously to (5.11) and (5.12), one defines the "linearly polarized", ˆ a ± j , and "circularly polarized", , creation-annihilation operators in terms of ˆ q i and ˆ p i , and we obtain where In this way we reproduce the results for the supercritical (chiral) phase of the NCLP in the way alternative to that presented in the previous section.

Conformal bridge transformation for the NCLP
Consider a quantum free particle system in 2D Euclidean space, which is described by the operators of canonical coordinates qi and momenta pi .Its generators of the conformal sl(2, R) symmetry are the 2D analogs of those considered in Appendix with restored and explicitly shown dependence on parameters m, Ω = B 2m and constant , ) Operators Ĥ+ and Ĥ− correspond to the Hamiltonians of the quantum 2D isotropic harmonic and inverted harmonic oscillators, respectively, while D is the generator of dilatations.A two-dimensional analog of the similarity transformation (A.7) is where Ĵ1 is given by Eqs.(5.20) and (5.21).Transformation (5.22) yields a mapping where â± j are the "linearly polarized" creation-annihilation operators of the form (5.11), and M = ǫ ik qj pk = i ǫ jk â− j â+ k .Introduce the complex canonical coordinates and momenta, with the nontrivial brackets {w ε , p ε } = { wε , pε } = 1.According to (5.22), their quantum analogs are transformed into the "circularly polarized" creation-annihilation operators, which are linear combinations of â± j analogous to (5.12).One also finds that (5.27) The angular momentum operator M commutes with the conformal symmetry generators (5.20).In the coordinate representation one has i D = (w ε ∂ wε + wε ∂ wε + 1) and M = ε (w ǫ ∂ wε − wε ∂ wε ).The set of functions is a set of formal eigenstates of the Wick rotated dilatation and momentum operators, , these states also are Jordan states of the free particle Hamiltonian corresponding to zero energy [65,66], ( Ĥ0 ) n + +n − +1 Φ n + ,n − = 0 .The state Φ 0,0 is annihilated by the operators pε and pε , and in accordance with (A.8), is transformed, up to a normalization, into the vacuum state of the operators b− , where Φ ′ 0,0 = ŜΦ 0,0 ∝ exp − m|Ω| w ε wε .In correspondence with (A.8), in the Fock representation for the "circularly polarized" creation-annihilation operators, the states |n + , n − are the common eigenstates of the number operators Nε and N−ε .In the holomorphic representation these are, up to a normalization, the transformed states (5.28), Φ ′ n + ,n − = ŜΦ n + ,n − , cf. (A.8).Identifying now the canonical variables (5.5) with (q i , p i ), we generate the non-chiral (subcritical) phase of the NCLP from the free particle in a plane by applying to the latter the CBT (5.22).The pre-images of the basic operators Pi and Pi of the NCLP are the corresponding operators of the free particle, (5.30) (5.31) The pre-images of the quadratic integrals are identified from the CBT map The linear combinations of Pj and Pj appearing in (5.30), (5.31) are represented in terms of the circularly polarized creation-annihilation operators according to Eq. (5.25).The pre-images of Xi and Ŷi are found from (5.30), (5.31) by using Eq.(4.14).
In the supercritical phase we identify the canonical variables q i and p i given by Eq. (5.14) with the canonical variables q i and p i of a free particle.Then the similarity transformation (5.22) yields us the correspondence of the form (5.25), (5.30), (5.31), with λ and Pi changed for λ and Pi .The pre-images of the operators Xi and Ŷi are found by using Eqs.(4.14) and (5.2).The map (5.32) is changed here for In both sub-and super-critical phases the last relations in (5.32) and (5.33) can be presented in a universal form While the angular momentum of the free particle M transforms into the angular momentum of the NCLP in the subcritical phase, and the Wick rotated generator of dilatations multiplied by ε transforms in the integral M, the CBT from the free particle into the chiral phase of the NCLP transmutes M and iε D into the integrals M and M, respectively.Since in the case of a free particle in non-commutative plane we have P i | B=0 = P i | B=0 = p i , and q i reduces to the coordinate X i defined in Eq. (4.20), the CBT can also be reinterpreted as a non-unitary mapping from a free particle system in non-commutative plane (B = 0, θ = 0) into the NCLP (B = 0, θ = 0) in non-chiral and chiral phases by changing the coordinate variable q i of the free particle for X i in the above relations.
As the CBT is an invertible non-unitary transformation, one can also relate the nonchiral and chiral phases of the NCLP by means of the composition of the corresponding inverse CBT from the subcritical phase into a "virtual" free particle with the direct CBT from the free particle into the supercritical phase.In such a composition, it is convenient to take in both phases the same value for magnetic field B, and choose the non-commutative parameter θ in the first, inverse CBT, such that β = Bθ < 1, and take θ ′ so that β ′ = Bθ ′ > 1 for the second, direct CBT from the "virtual" free particle in commutative (or non-commutative) plane into the chiral phase of the NCLP.
For completeness we just note that the critical phase β = 1 can be generated by the CBT from a 1D free particle, see Appendix.In this case the Wick rotated dilatation operator multiplied by −ε θ = −sgn θ is mapped into the angular momentum M.

Conformal bridge transformation for the two-vortex system
The CBT for the two-vortex system can be realized based on the correspondence with the NCLP established in Sec.4.1.We restrict ourselves here by presenting the basic relations necessary for the construction, and making brief comments on the transformation.
The correspondence (4.16), (4.17) between the NCLP and the two-vortex system is extended by relations for the integral M and spatial reflection of P i , We also denote the direct analog of the NCLP Hamiltonian, cf.Eq. (3.20), In the two-vortex system with strenths γ a of the opposite sign, the canonical variables (q i , p i ) can be defined based on the correspondence with the NCLP and Eq.(5.5), where λ = −γ 1 /γ 2 .In the limit Γ → 0, one has , where χ i and Π i are the canonical variables of the vortex-antivortex system, see Eq. (3.15).Inverting (5.37), one finds In terms of canonical variables (5.37), the integrals M Γ , MΓ and H Γ are presented in a form similar to (5.8), (5.9), The free particle dilatation generator is presented here in the form (5.42) In the two-vortex system with vorticities of the same sign, κ > 0, the canonical variables ( q i , p i ) can be defined by analogy with (5.14), where λ = γ 1 /γ 2 .For the integrals MΓ , M Γ and H Γ we obtain here a representation similar to (5.16), (5.17), where Ω is defined as in (5.41), Γ ǫ ij P i řj .Having the complete correspondence list for the dynamical variables and integrals in the NCLP and the two-vortex system in the cases of vorticities γ 1 and γ 2 of the opposite and equal signs, one can apply the CBT to a free particle to obtain the quantum system described by the auxiliary Hamiltonian ĤΓ with nonzero total vorticity Γ.The spectrum of the two-vortex system with Γ = 0 is obtained then in accordance with Eq. (5.36) by means of the relation ĤΓ = − 1 2 κ log 2γ −2 2 ĤΓ .One can also generate the two-vortex systems with κ = γ 1 γ 2 < 0, Γ = 0, and κ > 0 from the vortex-antivortex system with Γ = 0.In this case, the free particle coordinates q i are identified with the coordinates χ i of the vortex-antivortex system, while the canonical momenta p i of the former are identified with the variables Π i of the latter.Also, by means of the composition of the corresponding inverse and direct conformal bridge transformations, the two-vortex system with κ < 0, Γ = 0, and the system with κ > 0 can be related via the "virtual" free particle, or via the vortex-antivortex system with Γ = 0.

Summary, discussion and outlook
Let us first summarize and discuss the obtained results.
• We have established a correspondence between the non-commutative Landau problem and the system of two point vortices both at the classical and quantum levels.This is done for all three different cases of the NCLP: its sub-critical (non-chiral), super-critical (chiral), and a critical phases.
As both systems have phase spaces of the same dimension (equal four in non-critical cases) and their symplectic structures are similar, a priori one would immediately expect that there is a way to reinterpret one system in terms of another.However, in general case similarity of symplectic structures is not sufficient to establish a correspondence as classical systems with the same symplectic structure in phase space of dimension n ≥ 4 can still have different nature in the sense of integrability, that prevents the reinterpretation of one system in terms of another.On the other hand, sometimes distinct integrable systems with different symplectic structures can be related by means of a nontrivial mechanism of coupling constant metamorphosis, which employs a non-canonical transformation that exchanges the roles of a coupling constant and the energy in Hamiltonian systems while preserving integrability, and includes additionally a correction of order 2 at the quantum level [13].Finally, bi-Hamlitonian systems are described by different symplectic structures and different Hamiltonians.
The two considered systems not only have a similar symplectic structure, but both are superintegrable, and have the same symmetries being invariant under time and space translations, and rotations.Based also on the similarity of their dynamics, we introduced an imaginary mirror particle in the NCLP and put the vector coordinates of the particle, X, and of its imaginary partner, Y, in correspondence with coordinates of point vortices.In this way the total vorticity parameter Γ = γ 1 + γ 2 of the two-vortex system was set in accordance with the magnetic field value B, Γ ∼ −B, in the Landau problem.As a consequence, a strength of one of the vortices is mapped to −θ −1 , where θ is the noncommutativity parameter in the NCLP.
• We have found that the i) non-chiral (κ = γ 1 γ 2 < 0, Γ = 0), ii) chiral (κ > 0), and iii) stationary (when one of the vortex strengths γ a vanishes) cases of the two-vortex system correspond to the i) sub-(β = Bθ < 1), ii) super-(β > 1), and iii) critical (β = 1) phases in the NCLP.The angular momentum takes on the values of both signs in their non-chiral phase i), and of one sign in cases ii) and iii).In spite of the indicated correspondence, the frequency of the classical circular motion in non-chiral and chiral phases of the system of two vortices is energy-dependent in contrast with the NCLP.Another difference, associated with distinct form of Hamiltonians (3.1) and (4.12), is that the quantized energy levels in both non-critical phases of the NCLP are bounded from below, while in the two-vortex system with κ < 0, Γ = 0, and κ > 0 they are bounded from below and above, respectively.
• The vortex-antivortex system (Γ = 0) with its rectilinear trajectories and hidden (1 + 1)D "isospace" Lorentz symmetry corresponds to a free particle (B = 0, θ = 0) in the non-commutative plane.In this case, essential difference in classical dynamics of the systems is that with energy increasing of a free particle in non-commutative plane, its velocity increases as well as that of its imaginary partner, together with the increase of the distance between their parallel trajectories, while the velocities of the vortex and antivortex decrease when the vortex-anti-vortex energy and the distance between their parallel trajectories increase.
• The case of the ordinary (commutative) Landau problem (θ = 0, B = 0) is also covered by the correspondence with the two-vortex system through a limit procedure in which one of the vortex strengths tends to infinity.In this limit, the coordinates of the vortices coincide, x i 1 = x i 2 , and their components commute like this happens with the coordinates of the particle and its imaginary partner in the ordinary Landau problem, X i = Y i , {X i , X j } = 0.
• A non-trivial nature of the revealed correspondence is manifested in the fact that a simple permutation symmetry of the vortices generates a weak-strong coupling duality in the NCLP, I : H → (1 − β) 2 H, where H is the NCLP Hamiltonian.The particular chiral, β = 2, and non-chiral, β = 0, cases are stable under this weak-strong duality, and correspond to the cases of the vortices of equal strengths, γ 1 = γ 2 , and to the vortexantivortex system with zero total vorticity, Γ = 0, or to the indicated limit case to be similar to the ordinary Landau problem with θ = 0, B = 0.
We introduced a linear combination M of the angular momentum integral M and Hamiltonian H in the NCLP, which generates rotations of the vector integral of motion P i and of a spatially reflected non-commutative momentum Pi , and found the analogs of these objects in the two-vortex system.This allowed us to identify Darboux coordinates (canonical coordinates and momenta) in phase space, which are transformed as 2D vectors relative either to the angular momentum M or the modified integral M in the non-chiral or chiral phases, respectively.
Let us emphasize that the CBT construction for the NCLP in the chiral phase (β > 1) (and in the system of the two point vortices with the strengths γ a of the same sign) requires to introduce into consideration the two indicated additional objects M and Pi .This is necessary because the CBT implies the construction of the canonical coordinates and momenta in the phase space of the NCLP (and the two-vortex system) with their subsequent replacement with those of a free particle.By Darboux theorem, there is no problem to construct such variables in the phase space of the NCLP or the two-vortex system (see, for instance, ref. [46] where this was done for the NCLP by applying certain Bogolyubov transformations).However, the corresponding sets of canonical coordinates and momenta constructed as linear combinations of the NCLP vectors X and P (or of the vortex coordinates x a , a = 1, 2) cannot have a vector nature with respect to the angular momentum integral M (M Γ ) in the chiral phase, where the latter takes on the values of only one sign in contrast to the angular momentum of a free particle.On the other hand, the momentum integral P i for the NCLP and the Poisson-commuting with it Pi are transformed by M as the two-component vector objects.The canonical coordinates and momenta on the phase space are constructed as linear combinations of P i and Pi , and in their terms the integral M takes the form of the angular momentum of a 2D free particle.See Eqs.(5.14) and (5.16), and Eqs.(5.43), (5.44) for the corresponding chiral phase of the two-vortex system3 .
• Based on the two sets of canonical variables constructed separately for non-chiral and chiral phases, we built two forms of the conformal bridge transformation (CBT), which is a non-unitary similarity transformation that maps a 2D quantum free particle system into non-chiral and chiral phases of the two-vortex and NCLP systems.
• The generator of the CBT has a form of the evolution operator for the 2D inverted harmonic oscillator taken for a particular complex value of the time parameter t = iπ/4.It acts as the eighth-order root of an identity transformation in the phase space of a 2D free particle, which changes the topological nature of its sl(2, R) conformal symmetry generators, see Eq. (5.23) and Fig. 3 in Appendix.As a result, it produces the common eigenstates of the Hamiltonian and angular momentum of the quantum two-vortex and NCLP systems both in chiral and non-chiral phases from the simple monomial eigenfunctions of the angular momentum operator, which simultaneously are formal eigenfunctions of the Wick rotated dilatation generator of the 2D free particle.Such monomial in coordinates functions are simultaneously Jordan states corresponding to zero energy of the free particle and annihilated by its Hamiltonian Ĥ0 , or by higher powers of Ĥ0 [65,66].
It is interesting to note here some formal similarity of the CBT in a simpler case of the 2D isotropic harmonic oscillator [18], where the Wick rotated dilatation generator is transformed in Hamiltonian, with the reverse picture corresponding to the radial quantization of the closed string [68].In string theory, after the Wick rotation of the evolution parameter on the string surface and the subsequent mapping of the cylinder onto the complex plane, by means of an exponential conformal transformation, the generator of time translations on the cylinder is transformed into a dilatation generator, which takes on the role of the Hamiltonian.
• Here, unlike the case of the 2D isotropic harmonic oscillator, a linear combination of the Wick rotated dilatation generator and angular momentum operator of the free particle is transformed under the CBT into the Hamiltonian operators of the two quantum systems in corresponding non-chiral or chiral phases.In the case of the mapping by the CBT into the non-chiral phase, the angular momentum operator of the free particle is transformed into the angular momentum integral of the two-vortex or the NCLP systems, while the Wick rotated dilatation generator of the free particle is mapped into the integral M of the NCLP, or its analog in the two-vortex system.In the case of the transformation into the chiral phase of the two systems, the Wick rotated dilatation generator of the free particle transmutes into the angular momentum operator, while its angular momentum transmutes into the integral M of the NCLP and its analog in the two-vortex system.
Such a phenomenon of transmutation into and from the angular momentum, which occurs in the chiral phases of the two systems, has not previously been observed in any of other systems [18]- [21], [26]- [28], to which the CBT has been applied.Thus, the CBT highlights an essential difference between the non-chiral and chiral phases in both considered systems from a new perspective.
• The nature of the chiral and non-chiral phases in both quantum systems can be related to the outer Z 2 automorphism of the conformal sl(2, R) algebra, see Appendix.
• The CBT can also be applied to map the vortex-antivortex system (Γ = 0) and free particle in the non-commutative plane (B = 0, θ = 0) into non-chiral and chiral phases of the two-vortex and NCLP systems, respectively.
• Though the chiral and non-chiral phases of the systems are characterized by essentially different properties, they can be mutually mapped by a composition of the inverse and direct conformal bridge transformations.
The described picture of the mapping which transforms a linear combination of the Wick rotated dilatation generator and the angular momentum of the "source system" into the Hamiltonians of the "target systems" together with the mapping of the corresponding eigenstates of the symmetry generators represents some development of the idea of different forms of dynamics of Dirac [67] by incorporation into the construction of a kind of the Dyson map associated with the PT symmetry [24,26,27].
We expect that the established correspondences and mappings can be useful for the theory of anyons and fractional Hall effect, where the point vortices and non-commutative quantum mechanics play an important role.In this context it is interesting to mention a recent application of the point vortices in the rapidly developing topic of fracton phases in condensed matter physics, characterized by local excitations with restricted mobility [69].In particular, the single-vortex system (corresponding, as we showed, to critical phase in the NCLP) is identified there as an immobile fracton.The vortex-antivortex system (Γ = 0) is treated there as a "lineon", and a system of the two point vortices of the vorticity strengths of the same sign is considered in fractonic context as a 2-dipole.Because of the established correspondence of the two-vortex systems with different phases of the NCLP, the latters also can be related to simple fractonic systems.
We investigated the relationships between the classical and quantum dynamics of the planar two-vortex system, the noncommutative Landau problem, and the free particle in the commutative and noncommutative planes.It seems to be interesting to generalize the obtained results for the case of the spherical and hyperbolic geometries [46,70,71].The case of the hyperbolic geometry is of a particular interest in the light of the non-relativistic conformal-invariant Schwartzian mechanics associated with the low energy limit of the Sachdev-Ye-Kitaev model [38,39], which reveals a close relationship with the particle dynamics on AdS 2 and Landau problem on hyperbolic plane [40,41].transformation.For the sl(2, R) algebra, this acts as the fourth-order root of the identity transformation.The generator Ŝ of the similarity transformation (A.7) has a form of the evolution operator for complex time t = iπ/4 of the inverted harmonic oscillator.This is a non-unitary, non-local transformation, which changes the anti-hermitian operator i D = i 2 (q p + pq) = i 2 (p ′ 2 − q′ 2 ) = i Ĥ′ − into Hermitian Hamiltonian operator Ĥ+ = 1 2 (p 2 + q2 ) of the quantum harmonic oscillator, and corresponds to the Dyson map, to which the PT symmetry is intimately related, see refs.[26,27].At the level of the canonical operators q and p, this can be compared with transformation from the Schrödinger representation of the Heisenberg algebra to its Fock-Bargmann representation [18].The monomials φ n = x n , n = 0, 1, . .., are the formal eigenfunctions of the Wick rotated dilatation operator, i Dφ n = (n + 1  2 )φ n .At the same time, φ n are the Jordan states [65] of the free particle corresponding to its zero energy eigenvalue, ( Ĥ0 ) ⌊ n 2 ⌋ φ n = 0, where ⌊ n 2 ⌋ is the integer part of n/2.As a consequence, the CBT applied to φ n transforms them, up to a normalization, into eigenstates of the quantum harmonic oscillator, Ŝφ n (q) ∝ ψ n (q) = 1 2 n π 1/2 n! H n (q)e −q 2 /2 , (A.8) where H n (q) are the Hermite polynomials.At the same time, the free particle plane wave eigenstates and the Gaussian packet composed from them are transformed into coherent states and single-mode squeezed coherent states of the quantum harmonic oscillator respectively, see ref. [18].The described CBT can be generalized directly for the case of d > 1 dimensions by supplying the phase space coordinates q and p with index j = 1, . . ., d, and defining H ± = 1 2 (p 2 j ± q 2 j ), D = q j p j .In this way we obtain generators of the sl(2, R) algebra, which are invariant under the so(d) rotations.In the case of d = 2 dimensions one gets exp i π 4 H − ⋆ (iD + µM) = H + + µM, where M = ǫ ij q i p j is the angular momentum and µ is a constant.
The sl(2, R) algebra (A.1) has an outer Z 2 automorphism J µ → J ′ µ , (J ′ 0 , J ′ 1 , J ′ 2 ) = (−J 0 , J 2 , J 1 ) , (A.9) related to the existence of the infinite-dimensional unitary representations in which eigenvalues of the compact generator Ĵ0 are bounded from below or from above [72].Note that (A.9) represents a kind of the PT -inversion applied to J µ , which is a composition of the time, J 0 → −J 0 , and space, J i = (J 1 , J 2 ) → −ǫ ij Jj = (J 2 , J 1 ), cf.(5.35), inversions acting in the (2+1)D space with coordinates J µ .If one takes two copies of the sl(2, R) algebra generated by J µ , J ν } = 0, the three sets of the sl(2, R) generators can be constructed, J µ = J (1) 2) , J µ = J (1) 2) , where J ′ µ (a) are obtained from J (a) µ by applying the Z 2 automorphism (A.9).If J (a) µ are realized in terms of the onedimensional canonical variables (q (a) , p (a) ), in particular, J (a) 0 + , one finds that J 0 (J ′ 0 ) takes non-negative (non-positive) values, while J 0 takes values on the entire real line.At the quantum level, J 0 acts in the space which corresponds to the direct sum of the two infinite-dimensional (reducible) unitary representations of the sl(2, R) algebra mentioned above, each of which is, in turn, a direct sum of the two irreducible representations with eigenvalues of the corresponding compact generators shifted mutually in 1/2.In these representations, the Casimirs take the same value, Ĵ µ are treated as even generators, while ξ(1) α and ξ(2) α are its odd generators which mutually transform the corresponding states from the α = 1/4 and α = 3/4 subspaces [72].It is this picture related to the Z 2 automorphism of the sl(2, R) algebra that underlies different properties of the angular momentum in the sub-(non-chiral) and super-critical (chiral) phases of the NCLP and in the corresponding two-vortex systems with κ = γ 1 γ 2 < 0, Γ = 0, and κ > 0, see the second relation in Eq. (3.11).
Equations of motion generated by Hamiltonian (4.3) through DPBs (4.2) are which have a nature of the 2D vectors.Here and in what follows we restore the explicit dependence on .The "ciricularly polarized" creation-annihilation operators b

Figure 3 :
Figure 3: Action of the conformal bridge transformation.