Quantum groups as generalized gauge symmetries in WZNW models. Part II. The quantized model

This is the second part of a paper dealing with the “internal” (gauge) symmetry of the Wess–Zumino–Novikov–Witten (WZNW) model on a compact Lie group G. It contains a systematic exposition, for G = SU(n), of the canonical quantization based on the study of the classical model (performed in the first part) following the quantum group symmetric approach first advocated by L.D. Faddeev and collaborators. The internal symmetry of the quantized model is carried by the chiral WZNW zero modes satisfying quadratic exchange relations and an n-linear determinant condition. For generic values of the deformation parameter the Fock representation of the zero modes’ algebra gives rise to a model space of U _q (sl(n)). The relevant root of unity case is studied in detail for n = 2 when a “restricted” (finite dimensional) quotient quantum group is shown to appear in a natural way. The module structure of the zero modes’ Fock space provides a specific duality with the solutions of the Knizhnik–Zamolodchikov equation for the four point functions of primary fields suggesting the existence of an extended state space of logarithmic CFT type. Combining left and right zero modes (i.e., returning to the 2D model), the rational CFT structure shows up in a setting reminiscent to covariant quantization of gauge theories in which the restricted quantum group plays the role of a generalized gauge symmetry.     

Unitarity in “Quantization Commutes with Reduction”

Let M be a compact Kähler manifold equipped with a Hamiltonian action of a compact Lie group G. In this paper, we study the geometric quantization of the symplectic quotient M // G. Guillemin and Sternberg [Invent. Math. 67, 515–538 (1982)] have shown, under suitable regularity assumptions, that there is a natural invertible map between the quantum Hilbert space over M //G and the G-invariant subspace of the quantum Hilbert space over M.Reproducing other recent results in the literature, we prove that in general the natural map of Guillemin and Sternberg is not unitary, even to leading order in Planck’s constant. We then modify the quantization procedure by the “metaplectic correction” and show that in this setting there is still a natural invertible map between the Hilbert space over M // G and the G-invariant subspace of the Hilbert space over M. We then prove that this modified Guillemin–Sternberg map is asymptotically unitary to leading order in Planck’s constant. The analysis also shows a good asymptotic relationship between Toeplitz operators on M and on M // G.     

Chiral and deconfinement phase transitions in QED<Subscript>3</Subscript> with finite gauge boson mass

Based on the experimental observation that there is a coexisting region between the antiferromagnetic (AF) and d-wave superconducting (dSC) phases, the influences of gauge boson mass m _a on chiral symmetry restoration and deconfinement phase transitions in QED₃ are investigated simultaneously within a unified framework, i.e., Dyson–Schwinger equations. The results show that the chiral symmetry restoration phase transition in the presence of the gauge boson mass m _a is a typical second-order phase transition; the chiral symmetry restoration and deconfinement phase transitions are coincident; the critical number of fermion flavors N ^c _f decreases as the gauge boson mass m _a increases, which implies that there exists a boundary that separates the N ^c _f –m _a plane into chiral symmetry breaking/confinement region for (N ^c _f , m _a ) below the boundary and chiral symmetry restoration/deconfinement region for (N ^c _f , m _a ) above it.     

Low-temperature contribution to the resonant tunneling conductance of a disordered N–I–N junction

A formula for the contribution ΔG ^res(T) to the resonant tunneling conductance of the N–I–N junction (where N is a normal metal and I is an insulator) with a weak (low impurity concentrations) structural disorder in the I layer from the low-temperature “smearing” electron Fermi surfaces in its N shores is obtained. It is shown that the temperature dependence ΔG ^res(T) in such a “dirty” junction qualitatively differs from the corresponding dependence ΔG ₀(T) in a “pure” (without resonant impurities in the I layer) junction: ΔG ^res(T) < 0, d(ΔG ^res)/dT < 0; ΔG ₀(T) > 0, d(ΔG ₀)/dT > 0, which can serve as an experimental test of the presence of impurity tunneling resonances in the disordered I layer.     

Equivalent Theories and Changing Hamiltonian Observables in General Relativity

Change and local spatial variation are missing in Hamiltonian general relativity according to the most common definition of observables as having 0 Poisson bracket with all first-class constraints. But other definitions of observables have been proposed. In pursuit of Hamiltonian–Lagrangian equivalence, Pons, Salisbury and Sundermeyer use the Anderson–Bergmann–Castellani gauge generator G, a tuned sum of first-class constraints. Kucha? waived the 0 Poisson bracket condition for the Hamiltonian constraint to achieve changing observables. A systematic combination of the two reforms might use the gauge generator but permit non-zero Lie derivative Poisson brackets for the external gauge symmetry of General Relativity. Fortunately one can test definitions of observables by calculation using two formulations of a theory, one without gauge freedom and one with gauge freedom. The formulations, being empirically equivalent, must have equivalent observables. For de Broglie-Proca non-gauge massive electromagnetism, all constraints are second-class, so everything is observable. Demanding equivalent observables from gauge Stueckelberg–Utiyama electromagnetism, one finds that the usual definition fails while the Pons–Salisbury–Sundermeyer definition with G succeeds. This definition does not readily yield change in GR, however. Should GR’s external gauge freedom of general relativity share with internal gauge symmetries the 0 Poisson bracket (invariance), or is covariance (a transformation rule) sufficient? A graviton mass breaks the gauge symmetry (general covariance), but it can be restored by parametrization with clock fields. By requiring equivalent observables, one can test whether observables should have 0 or the Lie derivative as the Poisson bracket with the gauge generator G. The latter definition is vindicated by calculation. While this conclusion has been reported previously, here the calculation is given in some detail.     

Electronic transport through a Majorana bound state coupled to a T-shaped quantum-dot system with Coulomb interaction

Using the Green’s function technique, we respectively investigate the electron transport properties of two spin components through the system of a T-shaped double quantum dot structure coupled to a Majorana bound state, in which only one quantum dot is connected with two metallic leads. We explore the interplay between the Fano effect and the MBSs for different dot-MBS coupling strength λ, dot-dot coupling strength t, and MBS-MBS coupling strength ε_M in the noninteracting case. Then the Coulomb interaction and magnetic field effect on the conductance spectra are investigated. Our results indicate that G_↓(ω) is not affected by the Majorana bound states, but a “0.5” conductance signature occurs in the vicinities of Fermi level of G_↑(ω). This robust property persists for a wide range of dot-dot coupling strength and dot-MBS coupling strength, but it can be destroyed by Coulomb interaction in quantum dots. By adjusting the size and direction of magnetic field around the quantum dots, the “0.5” conductance signature damaged by U can be restored. At last, the spin magnetic moments of two dots by applying external magnetic field are also predicted.     

A Rigidity Result for Extensions of Braided Tensor <Emphasis Type="Italic">C</Emphasis>*–Categories Derived from Compact Matrix Quantum Groups

Let G be a classical compact Lie group and G _μ the associated compact matrix quantum group deformed by a positive parameter μ (or \({\mu\in{\mathbb R}\setminus\{0\}}\) in the type A case). It is well known that the category of unitary representations of G _μ is a braided tensor C*–category. We show that any braided tensor *–functor \({\rho: \text{Rep}(G_\mu)\to\mathcal{M}}\) to another braided tensor C*–category with irreducible tensor unit is full if |μ| ≠ 1. In particular, the functor of restriction RepG _μ → Rep(K) to a proper compact quantum subgroup K cannot be made into a braided functor. Our result also shows that the Temperley–Lieb category \({\mathcal{T}_{\pm d}}\) for d > 2 can not be embedded properly into a larger category with the same objects as a braided tensor C*–subcategory.     

Hidden conformal symmetry of rotating black holes in the five-dimensional Gödel universe

We have investigated the hidden conformal symmetry of generic non-extremal rotating black holes in the five-dimensional Gödel universe. In a range of parameters, the low-frequency massless scalar wave equation in the “near region” can be described by an SL(2, R) _L × SL(2, R) _R conformal symmetry. We further found that the microscopic entropy via Cardy formula matches the macroscopic Bekenstein-Hawking entropy and the absorption cross section for the massless scalar also agrees with the one for the two dimensional finite temperature conformal field theory (CFT). All these evidences support the conjecture that the generic non-extremal rotating black hole immersed in the Gödel universe can be dual to a two dimensional finite temperature CFT. In addition, we have reformulated the first laws of thermodynamics associated with the inner and outer horizons of the rotating Gödel-type black holes into the forms of conformal thermodynamics.     

Higgs Bundles,Gauge Theories and Quantum Groups

The appearance of the Bethe Ansatz equation for the Nonlinear Schrödinger equation in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding two-dimensional topological U(N) gauge theory reproduce quantum wave functions of the Nonlinear Schrödinger equation in the N-particle sector. This implies the full equivalence between the above gauge theory and the N-particle sub-sector of the quantum theory of the Nonlinear Schrödinger equation. This also implies the explicit correspondence between the gauge theory and the representation theory of the degenerate double affine Hecke algebra. We propose a similar construction based on the G/G gauged WZW model leading to the representation theory of the double affine Hecke algebra.     

Anti-Kählerian Geometry on Lie Groups

Let G be a Lie group of even dimension and let (g, J) be a left invariant anti-Kähler structure on G. In this article we study anti-Kähler structures considering the distinguished cases where the complex structure J is abelian or bi-invariant. We find that if G admits a left invariant anti-Kähler structure (g, J) where J is abelian then the Lie algebra of G is unimodular and (G, g) is a flat pseudo-Riemannian manifold. For the second case, we see that for any left invariant metric g for which J is an anti-isometry we obtain that the triple (G, g, J) is an anti-Kähler manifold. Besides, given a left invariant anti-Hermitian structure on G we associate a covariant 3-tensor ?? on its Lie algebra and prove that such structure is anti-Kähler if and only if ?? is a skew-symmetric and pure tensor. From this tensor we classify the real 4-dimensional Lie algebras for which the corresponding Lie group has a left invariant anti-Kähler structure and study the moduli spaces of such structures (up to group isomorphisms that preserve the anti-Kähler structures).     

<Emphasis Type="Italic">Q</Emphasis>-systems as Cluster Algebras II: Cartan Matrix of Finite Type and the Polynomial Property

We define the cluster algebra associated with the Q-system for the Kirillov–Reshetikhin characters of the quantum affine algebra \({U_q(\widehat{\mathfrak {g}})}\) for any simple Lie algebra \({\mathfrak {g}}\), generalizing the simply-laced case treated in (Kedem in Q-systems as cluster algebras. arXiv:0712.2695 [math.RT], 2007). We describe some special properties of this cluster algebra, and explain its relation to the deformed Q-systems which appeared on our proof of the combinatorial-KR conjecture. We prove that the polynomiality of the cluster variables in terms of the “initial cluster seeds”, including solutions of the Q-system, is a consequence of the Laurent phenomenon and the boundary conditions. We also define the cluster algebra associated with T-systems, or general systems which take the form of T-systems in the bipartite case. Such systems describe the recursion relations satisfied by the q-characters of Kirillov–Reshetikhin modules and also appear in the categorification picture in terms of preprojective algebras of Geiss, Leclerc and Schröer. We give a formulation of both Q-systems and generalized T-systems as cluster algebras with coefficients. This provides a proof of the polynomiality of solutions of all such “generalized T-systems” with appropriate boundary conditions.     

Restricting Positive Energy Representations of Diff<Superscript>+</Superscript>(<Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript>) to the Stabilizer of <Emphasis Type="Italic">n</Emphasis> Points

Let G _n ? Diff⁺(S ¹) be the stabilizer of n given points of S ¹. How much information do we lose if we restrict a positive energy representation \(U^c_h\) associated to an admissible pair (c, h) of the central charge and lowest energy, to the subgroup G _n ? The question, and a part of the answer originate in chiral conformal QFT. The value of c can be easily “recovered” from such a restriction; the hard question concerns the value of h. If c ≤ 1, then there is no loss of information, and accordingly, all of these restrictions are irreducible. In this work it is shown that \(U^c_{h}|_{G_n}\) is always irreducible for n =  1 and, if h =  0, it is irreducible at least up to n ≤  3. Moreover, an example is given for c >  2 and certain values of \(h \neq \tilde{h}\) such that \(U^c_{h}|_{G_1}\simeq U^c_{\tilde{h}}|_{G_1}\) . It is also concluded that for these values \(U^c_{h}|_{G_n}\) cannot be irreducible for n ≥  2. For further values of c, h and n, the question is left open. Nevertheless, the example already shows that, on the circle, there are conformal QFT models in which local and global intertwiners are not equivalent.     

Quantum signatures of chaos or quantum chaos?

A critical analysis of the present-day concept of chaos in quantum systems as nothing but a “quantum signature” of chaos in classical mechanics is given. In contrast to the existing semi-intuitive guesses, a definition of classical and quantum chaos is proposed on the basis of the Liouville–Arnold theorem: a quantum chaotic system featuring N degrees of freedom should have M < N independent first integrals of motion (good quantum numbers) specified by the symmetry of the Hamiltonian of the system. Quantitative measures of quantum chaos that, in the classical limit, go over to the Lyapunov exponent and the classical stability parameter are proposed. The proposed criteria of quantum chaos are applied to solving standard problems of modern dynamical chaos theory.     

A model of classical thermodynamics based on the partition theory of integers,Earth gravitation,and semiclassical asymptotics. I

In the paper, a new construction of the theory of partitions of integers is proposed. The author defines entropy as the natural logarithm of the number of partitions of a number M into natural summands with repetitions allowed p(M) and repetitions forbidden q(M). The passage from ln p(M) to lnq(M) through the mesoscopic values M → 0 is studied. The topological transition from the mesoscopic lower levels of the Bohr–Kalckar construction to the macroscopic levels corresponding to the critical number of neutrons according to the consequence of Einstein’s inequality M ≤ c N _c , where c is determined for the particles of the given atomic nucleus. The role of quantum mechanics in establishing the new world outlook in physics is analyzed. It is pointed out that the main equations of thermodynamics in the volume “Statistical Physics” of the Landau–Lifshits treatise are obtained without appealing to the so-called “three main principles of thermodynamics”. It is also pointed out that Niels Bohr’s liquid model of the nucleus does not involve any interaction of particles in the form of attraction and is based on the presence of a common potential trough for all elements of the nucleus. The author constructs a new approach to thermodynamics, using quantum mechanics and the Earth’s gravitational attraction as a common potential trough.     

Bad communities with high modularity

In this paper we discuss some problematic aspects of Newman and Girvan’s modularity function Q _N . Given a graph G, the modularity of G can be written as Q _N = Q _f ? Q ₀, where Q _f is the intracluster edge fraction of G and Q ₀ is the expected intracluster edge fraction of the null model, i.e., a randomly connected graph with same expected degree distribution as G. It follows that the maximization of Q _N must accomodate two factors pulling in opposite directions:Q _f favors a small number of clusters and Q ₀ favors many balanced (i.e., with approximately equal degrees) clusters. In certain cases the Q ₀ term can cause overestimation of the true cluster number; this is the opposite of the well-known underestimation effect caused by the “resolution limit” of modularity. We illustrate the overestimation effect by constructing families of graphs with a “natural” community structure which, however, does not maximize modularity. In fact, we show there exist graphs G with a “natural clustering” V of G and another, balanced clustering U of G such that (i) the pair (G, U) has higher modularity than (G, V) and (ii) V and U are arbitrarily different.     

Computational studies of third-order nonlinear optical properties of pyridine derivative 2-aminopyridinium p-toluenesulphonate crystal

In this note, method of Lie symmetries is applied to investigate symmetry properties of time-fractional K(m, n) equation with the Riemann–Liouville derivatives. Reduction of time-fractional K(m, n) equation is done by virtue of the Erdélyi–Kober fractional derivative which depends on a parameter α. Then soliton solutions are extracted by means of a transformation.     

Converting heat to electricity by a graphene stripe with heavy chiral fermions

A conversion of thermal energy into electricity is considered in the electricallypolarized graphene stripes with zigzag edges where the heavy chiral fermion (HCF) statesare formed. The stripes are characterized by a high electric conductance G _e and by a significantSeebeck coefficient S. The electric current in the stripes is induced due toa non-equilibrium thermal injection of “hot” electrons. This thermoelectric generationprocess might be utilized for building of thermoelectric generators with an exceptionallyhigh figure of merit ZδT ?1 and with an appreciable electric power densities ~1 MW/cm².     

Conformal Geometry of the Supercotangent and Spinor Bundles

We study the actions of local conformal vector fields \({X \in {\rm conf}(M,g)}\) on the spinor bundle of (M, g) and on its classical counterpart: the supercotangent bundle \({\mathcal{M}}\) of (M, g). We first deal with the classical framework and determine the Hamiltonian lift of conf (M, g) to \({\mathcal{M}}\) . We then perform the geometric quantization of the supercotangent bundle of (M, g), which constructs the spinor bundle as the quantum representation space. The Kosmann Lie derivative of spinors is obtained by quantization of the comoment map.The quantum and classical actions of conf (M, g) turn, respectively, the space of differential operators acting on spinor densities and the space of their symbols into conf (M, g)-modules. They are filtered and admit a common associated graded module. In the conformally flat case, the latter helps us determine the conformal invariants of both conf (M, g)-modules, in particular the conformally odd powers of the Dirac operator.