Models Of Conformal Field Theory Research Articles

‘Probabilistic’ approach to Richardson equations

It is known that solutions of Richardson equations can be represented as stationary points of the ‘energy’ of classical free charges on the plane. We suggest considering the ‘probabilities’ of the system of charges occupying certain states in the configurational space at the effective temperature given by the interaction constant, which goes to zero in the thermodynamical limit. It is quite remarkable that the expression of ‘probability’ has similarities with the square of the Laughlin wavefunction. Next, we introduce the ‘partition function’, from which the ground state energy of the initial quantum-mechanical system can be determined. The ‘partition function’ is given by a multidimensional integral, which is similar to the Selberg integrals appearing in conformal field theory and random-matrix models. As a first application of this approach, we consider a system with the constant density of energy states at arbitrary filling of the energy interval where potential acts. In this case, the ‘partition function’ is rather easily evaluated using properties of the Vandermonde matrix. Our approach thus yields a quite simple and short way to find the ground state energy, which is shown to be described by a single expression all over from the dilute to the dense regime of pairs. It also provides additional insight into the physics of Cooper-paired states.

Gravity dual of the Ising model

We evaluate the partition function of three-dimensional theories of gravity in the quantum regime, where the anti--de Sitter (AdS) radius is Planck scale and the central charge is of order one. The contribution from the AdS vacuum sector can---with certain assumptions---be computed and equals the vacuum character of a minimal model conformal field theory. The torus partition function is given by a sum over geometries which is finite and computable. For generic values of Newton's constant $G$ and the AdS radius $\ensuremath{\ell}$, the result has no Hilbert space interpretation, but in certain cases it agrees with the partition function of a known conformal field theory. For example, the partition function of pure Einstein gravity with $G=3\ensuremath{\ell}$ equals that of the Ising model, providing evidence that these theories are dual. We also present somewhat weaker evidence that the three-state and tricritical Potts models are dual to pure higher spin theories of gravity based on $SL(3)$ and ${E}_{6}$, respectively.

The large N ’t Hooft limit of coset minimal models

Recently, Gaberdiel and Gopakumar proposed that the two-dimensional WA_{N-1} minimal model conformal field theory in the large N 't Hooft limit is dual to the higher spin theories on the three-dimensional AdS space with two complex scalars. In this paper, we examine this proposal for the WD_{N/2} and WB_{(N-1)/2} minimal models initiated by Fateev and Lukyanov in 1988. By analyzing the renormalization group flows on these models, we find that the gravity duals in AdS space are higher spin theories coupled to two equally massive real scalar fields. We also describe the large N 't Hooft limit for the minimal model of the second parafermion theory.

Higher level twisted Zhu algebras

The study of twisted representations of graded vertex algebras is important for understanding orbifold models in conformal field theory. In this paper, we consider the general setup of a vertex algebra V, graded by \documentclass[12pt]{minimal}\begin{document}$\Gamma /\mathbb {Z}$\end{document}Γ/Z for some subgroup Γ of \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}$\end{document}R containing \documentclass[12pt]{minimal}\begin{document}$\mathbb {Z}$\end{document}Z, and with a Hamiltonian operator H having real (but not necessarily integer) eigenvalues. We construct the directed system of twisted level p Zhu algebras \documentclass[12pt]{minimal}\begin{document}$\operatorname{Zhu}_{p, \Gamma }(V)$\end{document}Zhup,Γ(V), and we prove the following theorems: For each p, there is a bijection between the irreducible \documentclass[12pt]{minimal}\begin{document}$\operatorname{Zhu}_{p, \Gamma }(V)$\end{document}Zhup,Γ(V)-modules and the irreducible Γ-twisted positive energy V-modules, and V is (Γ, H)-rational if and only if all its Zhu algebras \documentclass[12pt]{minimal}\begin{document}$\operatorname{Zhu}_{p, \Gamma }(V)$\end{document}Zhup,Γ(V) are finite dimensional and semisimple. The main novelty is the removal of the assumption of integer eigenvalues for H. We provide an explicit description of the level p Zhu algebras of a universal enveloping vertex algebra, in particular of the Virasoro vertex algebra \documentclass[12pt]{minimal}\begin{document}$\operatorname{Vir}^c$\end{document}Virc and the universal affine Kac-Moody vertex algebra \documentclass[12pt]{minimal}\begin{document}$V^k(\mathfrak {g})$\end{document}Vk(g) at non-critical level. We also compute the inverse limits of these directed systems of algebras.

Recursive algorithm and branching for nonmaximal embeddings

Recurrent relations for branching coefficients in affine Lie algebras integrable highest weight modules are studied. The decomposition algorithm based on the injection fan technique is developed for the case of an arbitrary reductive subalgebra. In particular, we consider the situation where the Weyl denominator becomes singular with respect to the subalgebra. We demonstrate that for any reductive subalgebra it is possible to define the injection fan and the analogue of the Weyl numerator—the tools that describe explicitly the recurrent properties of branching coefficients. Possible applications of the fan technique in conformal field theory models are considered.

Global Gauge Anomalies in Two-Dimensional Bosonic Sigma Models

We revisit the gauging of rigid symmetries in two-dimensional bosonic sigma models with a Wess-Zumino term in the action. Such a term is related to a background closed 3-form H on the target space. More exactly, the sigma-model Feynman amplitudes of classical fields are associated to a bundle gerbe with connection of curvature H over the target space. Under conditions that were unraveled more than twenty years ago, the classical amplitudes may be coupled to the topologically trivial gauge fields of the symmetry group in a way which assures infinitesimal gauge invariance. We show that the resulting gauged Wess-Zumino amplitudes may, nevertheless, exhibit global gauge anomalies that we fully classify. The general results are illustrated on the example of the WZW and the coset models of conformal field theory. The latter are shown to be inconsistent in the presence of global anomalies. We introduce a notion of equivariant gerbes that allow an anomaly-free coupling of the Wess-Zumino amplitudes to all gauge fields, including the ones in non-trivial principal bundles. Obstructions to the existence of equivariant gerbes and their classification are discussed. The choice of different equivariant structures on the same bundle gerbe gives rise to a new type of discrete-torsion ambiguities in the gauged amplitudes. An explicit construction of gerbes equivariant with respect to the adjoint symmetries over compact simply connected simple Lie groups is given.

Double affine Hecke algebra in logarithmic conformal field theory

We construct a representation of the double affine Hecke algebra. The symmetrization of this representation coincides with the center of the quantum group \( U_\mathfrak{q} \mathcal{S}\ell (2) \) and, by Kazhdan-Lusztig duality with the Verlinde algebra of the (1,p)-model of the logarithmic conformal field theory.

Gause Symmetry and Howe Duality in 4D Conformal Field Theory Models

It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space-time dimensions the associated dual pairs are constructed and classified.The paper reviews joint work of B. Bakalov, N. M. Nikolov, K.-H. Rehren and the author.

Gauge Symmetry and Howe Duality in 4D Conformal Field Theory Models

It is known that there are no local scalar Lie fields in more than two dimensions. Bilocal fields, however, which naturally arise in conformal operator product expansions, do generate infinite Lie algebras. It is demonstrated that these Lie algebras of local observables admit (highly reducible) unitary positive energy representations in a Fock space. The multiplicity of their irreducible components is governed by a compact gauge group. The mutually commuting observable algebra and gauge group form a dual pair in the sense of Howe. In a theory of local scalar fields of conformal dimension two in four space‐time dimensions the associated dual pairs are constructed and classified.

The 2×2 matrix Schlesinger system and the Belavin-Polyakov-Zamolodchikov system

We show that the Belavin-Polyakov-Zamolodchikov equation of the minimal model of conformal field theory with the central charge c = 1 for the Virasoro algebra is contained in a system of linear equations that generates the Schlesinger system with 2×2 tmatrices. This generalizes Suleimanov’s result on the Painleve equations. We consider the properties of the solutions, which are expressible in terms of the Riemann theta function.

A generalization of the Verlinde formula in logarithmic conformal field theory

We propose a generalized Verlinde formula associated with (1, p) logarithmic models of two-dimensional conformal field theories, which have applications in statistical physics problems such as the sand-pile model and phase transitions in polymers. This formula gives the integer structure constants in the whole (3p-1)-dimensional space of vacuum torus amplitudes in which the fusion algebra is a 2p-dimensional subalgebra.

A general CFT model for antiferromagneticspin-1/2 ladders with Mobius boundary conditions

We show how the low energy properties of the two-leg XXZspin-1/2 ladders with generalanisotropy parameter Δ on closed geometries can be accounted for in the framework of them-reduction procedure developed previously (Cristofano et al 2000 Mod. Phys. Lett. A 15 547;Cristofano et al 2000 Mod. Phys. Lett. A 15 1679; Cristofano et al 2002 Nucl. Phys. B 641547; Cristofano et al 2004 J. High Energy Phys. JHEP06(2004) 056). In the limitof quasi-decoupled chains, a conformal field theory (CFT) with central chargec = 2 is derived and its ability to describe the model with different boundary conditions is shown.Special emphasis is given to the Mobius boundary conditions which generate a topologicaldefect corresponding to non-trivial single-spinon excitations. Then, in the case of thetwo-leg XXX ladders we discuss in detail the role of various perturbations in determiningthe renormalization group flow starting from the ultraviolet (UV) critical point withc = 2.

Factorizable ribbon quantum groups in logarithmic conformal field theories

We review the properties of quantum groups occurring as the Kazhdan-Lusztig dual to logarithmic conformal field theory models. These quantum groups at even roots of unity are not quasitriangular but are factorizable and have a ribbon structure; the modular group representation on their center coincides with the representation on generalized characters of the chiral algebra in logarithmic conformal field models.

Representations of centrally-extended Lie algebras over differential operators and vertex algebras

We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries. The structures of vertex algebras associated with the vacuum representations of these algebras are determined. Moreover, we prove that under certain conditions, the highest-weight irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries naturally give rise to the irreducible modules of the simple quotients of these vertex algebras. From vertex algebra and its representation point of view, our results with positive integral central charge are high-order differential operator analogues of the well-known WZW models in conformal field theory associated with affine Kac–Moody algebras. Indeed, when the left ideals are the algebra of differential operators, our Lie algebras do contain affine Kac–Moody algebras as subalgebras and our results restricted on them are exactly the representation contents in WZW models. Similar results with negative central charge are also obtained.

Pure gauge SU(2) Seiberg-Witten theory and modular forms

We identify the spectral curve of pure gauge SU(2) Seiberg-Witten theory with the Weierstrass curve C∕L∍z↦(1,℘(z),℘(z)′) and thereby obtain explicitely a modular form from which the moduli space parameter u and lattice parameters a and aD can be derived in terms of modular, respectively, theta functions. We further discuss its relationship with the c=−2 triplet model conformal field theory.

Kazhdan-Lusztig-dual quantum group for logarithimic extensions of Virasoro minimal models

We derive and study a quantum group gp,q that is Kazhdan-Lusztig dual to the W-algebra Wp,q of the logarithmic (p,q) conformal field theory model. The algebra Wp,q is generated by two currents W+(z) and W−(z) of dimension (2p−1)(2q−1) and the energy-momentum tensor T(z). The two currents generate a vertex-operator ideal R with the property that the quotient Wp,q∕R is the vertex-operator algebra of the (p,q) Virasoro minimal model. The number (2pq) of irreducible gp,q representations is the same as the number of irreducible Wp,q representations on which R acts nontrivially. We find the center of gp,q and show that the modular group representation on it is equivalent to the modular group representation on the Wp,q characters and “pseudocharacters.” The factorization of the gp,q ribbon element leads to a factorization of the modular group representation on the center. We also find the gp,q Grothendieck ring, which is presumably the “logarithmic” fusion of the (p,q) model.

Construction of a paired wave function for spinless electrons at filling fractionν=2∕5

We construct a wave function, generalizing the well-known Moore-Read Pfaffian, that describes spinless electrons at filling fraction $\ensuremath{\nu}=2∕5$ (or bosons at filling fraction $\ensuremath{\nu}=2∕3$) as the ground state of a very simple three body potential. We find, analogous to the Pfaffian, that when quasiholes are added there is a ground state degeneracy which can be identified as zero modes of the quasiholes. The zero modes are identified as having semionic statistics. We write this wave function as a correlator of the Virasoro minimal model conformal field theory $\mathcal{M}(5,3)$. Since this model is nonunitary, we conclude that this wave function is likely a quantum critical state. Nonetheless, we find that the overlaps of this wave function with exact diagonalizations in the lowest and first excited Landau level are very high, suggesting that this wave function may have experimental relevance for some transition that may occur in that regime.

Fermionic expressions for the characters of [formula omitted] logarithmic conformal field theories

We present fermionic quasi-particle sum representations consisting of a single fundamental fermionic form for all characters of the logarithmic conformal field theory models with central charge c p , 1 , p ⩾ 2 , and suggest a physical interpretation. We also show that it is possible to correctly extract dilogarithm identities.

JORDAN CELLS IN LOGARITHMIC LIMITS OF CONFORMAL FIELD THEORY

It is discussed how a limiting procedure of conformal field theories may result in logarithmic conformal field theories with Jordan cells of arbitrary rank. This extends our work on rank-two Jordan cells. We also consider the limits of certain three-point functions and find that they are compatible with known results. The general construction is illustrated by logarithmic limits of minimal models in conformal field theory. Characters of quasirational representations are found to emerge as the limits of the associated irreducible Virasoro characters.

Kazhdan-Lusztig correspondence for the representation category of the triplet W-algebra in logarithmic CFT

To study the representation category of the triplet W-algebra \(\mathcal{W}\left( p \right)\) that is the symmetry of the (1, p) logarithmic conformal field theory model, we propose the equivalent category Cp of finite-dimensional representations of the restricted quantum group Ūqsl(2) at \(\mathfrak{q} = e^{{{i\pi } \mathord{\left/ {\vphantom {{i\pi } p}} \right. \kern-\nulldelimiterspace} p}} \). We fully describe the category Cp by classifying all indecomposable representations. These are exhausted by projective modules and three series of representations that are essentially described by indecomposable representations of the Kronecker quiver. The equivalence of the \(\mathcal{W}\left( p \right)\)-and Ūqsl(2)-representation categories is conjectured for all p = 2 and proved for p = 2. The implications include identifying the quantum group center with the logarithmic conformal field theory center and the universal R-matrix with the braiding matrix.