Indecomposability parameters in chiral Logarithmic Conformal Field Theory

Work of the last few years has shown that the key algebraic features of Logarithmic Conformal Field Theories (LCFTs) are already present in some finite lattice systems (such as the XXZ spin-1/2 chain) before the continuum limit is taken. This has provided a very convenient way to analyze the structure of indecomposable Virasoro modules and to obtain fusion rules for a variety of models such as (boundary) percolation etc. LCFTs allow for additional quantum numbers describing the fine structure of the indecomposable modules, and generalizing the `b-number' introduced initially by Gurarie for the c=0 case. The determination of these indecomposability parameters has given rise to a lot of algebraic work, but their physical meaning has remained somewhat elusive. In a recent paper, a way to measure b for boundary percolation and polymers was proposed. We generalize this work here by devising a general strategy to compute matrix elements of Virasoro generators from the numerical analysis of lattice models and their continuum limit. The method is applied to XXZ spin-1/2 and spin-1 chains with open (free) boundary conditions. They are related to gl(n+m|m) and osp(n+2m|2m)-invariant superspin chains and to nonlinear sigma models with supercoset target spaces. These models can also be formulated in terms of dense and dilute loop gas. We check the method in many cases where the results were already known analytically. Furthermore, we also confront our findings with a construction generalizing Gurarie's, where logarithms emerge naturally in operator product expansions to compensate for apparently divergent terms. This argument actually allows us to compute indecomposability parameters in any logarithmic theory. A central result of our study is the construction of a Kac table for the indecomposability parameters of the logarithmic minimal models LM(1,p) and LM(p,p+1).Comment: 32 pages, 2 figures, Published Versio

Associative-algebraic approach to logarithmic conformal field theories

We set up a strategy for studying large families of logarithmic conformal field theories by using the enlarged symmetries and non--semi-simple associative algebras appearing in their lattice regularizations (as discussed in a companion paper). Here we work out in detail two examples of theories derived as the continuum limit of XXZ spin-1/2 chains, which are related to spin chains with supersymmetry algebras gl($n|n$) and gl($n+1|n$), respectively, with open (or free) boundary conditions in all cases. These theories can also be viewed as vertex models, or as loop models. Their continuum limits are boundary conformal field theories (CFTs) with central charge $c=-2$ and $c=0$ respectively, and in the loop interpretation they describe dense polymers and the boundaries of critical percolation clusters, respectively. We also discuss the case of dilute (critical) polymers as another boundary CFT with $c=0$. Within the supersymmetric formulations, these boundary CFTs describe the fixed points of certain nonlinear sigma models that have a supercoset space as the target manifold, and of Landau-Ginzburg field theories. The submodule structures of indecomposable representations of the Virasoro algebra appearing in the boundary CFT, representing local fields, are derived from the lattice. A central result is the derivation of the fusion rules for these fields

Edge states and conformal boundary conditions in super spin chains and super sigma models

The sigma models on projective superspaces CP^{N+M-1|N} with topological angle theta=pi mod 2pi flow to non-unitary, logarithmic conformal field theories in the low-energy limit. In this paper, we determine the exact spectrum of these theories for all open boundary conditions preserving the full global symmetry of the model, generalizing recent work on the particular case M=0 [C. Candu et al, JHEP02(2010)015]. In the sigma model setting, these boundary conditions are associated with complex line bundles, and are labelled by an integer, related with the exact value of theta. Our approach relies on a spin chain regularization, where the boundary conditions now correspond to the introduction of additional edge states. The exact values of the exponents then follow from a lengthy algebraic analysis, a reformulation of the spin chain in terms of crossing and non-crossing loops (represented as a certain subalgebra of the Brauer algebra), and earlier results on the so-called one- and two-boundary Temperley Lieb algebras (also known as blob algebras). A remarkable result is that the exponents, in general, turn out to be irrational. The case M=1 has direct applications to the spin quantum Hall effect, which will be discussed in a sequel.Comment: 50 pages, 18 figure

Kazhdan-Lusztig equivalence and fusion of Kac modules in Virasoro logarithmic models

The subject of our study is the Kazhdan-Lusztig (KL) equivalence in the context of a one-parameter family of logarithmic CFTs based on Virasoro symmetry with the (1,p) central charge. All finite-dimensional indecomposable modules of the KL-dual quantum group - the "full" Lusztig quantum sl(2) at the root of unity - are explicitly described. These are exhausted by projective modules and four series of modules that have a functorial correspondence with any quotient or a submodule of Feigin-Fuchs modules over the Virasoro algebra. Our main result includes calculation of tensor products of any pair of the indecomposable modules. Based on the Kazhdan-Lusztig equivalence between quantum groups and vertex-operator algebras, fusion rules of Kac modules over the Virasoro algebra in the (1,p) LCFT models are conjectured.Comment: 40pp. V2: a new introduction, corrected typos, some explanatory comments added, references adde

Integrable quantum field theories with OSP(m/2n) symmetries

We conjecture the factorized scattering description for OSP(m/2n)/OSP(m-1/2n) supersphere sigma models and OSP(m/2n) Gross Neveu models. The non-unitarity of these field theories translates into a lack of `physical unitarity' of the S matrices, which are instead unitary with respect to the non-positive scalar product inherited from the orthosymplectic structure. Nevertheless, we find that formal thermodynamic Bethe ansatz calculations appear meaningful, reproduce the correct central charges, and agree with perturbative calculations. This paves the way to a more thorough study of these and other models with supergroup symmetries using the S matrix approach.Comment: 32 pages, 9 figure

Associative algebraic approach to logarithmic CFT in the bulk: the continuum limit of the gl(1|1) periodic spin chain, Howe duality and the interchiral algebra

We develop in this paper the principles of an associative algebraic approach to bulk logarithmic conformal field theories (LCFTs). We concentrate on the closed $gl(1|1)$ spin-chain and its continuum limit - the $c=-2$ symplectic fermions theory - and rely on two technical companion papers, "Continuum limit and symmetries of the periodic gl(1|1) spin chain" [Nucl. Phys. B 871 (2013) 245-288] and "Bimodule structure in the periodic gl(1|1) spin chain" [Nucl. Phys. B 871 (2013) 289-329]. Our main result is that the algebra of local Hamiltonians, the Jones-Temperley-Lieb algebra JTL_N, goes over in the continuum limit to a bigger algebra than the product of the left and right Virasoro algebras. This algebra, S - which we call interchiral, mixes the left and right moving sectors, and is generated, in the symplectic fermions case, by the additional field $S(z,\bar{z})=S_{ab}\psi^a(z)\bar{\psi}^b(\bar{z})$, with a symmetric form $S_{ab}$ and conformal weights (1,1). We discuss in details how the Hilbert space of the LCFT decomposes onto representations of this algebra, and how this decomposition is related with properties of the finite spin-chain. We show that there is a complete correspondence between algebraic properties of finite periodic spin chains and the continuum limit. An important technical aspect of our analysis involves the fundamental new observation that the action of JTL_N in the $gl(1|1)$ spin chain is in fact isomorphic to an enveloping algebra of a certain Lie algebra, itself a non semi-simple version of $sp(N-2)$. The semi-simple part of JTL_N is represented by $Usp(N-2)$, providing a beautiful example of a classical Howe duality, for which we have a non semi-simple version in the full JTL image represented in the spin-chain. On the continuum side, simple modules over the interchiral algebra S are identified with "fundamental" representations of $sp(\infty)$.Comment: 69 pp., 10 figs, v2: the paper has been substantially modified - new proofs, new refs, new App C with inductive limits construction, et

Differential (2+1) Jet Event Rates and Determination of alpha_s in Deep Inelastic Scattering at HERA

Events with a (2+1) jet topology in deep-inelastic scattering at HERA are studied in the kinematic range 200 < Q^2< 10,000 GeV^2. The rate of (2+1) jet events has been determined with the modified JADE jet algorithm as a function of the jet resolution parameter and is compared with the predictions of Monte Carlo models. In addition, the event rate is corrected for both hadronization and detector effects and is compared with next-to-leading order QCD calculations. A value of the strong coupling constant of alpha_s(M_Z^2)= 0.118+- 0.002 (stat.)^(+0.007)_(-0.008) (syst.)^(+0.007)_(-0.006) (theory) is extracted. The systematic error includes uncertainties in the calorimeter energy calibration, in the description of the data by current Monte Carlo models, and in the knowledge of the parton densities. The theoretical error is dominated by the renormalization scale ambiguity.Comment: 25 pages, 6 figures, 3 tables, submitted to Eur. Phys.

Hadron Production in Diffractive Deep-Inelastic Scattering

Characteristics of hadron production in diffractive deep-inelastic positron-proton scattering are studied using data collected in 1994 by the H1 experiment at HERA. The following distributions are measured in the centre-of-mass frame of the photon dissociation system: the hadronic energy flow, the Feynman-x (x_F) variable for charged particles, the squared transverse momentum of charged particles (p_T^{*2}), and the mean p_T^{*2} as a function of x_F. These distributions are compared with results in the gamma^* p centre-of-mass frame from inclusive deep-inelastic scattering in the fixed-target experiment EMC, and also with the predictions of several Monte Carlo calculations. The data are consistent with a picture in which the partonic structure of the diffractive exchange is dominated at low Q^2 by hard gluons.Comment: 16 pages, 6 figures, submitted to Phys. Lett.

Measurement of D* Meson Cross Sections at HERA and Determination of the Gluon Density in the Proton using NLO QCD

With the H1 detector at the ep collider HERA, D* meson production cross sections have been measured in deep inelastic scattering with four-momentum transfers Q^2>2 GeV2 and in photoproduction at energies around W(gamma p)~ 88 GeV and 194 GeV. Next-to-Leading Order QCD calculations are found to describe the differential cross sections within theoretical and experimental uncertainties. Using these calculations, the NLO gluon momentum distribution in the proton, x_g g(x_g), has been extracted in the momentum fraction range 7.5x10^{-4}< x_g <4x10^{-2} at average scales mu^2 =25 to 50 GeV2. The gluon momentum fraction x_g has been obtained from the measured kinematics of the scattered electron and the D* meson in the final state. The results compare well with the gluon distribution obtained from the analysis of scaling violations of the proton structure function F_2.Comment: 27 pages, 9 figures, 2 tables, submitted to Nucl. Phys.

Multiplicity Structure of the Hadronic Final State in Diffractive Deep-Inelastic Scattering at HERA

The multiplicity structure of the hadronic system X produced in deep-inelastic processes at HERA of the type ep -> eXY, where Y is a hadronic system with mass M_Y< 1.6 GeV and where the squared momentum transfer at the pY vertex, t, is limited to |t|<1 GeV^2, is studied as a function of the invariant mass M_X of the system X. Results are presented on multiplicity distributions and multiplicity moments, rapidity spectra and forward-backward correlations in the centre-of-mass system of X. The data are compared to results in e+e- annihilation, fixed-target lepton-nucleon collisions, hadro-produced diffractive final states and to non-diffractive hadron-hadron collisions. The comparison suggests a production mechanism of virtual photon dissociation which involves a mixture of partonic states and a significant gluon content. The data are well described by a model, based on a QCD-Regge analysis of the diffractive structure function, which assumes a large hard gluonic component of the colourless exchange at low Q^2. A model with soft colour interactions is also successful.Comment: 22 pages, 4 figures, submitted to Eur. Phys. J., error in first submission - omitted bibliograph