Bits and Pieces in Logarithmic Conformal Field Theory

Journal of Mathematical Physics, 2014

We consider C-graded vertex algebras, which are vertex algebras V with a Cgrading such that V is an admissible V -module generated by 'lowest weight vectors'. We show that such vertex algebras have a 'good' representation theory in the sense that there is a Zhu algebra A(V ) and a bijection between simple admissible V -modules and simple A(V )-modules. We also consider pseudo vertex operator algebras, which are C-graded vertex algebras with a conformal vector such that the homogeneous subspaces of V are generalized eigenspaces for L(0); essentially, these are VOAs that lack any semisimplicity or integrality assumptions on L(0). As a motivating example, we show that deformation of the conformal structure (conformal flow) of a strongly regular VOA (eg a lattice theory, or WZW model) is a path in a space whose points are PVOAs. MSC(2012): 17B69. * Supported by NSF

Physical Review D, 2006

We discuss two classes of semi-microscopic theoretical models of stochastic space-time foam in quantum gravity and the associated effects on entangled states of neutral mesons, signalling an intrinsic breakdown of CPT invariance. One class of models deals with a specific model of foam, initially constructed in the context of non-critical (Liouville) string theory, but viewed here in the more general context of effective quantum-gravity models. The relevant Hamiltonian perturbation, describing the interaction of the meson with the foam medium, consists of off-diagonal stochastic metric fluctuations, connecting distinct mass eigenstates (or the appropriate generalisation thereof in the case of K-mesons), and it is proportional to the relevant momentum transfer (along the direction of motion of the meson pair). There are two kinds of CPT-violating effects in this case, which can be experimentally disentangled: one (termed "ω-effect") is associated with the failure of the indistinguishability between the neutral meson and its antiparticle, and affects certain symmetry properties of the initial state of the two-meson system; the second effect is generated by the time evolution of the system in the medium of the space-time foam, and can result in time-dependent contributions of the ω-effect type in the time profile of the two meson state. Estimates of both effects are given, which show that, at least in certain models, such effects are not far from the sensitivity of experimental facilities available currently or in the near future. The other class of quantum gravity models involves a medium of gravitational fluctuations which behaves like a "thermal bath". In this model both of the above-mentioned intrinsic CPT violation effects are not valid.

Physical Review D, 2005

We study in some detail the master equation, and its solution in a simplified case modelling flavour oscillations of a two-level system, stemming from the Liouvillestring approach to quantum space time foam. In this framework we discuss the appearance of diffusion terms and decoherence due to the interaction of low-energy string matter with space-time defects, such as D-particles in the specific model of "D-particle foam", as well as dark energy contributions. We pay particular attention to contrasting the decoherent role of a cosmological constant in inducing exponential quantum damping in the evolution of low-energy observables, such as the probability of flavour oscillations, with the situation where the dark energy relaxes to zero for asymptotically large times, in which case such a damping is absent. Our findings may be of interest to (astrophysical) tests of quantum space-time foam models in the not-so-distant future.

Journal of High Energy Physics

We propose a non-unitary example of holography for the family of two-dimensional logarithmic conformal field theories with negative central charge c = cp,1 = −6p + 13 − 6p−1. We argue that at large p, these models have a semiclassical gravity-like description which contains, besides the global AdS3 spacetime, a tower of solitonic solutions describing conical excess angles. Evidence comes from the fact that the central charge and the natural modular invariant partition function of such a theory coincide with those of the cp,1 model. These theories have an extended chiral W-algebra whose currents have large spin of order |c|, and which in the bulk are realized as spinning conical solutions. As a by-product we also find a direct link between geometric actions for exceptional Virasoro coadjoint orbits, which describe fluctuations around the conical spaces, and Felder’s free field construction of degenerate representations.

Journal of High Energy Physics, 2004

Amongst conformal field theories, there exist logarithmic conformal field theories such as c p,1 models, various WZNW models, and a large variety of statistical models. It is well known that these theories generally contain a Jordan cell structure, which is a reducible but indecomposable representation. Our main aim in this thesis is to address the results and prospects of boundary logarithmic conformal field theories: theories with boundaries that contain the above Jordan cell structure. In this thesis, we briefly review conformal field theory and the appearance of logarithmic conformal field theories in the literature in the chronological order. Thereafter, we introduce the conventions and basic facts of logarithmic conformal field theory, and sketch an essential note on boundary conformal field theory. We have investigated c p,q boundary theory in search of logarithmic theories and have found logarithmic solutions of two-point functions in the context of the Coulomb gas picture. Other two-point functions have also been studied in the free boson construction of BCFT with SU (2) k symmetry. In addition, we have analyzed and obtained the boundary Ishibashi state for a rank-2 Jordan cell structure [22]. We have also examined the (generalised) Ishibashi state construction and the symplectic fermion construction at c = −2 for boundary states in the context of the c = −2 triplet model [23, 24]. It is also presented how the differences between two constructions should be interpreted, resolved and extended beyond each case. Some discussions on possible applications are given in the final chapter.

Journal of High Energy Physics

We consider the Ω-deformed N = 2 SU(2) gauge theory in four dimensions with N f = 4 massive fundamental hypermultiplets. The low energy effective action depends on the deformation parameters ε 1 , ε 2 , the scalar field expectation value a, and the hypermultiplet masses m = (m 1 , m 2 , m 3 , m 4). Motivated by recent findings in the N = 2 * theory, we explore the theories that are characterized by special fixed ratios ε 2 /ε 1 and m/ε 1 and propose a simple condition on the structure of the multi-instanton contributions to the prepotential determining the effective action. This condition determines a finite set Π N of special points such that the prepotential has N poles at fixed positions independent on the instanton number. In analogy with what happens in the N = 2 * gauge theory, the full prepotential of the Π N theories may be given in closed form as an explicit function of a and the modular parameter q appearing in special combinations of Eisenstein series and Jacobi theta functions with well defined modular properties. The resulting finite pole partition functions are related by AGT correspondence to special 4-point spherical conformal blocks of the Virasoro algebra. We examine in full details special cases where the closed expression of the block is known and confirms our Ansatz. We systematically study the special features of Zamolodchikov's recursion for the Π N conformal blocks. As a result, we provide a novel effective recursion relation that can be exactly solved and allows to prove the conjectured closed expressions analytically in the case of the Π 1 and Π 2 conformal blocks.

Journal of Physics A: Mathematical and Theoretical, 2013

In this article, we review some aspects of logarithmic conformal field theories which can be inferred from the characters of irreducible submodules of indecomposable modules. We will mainly consider the W(2, 2p − 1, 2p − 1, 2p − 1) series of triplet algebras and a bit logarithmic extensions of the minimal Virasoro models. Since in all known examples of logarithmic conformal field theories the vacuum representation of the maximally extended chiral symmetry algebra is an irreducible submodule of a larger, indecomposable module, its character provides a lot of non-trivial information about the theory such as a set of functions which spans the space of all torus amplitudes. Despite such characters being modular forms of inhomogeneous weight, they fit in the ADET-classification of fermionic sum representations. Thus, they show that logarithmic conformal field theories naturally have to be taken into account when attempting to classify rational conformal field theories.

Physical Review D, 2019

Ghost-spins, 2-level spin-like variables with indefinite norm have been studied in previous work. Here we explore various $N$-level generalizations of ghost-spins. First we discuss a flavoured generalization comprising $N$ copies of the ghost-spin system, as well as certain ghost-spin chains which in the continuum limit lead to 2-dim $bc$-ghost CFTs with $O(N)$ flavour symmetry. Then we explore a symplectic generalization that involves antisymmetric inner products, and finally a ghost-spin system exhibiting $N$ irreducible levels. We also study entanglement properties. In all these cases, we show the existence of positive norm "correlated ghost-spin" states in two copies of ghost-spin ensembles obtained by entangling identical ghost-spins from each copy: these exhibit positive entanglement entropy.

Journal of High Energy Physics, 2017

We continue the study of the gl(1|1) Wess-Zumino-Witten model. The Knizhnik-Zamolodchikov equations for the one, two, three and four point functions are analyzed, for vertex operators corresponding to typical and projective representations. We demonstrate their interplay with the logarithmic global conformal Ward identities. We compute the four point function for one projective and three typical representations. Three coupled first order Knizhnik-Zamolodchikov equations are integrated consecutively in terms of generalized hypergeometric functions, and we assemble the solutions into a local correlator. Moreover, we prove crossing symmetry of the four point function of four typical representations at generic momenta. Throughout, the map between the gl(1|1) Wess-Zumino-Witten model and symplectic fermions is exploited and extended.

Journal of Mathematical Physics, 2012

We study the Yangian of the sl(2|1) Lie superalgebra in a multi-parametric fourdimensional representation. We use Drinfeld's second realization to independently rederive the R-matrix, and to obtain the antiparticle representation, the crossing and the unitarity condition. We consistently apply the Yangian antipode and its inverse to the individual particles involved in the scattering. We explicitly find a scalar factor solving the crossing and unitarity conditions, and study the analytic structure of the resulting dressed R-matrix. The formulas we obtain bear some similarities with those familiar from the study of integrable structures in the AdS/CFT correspondence, although they present obvious crucial differences.

Journal of Mathematical Physics, 2010

We show how logarithmic terms may arise in the correlators of fields which belong to the representation of the Schrödinger-Virasoro algebra (SV) or the affine Galilean Conformal Algebra (GCA). We show that in GCA, only scaling operator can have a Jordan form and rapidity cannot. We observe that in both algebras logarithmic dependence appears along the time direction alone.

Journal of Mathematical Physics, 2011

We study 2-dimensional Logarithmic Galilean Conformal Algebra (LGCA) by making use of a contraction of Topologically Massive Gravity at critical point. We observe that using a naive contraction at the critical point fails to give a well defined theory, though contracting the theory while we are approaching the critical point leads to a well behaved expression for two point functions of the energy-momentum tensors of LGCA.

Journal of Physics A: Mathematical and General, 2006

The Raise and Peel model is a recently proposed one-dimensional statistical model describing a fluctuating interface. The evolution of the model follows from the competition between adsorption and desorption processes. The model is non-local due to the possible occurrence of avalanches. At a special ratio of the adsorption-desorption rates the model is integrable and many rigorous results are known. Off the critical point, the phase diagram and scaling properties are not known. In this paper, we search for indications of phase transition studying the gap in the spectrum of the nonhermitian generator of the stochastic interface evolution. We present results for the gap obtained from exact diagonalization and from Monte Carlo estimates derived from temporal correlations of various observables.

Physical Review D

We use the isomorphism between the BMS 3 and the W (2, 2) algebras to reconsider some generic aspects of CFTs with the BMS 3 algebra defined as a chiral symmetry. For unitarity theories, it is known that the extended symmetry generator acts trivially, and the resulting theory is equivalent to a CFT with a Virasoro symmetry only. For nonunitary CFTs, we define an operator depending on a nilpotent variable, and we organize the Verma module through the action of this new operator. Finally, we find the conditions imposed by the modified Ward identity.

Physical Review E, 2009

Appreciation of Stochastic Loewner evolution (SLEκ), as a powerful tool to check for conformal invariant properties of geometrical features of critical systems has been rising. In this paper we use this method to check conformal invariance in sandpile models. Avalanche frontiers in Abelian sandpile model (ASM) are numerically shown to be conformally invariant and can be described by SLE with diffusivity κ = 2. This value is the same as value obtained for loop erased random walks (LERW). The fractal dimension and Schramm's formula for left passage probability also suggest the same result. We also check the same properties for Zhang's sandpile model.

Physical Review D, 2012

We consider counterterms for odd dimensional holographic CFTs. These counterterms are derived by demanding cut-off independence of the CFT partition function on S d and S 1 × S d-1 . The same choice of counterterms leads to a cut-off independent Schwarzschild black hole entropy. When treated as independent actions, these counterterm actions resemble critical theories of gravity, i.e., higher curvature gravity theories where the additional massive spin-2 modes become massless. Equivalently, in the context of AdS/CFT, these are theories where at least one of the central charges associated with the trace anomaly vanishes. Connections between these theories and logarithmic CFTs are discussed. For a specific choice of parameters, the theories arising from counterterms are non-dynamical and resemble a DBI generalization of gravity. For even dimensional CFTs, analogous counterterms cancel log-independent cut-off dependence.