Non-hermitian random matrix models

We introduce an extension of the diagrammatic rules in random matrix theory and apply it to non-hermitian random matrix models using the 1/N approximation. A number of one-and two-point functions are evaluated on their holomorphic and non-holomorphic supports to leading order in 1/N. The one-point functions describe the distribution of eigenvalues, while the two-point functions characterize their macroscopic cotrelations. The generic form for the two-point functions is obtained, generalizing the concept of macroscopic universality to non-hermitian random matrices. We show that the holomorphic and non-holomorphic one- and two-point functions condition the behavior of pertinent partition functions to order O(1/N). We derive explicit conditions for the location and distribution of their singularities. Most of our analytical results are found to be in good agreement with numerical calculations using large ensembles of complex matrices.     

TFT construction of RCFT correlators I: partition functions

We formulate rational conformal field theory in terms of a symmetric special Frobenius algebra A and its representations. A is an algebra in the modular tensor category of Moore–Seiberg data of the underlying chiral CFT. The multiplication on A corresponds to the OPE of boundary fields for a single boundary condition. General boundary conditions are A-modules, and (generalised) defect lines are A–A-bimodules.The relation with three-dimensional TFT is used to express CFT data, like structure constants or torus and annulus coefficients, as invariants of links in three-manifolds. We compute explicitly the ordinary and twisted partition functions on the torus and the annulus partition functions. We prove that they satisfy consistency conditions, like modular invariance and NIM-rep properties.We suggest that our results can be interpreted in terms of non-commutative geometry over the modular tensor category of Moore–Seiberg data.     

One-point functions in perturbed boundary conformal field theories

We consider the one-point functions of bulk and boundary fields in the scaling Lee–Yang model for various combinations of bulk and boundary perturbations. The one-point functions of the bulk fields are analysed using the truncated conformal space approach and the form-factor expansion. Good agreement is found between the results of the two methods, though we find that the expression for the general boundary state given by Ghoshal and Zamolodchikov has to be corrected slightly. For the boundary fields we use thermodynamic Bethe ansatz equations to find exact expressions for the strip and semi-infinite cylinder geometries. We also find a novel off-critical identity between the cylinder partition functions of models with differing boundary conditions, and use this to investigate the regions of boundary-induced instability exhibited by the model on a finite strip.     

Boundary Correlation Functions of the gl(1|1) Supersymmetric Vertex Model

The gl(1|1) supersymmetric vertex model with domain wall boundary conditions (DWBC) on an N×N square lattice is considered. We derive the reduction formulae for the one-point boundary correlation functions of the model. The determinant representation for the boundary correlation functions is also obtained.     

Conformal Loop Ensembles and the Stress–Energy Tensor

We give a construction of the stress–energy tensor of conformal field theory (CFT) as a local “object” in conformal loop ensembles CLE _κ , for all values of κ in the dilute regime 8/3 < κ ≤ 4 (corresponding to the central charges 0 < c ≤ 1 and including all CFT minimal models). We provide a quick introduction to CLE, a mathematical theory for random loops in simply connected domains with properties of conformal invariance, developed by Sheffield and Werner (Ann Math 176, 1827–1917, 2012). We consider its extension to more general regions of definition and make various hypotheses that are needed for our construction and expected to hold for CLE in the dilute regime. Using this, we identify the stress–energy tensor in the context of CLE. This is done by deriving its associated conformal Ward identities for single insertions in CLE probability functions, along with the appropriate boundary conditions on simply connected domains; its properties under conformal maps, involving the Schwarzian derivative; and its one-point average in terms of the “relative partition function”. Part of the construction is in the same spirit as, but widely generalizes, that found in the context of SLE_8/3 by the author, Riva and Cardy (Commun Math Phys 268, 687–716, 2006), which only dealt with the case of zero central charge in simply connected hyperbolic regions. We do not use the explicit construction of the CLE probability measure, but only its defining and expected general properties.     

Complex random matrix models with possible applications to spin-impurity scattering in quantum Hall fluids

We study the one-point and two-point Green functions in a complex random matrix model to sub-leading orders in the large-N limit. We take this complex matrix model as a model for the two-state scattering problem, as applied to spin-dependent scattering of impurities in quantum Hall fluids. The density of state shows a singularity at the band center due to reflection symmetry. We also compute the one-point Green function for a generalized situation by putting random matrices on a lattice of arbitrary dimensions.     

Quantum brownian motion on a triangular lattice and c=2 boundary conformal field theory

We study a single particle diffusing on a triangular lattice and interacting with a heat bath, using boundary conformal field theory (CFT) and exact integrability techniques. We derive a correspondence between the phase diagram of this problem and that recently obtained for the 2-dimensional 3-state Potts model with a boundary. Exact results are obtained on phases with intermediate mobilities. These correspond to nontrivial boundary states in a conformal field theory with 2 free bosons which we explicitly construct for the first time. These conformally invariant boundary conditions are not simply products of Dirichlet and Neumann ones and unlike those trivial boundary conditions, are not invariant under a Heisenberg algebra.     

On a2(1) reflection matrices and affine Toda theories

We construct new non-diagonal solutions to the boundary Yang-Baxter equation corresponding to a two-dimensional field theory with U_q(a₂⁽¹⁾) quantum affine symmetry on a half-line. The requirements of boundary unitarity and boundary crossing symmetry are then used to find overall scalar factors which lead to consistent reflection matrices. Using the boundary bootstrap equations we also compute the reflection factors for scalar bound states (breathers). These breathers are expected to be identified with the fundamental quantum particles in a₂⁽¹⁾ affine Toda field theory and we therefore obtain a conjecture for the affine Toda reflection factors. We compare these factors with known classical results and discuss their duality properties and their connections with particular boundary conditions.     

Hydrodynamic Limit of a B.G.K. Like Model on Domains with Boundaries and Analysis of Kinetic Boundary Conditions for Scalar Multidimensional Conservation Laws

In this paper we study the hydrodynamic limit of a B.G.K. like kinetic model on domains with boundaries via BV _loc theory. We obtain as a consequence existence results for scalar multidimensional conservation laws with kinetic boundary conditions. We require that the initial and boundary data satisfy the optimal assumptions that they all belong to L ¹∩L ^∞ with the additional regularity assumptions that the initial data are in BV _loc . We also extend our hydrodynamic limit analysis to the case of a generalized kinetic model to account for forces effects and we obtain as a consequence the existence theory for conservation laws with source terms and kinetic boundary conditions.     

Revisiting Cardassian model and cosmic constraint

Boundary conformal field theory (BCFT) is the study of conformal field theory (CFT) in semi-infinite space-time. In the non-relativistic limit (x???x,t??t,???0), the boundary conformal algebra changes to boundary Galilean conformal algebra (BGCA). In this work, some aspects of AdS/BCFT in the non-relativistic limit were explored. We constrain correlation functions of Galilean conformal invariant fields with BGCA generators. For a situation with a boundary condition at surface x=0 ( $z=\overline{z}$ ), our result agrees with the non-relativistic limit of the BCFT two-point function. We also introduce the holographic dual of boundary Galilean conformal field theory.     

Finite size effects in the spin-1 XXZ and supersymmetric sine-Gordon models with Dirichlet boundary conditions

Starting from the Bethe Ansatz solution of the open integrable spin-1 XXZ quantum spin chain with diagonal boundary terms, we derive a set of nonlinear integral equations (NLIEs), which we propose to describe the boundary supersymmetric sine-Gordon model BSSG⁺ with Dirichlet boundary conditions on a finite interval. We compute the corresponding boundary S matrix, and find that it coincides with the one proposed by Bajnok, Palla and Takács for the Dirichlet BSSG⁺ model. We derive a relation between the (UV) parameters in the boundary conditions and the (IR) parameters in the boundary S matrix. By computing the boundary vacuum energy, we determine a previously unknown parameter in the scattering theory. We solve the NLIEs numerically for intermediate values of the interval length, and find agreement with our analytical result for the effective central charge in the UV limit and with boundary conformal perturbation theory.     

Boundary Conditions for Topological Quantum Field Theories,Anomalies and Projective Modular Functors

We study boundary conditions for extended topological quantum field theories (TQFTs) and their relation to topological anomalies. We introduce the notion of TQFTs with moduli level m, and describe extended anomalous theories as natural transformations of invertible field theories of this type. We show how in such a framework anomalous theories give rise naturally to homotopy fixed points for n-characters on ∞-groups. By using dimensional reduction on manifolds with boundaries, we show how boundary conditions for n + 1-dimensional TQFTs produce n-dimensional anomalous field theories. Finally, we analyse the case of fully extended TQFTs, and show that any fully extended anomalous theory produces a suitable boundary condition for the anomaly field theory.     

Vacuum Stability of the Wrong Sign (−ϕ 6) Scalar Field Theory

We apply the effective potential method to study the vacuum stability of the bounded from above (?? ⁶) (unstable) quantum field potential. The stability (?E/?b = 0) and the mass renormalization (? ² E/?b ² = M ²) conditions force the effective potential of this theory to be bounded from below (stable). Since bounded from below potentials are always associated with localized wave functions, the algorithm we use replaces the boundary condition applied to the wave functions in the complex contour method by two stability conditions on the effective potential obtained. To test the validity of our calculations, we show that our variational predictions can reproduce exactly the results in the literature for the \(\mathcal {PT}\) -symmetric ? ⁴ theory. We then extend the applications of the algorithm to the unstudied stability problem of the bounded from above (?? ⁶) scalar field theory where classical analysis prohibits the existence of a stable spectrum. Concerning this, we calculated the effective potential up to first order in the couplings in d space-time dimensions. We find that a Hermitian effective theory is instable while a non-Hermitian but \(\mathcal {PT}\) -symmetric effective theory characterized by a pure imaginary vacuum condensate is stable (bounded from below) which is against the classical predictions of the instability of the theory. We assert that the work presented here represents the first calculations that advocates the stability of the (?? ⁶) scalar potential.     

Boundary energy and boundary states in integrable quantum field theories

We study the ground-state energy of integrable 1 + 1 quantum field theories with boundaries (the genuine Casimir effect). In the scalar case, this is done by introducing a new “R-channel TBA”, where the boundary is represented by a boundary state, and the thermodynamics involves evaluating scalar products of boundary states with all the states of the theory. In the non-scalar, sine-Gordon case, this is done by generalizing the method of Destri and De Vega. The two approaches are compared. Miscellaneous other results are obtained, in particular formulas for the overall normalization and scalar products of boundary states, exact partition functions for the critical Ising model in a boundary magnetic field, and also results for the energy, excited states and boundary S-matrix of O(n) and minimal models.     

Order statistics for first passage times in diffusion processes

We consider the problem of the first passage times for absorption (trapping) of the firstj (j = 1,2, ....) ofk, j <k, identical and independent diffusing particles for the asymptotic case k?>1. Our results are a special case of the theory of order statistics. We show that in one dimension the mean time to absorption at a boundary for the first ofk diffusing particles, μ_1,k , goes as (lnk)^?1 for the set of initial conditions in which none of thek particles is located at a boundary and goes ask ^?2 for the set of initial conditions in which some of thek particles may be located at the boundary. We demonstrate that in one dimension our asymptotic results (k21) are independent of the potential field in which the diffusion takes place for a wide class of potentials. We conjecture that our results are independent of dimension and produce some evidence supporting this conjecture. We conclude with a discussion of the possible import of these results on diffusion-controlled rate processes.     

On the vacuum structure of an SU(2) Yang-Mills theory

By using a compactification of the spatial part R³ of Minkowski-space different from the one-point compactification to S³, we get a new classification of the vacua for an SU(2) gauge theory. It contains, besides the vacua arising in the S³ compactification, the Gribov vacua as new classes. We discuss the role of pseudoparticle solutions within this framework and comment on the problem of the Coulomb gauge degeneracy.     

On SLE Martingales in Boundary WZW Models

Following Bettelheim et al. (Phys Rev Lett 95:251601, 2005), we consider the boundary WZW model on a half-plane with a cut growing according to the Schramm–Loewner stochastic evolution and the boundary fields inserted at the tip of the cut and at infinity. We study necessary and sufficient conditions for boundary correlation functions to be SLE martingales. Necessary conditions come from the requirement for the boundary field at the tip of the cut to have a depth two null vector. Sufficient conditions are established using Knizhnik–Zamolodchikov equations for boundary correlators. Combining these two approaches, we show that in the case of G = SU(2) the boundary correlator is an SLE martingale if and only if the boundary field carries spin 1/2. In the case of G = SU(n) and the level k = 1, there are several situations when boundary one-point correlators are SLE _κ -martingales. If the boundary field is labelled by the defining n-dimensional representation of SU(n), we obtain \varkappa = 2{\varkappa=2} . For n even, by choosing the boundary field labelled by the (unique) self-adjoint fundamental representation, we get \varkappa = 8/(n + 2){\varkappa=8/(n {+} 2)} . We also study the situation when the distance between the two boundary fields is finite, and we show that in this case the SLE_\varkappa{{\rm SLE}_\varkappa} evolution is replaced by SLE_\varkappa,r{{\rm SLE}_{\varkappa,\rho}} with r = \varkappa -6{\rho=\varkappa -6} .     

Operator approach to boundary Liouville theory

We propose new methods for calculation of the discrete spectrum, the reflection amplitude and the correlation functions of boundary Liouville theory on a strip with Lorentzian signature. They are based on the structure of the vertex operator V=e^-φ in terms of the asymptotic operators. The methods first are tested for the particle dynamics in the Morse potential, where similar structures appear. Application of our methods to boundary Liouville theory reproduces the known results obtained earlier in the bootstrap approach, but there can arise a certain extension when the boundary parameters are near to critical values. Namely, in this case we have found up to four different equidistant series of discrete spectra, and the reflection amplitude is modified, respectively.     

Spectrum of boundary states in N=1 SUSY sine-Gordon theory

We consider N=1 supersymmetric sine-Gordon theory (SSG) with supersymmetric integrable boundary conditions (boundary SSG=BSSG). We find two possible ways to close the boundary bootstrap for this model, corresponding to two different choices for the boundary supercharge. We argue that these two bootstrap solutions should correspond to the two integrable Lagrangian boundary theories considered recently by Nepomechie.