Conformal Boundary Conditions Research Articles

On fusing matrices associated with conformal boundary conditions

In the context of rational conformal field theories (RCFT) we look at the fusing matrices that arise when a topological defect is attached to a conformal boundary condition. We call such junctions open topological defects. One type of fusing matrices arises when two open defects fuse while another arises when an open defect passes through a boundary operator. We use the topological field theory approach to RCFTs based on Frobenius algebra objects in modular tensor categories to describe the general structure associated with such matrices and how to compute them from a given Frobenius algebra object and its representation theory. We illustrate the computational process on the rational free boson theories. Applications to boundary renormalisation group flows are briefly discussed.

Boundary Liouville conformal field theory in four dimensions

Higher dimensional Euclidean Liouville conformal field theories (LCFTs) consist of a log-correlated real scalar field with a background charge and an exponential potential. We analyse the LCFT on a four-dimensional manifold with a boundary. We extend to the boundary, the conformally covariant GJMS operator and the Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{Q} $$\\end{document}-curvature term in the LCFT action and classify the conformal boundary conditions. Working on a flat space with plate boundary, we calculate the dimensions of the boundary conformal primary operators, the two- and three-point functions of the displacement operator and the boundary conformal anomaly coefficients.

Cosmological observatories

Abstract We study the static patch of de Sitter space in the presence of a timelike boundary. We impose that the conformal class of the induced metric and the trace of the extrinsic curvature, K, are fixed at the boundary. We present the thermodynamic structure of de Sitter space subject to these boundary conditions, for static and spherically symmetric configurations to leading order in the semiclassical approximation. In three spacetime dimensions, and taking K constant on a toroidal Euclidean boundary, we find that the spacetime is thermally stable for all K. In four spacetime dimensions, the thermal stability depends on the value of K. It is established that for sufficiently large K, the de Sitter static patch subject to conformal boundary conditions is thermally stable. This contrasts the Dirichlet problem for which the region encompassing the cosmological horizon has negative specific heat. We present an analysis of the linearised Einstein equations subject to conformal boundary conditions. In the worldline limit of the timelike boundary, the underlying modes are linked to the quasinormal modes of the static patch. In the limit where the timelike boundary approaches the cosmological event horizon, the linearised modes are interpreted in terms of the shear and sound modes of a fluid dynamical system. Additionally, we find modes with a frequency of positive imaginary part. Measured in a local inertial reference frame, and taking the stretched cosmological horizon limit, these modes grow at most polynomially.

Universal Entanglement Spectrum in One-Dimensional Gapless Symmetry Protected Topological States.

Quantum entanglement marks a definitive feature of topological states. However, the entanglement spectrum remains insufficiently explored for topological states without a bulk energy gap. Using a combination of field theory and numerical techniques, we accurately calculate and analyze the entanglement spectrum of gapless symmetry protected topological states in one dimension. We highlight that the universal entanglement spectrum not only encodes the nontrivial edge degeneracy, generalizing the Li-Haldane conjecture to gapless topological states, but also contains the operator content of the underlying boundary conformal field theory. This implies that the bulk wave function can act as a fingerprint of both quantum criticality and topology in gapless symmetry protected topological states. We also identify a symmetry enriched conformal boundary condition that goes beyond the conventional conformal boundary condition.

Gauge symmetries and conserved currents in AdS/BCFT

In this paper, we study massless/massive vector and p-form field perturbations in AdS spacetime with an end-of-the-world brane. By imposing Υ(1) preserving Neumann boundary condition on the end-of-the-world brane, we study their spectrum and discuss their implications for dual BCFT operators. When the perturbation is massless, the dual BCFT operator is a conserved current and we show that such an operator indeed satisfies the Υ(1) preserving conformal boundary condition. On the other hand, when the perturbation is massive, in general there exists non-vanishing perpendicular components of the dual BCFT operator, even in the massless limit. We explain this difference between massless and massive perturbations from the point of view of the bulk gauge symmetry, or equivalently from different structure of equations of motion. We also find several brane-tension-independent modes in massless perturbations, and these are understood as boundary-condition-independent modes from the dual BCFT point of view.

Moving interfaces and two-dimensional black holes

Conformal field theories can exchange energy through a boundary interface. Imposing conformal boundary conditions for static interfaces implies energy conservation at the interface. Recently, the reflective and transmittive properties of such static conformal interfaces have been studied in two dimensions by scattering matter at the interface impurity. In this note, we generalize this to the case of dynamic interfaces. Motivated by the connections between the moving mirror and the black hole, we choose a particular profile for the dynamical interface. We show that a part of the total energy of each side will be lost in the interface. In other words, a time-dependent interface can accumulate or absorb energy. While, in general, the interface follows a time-like trajectory, one can take a particular limit of a profile parameter(β), such that the interface approaches a null line asymptotically(β → 0). In this limit, we show that for a class of boundary conditions, the interface behaves like a semipermeable membrane - it behaves like a (partially) reflecting mirror from one side and is (partially) transparent from the other side. We also consider another set of conformal boundary conditions for which, in the null line limit, the interface mimics the properties expected of a horizon. In this case, we devise a scattering experiment, where (zero-point subtracted) energy from one CFT is fully transmitted to the other CFT, while from the other CFT, energy can neither be transmitted nor reflected, i.e., it gets lost in the interface. This boundary condition is also responsible for the thermal energy spectrum which mimics Hawking radiation. This is analogous to the black hole where the horizon plays the role of a one-sided ‘membrane’, which accumulates all the interior degrees of freedom and radiates thermally in the presence of quantum fluctuation. Stimulated by this observation, we comment on some plausible construction of wormhole analogues.

Categorical-symmetry resolved entanglement in conformal field theory

We propose a symmetry resolution of entanglement for categorical noninvertible symmetries (CaT-SREE) in (1+1)-dimensional conformal field theories. The definition parallels that of grouplike invertible symmetries, employing the concept of symmetric boundary states with respect to a categorical symmetry. Our examination extends to rational conformal field theories, where the behavior of CaT-SREE mirrors that of grouplike invertible symmetries. We find that CaT-SREE can be defined if there is no obstruction to gauging the categorical symmetry, as happens in the case of grouplike symmetries. We also provide instances of the breakdown of entanglement equipartition at the next-to-leading order in the cutoff expansion. Our findings shed light on how the interplay between conformal boundary conditions and categorical symmetries lead to specific patterns in the entanglement entropy. Published by the American Physical Society 2024

Conformal boundary conditions for a 4d scalar field

We construct unitary, stable, and interacting conformal boundary conditions for a free massless scalar in four dimensions by coupling it to edge modes living on a boundary. The boundary theories we consider are bosonic and fermionic QED_33 with N_fNf flavors and a Chern-Simons term at level kk, in the large-N_fNf limit with fixed k/N_fk/Nf. We find that interacting boundary conditions only exist when k≠ 0k≠0. To obtain this result we compute the \betaβ functions of the classically marginal couplings at the first non-vanishing order in the large-N_fNf expansion, and to all orders in k/N_fk/Nf and in the couplings. To check vacuum stability we also compute the large-N_fNf effective potential. We compare our results with the the known conformal bootstrap bounds.

The Virasoro minimal string

We introduce a critical string theory in two dimensions and demonstrate that this theory, viewed as two-dimensional quantum gravity on the worldsheet, is equivalent to a double-scaled matrix integral. The worldsheet theory consists of Liouville CFT with central charge c≥≥ 25 coupled to timelike Liouville CFT with central charge 26-c. The double-scaled matrix integral has as its leading density of states the universal Cardy density of primaries in a two-dimensional CFT, thus motivating the name Virasoro minimal string. The duality holds for any value of the continuous parameter cc and reduces to the JT gravity/matrix integral duality in the large central charge limit. It thus provides a precise stringy realization of JT gravity. The main observables of the Virasoro minimal string are quantum analogues of the Weil-Petersson volumes, which are computed as absolutely convergent integrals of worldsheet CFT correlators over the moduli space of Riemann surfaces. By exploiting a relation of the Virasoro minimal string to three-dimensional gravity and intersection theory on the moduli space of Riemann surfaces, we are able to give a direct derivation of the duality. We provide many checks, such as explicit numerical - and in special cases, analytic - integration of string diagrams, the identification of the CFT boundary conditions with asymptotic boundaries of the two-dimensional spacetime, and the matching between the leading non-perturbative corrections of the worldsheet theory and the matrix integral. As a byproduct, we discover natural conformal boundary conditions for timelike Liouville CFT.

Emergent Conformal Boundaries from Finite-Entanglement Scaling in Matrix Product States.

The use of finite entanglement scaling with matrix product states (MPS) has become a crucial tool for studying one-dimensional critical lattice theories, especially those with emergent conformal symmetry. We argue that finite entanglement introduces a relevant deformation in the critical theory. As a result, the bipartite entanglement Hamiltonian defined from the MPS can be understood as a boundary conformal field theory with a physical and an entanglement boundary. We are able to exploit the symmetry properties of the MPS to engineer the physical conformal boundary condition. The entanglement boundary, on the other hand, is related to the concrete lattice model and remains invariant under this relevant perturbation. Using critical lattice models described by the Ising, Potts, and free compact boson conformal field theories, we illustrate the influence of the symmetry and the relevant deformation on the conformal boundaries in the entanglement spectrum.

General gravitational charges on null hypersurfaces

We perform a detailed study of the covariance properties of the symplectic potential of general relativity on a null hypersurface, and of the different polarizations that can be used to study conservative as well as leaky boundary conditions. This allows us to identify a one-parameter family of covariant symplectic potentials. We compute the charges and fluxes for the most general phase space with arbitrary variations. We study five symmetry groups that arise when different restrictions on the variations are included. Requiring stationarity as in the original Wald-Zoupas prescription selects a unique member of the family of symplectic potentials, the one of Chandrasekaran, Flanagan and Prabhu. The associated charges are all conserved on non-expanding horizons, but not on flat spacetime. We show that it is possible to require a weaker notion of stationarity which selects another symplectic potential, again in a unique way, and whose charges are conserved on both non-expanding horizons and flat light-cones. Furthermore, the flux of future-pointing diffeomorphisms at leading-order around an outgoing flat light-cone is positive and reproduces a tidal heating plus a memory term. We also study the conformal conservative boundary conditions suggested by the alternative polarization and identify under which conditions they define a non-ambiguous variational principle. Our results have applications for dynamical notions of entropy, and are useful to clarify the interplay between different boundary conditions, charge prescriptions, and symmetry groups that can be associated with a null boundary.

Gravitational observatories

We consider four-dimensional general relativity with vanishing cosmological constant defined on a manifold with a boundary. In Lorentzian signature, the timelike boundary is of the form σ × ℝ, with σ a spatial two-manifold that we take to be either flat or S2. In Euclidean signature we take the boundary to be S2 × S1. We consider conformal boundary conditions, whereby the conformal class of the induced metric and trace K of the extrinsic curvature are fixed at the timelike boundary. The problem of linearised gravity is analysed using the Kodama-Ishibashi formalism. It is shown that for a round metric on S2 with constant K, there are modes that grow exponentially in time. We discuss a method to control the growing modes by varying K. The growing modes are absent for a conformally flat induced metric on the timelike boundary. We provide evidence that the Dirichlet problem for a spherical boundary does not suffer from non-uniqueness issues at the linearised level. We consider the extension of black hole thermodynamics to the case of conformal boundary conditions, and show that the form of the Bekenstein-Hawking entropy is retained.

RG boundaries and Cardy’s variational ansatz for multiple perturbations

We consider perturbations of 2D CFTs by multiple relevant operators. The massive phases of such perturbations can be labeled by conformal boundary conditions. Cardy’s variational ansatz approximates the vacuum state of the perturbed theory by a smeared conformal boundary state. In this paper we study the limitations and propose generalisations of this ansatz using both analytic and numerical insights based on TCSA. In particular we analyse the stability of Cardy’s ansatz states with respect to boundary relevant perturbations using bulk-boundary OPE coefficients. We show that certain transitions between the massive phases arise from a pair of boundary RG flows. The RG flows start from the conformal boundary on the transition surface and end on those that lie on the two sides of it. As an example we work out the details of the phase diagram for the Ising field theory and for the tricritical Ising model perturbed by the leading thermal and magnetic fields. For the latter we find a pair of novel transition lines that correspond to pairs of RG flows. Although the mass gap remains finite at the transition lines, several one-point functions change their behaviour. We discuss how these lines fit into the standard phase diagram of the tricritical Ising model. We show that each line extends to a two-dimensional surface ξσ,c in a three coupling space when we add perturbations by the subleading magnetic field. Close to this surface we locate symmetry breaking critical lines leading to the critical Ising model. Near the critical lines we find first order phase transition lines describing two-phase coexistence regions as predicted in Landau theory. The surface ξσ,c is determined from the CFT data using Cardy’s ansatz and its properties are checked using TCSA numerics.

On the boundary conformal field theory approach to symmetry-resolved entanglement

We study the symmetry resolution of the entanglement entropy of an interval in two-dimensional conformal field theories (CFTs), by relating the bipartition to the geometry of an annulus with conformal boundary conditions. In the presence of extended symmetries such as Kac-Moody type current algebrae, symmetry resolution is possible only if the boundary conditions on the annulus preserve part of the symmetry group, i.e. if the factorization map associated with the spatial bipartition is compatible with the symmetry in question. The partition function of the boundary CFT (BCFT) is then decomposed in terms of the characters of the irreducible representations of the symmetry group preserved by the boundary conditions. We demonstrate that this decomposition already provides the symmetry resolution of the entanglement spectrum of the corresponding bipartition. Considering the various terms of the partition function associated with the same representation, or charge sector, the symmetry-resolved Rényi entropies can be derived to all orders in the UV cutoff expansion without the need to compute the charged moments. We apply this idea to the theory of a free massless boson with U(1)U(1), \mathbb{R}ℝ and \mathbb{Z}_2ℤ2 symmetry.

Boundaries in free higher derivative conformal field theories

We consider free higher derivative theories of scalars and Dirac fermions in the presence of a boundary in general dimension. We establish a method for finding consistent conformal boundary conditions in these theories by removing certain boundary primaries from the spectrum. A rich set of renormalization group flows between various conformal boundary conditions is revealed, triggered by deformations quadratic in the boundary primaries. We compute the free energy of these theories on a hemisphere, and show that the boundary a-theorem is generally violated along boundary flows as a consequence of bulk non-unitarity. We further characterize the boundary theory by computing the two-point function of the displacement operator.

Bootstrapping boundaries and branes

The study of conformal boundary conditions for two-dimensional conformal field theories (CFTs) has a long history, ranging from the description of impurities in one-dimensional quantum chains to the formulation of D-branes in string theory. Nevertheless, the landscape of conformal boundaries is largely unknown, including in rational CFTs, where the local operator data is completely determined. We initiate a systematic bootstrap study of conformal boundaries in 2d CFTs by investigating the bootstrap equation that arises from the open-closed consistency condition of the annulus partition function with identical boundaries. We find that this deceivingly simple bootstrap equation, when combined with unitarity, leads to surprisingly strong constraints on admissible boundary states. In particular, we derive universal bounds on the tension (boundary entropy) of stable boundary conditions, which provide a rigorous diagnostic for potential D-brane decays. We also find unique solutions to the bootstrap problem of stable branes in a number of rational CFTs. Along the way, we observe a curious connection between the annulus bootstrap and the sphere packing problem, which is a natural extension of previous work on the modular bootstrap. We also derive bounds on the boundary entropy at large central charge. These potentially have implications for end-of-the-world branes in pure gravity on AdS3.

Conformal Boundary Conditions of Symmetry-Enriched Quantum Critical Spin Chains.

Some quantum critical states cannot be smoothly deformed into each other without either crossing some multicritical points or explicitly breaking certain symmetries even if they belong to the same universality class. This brings up the notion of "symmetry-enriched" quantum criticality. While recent works in the literature focused on critical states with robust degenerate edge modes, we propose that the conformal boundary condition (B.C.) is a more generic characteristic of such quantum critical states. We show that in two families of quantum spin chains, which generalize the Ising and the three-state Potts models, the quantum critical point between a symmetry-protected topological phase and a symmetry-breaking order realizes a conformal B.C. distinct from the simple Ising and Potts chains. Furthermore, we argue that the conformal B.C. can be derived from the bulk effective field theory, which realizes a novel bulk-boundary correspondence in symmetry-enriched quantum critical states.

Fermions in AdS and Gross-Neveu BCFT

We study the boundary critical behavior of conformal field theories of interacting fermions in the Gross-Neveu universality class. By a Weyl transformation, the problem can be studied by placing the CFT in an anti de Sitter space background. After reviewing some aspects of free fermion theories in AdS, we use both large N methods and the epsilon expansion near 2 and 4 dimensions to study the conformal boundary conditions in the Gross-Neveu CFT. At large N and general dimension d, we find three distinct boundary conformal phases. Near four dimensions, where the CFT is described by the Wilson-Fisher fixed point of the Gross-Neveu-Yukawa model, two of these phases correspond respectively to the choice of Neumann or Dirichlet boundary condition on the scalar field, while the third one corresponds to the case where the bulk scalar field acquires a classical expectation value. One may flow between these boundary critical points by suitable relevant boundary deformations. We compute the AdS free energy on each of them, and verify that its value is consistent with the boundary version of the F-theorem. We also compute some of the BCFT observables in these theories, including bulk two-point functions of scalar and fermions, and four-point functions of boundary fermions.

Lattice models from CFT on surfaces with holes: I. Torus partition function via two lattice cells

We construct a one-parameter family of lattice models starting from a two-dimensional rational conformal field theory on a torus with a regular lattice of holes, each of which is equipped with a conformal boundary condition. The lattice model is obtained by cutting the surface into triangles with clipped-off edges using open channel factorisation. The parameter is given by the hole radius. At finite radius, high energy states are suppressed and the model is effectively finite. In the zero-radius limit, it recovers the CFT amplitude exactly. In the touching hole limit, one obtains a topological field theory. If one chooses a special conformal boundary condition which we call ‘cloaking boundary condition’, then for each value of the radius the fusion category of topological line defects of the CFT is contained in the lattice model. The fact that the full topological symmetry of the initial CFT is realised exactly is a key feature of our lattice models. We provide an explicit recursive procedure to evaluate the interaction vertex on arbitrary states. As an example, we study the lattice model obtained from the Ising CFT on a torus with one hole, decomposed into two lattice cells. We numerically compare the truncated lattice model to the CFT expression obtained from expanding the boundary state in terms of the hole radius and we find good agreement at intermediate values of the radius.

D-Branes in Para-Hermitian Geometries

We introduce T-duality invariant versions of D-branes in doubled geometry using a global covariant framework based on para-Hermitian geometry and metric algebroids. We define D-branes as conformal boundary conditions for the open string version of the Born sigma-model, where they are given by maximally isotropic vector bundles which do not generally admit the standard geometric picture in terms of submanifolds. When reduced to the conventional sigma-model description of a physical string background as the leaf space of a foliated para-Hermitian manifold, integrable branes yield D-branes as leaves of foliations which are interpreted as Dirac structures on the physical spacetime. We define a notion of generalised para-complex D-brane, which realises our D-branes as para-complex versions of topological A/B-branes. We illustrate how our formalism recovers standard D-branes in the explicit example of reductions from doubled nilmanifolds.