Symmetry-resolved entanglement for excited states and two entangling intervals in AdS3/CFT2

We test the proposal of [1] for the holographic computation of the charged moments and the resulting symmetry-resolved entanglement entropy in different excited states, as well as for two entangling intervals. Our holographic computations are performed in U(1) Chern-Simons-Einstein-Hilbert gravity, and are confirmed by independent results in a conformal field theory at large central charge. In particular, we consider two classes of excited states, corresponding to charged and uncharged conical defects in AdS3. In the conformal field theory, these states are generated by the insertion of charged and uncharged heavy operators. We employ the monodromy method to calculate the ensuing four-point function between the heavy operators and the twist fields. For the two-interval case, we derive our results on the AdS and the conformal field theory side, respectively, from the generating function method of [1], as well as the vertex operator algebra. In all cases considered, we find equipartition of entanglement between the different charge sectors. We also clarify an aspect of conformal field theories with a large central charge and $$ \hat{\mathfrak{u}}{(1)}_k $$ u ̂ 1 k Kac-Moody symmetry used in our calculations, namely the factorization of the Hilbert space into a gravitational Virasoro sector with large central charge, and a $$ \hat{\mathfrak{u}}{(1)}_k $$ u ̂ 1 k Kac-Moody sector.

One-loop masses of Neumann-Dirichlet open strings and boundary-changing vertex operators

We derive the masses acquired at one loop by massless scalars in the Neumann-Dirichlet sector of open strings, when supersymmetry is spontaneously broken. It is done by computing two-point functions of “boundary-changing vertex operators” inserted on the boundaries of the annulus and Möbius strip. This requires the evaluation of correlators of “excited boundary-changing fields,” which are analogous to excited twist fields for closed strings. We work in the type IIB orientifold theory compactified on T2× T4/ℤ2, where $$ \mathcal{N} $$ N = 2 supersymmetry is broken to $$ \mathcal{N} $$ N = 0 by the Scherk-Schwarz mechanism implemented along T2. Even though the full expression of the squared masses is complicated, it reduces to a very simple form when the lowest scale of the background is the supersymmetry breaking scale M3/2. We use our results to analyze in this regime the stability at the quantum level of the moduli fields arising in the Neumann-Dirichlet sector. This completes the study of ref. [32], where the quantum masses of all other types of moduli arising in the open- or closed-string sectors are derived. Ultimately, we identify all brane configurations that produce backgrounds without tachyons at one loop and yield an effective potential exponentially suppressed, or strictly positive with runaway behavior of M3/2.

Two-point functions of Neumann–Dirichlet open-string sector moduli

In this paper, we compute at one loop the two-point functions of massless scalars in the Neumann–Dirichlet open-string sector of the type IIB orientifold compactified on [Formula: see text], when [Formula: see text] supersymmetry is spontaneously broken. This is done by evaluating correlation functions of “boundary-changing vertex operators” which are analogous to correlators of twist fields for closed strings. We use our results to compute the mass developed at one loop by the moduli fields arising in the Neumann–Dirichlet sector.

Analytic conformal bootstrap and Virasoro primary fields in the Ashkin-Teller model

We revisit the critical two-dimensional Ashkin–Teller model, i.e. the \mathbb{Z}_2ℤ2 orbifold of the compactified free boson CFT at c=1c=1. We solve the model on the plane by computing its three-point structure constants and proving crossing symmetry of four-point correlation functions. We do this not only for affine primary fields, but also for Virasoro primary fields, i.e. higher twist fields and degenerate fields. This leads us to clarify the analytic properties of Virasoro conformal blocks and fusion kernels at c=1c=1. We show that blocks with a degenerate channel field should be computed by taking limits in the central charge, rather than in the conformal dimension. In particular, Al. Zamolodchikov’s simple explicit expression for the blocks that appear in four-twist correlation functions is only valid in the non-degenerate case: degenerate blocks, starting with the identity block, are more complicated generalized theta functions.

Lorentzian dynamics and factorization beyond rationality

We investigate the emergence of topological defect lines in the conformal Regge limit of two-dimensional conformal field theory. We explain how a local operator can be factorized into a holomorphic and an anti-holomorphic defect operator connected through a topological defect line, and discuss implications on analyticity and Lorentzian dynamics including aspects of chaos. We derive a formula relating the infinite boost limit, which holographically encodes the “opacity” of bulk scattering, to the action of topological defect lines on local operators. Leveraging the unitary bound on the opacity and the positivity of fusion coefficients, we show that the spectral radii of a large class of topological defect lines are given by their loop expectation values. Factorization also gives a formula relating the local and defect operator algebras and fusion categorical data. We then review factorization in rational conformal field theory from a defect perspective, and examine irrational theories. On the orbifold branch of the c = 1 free boson theory, we find a unified description for the topological defect lines through which the twist fields are factorized; at irrational points, the twist fields factorize through “non-compact” topological defect lines which exhibit continuous defect operator spectra. Along the way, we initiate the development of a formalism to characterize non-compact topological defect lines.

Symmetry resolved relative entropies and distances in conformal field theory

We develop a systematic approach to compute the subsystem trace distances and relative entropies for subsystem reduced density matrices associated to excited states in different symmetry sectors of a 1+1 dimensional conformal field theory having an internal U(1) symmetry. We provide analytic expressions for the charged moments corresponding to the resolution of both relative entropies and distances for general integer n. For the relative entropies, these formulas are manageable and the analytic continuation to n = 1 can be worked out in most of the cases. Conversely, for the distances the corresponding charged moments become soon untreatable as n increases. A remarkable result is that relative entropies and distances are the same for all symmetry sectors, i.e. they satisfy entanglement equipartition, like the entropies. Moreover, we exploit the OPE expansion of composite twist fields, to provide very general results when the subsystem is a single interval much smaller than the total system. We focus on the massless compact boson and our results are tested against exact numerical calculations in the XX spin chain.

U(1) symmetry resolved entanglement in free 1+1 dimensional field theories via form factor bootstrap

We generalise the form factor bootstrap approach to integrable field theories with U(1) symmetry to derive matrix elements of composite branch-point twist fields associated with symmetry resolved entanglement entropies. The bootstrap equations are solved for the free massive Dirac and complex boson theories, which are the simplest theories with U(1) symmetry. We present the exact and complete solution for the bootstrap, including vacuum expectation values and form factors involving any type and arbitrarily number of particles. The non-trivial solutions are carefully cross-checked by performing various limits and by the application of the ∆-theorem. An alternative and compact determination of the novel form factors is also presented. Based on the form factors of the U(1) composite branch-point twist fields, we re-derive earlier results showing entanglement equipartition for an interval in the ground state of the two models.

Symmetry resolved entanglement in integrable field theories via form factor bootstrap

We consider the form factor bootstrap approach of integrable field theories to derive matrix elements of composite branch-point twist fields associated with symmetry resolved entanglement entropies. The bootstrap equations are determined in an intuitive way and their solution is presented for the massive Ising field theory and for the genuinely interacting sinh-Gordon model, both possessing a ℤ2 symmetry. The solutions are carefully cross-checked by performing various limits and by the application of the ∆-theorem. The issue of symmetry resolution for discrete symmetries is also discussed. We show that entanglement equipartition is generically expected and we identify the first subleading term (in the UV cutoff) breaking it. We also present the complete computation of the symmetry resolved von Neumann entropy for an interval in the ground state of the paramagnetic phase of the Ising model. In particular, we compute the universal functions entering in the charged and symmetry resolved entanglement.

Entanglement entropies of minimal models from null-vectors

We present a new method to compute Rényi entropies in one-dimensional critical systems. The null-vector conditions on the twist fields in the cyclic orbifold allow us to derive a differential equation for their correlation functions. The latter are then determined by standard bootstrap techniques. We apply this method to the calculation of various Rényi entropies.