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arXiv:1305.2122

Precise lower bound on Monster brane boundary entropy

Daniel Harry Friedan, Anatoly Konechny, Cornelius Schmidt-Colinet

Published May 9, 2013• 1 citation

Abstract

In this paper we develop further the linear functional method of deriving lower bounds on the boundary entropy of conformal boundary conditions in 1+1 dimensional conformal field theories (CFTs). We show here how to use detailed knowledge of the bulk CFT spectrum. Applying the method to the Monster CFT with c=\bar c=24 we derive a lower bound s > - 3.02 x 10^{-19} on the boundary entropy s=ln g, and find compelling evidence that the optimal bound is s>= 0. We show that all g=1 branes must have the same low-lying boundary spectrum, which matches the spectrum of the known g=1 branes, suggesting that the known examples comprise all possible g=1 branes, and also suggesting that the bound s>= 0 holds not just for critical boundary conditions but for all boundary conditions in the Monster CFT. The same analysis applied to a second bulk CFT -- a certain c=2 Gaussian model -- yields a less strict bound, suggesting that the precise linear functional bound on s for the Monster CFT is exceptional.

DOI: 10.48550/arxiv.1305.2122

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In this paper we develop further the linear functional method of deriving lower bounds on the boundary entropy of conformal boundary conditions in 1+1 dimensional conformal field theories (CFTs). We show here how to use detailed knowledge of the bulk CFT spectrum. Applying the method to the Monster CFT with c=\bar c=24 we derive a lower bound s > - 3.02 x 10^{-19} on the boundary entropy s=ln g, and find compelling evidence that the optimal bound is s>= 0. We show that all g=1 branes must have the same low-lying boundary spectrum, which matches the spectrum of the known g=1 branes, suggesting that the known examples comprise all possible g=1 branes, and also suggesting that the bound s>= 0 holds not just for critical boundary conditions but for all boundary conditions in the Monster CFT. The same analysis applied to a second bulk CFT -- a certain c=2 Gaussian model -- yields a less strict bound, suggesting that the precise linear functional bound on s for the Monster CFT is exceptional.

Abstract 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.

A lower bound is derived for the boundary entropy

s=\mathrm{ln}﻿g

of a (

1+1

)-dimensional quantum critical system with boundary under the conditions $c\ensuremath{\ge}1$ on the bulk conformal central charge and ${\ensuremath{\Delta}}_{1}>(c\ensuremath{-}1)/12$ on the most relevant bulk scaling dimension. This is the first general restriction on the possible values of

g

for bulk critical systems with $c\ensuremath{\ge}1$.

Entanglement entropy in holographic conformal field theories with boundaries of generic spacetime signature cannot be described entirely within the standard holographic duality to a real bulk spacetime. We show, both in static and time-dependent setups, that an exact match with the conformal field theory predictions in two boundary dimensions entails an extension of the dual spacetime to complex coordinates. As a consequence, relevant bulk objects such as Ryu-Takayanagi surfaces and branes are in general complex, yet anchored on a real asymptotic boundary preserving the physical meaning of field-theoretical quantities. We also provide a third numerical check of the holographic prediction with a two-dimensional system of free Gaussian fermions exhibiting the same asymptotic regimes across different entanglement phases.

The boundary entropy log(g) of a critical one-dimensional quantum system (or two-dimensional conformal field theory) is known to decrease under renormalization group (RG) flow of the boundary theory. We study instead the behavior of the boundary entropy as the bulk theory flows between two nearby critical points. We use conformal perturbation theory to calculate the change in g due to a slightly relevant bulk perturbation and find that it has no preferred sign. The boundary entropy log(g) can therefore increase during appropriate bulk flows. This is demonstrated explicitly in flows between minimal models. We discuss the applications of this result to D-branes in string theory and to impurity problems in condensed matter.

The boundary entropy log(g) of a critical one-dimensional quantum system (or two-dimensional conformal field theory) is known to decrease under renormalization group (RG) flow of the boundary theory. We study instead the behavior of the boundary entropy as the bulk theory flows between two nearby critical points. We use conformal perturbation theory to calculate the change in g due to a slightly relevant bulk perturbation and find that it has no preferred sign. The boundary entropy log(g) can therefore increase during appropriate bulk flows. This is demonstrated explicitly in flows between minimal models. We discuss the applications of this result to D-branes in string theory and to impurity problems in condensed matter.