arXiv subjects

Henry Maxfield

Publications and source records attributed to Henry Maxfield.

32 records · Page 2Linked to original sources

arXiv2019

Quantum corrections to the BTZ black hole extremality bound from the conformal bootstrap

Any unitary compact two-dimensional CFT with $c>1$ and no symmetries beyond Virasoro has a parametrically large density of primary states at large spin for $\bar{h}>\bar{h}_\text{extr}\sim \frac{c-1}{24}$, of a universal form determined by modular invariance. By including the contribution of light primary operators and multi-twist composites constructed from them in the modular bootstrap, we find that $\bar{h}_\text{extr}$ receives corrections in a large spin expansion, which we compute at finite $c$. The analysis uses a formulation of the modular S-transform as a Fourier transform acting on the density of primary states. For theories with gravitational duals, $\bar{h}_\text{extr}$ is interpreted as the extremality bound of rotating BTZ black holes, receiving quantum corrections which we compute at one loop by prohibiting naked singularities in the quantum-corrected geometry. This gravity result is reproduced by modular bootstrap in a semiclassical $c\to\infty$ limit.

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arXiv2019

Quantum Regge Trajectories and the Virasoro Analytic Bootstrap

Every conformal field theory (CFT) above two dimensions contains an infinite set of Regge trajectories of local operators which, at large spin, asymptote to "double-twist" composites with vanishing anomalous dimension. In two dimensions, due to the existence of local conformal symmetry, this and other central results of the conformal bootstrap do not apply. We incorporate exact stress tensor dynamics into the CFT$_2$ analytic bootstrap, and extract several implications for AdS$_3$ quantum gravity. Our main tool is the Virasoro fusion kernel, which we newly analyze and interpret in the bootstrap context. The contribution to double-twist data from the Virasoro vacuum module defines a "Virasoro Mean Field Theory" (VMFT), its spectrum includes a finite number of discrete Regge trajectories, whose dimensions obey a simple formula exact in the central charge $c$ and external operator dimensions. We then show that VMFT provides a baseline for large spin universality in two dimensions: in every unitary compact CFT$_2$ with $c > 1$ and a twist gap above the vacuum, the double-twist data approaches that of VMFT at large spin $\ell$. Corrections to the large spin spectrum from individual non-vacuum primaries are exponentially small in $\sqrt{\ell}$ for fixed $c$. We analyze our results in various large $c$ limits. Further applications include a derivation of the late-time behavior of Virasoro blocks at generic $c$, a refined understanding and new derivation of heavy-light blocks, and the determination of the cross-channel limit of generic Virasoro blocks. We deduce non-perturbative results about the bound state spectrum and dynamics of quantum gravity in AdS$_3$.

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arXiv2018

Phase transitions in 3D gravity and fractal dimension

We show that for three dimensional gravity with higher genus boundary conditions, if the theory possesses a sufficiently light scalar, there is a second order phase transition where the scalar field condenses. This three dimensional version of the holographic superconducting phase transition occurs even though the pure gravity solutions are locally AdS$_3$. This is in addition to the first order Hawking-Page-like phase transitions between different locally AdS$_3$ handlebodies. This implies that the Rényi entropies of holographic CFTs will undergo phase transitions as the Rényi parameter is varied, as long as the theory possesses a scalar operator which is lighter than a certain critical dimension. We show that this critical dimension has an elegant mathematical interpretation as the Hausdorff dimension of the limit set of a quotient group of AdS$_3$, and use this to compute it, analytically near the boundary of moduli space and numerically in the interior of moduli space. We compare this to a CFT computation generalizing recent work of Belin, Keller and Zadeh, bounding the critical dimension using higher genus conformal blocks, and find a surprisingly good match.

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arXiv2018

Holographic complexity is nonlocal

We study the "complexity equals volume" (CV) and "complexity equals action" (CA) conjectures by examining moments of of time symmetry for $\rm AdS_3$ wormholes having $n$ asymptotic regions and arbitrary (orientable) internal topology. For either prescription, the complexity relative to $n$ copies of the $M=0$ BTZ black hole takes the form $ΔC = αc χ$, where $c$ is the central charge and $χ$ is the Euler character of the bulk time-symmetric surface. The coefficients $α_V = -4π/3$, $α_A = 1/6 $ defined by CV and CA are independent of both temperature and any moduli controlling the geometry inside the black hole. Comparing with the known structure of dual CFT states in the hot wormhole limit, the temperature and moduli independence of $α_V$, $α_A$ implies that any CFT gate set defining either complexity cannot be local. In particular, the complexity of an efficient quantum circuit building local thermofield-double-like entanglement of thermal-sized patches does not depend on the separation of the patches so entangled. We also comment on implications of the (positive) sign found for $α_A$, which requires the associated complexity to decrease when handles are added to our wormhole.

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arXiv2017

A view of the bulk from the worldline

A new method to compute correlation functions in AdS$_{d+1}$ in general dimension is introduced, considering a particle quantised in the worldline formalism of quantum field theory, coupled to bulk fields, in particular gravity, quantised in the standard manner. This gives a systematic two-parameter perturbative expansion organised by small parameters $G_N$ and the inverse mass of the particle, complementary to the usual Witten diagram expansion. This connects closely to CFT language, with the geodesic Witten diagram representation of global conformal blocks emerging naturally, and for two dimensions gives a bulk representation of the Virasoro block, with a systematic method for computing quantum corrections. The global conformal block and other contributions are shown to exponentiate in correlation functions in any dimension, corresponding to pieces of Witten diagrams at arbitrary loop order.

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arXiv2017

A conformal block Farey tail

We investigate the constraints of crossing symmetry on CFT correlation functions. Four point conformal blocks are naturally viewed as functions on the upper-half plane, on which crossing symmetry acts by PSL(2,Z) modular transformations. This allows us to construct a unique, crossing symmetric function out of a given conformal block by averaging over PSL(2,Z). In some two dimensional CFTs the correlation functions are precisely equal to the modular average of the contributions of a finite number of light states. For example, in the two dimensional Ising and tri-critical Ising model CFTs, the correlation functions of identical operators are equal to the PSL(2,Z) average of the Virasoro vacuum block; this determines the 3 point function coefficients uniquely in terms of the central charge. The sum over PSL(2,Z) in CFT2 has a natural AdS3 interpretation as a sum over semi-classical saddle points, which describe particles propagating along rational tangles in the bulk. We demonstrate this explicitly for the correlation function of certain heavy operators, where we compute holographically the semi-classical conformal block with a heavy internal operator.

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arXiv2017

Witten Diagrams for Torus Conformal Blocks

We give a holographic description of global conformal blocks in two dimensional conformal field theory on the sphere and on the torus. We show that the conformal blocks for one-point functions on the torus can be written as Witten diagrams in thermal AdS. This is accomplished by deriving a general conformal Casimir equation for global conformal blocks, and showing that Witten diagrams obey the same equation. We study the semi-classical limit of n-point conformal blocks, and show that these equal the action of a network of bulk world-lines obeying appropriate geodesic equations. We give an alternate description in the Chern-Simons formulation of 3D gravity, where the conformal blocks are described by networks of Wilson lines, and argue that these formulations are equivalent.

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arXiv2017

A new handle on three-point coefficients: OPE asymptotics from genus two modular invariance

We derive an asymptotic formula for operator product expansion coefficients of heavy operators in two dimensional conformal field theory. This follows from modular invariance of the genus two partition function, and generalises the asymptotic formula for the density of states from torus modular invariance. The resulting formula is universal, depending only on the central charge, but involves the asymptotic behaviour of genus two conformal blocks. We use monodromy techniques to compute the asymptotics of the relevant blocks at large central charge to determine the behaviour explicitly.

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arXiv2016

Holographic partition functions and phases for higher genus Riemann surfaces

We describe a numerical method to compute the action of Euclidean saddlepoints for the partition function of a two-dimensional holographic CFT on a Riemann surface of arbitrary genus, with constant curvature metric. We explicitly evaluate the action for the saddles for genus two and map out the phase structure of dominant bulk saddles in a two-dimensional subspace of the moduli space. We discuss spontaneous breaking of discrete symmetries, and show that the handlebody bulk saddles always dominate over certain non-handlebody solutions.

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arXiv2015

Hot multiboundary wormholes from bipartite entanglement

We analyze the 1+1 CFT states dual to hot (time-symmetric) 2+1 multiboundary AdS wormholes. These are black hole geometries with high local temperature, $n \ge 1$ asymptotically-AdS$_3$ regions, and arbitrary internal topology. The dual state at $t=0$ is defined on $n$ circles. We show these to be well-described by sewing together tensor networks corresponding to thermofield double states. As a result, the entanglement is spatially localized and bipartite: away from particular boundary points ("vertices") any small connected region $A$ of the boundary CFT is entangled only with another small connected region $B$, where $B$ may lie on a different circle or may be a different part of the same circle. We focus on the pair-of-pants case, from which more general cases may be constructed. We also discuss finite-temperature corrections, where we note that the states involve a code subspace in each circle.

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arXiv2015

Topological aspects of generalized gravitational entropy

The holographic prescription for computing entanglement entropy requires that the bulk extremal surface, whose area encodes the amount of entanglement, satisfies a homology constraint. Usually this is stated as the requirement of a (spacelike) interpolating surface that connects the region of interest and the extremal surface. We investigate to what extent this constraint is upheld by the generalized gravitational entropy argument, which relies on constructing replica symmetric q-fold covering spaces of the bulk, branched at the extremal surface. We prove (at the level of topology) that the putative extremal surface satisfies the homology constraint if and only if the corresponding branched cover can be constructed for every positive integer q. We give simple examples to show that homology can be violated if the cover exists for some values of q but not others, along with some other issues.

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arXiv2015

Entanglement entropy in three dimensional gravity

The Ryu-Takayanagi and covariant Hubeny-Rangamani-Takayanagi proposals relate entanglement entropy in CFTs with holographic duals to the areas of minimal or extremal surfaces in the bulk geometry. We show how, in three dimensional pure gravity, the relevant regulated geodesic lengths can be obtained by writing a spacetime as a quotient of AdS3, with the problem reduced to a simple purely algebraic calculation. We explain how this works in both Lorentzian and Euclidean formalisms, before illustrating its use to obtain novel results in a number of examples, including rotating BTZ, the RP2 geon, and several wormhole geometries. This includes spatial and temporal dependence of single-interval entanglement entropy, despite these symmetries being broken only behind an event horizon. We also discuss considerations allowing HRT to be derived from analytic continuation of Euclidean computations in certain contexts, and a related class of complexified extremal surfaces.

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arXiv2014

Holographic probes of collapsing black holes

We continue the programme of exploring the means of holographically decoding the geometry of spacetime inside a black hole using the gauge/gravity correspondence. To this end, we study the behaviour of certain extremal surfaces (focusing on those relevant for equal-time correlators and entanglement entropy in the dual CFT) in a dynamically evolving asymptotically AdS spacetime, specifically examining how deep such probes reach. To highlight the novel effects of putting the system far out of equilibrium and at finite volume, we consider spherically symmetric Vaidya-AdS, describing black hole formation by gravitational collapse of a null shell, which provides a convenient toy model of a quantum quench in the field theory. Extremal surfaces anchored on the boundary exhibit rather rich behaviour, whose features depend on dimension of both the spacetime and the surface, as well as on the anchoring region. The main common feature is that they reach inside the horizon even in the post-collapse part of the geometry. In 3-dimensional spacetime, we find that for sub-AdS-sized black holes, the entire spacetime is accessible by the restricted class of geodesics whereas in larger black holes a small region near the imploding shell cannot be reached by any boundary-anchored geodesic. In higher dimensions, the deepest reach is attained by geodesics which (despite being asymmetric) connect equal time and antipodal boundary points soon after the collapse; these can attain spacetime regions of arbitrarily high curvature and simultaneously have smallest length. Higher-dimensional surfaces can penetrate the horizon while anchored on the boundary at arbitrarily late times, but are bounded away from the singularity. We also study the details of length or area growth during thermalization. While the area of extremal surfaces increases monotonically, geodesic length is neither monotonic nor continuous.

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arXiv2013

Holographic entanglement plateaux

We consider the entanglement entropy for holographic field theories in finite volume. We show that the Araki-Lieb inequality is saturated for large enough subregions, implying that the thermal entropy can be recovered from the knowledge of the region and its complement. We observe that this actually is forced upon us in holographic settings due to non-trivial features of the causal wedges associated with a given boundary region. In the process, we present an infinite set of extremal surfaces in Schwarzschild-AdS geometry anchored on a given entangling surface. We also offer some speculations regarding the homology constraint required for computing holographic entanglement entropy.

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