arXiv subjects

Thomas Hartman

Publications and source records attributed to Thomas Hartman.

At least 19 recordsLinked to original sources

arXiv2025

Triangulating quantum gravity in AdS$_3$

The path integral of pure 3D gravity with negative cosmological constant is formulated on a finite region of spacetime $M$, with boundary conditions that fix geodesic lengths or dihedral angles on $\partial M$. In the dual CFT, this quasi-local amplitude calculates corrections to the Gaussian ensemble of OPE coefficients for black hole states. By triangulating $M$ with generalized tetrahedra, we develop a general method to construct semiclassical geometries and to calculate the exact gravitational path integral on a fixed hyperbolic topology. The path integral with fixed-length boundary conditions is a Virasoro TQFT amplitude-squared, and with fixed-angle boundary conditions it is a partition function of Conformal Turaev-Viro theory. The two are related by a modular S-transform. In addition, we show how to translate the calculation of OPE statistics from Virasoro TQFT to the metric formalism, on general topologies. These results are derived exactly, and some examples are also checked semiclassically, including the geometries dual to the Virasoro 6j-symbol and the modular S-matrix. The classical saddlepoint geometries are finite-volume hyperbolic 3-manifolds ending on pleated Riemann surfaces, which have vanishing extrinsic curvature except on geodesics where they can bend into corners. The hyperbolic volumes of these geometries match the predictions of Conformal Turaev-Viro theory and the dual CFT.

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arXiv2025

Conformal Turaev-Viro Theory

We define and study Conformal Turaev-Viro (CTV) theory, a dual formulation of Virasoro TQFT based on triangulating 3-manifolds with tetrahedra. Edges of the triangulation are labeled by continuous conformal weights, and tetrahedra are glued together weighted by the Cardy density of states. We demonstrate that the CTV partition function is equal to the modular S-transform of the Virasoro TQFT amplitude-squared, $|Z_{Vir}|^2$. This is analogous to a known result for discrete spin networks. The derivation uses a variant of the chain-mail formalism, adapted to the Virasoro context. As a CFT application, we derive formulae for the S-transforms of the squared Virasoro crossing kernels. These results lay the topological foundation to study the exact path integral of pure AdS$_3$ quantum gravity by triangulations.

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arXiv2025

Replica manifolds, pole skipping, and the butterfly effect

The black hole butterfly effect is a signal of quantum chaos in holographic theories that can be probed in different ways, including out-of-time-order correlators (OTOCs), pole skipping (PS), and entanglement wedge (EW) reconstruction. Each of these three phenomena can be used to define a butterfly velocity that measures the speed at which chaos spreads. In a general quantum system the three velocities $v_B^{\text{OTOC}}$, $v_B^{\text{PS}}$, and $v_B^{\text{EW}}$ can be different, but it is known from explicit calculations that they are all equal in certain holographic theories dual to Einstein gravity plus higher-curvature corrections. A conceptual explanation for this apparent coincidence is lacking. We show that it follows from a deeper relationship: The pole-skipping mode, added to the black hole background, can be reinterpreted as the gravitational replica manifold for the late-time entanglement wedge, and its imaginary part is the shockwave that computes the OTOC. Thus pole skipping is directly related to entanglement dynamics in holographic theories, and the origin of the pole-skipping mode is an extremal surface on the horizon. This explains the coincidence $v_B^{\text{OTOC}} = v_B^{\text{PS}} = v_B^{\text{EW}}$ in known cases, and extends it to general theories of gravity with a pole-skipping mode having the usual behavior.

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arXiv2025

Rademacher expansion of modular integrals

We develop a method to evaluate integrals of non-holomorphic modular functions over the fundamental domain of the torus with modular parameter $\tau$ analytically. It proceeds in two steps: first the integral is transformed to a Lorentzian contour by the same strategy that leads to the Lorentzian inversion formula in CFT, and then we apply a two-dimensional version of the Rademacher expansion. This computes the integral in terms of an expansion sensitive to the singular behaviour of the integrand near all the Lorentzian cusps $\tau \to i \infty$, $\bar{\tau} \to x \in \mathbb{Q}$. We apply this technique to a variety of examples such as the evaluation of string one-loop partition functions, where it leads to the first analytic formula for the cosmological constants of the bosonic string and the $\mathrm{SO}(16) \times \mathrm{SO}(16)$ string.

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arXiv2024

Light-ray sum rules and the c-anomaly

In a four-dimensional quantum field theory that flows between two fixed points under the renormalization group, the change in the conformal anomaly $\Delta a$ has been related to the average null energy. We extend this result to derive a sum rule for the other anomaly coefficient, $\Delta c$, in terms of the stress tensor three-point function. While the sum rule for $\Delta a$ is an expectation value of the averaged null energy operator, and therefore positive, the result for $\Delta c$ involves the off-diagonal matrix elements, so it does not have a fixed sign.

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arXiv2024

Statistics of three-dimensional black holes from Liouville line defects

Black holes and wormholes in the gravitational path integral can be used to calculate the statistics of heavy operators. An explicit example in higher dimensions is provided by thin shells of matter. We study these solutions in 3D gravity, and reproduce the behavior of black holes and wormholes from the dual CFT using the large-$c$ conformal bootstrap. The CFT operator that creates a thin shell black hole is a line defect, so we begin by using the bootstrap to study the statistics of line defects, both at finite $c$ and in the holographic large-$c$ limit. The crossing equation leads to a universal formula for the average high-energy matrix elements of the line defect in any compact, unitary 2d CFT with $c>1$. The asymptotics are controlled by a line defect in Liouville CFT at the same value of the central charge. At large $c$, three distinct quantities are related: The statistics of line defects in holographic CFTs, the individual matrix elements of a line defect in Liouville CFT, and the on-shell action of black holes and wormholes in 3D gravity. The three calculations match for black holes, and if the statistics of the line defects are assumed to be approximately Gaussian, then a class of wormholes is also reproduced by the dual CFT.

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arXiv2023

Null energy constraints on two-dimensional RG flows

We study applications of spectral positivity and the averaged null energy condition (ANEC) to renormalization group (RG) flows in two-dimensional quantum field theory. We find a succinct new proof of the Zamolodchikov $c$-theorem, and derive further independent constraints along the flow. In particular, we identify a natural $C$-function that is a completely monotonic function of scale, meaning its derivatives satisfy the alternating inequalities $(-1)^nC^{(n)}(\mu^2) \geq 0$. The completely monotonic $C$-function is identical to the Zamolodchikov $C$-function at the endpoints, but differs along the RG flow. In addition, we apply Lorentzian techniques that we developed recently to study anomalies and RG flows in four dimensions, and show that the Zamolodchikov $c$-theorem can be restated as a Lorentzian sum rule relating the change in the central charge to the average null energy. This establishes that the ANEC implies the $c$-theorem in two dimensions, and provides a second, simpler example of the Lorentzian sum rule.

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arXiv2023

Averaged Null Energy and the Renormalization Group

We establish a connection between the averaged null energy condition (ANEC) and the monotonicity of the renormalization group, by studying the light-ray operator $\int du T_{uu}$ in quantum field theories that flow between two conformal fixed points. In four dimensions, we derive an exact sum rule relating this operator to the Euler coefficient in the trace anomaly, and show that the ANEC implies the a-theorem. The argument is based on matching anomalies in the stress tensor 3-point function, and relies on special properties of contact terms involving light-ray operators. We also illustrate the sum rule for the example of a free massive scalar field. Averaged null energy appears in a variety of other applications to quantum field theory, including causality constraints, Lorentzian inversion, and quantum information. The quantum information perspective provides a new derivation of the $a$-theorem from the monotonicity of relative entropy. The equation relating our sum rule to the dilaton scattering amplitude in the forward limit suggests an inversion formula for non-conformal theories.

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arXiv2023

Black hole wavefunctions and microcanonical states

We consider the problem of defining a microcanonical thermofield double state at fixed energy and angular momentum from the gravitational path integral. A semiclassical approximation to this state is obtained by imposing a mixed boundary condition on an initial time surface. We analyze the corresponding boundary value problem and gravitational action. The overlap of this state with the canonical thermofield double state, which is interpreted as the Hartle-Hawking wavefunction of an eternal black hole in a mini-superspace approximation, is calculated semiclassically. The relevant saddlepoint is a higher-dimensional, rotating generalization of the wedge geometry that has been studied in two-dimensional gravity.

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arXiv2023

Toward random tensor networks and holographic codes in CFT

In holographic CFTs satisfying eigenstate thermalization, there is a regime where the operator product expansion can be approximated by a random tensor network. The geometry of the tensor network corresponds to a spatial slice in the holographic dual, with the tensors discretizing the radial direction. In spherically symmetric states in any dimension and more general states in 2d CFT, this leads to a holographic error-correcting code, defined in terms of OPE data, that can be systematically corrected beyond the random tensor approximation. The code is shown to be isometric for light operators outside the horizon, and non-isometric inside, as expected from general arguments about bulk reconstruction. The transition at the horizon occurs due to a subtle breakdown of the Virasoro identity block approximation in states with a complex interior.

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arXiv2022

Focusing bounds for CFT correlators and the S-matrix

The focusing theorem in General Relativity underlies causality, singularity theorems, entropy inequalities, and more. In AdS/CFT, we show that focusing in the bulk leads to a bound on CFT $n$-point functions that is generally stronger than causality. Causality is related to the averaged null energy condition (ANEC) on the boundary, while focusing is related to the ANEC in the bulk. The bound is derived by translating the Einstein equations into a relation between bulk and boundary light-ray operators. We also discuss the consequences of focusing for the flat space $S$-matrix, which satisfies a similar inequality, and give a new derivation of bounds on higher derivative operators in effective field theories. The string theory $S$-matrix and CFT correlators in conformal Regge theory also satisfy the focusing bound, even though in these cases it cannot be derived from the standard focusing theorem.

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arXiv2022

Report of the Snowmass 2021 Theory Frontier Topical Group on Quantum Information Science

We summarize current and future applications of quantum information science to theoretical high energy physics. Three main themes are identified and discussed; quantum simulation, quantum sensors and formal aspects of the connection between quantum information and gravity. Within these themes, there are important research questions and opportunities to address them in the years and decades ahead. Efforts in developing a diverse quantum workforce are also discussed. This work summarizes the subtopical area Quantum Information for High Energy Physics TF10 which forms part of the Theory Frontier report for the Snowmass 2021 planning process.

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arXiv2022

Coarse graining pure states in AdS/CFT

We construct new Euclidean wormhole solutions in AdS(d+1) and discuss their role in UV-complete theories, without ensemble averaging. The geometries are interpreted as overlaps of GHZ-like entangled states, which arise naturally from coarse graining the density matrix of a pure state in the dual CFT. In several examples, including thin-shell collapsing black holes and pure black holes with an end-of-the-world brane behind the horizon, the coarse-graining map is found explicitly in CFT terms, and used to define a coarse-grained entropy that is equal to one quarter the area of a time-symmetric apparent horizon. Wormholes are used to derive the coarse-graining map and to study statistical properties of the quantum state. This reproduces aspects of the West Coast model of 2D gravity and the large-c ensemble of 3D gravity, including a Page curve, in a higher-dimensional context with generic matter fields.

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arXiv2022

Snowmass white paper: Quantum information in quantum field theory and quantum gravity

We present a summary of recent progress and remaining challenges in applying the methods and ideas of quantum information theory to the study of quantum field theory and quantum gravity. Important topics and themes include: entanglement entropy in QFTs and what it reveals about RG flows, symmetries, and phases; scrambling, information spreading, and chaos; state preparation and complexity; classical and quantum simulation of QFTs; and the role of information in holographic dualities. We also highlight the ways in which quantum information science benefits from the synergy between the fields.

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arXiv2022

Semiclassical 3D gravity as an average of large-c CFTs

A two-dimensional CFT dual to a semiclassical theory of gravity in three dimensions must have a large central charge $c$ and a sparse low energy spectrum. This constrains the OPE coefficients and density of states of the CFT via the conformal bootstrap. We define an ensemble of CFT data by averaging over OPE coefficients subject to these bootstrap constraints, and show that calculations in this ensemble reproduce semiclassical 3D gravity. We analyze a wide variety of gravitational solutions, both in pure Einstein gravity and gravity coupled to massive point particles, including Euclidean wormholes with multiple boundaries and higher topology spacetimes with a single boundary. In all cases we find that the on-shell action of gravity agrees with the ensemble-averaged CFT at large $c$. The one-loop corrections also match in the cases where they have been computed. We also show that the bulk effective theory has random couplings induced by wormholes, providing a controlled, semiclassical realization of the mechanism of Coleman, Giddings, and Strominger.

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arXiv2022

Snowmass White Paper: The Analytic Conformal Bootstrap

The analytic conformal bootstrap is an array of techniques to characterize, constrain, and solve strongly interacting quantum field theories using symmetries, causality, unitarity, and other general principles. In the last decade, bolstered by the development of new Lorentzian methods, it has been used to solve conformal field theories at large spin; to place bounds on energy distributions, event shapes, operator product coefficients, and other observables; and to understand aspects of quantum gravity in anti-de Sitter space. We review these advances and highlight several promising areas for future exploration. Targets include developing new methods to close the gap between numerical and analytic bounds, extending the bootstrap beyond conformal fixed points, applications to quantum gravity and cosmology, and building on ties to condensed matter theory and mathematics.

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arXiv2022

Snowmass White Paper: Quantum Aspects of Black Holes and the Emergence of Spacetime

Black holes provide a window into the microscopic structure of spacetime in quantum gravity. Recently the quantum information contained in Hawking radiation has been calculated, verifying a key aspect of the consistency of black hole evaporation with quantum mechanical unitarity. This calculation relied crucially on recent progress in understanding the emergence of bulk spacetime from a boundary holographic description. Spacetime wormholes have played an important role in understanding the underpinnings of this result, and the precision study of such wormholes, in this and other contexts, has been enabled by the development of low-dimensional models of holography. In this white paper we review these developments and describe some of the deep open questions in this subject. These include the nature of the black hole interior, potential applications to quantum cosmology, the gravitational explanation of the fine structure of black holes, and the development of further connections to quantum information and laboratory quantum simulation.

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arXiv2021

Averaging over moduli in deformed WZW models

WZW models live on a moduli space parameterized by current-current deformations. The moduli space defines an ensemble of conformal field theories, which generically have $N$ abelian conserved currents and central charge $c > N$. We calculate the average partition function and show that it can be interpreted as a sum over 3-manifolds. This suggests that the ensemble-averaged theory has a holographic dual, generalizing recent results on Narain CFTs. The bulk theory, at the perturbative level, is identified as $U(1)^{2N}$ Chern-Simons theory coupled to additional matter fields. From a mathematical perspective, our principal result is a Siegel-Weil formula for the characters of an affine Lie algebra.

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