arXiv subjects

Alan Garbarz

Publications and source records attributed to Alan Garbarz.

At least 19 recordsLinked to original sources

arXiv2026

Bases of Steerable Kernels for Equivariant CNNs: From 2D Rotations to the Lorentz Group

We present an alternative way of solving the steerable kernel constraint that appears in the design of steerable equivariant convolutional neural networks. We find explicit real and complex bases which are ready to use, for different symmetry groups and for feature maps of arbitrary tensor type. A major advantage of this method is that it bypasses the need to numerically or analytically compute Clebsch-Gordan coefficients and works directly with the representations of the input and output feature maps. The strategy is to find a basis of kernels that respect a simpler invariance condition at some point $x_0$, and then \textit{steer} it with the defining equation of steerability to move to some arbitrary point $x=g\cdot x_0$. This idea has already been mentioned in the literature before, but not advanced in depth and with some generality. Here we describe how it works with minimal technical tools to make it accessible for a general audience.

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arXiv2025

Denoising Hamiltonian Network for Physical Reasoning

Machine learning frameworks for physical problems must capture and enforce physical constraints that preserve the structure of dynamical systems. Many existing approaches achieve this by integrating physical operators into neural networks. While these methods offer theoretical guarantees, they face two key limitations: (i) they primarily model local relations between adjacent time steps, overlooking longer-range or higher-level physical interactions, and (ii) they focus on forward simulation while neglecting broader physical reasoning tasks. We propose the Denoising Hamiltonian Network (DHN), a novel framework that generalizes Hamiltonian mechanics operators into more flexible neural operators. DHN captures non-local temporal relationships and mitigates numerical integration errors through a denoising mechanism. DHN also supports multi-system modeling with a global conditioning mechanism. We demonstrate its effectiveness and flexibility across three diverse physical reasoning tasks with distinct inputs and outputs.

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arXiv2024

Entanglement of a chiral scalar on the torus

We compute the entanglement entropy of an interval for a chiral scalar on a circle at an arbitrary temperature. We use the resolvent method, which involves expressing the entropy in terms of the resolvent of a certain operator, and we compute that resolvent by solving a problem that entails finding an analytic function on the complex torus with certain jump conditions at the interval. The resolvent is relevant by itself, since it can be used to compute any function of the reduced density matrix. We illustrate that by also computing all the R\'enyi entropies for the model.

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arXiv2022

Relative entropy of an interval for a massless boson at finite temperature

We compute Araki's relative entropy associated to a bounded interval $I=(a,b)$ between a thermal state and a coherent excitation of itself in the bosonic U(1)-current model, namely the (derivative of the) chiral boson. For this purpose we briefly review some recent results on the entropy of standard subspaces and on the relative entropy of non-pure states such as thermal states. In particular, recently Bostelmann, Cadamuro and Del Vecchio have obtained the relative entropy at finite temperature for the unbounded interval $(-\infty,t)$, using previous results of Borchers and Yngvason, mainly a unitary dilation that provides the modular evolution in the negative half-line. Here we find a unitary rotation in order to make use of the full PSL$(2,\mathbb{R})$ symmetries and obtain the modular group, modular Hamiltonian and the relative entropy $S$ of a bounded interval at finite temperature. Such relative entropy entails both a Bekenstein-like bound and a QNEC-like bound, but violates $S''\geq 0$. Finally, we extend the results to the free massless boson in $1+1$ dimensions with analogous bounds.

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arXiv2022

A note on Haag duality

Haag duality is a remarkable property in QFT stating that the commutant of the algebra of observables localized in some region of spacetime is exactly the algebra associated to the causally disconnected region. It is a strong condition on the local structure and has direct consequences on entanglement measures. It was first shown to hold for a free scalar field and causal diamonds by Araki in 1964 and later by many authors in different ways. In particular, Eckmann and Osterwalder (EO) used Tomita-Takesaki modular theory to give a direct proof. However, it is not straightforward to relate this proof to the works of Araki, since they rely on two forms of the canonical commutation relations (CCR), called Segal and Weyl formulations, while EO work as starting point assumes that duality holds in the so-called ``first quantization'' in the Weyl formulation. It is our purpose to first introduce the works of Araki in a more easy-to-read but still rigorous and self-contained fashion, and show how Haag duality is stated in the Segal and Weyl formulations and in both first and second quantizations (and their immediate combination). This permits to understand the setting of the EO proof of Haag duality. There is nothing essentially new in this manuscript, with the exception of what we consider a simplification of EO proof that uses the adjoint $S^*$ of the Tomita operator $S$ instead of introducing several auxiliary operators. We hope this note will be useful for those seeking to understand where Haag duality comes from in a free scalar QFT.

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arXiv2021

One-Loop Partition Function, Gauge Accessibility and Spectra in AdS$_3$ Gravity

We continue the study of the one-loop partition function of AdS$_3$ gravity with focus on the square-integrability condition on the fluctuating fields. In a previous work we found that the Brown-Henneaux boundary conditions follow directly from the $L^2$ condition. Here we rederive the partition function as a ratio of Laplacian determinants by performing a suitable decomposition of the metric fluctuations. We pay special attention to the asymptotics of the fields appearing in the partition function. We also show that in the usual computation using ghost fields for the de Donder gauge, such gauge condition is accessible precisely for square-integrable ghost fields. Finally, we compute the spectrum of the relevant Laplacians in thermal AdS$_3$, in particular noticing that there are no isolated eigenvalues, only essential spectrum. This last result supports the analytic continuation approach of David, Gaberdiel and Gopakumar. The purely essential spectra found are consistent with the independent results of Lee and Delay of the essential spectrum of the TT rank-2 tensor Lichnerowickz Laplacian on asymptotically hyperbolic spaces.

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arXiv2020

Cylinder transition amplitudes in pure AdS$_3$ Gravity

A spacelike surface with cylinder topology can be described by various sets of canonical variables within pure AdS3 gravity. Each is made of one real coordinate and one real momentum. The Hamiltonian can be either $H=0$ or it can be nonzero and we display the canonical transformations that map one into the other, in two relevant cases. In a choice of canonical coordinates, one of them is the cylinder aspect $q$, which evolves nontrivially in time. The time dependence of the aspect is an analytic function of time $t$ and an "angular momentum" $J$. By analytic continuation in both $t$ and $J$ we obtain a Euclidean evolution that can be described geometrically as the motion of a cylinder inside the region of the 3D hyperbolic space bounded by two "domes" (i.e. half spheres), which is topologically a solid torus. We find that for a given $J$ the Euclidean evolution cannot connect an initial aspect to an arbitrary final aspect; moreover, there are infinitely many Euclidean trajectories that connect any two allowed initial and final aspects. We compute the transition amplitude in two independent ways; first by solving exactly the time-dependent Schrödinger equation, then by summing in a sensible way all the saddle contributions, and we discuss why both approaches are mutually consistent.

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arXiv2020

Asymptotic Boundary Conditions and Square Integrability in the Partition Function of AdS Gravity

There has been renewed interest in the path-integral computation of the partition function of AdS3 gravity, both in the metric and Chern-Simons formulations. The one-loop partition function around Euclidean AdS3 turns out to be given by the vacuum character of Virasoro group. This stems from the work of Brown and Henneaux (BH) who showed that, in AdS3 gravity with sensible asymptotic boundary conditions, an infinite group of (improper) diffeomorphisms arises which acts canonically on phase space as two independent Virasoro symmetries. The gauge group turns out to be composed of so-called ``proper'' diffeomorphisms which approach the identity at infinity fast enough. However, it is sometimes far from evident to identify where BH boundary conditions enter in the path integral, and much more difficult to see how the improper diffeomorphisms are left out of the gauge group. In particular, in the metric formulation, Giombi, Maloney and Yin obtained the one-loop partition function around thermal AdS3 resorting to the heat kernel method to compute the determinants coming from the path integral. Here we identify how BH boundary conditions follow naturally from the usual requirement of square-integrability of the metric perturbations. Also, and equally relevant, we clarify how the quotient by only proper diffeomorphisms is implemented, promoting the improper diffeomorphisms to symmetries in the path integral. Our strategy is general enough to apply to other approaches where square integrability is assumed. Finally, we show that square integrability implies that the asymptotic symmetries in higher dimensional AdS gravity are just isometries.

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arXiv2019

Entanglement of a chiral fermion on the torus

In this paper we present the detailed calculation of a new modular Hamiltonian, namely that of a chiral fermion on a circle at non-zero temperature. We provide explicit results for an arbitrary collection of intervals, which we discuss at length by checking against known results in different limits and by computing the associated modular flow. We also compute the entanglement entropy, and we obtain a simple expression for it which appears to be more manageable than those already existing in the literature.

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arXiv2019

Boundary-to-bulk maps for AdS causal wedges and RG flow

We consider the problem of defining spacelike-supported boundary-to-bulk propagators in AdS$_{d+1}$ down to the unitary bound $Δ=(d-2)/2$. That is to say, we construct the `smearing functions' $K$ of HKLL but with different boundary conditions where both dimensions $Δ_+$ and $Δ_-$ are taken into account. More precisely, we impose Robin boundary conditions, which interpolate between Dirichlet and Neumann boundary conditions and we give explicit expressions for the distributional kernel $K$ with spacelike support. This flow between boundary conditions is known to be captured in the boundary by adding a double-trace deformation to the CFT. Indeed, we explicitly show that using $K$ there is a consistent and explicit map from a Wightman function of the boundary QFT to a Wightman function of the bulk theory. In order to accomplish this we have to study first the microlocal properties of the boundary two-point function of the perturbed CFT and prove its wavefront set satisfies the microlocal spectrum condition. This permits to assert that $K$ and the boundary two-point function can be multiplied as distributions.

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arXiv2017

Scalar field dynamics in a BTZ background with generic boundary conditions

The revisit the dynamics of a massive scalar field in a BTZ background taking into account the lack of global hyperbolicity of the spacetime. We approach this issue using the strategy of Ishibashi and Wald which finds a unique smooth solution as the causal evolution of initial data, each possible evolution corresponding to a positive self-adjoint extension of certain operator in a Hilbert space on the initial surface. Moreover, solutions obtained this way are the most general ones satisfying a few physically-sensible requirements. This procedure is intimately related to the choice of boundary conditions and the existence of bound states. We find that the scalar field dynamics in the (effective) mass window $-3/4\leq m_e^2\ell^2<0$ can be well-defined within a one-parametric family of distinct boundary conditions ($-3/4$ being the conformally-coupled case), while for $m_e^2\ell^2\geq 0$ the boundary condition is unique (only one self-adjoint extension is possible). It is argued that there is no sensible evolution possible for $m_e^2\ell^2<-1$, and also shown that in the range $m_e^2\ell^2 \in [-1,-3/4)$ there is a U$(1)$ family of allowed boundary conditions, however, the positivity of the self-adjoint extensions is only motivated but not proven. We focus mainly in describing the dynamics of such evolutions given the initial data and all possible boundary conditions, and in particular we show the energy is always positive and conserved.

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arXiv2016

The Virasoro Gibbs state and BTZ black holes

We show that the Virasoro Gibbs state accurately describes the thermodynamics of BTZ black holes at large temperatures and in the strong-coupling regime $c<1$. We first give a simple heuristic argument by showing that at high temperatures and arbitrary $c$, the energies are much larger than Planck mass. Then we give a detailed analysis of the quantum fluctuations of the Gibbs state on unitary irreducible representations of Virasoro group and explicitly show that they go to zero in the high-temperature limit by means of representation theory results. This implies the state has a sensible thermodynamic limit which actually holds for any $c$. Finally, the matching with BTZ thermodynamics for $c<1$ is obtained by using the known asymptotic behaviour of the characters of the Virasoro discrete series. This result supports the idea that minimal models could describe quantum gravity at strong coupling. We make no use of the Euclidean path integral nor assume modular invariance of the partition functions, although we comment on them along the way.

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arXiv2016

Quantization of BMS$_3$ orbits: a perturbative approach

We compute characters of the BMS group in three dimensions. The approach is the same as that performed by Witten in the case of coadjoint orbits of the Virasoro group in the eighties, within the large central charge approximation. The procedure involves finding a Poisson bracket between classical variables and the corresponding commutator of observables in a Hilbert space, explaining why we call this a quantization. We provide first a pedagogical warm up by applying the method to both SL(2,R) and Poincaré3 groups. As for BMS3, our results coincide with the characters of induced representations recently studied in the literature. Moreover, we relate the 'coadjoint representations' to the induced representations.

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arXiv2014

Classification of Boundary Gravitons in AdS$_3$ Gravity

We revisit the description of the space of asymptotically AdS3 solutions of pure gravity in three dimensions with a negative cosmological constant as a collection of coadjoint orbits of the Virasoro group. Each orbit corresponds to a set of metrics related by diffeomorphisms which do not approach the identity fast enough at the boundary. Orbits contain more than a single element and this fact manifests the global degrees of freedom of AdS3 gravity, being each element of an orbit what we call boundary graviton. We show how this setup allows to learn features about the classical phase space that otherwise would be quite difficult. Most important are the proof of energy bounds and the characterization of boundary gravitons unrelated to BTZs and AdS3. In addition, it makes manifest the underlying mathematical structure of the space of solutions close to infinity. Notably, because of the existence of a symplectic form in each orbit, being this related with the usual Dirac bracket of the asymptotic charges, this approach is a natural starting point for the quantization of different sectors of AdS3 gravity. We finally discuss previous attempts to quantize coadjoint orbits of the Virasoro group and how this is relevant for the formulation of AdS3 quantum gravity.

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arXiv2014

Quasilocal energy for three-dimensional massive gravity solutions with chiral deformations of AdS boundary conditions

We consider critical gravity in three dimensions; that is, the New Massive Gravity theory formulated about Anti-de Sitter (AdS) space with the specific value of the graviton mass for which it results dual to a two-dimensional conformal field theory with vanishing central charge. As it happens with Kerr black holes in four-dimensional critical gravity, in three-dimensional critical gravity the Banados-Teitelboim-Zanelli black holes have vanishing mass and vanishing angular momentum. However, provided suitable asymptotic conditions are chosen, the theory may also admit solutions carrying non-vanishing charges. Here, we give simple examples of exact solutions that exhibit falling-off conditions that are even weaker than those of the so-called Log-gravity. For such solutions, we define the quasilocal stress-tensor and use it to compute conserved charges. Despite the drastic deformation of AdS3 asymptotic, these solutions have finite mass and angular momentum.

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arXiv2011

Geometry and stability of spinning branes in AdS gravity

The geometry of spinning codimension-two branes in AdS spacetime is analyzed in three and higher dimensions. The construction of non-extremal solutions is based on identifications in the covering of AdS space by isometries that have fixed points. The discussion focuses on the cases where the parameters of spinning states can be related to the velocity of a boosted static codimension-two brane. The resulting configuration describes a single spinning brane, or a set of intersecting branes, each one produced by an independent identification. The nature of the singularity is also examined, establishing that the AdS curvature acquires one in the form of a Dirac delta distribution. The stability of the branes is studied in the framework of Chern-Simons AdS supergravity. A class of branes, characterized by one free parameter, are shown to be stable when the BPS conditions are satisfied. In 3D, these stable branes are extremal, while in higher dimensions, the BPS branes are not the extremal ones.

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arXiv2011

On the addition of torsion to chiral gravity

Three-dimensional gravity in Anti-de Sitter space is considered, including torsion. The derivation of the central charges of the algebra that generates the asymptotic isometry group of the theory is reviewed, and a special point of the theory, at which one of the central charges vanishes, is compared with the chiral point of topologically massive gravity. This special point corresponds to a singular point in Chern-Simons theory, where one of the two coupling constants of the SL(2,R) actions vanishes. A prescription to approach this point in the space of parameters is discussed, and the canonical structure of the theory is analyzed.

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arXiv2010

Naked Singularities, Topological Defects and Brane Couplings

A conical defect in (2+1) anti-de Sitter space is a BTZ solution with a negative mass parameter. This is a naked singularity, but a rather harmless one: it is a point particle. Naturally, the energy density and the spacetime curvature have a delta-like singularity at the apex of the conical defect, but that doesn't give rise to any unphysical situations. Since the conical solution implies the presence of a source, applying reverse enginnering, one can identify the coupling term that is required in the action to account for that source. In that way, a relation is established between the identification operation that gives rise to the topological defect and the interaction term in the action that produces it. This idea has a natural extension to higher dimensions, where instead of a point particle (zero-brane) one finds membranes of even spatial dimensions (p-branes, with p=2n). The generalization to other abelian and nonabelian gauge theories --including (super-) gravities-- is fairly straightforward: the 2n-brane couple to a (2n+1) Chern-Simons form. The construction suggests a generic role for Chern-Simons forms as the natural way to couple a gauge connection to a brane and avoids the inconsistency that results from the minimal coupling between a brane and a fundamental p-form field.

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