arXiv subjects

Uri Kol

Publications and source records attributed to Uri Kol.

At least 19 recordsLinked to original sources

arXiv2026

A Machine-Verified Proof of a Quantum-Optimization Conjecture

We report a machine-verified resolution of a problem open for over a decade in quantum optimization: the Farhi, Goldstone and Gutmann (FGG) conjecture that depth-$p$ Quantum Approximate Optimization Algorithm (QAOA) on the ring of disagrees attains approximation ratio $(2p+1)/(2p+2)$ exactly. We found the proof using a large language model, Claude Fable 5, and verified its correctness end-to-end by the Lean 4 proof assistant. Our methodology includes several ingredients: building on a substantial Lean library of quantum information, we formalized the QAOA components and the known parts of the problem, and reduced the conjecture to a single open mathematical statement. The model was then handed the library and our agentic toolkit, and tasked with closing that gap by constructing a proof in Lean. The resulting process is a feedback loop between the model's natural-language reasoning and Lean's mechanical verification, which converged to a machine-verified proof. Human verification is required only for the structural scaffolding - that the formal statement faithfully encodes the intended claim - while the proof itself is supplied by the model and certified mechanically by Lean. The proof is nevertheless striking - the model uncovered a hidden dynamical symmetry of the problem and exploited it, borrowing tools and machinery from an adjacent field to turn a hard existence problem into an explicit construction. This work paves the way for resolving open conjectures in quantum information science and beyond.

quant-ph↗

arXiv2025

New Near Extremal Black Holes and Love Symmetry

Rotating and charged black holes are known to exhibit remarkable properties close to extremality, including emergent hidden symmetries and holographic duality to 2D theories. In this note, we introduce a new class of near-extremal black holes living in $(2,2)$ signature, strongly resembling the Lorentzian ones but with an exact integrable structure $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$. The exterior of the black hole is a self-dual solution with a photon ring. It develops an infinite near-horizon throat described by the Eguchi-Hanson instanton. An exact version of the so-called Love symmetry controls perturbations on the throat. Furthermore, the normalizable spectrum of this black hole provides quasinormal modes in Lorentzian signature.

hep-th↗

arXiv2025

Fokker-Planck to Callan-Symanzik: evolution of weight matrices under training

The dynamical evolution of a neural network during training has been an incredibly fascinating subject of study. First principal derivation of generic evolution of variables in statistical physics systems has proved useful when used to describe training dynamics conceptually, which in practice means numerically solving equations such as Fokker-Planck equation. Simulating entire networks inevitably runs into the curse of dimensionality. In this paper, we utilize Fokker-Planck to simulate the probability density evolution of individual weight matrices in the bottleneck layers of a simple 2-bottleneck-layered auto-encoder and compare the theoretical evolutions against the empirical ones by examining the output data distributions. We also derive physically relevant partial differential equations such as Callan-Symanzik and Kardar-Parisi-Zhang equations from the dynamical equation we have.

cs.LG↗

arXiv2024

An Exact Black Hole Scattering Amplitude

General Relativity famously predicts precession of orbital motions in the Schwarzschild metric. In this paper we show that by adding a NUT charge $N = iM$ the precession vanishes to all orders in $G$ even for rotating black holes. Moreover, we conjecture a generalization of the eikonal formula and show that the classical integrable trajectories determine the full quantum amplitude for this black hole, by means of exponentiation of the Post-Minkowskian radial action. Several consequences of integrability in self-dual gravity are discussed.

hep-th↗

arXiv2024

Quantum Criticality in Black Hole Scattering

The Teukolsky equation describing scattering from Kerr black holes captures a few important effects in the process of binary mergers, such as tidal deformations and the decay of ringdown modes, thereby raising interest in the structure of its solutions. In this letter we identify critical phenomena emerging in the corresponding phase space. One special point exists in this phase space, where the black hole is extremal and the scattered wave lies exactly at the superradiant bound, at which the physics simplifies considerably. We provide an indirect realization of a conformal symmetry emerging at this configuration, which leads to its interpretation as a critical point. Away from the critical point conformal symmetry is broken, but it is shown that critical fluctuations continue to be dominant in a wide range of parameters and at finite black hole temperatures. As in quantum many-body systems, the physics in this regime is described exclusively by the temperature and a set of critical exponents, therefore leading to robust predictions that are unique to the Kerr metric.

hep-th↗

arXiv2023

Duality in Einstein's Gravity

We show that the Einstein equations in the vacuum are invariant under an $SO(2)$ duality symmetry which rotates the curvature 2-form into its tangent space Hodge dual. Akin to electric-magnetic duality in gauge theory, the duality operation maps classical solutions into each other. As an example, we demonstrate that the Kerr solution is non-linearly mapped by duality into Kerr-Taub-NUT.

hep-th↗

arXiv2023

Self Dual Black Holes as the Hydrogen Atom

Rotating black holes exhibit a remarkable set of hidden symmetries near their horizon. These hidden symmetries have been shown to determine phenomena such as absorption scattering, superradiance and more recently tidal deformations, also known as Love numbers. They have also led to a proposal for a dual thermal CFT with left and right movers recovering the entropy of the black hole. In this work we provide a constructive explanation of these hidden symmetries via analytic continuation to Klein signature. We first show that the near-horizon region of extremal black holes is a Kleinian static solution with mass $M$ and NUT charge $N$. We then analyze the self-dual solution, namely a Kerr black hole with a NUT charge $N=\pm M$. Remarkably, the self-dual solution is self-similar to its near-horizon region and hence approximate symmetries become exact: in particular, the original two isometries of Kerr are promoted to seven exact symmetries embedded in a conformal algebra. We analyze its full conformal group in Kleinian twistor space, where a breaking $SO(4,2) \to SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ occurs due to the insertion of a preferred time direction for the black hole. Finally, we show that the spectrum of the self-dual black hole is integrable and that the eigenvalue problem can be mapped exactly to the Hydrogen atom where the wavefunction is solved in terms of elementary polynomials. Perturbing to astrophysical black holes with $N=0$, we obtain a hyperfine splitting structure.

hep-th↗

arXiv2023

Duality in Gauge Theory, Gravity and String Theory

Einstein's theory in the vacuum was recently shown to possess an $SO(2)$ duality invariance, which is broken by coupling to matter. Duality invariance can be restored by enlarging the phase space of the theory to allow for violations of the algebraic Bianchi identity. We show that in cases where the matter content can be understood as a component of the torsion tensor duality can be restored and we compute the corresponding duality current. We consider the case of NS-NS gravity as an example and find that the duality current is given by the divergence of the axion. In the linearized approximation of the low energy heterotic string theory these results imply that duality of the generalized Riemann curvature tensor implements Riemannian, axion-dilaton and electro-magnetic dualities simultaneously.

hep-th↗

arXiv2022

Supertranslation-Invariant Dressed Lorentz Charges

We present an explicit formula for Lorentz boosts and rotations that commute with BMS supertranslations in asymptotically flat spacetimes. Key to the construction is the use of infrared regularizations and of a unitary transformation that makes observables commute with the soft degrees of freedom. We explicitly verify that our charges satisfy the Lorentz algebra and we check that they are consistent with expectations by evaluating them on the supertranslated Minkowski space and on the boosted Kerr black hole.

hep-th↗

arXiv2021

Classical double copy at null infinity

We give two double copy prescriptions which construct asymptotically flat solutions in gravity from asymptotically flat gauge fields. The first prescription applies to radiative fields, which are non-linear vacuum solutions determined by characteristic data at null infinity. For any two such radiative gauge fields (linear or non-linear), the characteristic data of a radiative metric, dilaton and axion is constructed by a simple `squaring' procedure, giving a classical double copy at the level of radiation fields. We demonstrate the procedure with several examples where the characteristic data can be explicitly integrated; for linear fields this also sheds light on the twistorial description of Weyl double copy. Our second prescription applies to all asymptotically flat fields at the level of their asymptotic equations of motion: we give a map between any solution of the asymptotic Maxwell equations and any solution of the asymptotic Einstein equations at null infinity. This also extends to the asymptotic charges and their duals, preserves the soft and hard sectors between gauge theory and gravity, and is related to the usual notion of double copy in scattering amplitudes.

hep-th↗

arXiv2021

The Radial Action from Probe Amplitudes to All Orders

We extract the relativistic classical radial action from scattering amplitudes, to all orders in perturbation theory, in the probe limit. Our sources include point charges and monopoles, as well as the Schwarzschild and pure-NUT gravitational backgrounds. A characteristic relativistic effect, that scattering trajectories may wind around these sources any number of times, can be recovered when all-order amplitudes are available. We show that the amplitude for scattering a probe off a pure NUT is given by the solution of a transcendental equation involving continued fractions, and explain how to solve this equation to any desired loop order.

hep-th↗

arXiv2021

Subleading BMS Charges and The Lorentz Group

The extended BMS group includes supertranslation, dual supertranslation and Lorentz transformations. The generators of these symmetries can be classified according to their parity into "electric" and "magnetic" types. Using a multipole expansion of gravitational sources in the Newman-Penrose formalism, we associate each one of these charges with a particular moment. At leading order, the "electric" and "magnetic" monopole moments correspond respectively to supertranslations and dual supertranslations. At the first subleading order, the "electric" and "magnetic" dipole moments correspond respectively to boosts and rotations, therefore comprising the entire Lorentz group. Electric-magnetic type of duality then rotates the boost and rotation generators into each other.

hep-th↗

arXiv2020

Amplitudes from Coulomb to Kerr-Taub-NUT

Electric-magnetic duality, the Newman-Janis shift, and the double copy all act by elementary operations on three-point amplitudes. At the same time, they generate a network of interesting classical solutions spanning from the Coulomb charge via the dyon to the Kerr-Taub-NUT spacetime. We identify the amplitudes corresponding to each of these solutions, working to all orders in spin, but to leading perturbative order. We confirm that the amplitudes double-copy when the solutions are related by the classical double copy. Along the way we show that the Kerr-Taub-NUT solution corresponds to a gravitational electric-magnetic duality rotation acting on the Kerr solution, again to all orders in spin, and demonstrate that the asymptotic charges also transform simply under our operations.

hep-th↗

arXiv2020

Dual Komar Mass, Torsion and Riemann-Cartan Manifolds

The dual Komar mass generalizes the concept of the NUT parameter and is akin to the magnetic charge in electrodynamics. In asymptotically flat spacetimes it coincides with the dual supertranslation charge. The dual mass vanishes identically on Riemannian manifolds in General Relativity unless conical singularities corresponding to Misner strings are introduced. In this paper we propose an alternative way to source the dual mass locally. We show that this can be done by enlarging the phase space of the theory to allow for a violation of the algebraic Bianchi identity using local fields. A minimal extension of Einstein's gravity that meets this requirement is known as the Einstein-Cartan theory. Our main result is that on Riemann-Cartan manifolds the dual Komar mass does not vanish and is given by a volume integral over a local 1-form gravitational-magnetic current that is a function of the torsion.

hep-th↗

arXiv2020

The Double Copy of Electric-Magnetic Duality

We argue that the complex transformation relating the Schwarzschild to the Taub-NUT metric, introduced by Talbot, is in fact an electric-magnetic duality transformation. We show that at null infinity, the complex transformation is equivalent to a complexified BMS supertranslation, which rotates the supertranslation and the dual (magnetic) supertranslation charges. This can also be seen from the cubic coupling between the classical source and its background, which for Taub-NUT is given by a complex phase rotation acting on gravitational minimal couplings. The same phase rotation generates dyons from electrons at the level of minimally coupled amplitudes, manifesting the double copy relation between the two solutions.

hep-th↗

arXiv2020

Properties of Dual Supertranslation Charges in Asymptotically Flat Spacetimes

We study several properties of some new charges of asymptotically flat spacetimes. These dual supertranslation charges are akin to the magnetic large $U(1)$ charges in QED. In this paper we find the symmetries associated with these charges and show that the global dual supertranslation charge is topological because it is invariant under globally defined, smooth variations of the asymptotic metric. We also exhibit spacetimes where the charge does not vanish and we find dynamical processes that interpolate between regions with different values of these charges.

hep-th↗

arXiv2020

Gravitational Wu-Yang Monopoles

We show that in an asymptotically flat space where an S-Matrix can be defined, dual supertranslations leave all its matrix elements invariant and the Hilbert space of asymptotic states factorizes into distinct super-selection sectors, labeled by their dual supertranslation charges. These results suggest that dual supertranslation may be interpreted as a redundant gauge symmetry of asymptotically flat spacetimes. This would allow to recast general relativity as a theory of diffeomorphisms possessing an additional asymptotic gauge symmetry. We then use the conjectured dual supertranslation gauge symmetry to construct a gravitational equivalent of the Wu-Yang monopole solution. The metric describing the solution is defined using two overlapping patches on the celestial sphere. The solution is regular on each one of the patches separately and differentiable in the overlap region, where the two descriptions are identical by virtue of a dual supertranslation gauge transformation. Our construction provides an alternative to Misner's interpretation of the Taub-NUT metric. In particular, we find that using our approach the Taub-NUT metric can be made regular everywhere on the celestial sphere and at the same time it is devoid of closed timelike curves, provided that the bound $\frac{m}{\ell} \leq \sqrt{\frac{5}{27}}$ on the ratio of mass to NUT charge is obeyed.

hep-th↗

arXiv2019

Comments on Lorentz Transformations, Dressed Asymptotic States and Hawking Radiation

We consider two applications of the factorization of infrared dynamics in QED and gravity. The first is a redefinition of the Lorentz transformations that makes them commute with supertranslations. The other is the process of particle creation near a black hole horizon. For the latter we show that the emission of soft particles factors out of the S-matrix in the fixed-background approximation and to leading order in the soft limit. The factorization is implemented by dressing the incoming and outgoing asymptotic states with clouds of soft photons and soft gravitons. We find that while the soft photon cloud has no effect, the soft graviton cloud induces a phase shift in the Bogolyubov coefficients relating the incoming and outgoing modes. However, the flux of outgoing particles, given by the absolute value of the Bogolyubov coefficient, is insensitive to this phase.

hep-th↗