William Allan Bardeen (September 15, 1941 − November 18, 2025) was an American theoretical physicist who worked at the Fermi National Accelerator Laboratory. He is renowned for his foundational work on the chiral anomaly, the Adler-Bardeen theorem, the non-Abelian anomaly and gravitational anomalies. He was instrumental in the development of quantum chromodynamics and its applications, such as semileptonic decays and the <math altimg='si1.svg'><msub><mstyle mathvariant='normal'><mi>&#923;</mi></mstyle><mover accent='true'><mrow><mi>M</mi><mi>S</mi></mrow><mo>&#175;</mo></mover></msub></math> scheme frequently used in perturbative analysis of high energy processes involving strong interactions. Bardeen also played a major role in developing a theory of dynamical breaking of electroweak symmetry via top quark condensates, leading to one of the first composite Brout-Englert-Higgs boson models. His work on the chiral symmetry dynamics of heavy-light quark bound states correctly predicted abnormally long-lived resonances which are chiral symmetry partners of the ground state.

United States

Christopher T. Hill

William A. Bardeen: A life in physics and the legacy of the chiral anomaly <cross-ref id='crf0001' refid='bib0001'>[1]</cross-ref>

Gravitational subsystems with boundaries carry the action of an infinite-dimensional symmetry algebra, with potentially profound implications for the quantum theory of gravity. We initiate an investigation into the quantization of this corner symmetry algebra for the phase space of gravity localized to a region bounded by a 2-dimensional sphere. Starting with the observation that the algebra   <math> <mtext>sdiff</mtext> </math>  $$ \mathfrak{sdiff} $$ (S 2) of area-preserving diffeomorphisms of the 2-sphere admits a deformation to the finite-dimensional algebra   <math> <mi>su</mi> </math>  $$ \mathfrak{su} $$ (N), we derive novel finite-N deformations for two important subalgebras of the gravitational corner symmetry algebra. Specifically, we find that the area-preserving hydrodynamical algebra   <math> <mtext>sdiff</mtext> <mfenced> <msup> <mi>S</mi> <mn>2</mn> </msup> </mfenced> <msub> <mo>⊕</mo> <mi>L</mi> </msub> <msup> <mi>ℝ</mi> <msup> <mi>S</mi> <mn>2</mn> </msup> </msup> </math>  $$ \mathfrak{sdiff}\left({S}^2\right){\oplus}_{\mathcal{L}}{\mathbb{R}}^{S^2} $$  arises as the large-N limit of   <math> <mi>sl</mi> <mfenced> <mi>N</mi> <mi>ℂ</mi> </mfenced> <mo>⊕</mo> <mi>ℝ</mi> </math>  $$ \mathfrak{sl}\left(N,\mathbb{C}\right)\oplus \mathbb{R} $$  and that the full area-preserving corner symmetry algebra   <math> <mtext>sdiff</mtext> <mfenced> <msup> <mi>S</mi> <mn>2</mn> </msup> </mfenced> <msub> <mo>⊕</mo> <mi>L</mi> </msub> <mi>sl</mi> <msup> <mfenced> <mn>2</mn> <mi>ℝ</mi> </mfenced> <msup> <mi>S</mi> <mn>2</mn> </msup> </msup> </math>  $$ \mathfrak{sdiff}\left({S}^2\right){\oplus}_{\mathcal{L}}\mathfrak{sl}{\left(2,\mathbb{R}\right)}^{S^2} $$  is the large-N limit of the pseudo-unitary group   <math> <mi>su</mi> </math>  $$ \mathfrak{su} $$ (N, N). We find matching conditions for the Casimir elements of the deformed and continuum algebras and show how these determine the value of the deformation parameter N as well as the representation of the deformed algebra associated with a quantization of the local gravitational phase space. Additionally, we present a number of novel results related to the various algebras appearing, including a detailed analysis of the asymptotic expansion of the   <math> <mi>su</mi> </math>  $$ \mathfrak{su} $$ (N) structure constants, as well as an explicit computation of the full   <math> <mtext>diff</mtext> </math>  $$ \mathfrak{diff} $$ (S 2) structure constants in the spherical harmonic basis. A consequence of our work is the definition of an area operator which is compatible with the deformation of the area-preserving corner symmetry at finite N.

Canada

William Donnelly

Laurent Freidel

Seyed Moosavian

Antony Speranza

Matrix quantization of gravitational edge modes

Progress in identifying the bulk microstate interpretation of the Ryu-Takayanagi formula requires understanding how to define entanglement entropy in the bulk closed string theory. Unfortunately, entanglement and Hilbert space factorization remains poorly understood in string theory. As a toy model for AdS/CFT, we study the entanglement entropy of closed strings in the topological A-model in the context of Gopakumar-Vafa duality. We will present our results in two separate papers. In this work, we consider the bulk closed string theory on the resolved conifold and give a self-consistent factorization of the closed string Hilbert space using extended TQFT methods. We incorporate our factorization map into a Frobenius algebra describing the fusion and splitting of Calabi-Yau manifolds, and find string edge modes transforming under a q-deformed surface symmetry group. We define a string theory analogue of the Hartle-Hawking state and give a canonical calculation of its entanglement entropy from the reduced density matrix. Our result matches with the geometrical replica trick calculation on the resolved conifold, as well as a dual Chern-Simons theory calculation which will appear in our next paper [1]. We find a realization of the Susskind-Uglum proposal identifying the entanglement entropy of closed strings with the thermal entropy of open strings ending on entanglement branes. We also comment on the BPS microstate counting of the entanglement entropy. Finally we relate the nonlocal aspects of our factorization map to analogous phenomenon recently found in JT gravity.

Yikun Jiang

Manki Kim

China

Gabriel Wong

Entanglement entropy and edge modes in topological string theory. Part I. Generalized entropy for closed strings

The phase space of general relativity in a finite subregion is characterized by edge modes localized at the codimension-2 boundary, transforming under an infinite-dimensional group of symmetries. The quantization of this symmetry algebra is conjectured to be an important aspect of quantum gravity. As a step towards quantization, we derive a complete classification of the positive-area coadjoint orbits of this group for boundaries that are topologically a 2-sphere. This classification parallels Wigner’s famous classification of representations of the Poincaré group since both groups have the structure of a semidirect product. We find that the total area is a Casimir of the algebra, analogous to mass in the Poincaré group. A further infinite family of Casimirs can be constructed from the curvature of the normal bundle of the boundary surface. These arise as invariants of the little group, which is the group of area-preserving diffeomorphisms, and are the analogues of spin. Additionally, we show that the symmetry group of hydrodynamics appears as a reduction of the corner symmetries of general relativity. Coadjoint orbits of both groups are classified by the same set of invariants, and, in the case of the hydrodynamical group, the invariants are interpreted as the generalized enstrophies of the fluid.

Gravitational edge modes, coadjoint orbits, and hydrodynamics

We calculate quantum corrections to holographic entanglement entropy in the proposed duality between     <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math> $$ T\overline{T} $$ -deformed holographic 2D CFTs and gravity in AdS3 with a finite cutoff. We first establish the dictionary between the two theories by mapping the flow equation of the deformed CFT to the bulk Wheeler-DeWitt equation. The latter reduces to an ordinary differential equation for the sphere partition function, which we solve to find the entanglement entropy for an entangling surface consisting of two antipodal points on a sphere. The entanglement entropy in the inverse central charge expansion yields the expectation value of the bulk length operator plus the entropy of length fluctuations, in accordance with the Ryu-Takayanagi formula and its generalization due to Faulkner, Lewkowycz, and Maldacena. Special attention is paid to the conformal mode problem and its resolution by a choice of contour for the gravitational path integral.

Elise LePage

Yan-Yan Li

Brazil

Andre Pereira

Vasudev Shyam

Quantum corrections to finite radius holography and holographic entanglement entropy

Two-dimensional Yang-Mills theory is a useful model of an exactly solvable gauge theory with a string theory dual at large . We calculate entanglement entropy in the 1 expansion by mapping the theory to a system of fermions interacting via a repulsive entropic force. The entropy is a sum of two terms: the “Boltzmann entropy,” log dim( ) per point of the entangling surface, which counts the number of distinct microstates, and the “Shannon entropy,” Σ log , which captures fluctuations of the macroscopic state. We find that the entropy scales as in the large limit, and that at this order only the Boltzmann entropy contributes. We further show that the Shannon entropy scales linearly with , and confirm this behaviour with numerical simulations. While the term of order is surprising from the point of view of the string dual — in which only even powers of appear in the partition function — we trace it to a breakdown of large counting caused by the replica trick. This mechanism could lead to corrections to holographic entanglement entropy larger than expected from semiclassical field theory.

Sydney Timmerman

Chile

Nicolás Valdés-Meller

Entanglement entropy and the large

Entanglement entropy is an important quantity in field theory, but its definition poses some challenges. The naive definition involves an extension of quantum field theory in which one assigns Hilbert spaces to spatial sub-regions. For two-dimensional topological quantum field theory we show that the appropriate extension is the open-closed topological quantum field theory of Moore and Segal. With the addition of one additional axiom characterizing the “entanglement brane” we show how entanglement calculations can be cast in this framework. We use this formalism to calculate modular Hamiltonians, entanglement entropy and negativity in two-dimensional Yang-Mills theory and relate these to singularities in the modular ow. As a byproduct we find that the negativity distinguishes between the “log dim R” edge term and the “Shannon” edge term. We comment on the possible application to understanding the Bekenstein-Hawking entropy in two-dimensional gravity.

Entanglement branes, modular flow, and extended topological quantum field theory

We explore the important fundamental question of how quantum information is localized in quantum gravity, in a perturbative approach. Familiar descriptions of localization of information, such as via tensor factorization of the Hilbert space or a net of commuting subalgebras of operators, conflict with basic gravitational properties—specifically gauge invariance—already at leading order in perturbation theory. However, previous work found that information can be classically localized in a region in a way such that measurements, including those of the gravitational field, outside the region are insensitive to that information and only measure total Poincaré charges. This paper shows that, working to leading order in the gravitational coupling, a similar quantum result holds, leading to a definition of a “gravitational splitting” on the Hilbert space for gravity. Such localization of information also argues against a role for “soft hair” in resolving the information problem for black holes. This basic mathematical structure plausibly plays a foundational role in the quantum description of gravity.

Steven B. Giddings

Gravitational splitting at first order: Quantum information localization in gravity

Quantum gravity in a finite region of spacetime is conjectured to be dual to a conformal field theory (CFT) deformed by the irrelevant operator <math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math>. We test this conjecture with entanglement entropy, which is sensitive to ultraviolet physics on the boundary, while also probing the bulk geometry. We find that the entanglement entropy for an entangling surface consisting of two antipodal points on a sphere is finite and precisely matches the Ryu-Takayanagi formula applied to a finite region consistent with the conjecture of McGough et al. We also consider a one-parameter family of conical entropies, which are finite and verify a conjecture due to Dong. Since ultraviolet divergences are local, we conclude that the <math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math> deformation acts as an ultraviolet cutoff on the entanglement entropy. Our results support the conjecture that the <math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math>-deformed CFT is the holographic dual of a finite region of spacetime.

Entanglement Entropy and <math><mi>T</mi><mover><mi>T</mi><mo>¯</mo></mover></math> Deformation

What is the meaning of entanglement in a theory of extended objects such as strings? To address this question we consider the spatial entanglement between two intervals in the Gross-Taylor model, the string theory dual to two-dimensional Yang-Mills theory at large N . The string diagrams that contribute to the entanglement entropy describe open strings with endpoints anchored to the entangling surface, as first argued by Susskind. We develop a canonical theory of these open strings, and describe how closed strings are divided into open strings at the level of the Hilbert space. We derive the modular Hamiltonian for the Hartle-Hawking state and show that the corresponding reduced density matrix describes a thermal ensemble of open strings ending on an object at the entangling surface that we call an entanglement brane, or E-brane.