Soft Charges Research Articles

A comment on loop corrections to the celestial stress tensor

In this note we show how the 1-loop exact correction to the subleading soft graviton theorem arising from IR divergences of scattering amplitudes matches onto the quadratic corrections to the soft charges computed from the BMS flux algebra. In the process, we examine how the BMS flux construction extends the celestial diamond framework to non-linear order and non-trivial vacua, and provides the natural symmetry generators for Celestial CFT.

Asymptotic symmetries and soft charges of fractons

The asymptotic structure of gauge theories describing fracton interactions is analyzed. Two sets of asymptotic conditions are proposed. Both encompass all known solutions, lead to finite charges and resolve the problem of the divergent energy coming from the monopole contribution. While the first set leads to the expected fracton symmetry algebra, including a dipole charge, the second set provides a soft infinite-dimensional extension of it. These soft charges provide evidence of a rich infrared structure for fractonlike theories and provide one corner of a possible fracton infrared triangle.

Conformally soft fermions

Celestial diamonds encode the global conformal multiplets of the conformally soft sector, elucidating the role of soft theorems, symmetry generators and Goldstone modes. Upon adding supersymmetry they stack into a pyramid. Here we treat the soft charges associated to the fermionic layers that tie this structure together. This extends the analysis of conformally soft currents for photons and gravitons which have been shown to generate asymptotic symmetries in gauge theory and gravity to infinite-dimensional fermionic symmetries. We construct fermionic charge operators in 2D celestial CFT from a suitable inner product between 4D bulk field operators and spin s = frac{1}{2} and frac{3}{2} conformal primary wavefunctions with definite SL(2, ℂ) conformal dimension ∆ and spin J where |J| ≤ s. The generator for large supersymmetry transformations is identified as the conformally soft gravitino primary operator with ∆ = frac{1}{2} and its shadow with ∆ = frac{3}{2} which form the left and right corners of the celestial gravitino diamond. We continue this analysis to the subleading soft gravitino and soft photino which are captured by degenerate celestial diamonds. Despite the absence of a gauge symmetry in these cases, they give rise to conformally soft factorization theorems in celestial amplitudes and complete the celestial pyramid.

Revisiting the conformally soft sector with celestial diamonds

Celestial diamonds encode the structure of global conformal multiplets in 2D celestial CFT and offer a natural language for describing the conformally soft sector. The operators appearing at their left and right corners give rise to conformally soft factorization theorems, the bottom corners correspond to conserved charges, and the top corners to conformal dressings. We show that conformally soft charges can be expressed in terms of light ray integrals that select modes of the appropriate conformal weights. They reside at the bottom corners of memory diamonds, and ascend to generalized currents. We then identify the top corners of the associated Goldstone diamonds with conformal Faddeev-Kulish dressings and compute the sub-leading conformally soft dressings in gauge theory and gravity which are important for finding nontrivial central extensions. Finally, we combine these ingredients to speculate on 2D effective descriptions for the conformally soft sector of celestial CFT.

Dressings and Asymptotic Symmetries in Quantum Field Theory

In quantum field theories with massless gauge bosons, the conventional formulation of scattering amplitudes in terms of momentum eigenstates lead to infrared divergences. To resolve this one should dress charged particles with an infinite number of low-energy (soft) gauge bosons. These dressings are closely related to the asymptotic symmetries of the the- ory. The aim of this dissertation is to investigate this relation in detail, and study the implications of dressings in a quantum field theoretical context in asymptotically flat and Schwarzschild spacetimes. In particular, the asymptotic symmetry of interest in gravity is the BMS supertranslation. First, we demonstrate in perturbative quantum gravity that the dressings facilitate charge conservation by carrying a definite supertranslation charge. We then use this property to derive dressed states from the charge conservation laws of asymp- totic symmetries. We develop a unified quantum mechanical framework for the construction of dressings at the null infinity and the horizon of a Schwarzschild black hole, and show that the black hole soft hairs are planted on the horizon by dressed particles falling into the black hole. This framework is then extended to the magnetic parity soft charges of electro- magnetism and gravity. It is shown that one can construct ’t Hooft line dressings at the asymptotic boundaries, which are charged under the magnetic large gauge transformations and the dual supertranslations. Finally, we study the standard and dual supertranslation charges at the black hole horizon. We find central terms in their algebra in the presence of singularities in the parameter functions, which hints at further interesting soft structures yet to be revealed.

Asymptotic symmetries and celestial CFT

We provide a unified treatment of conformally soft Goldstone modes which arise when spin-one or spin-two conformal primary wavefunctions become pure gauge for certain integer values of the conformal dimension ∆. This effort lands us at the crossroads of two ongoing debates about what the appropriate conformal basis for celestial CFT is and what the asymptotic symmetry group of Einstein gravity at null infinity should be. Finite energy wavefunctions are captured by the principal continuous series ∆ ∈ 1 + iℝ and form a complete basis. We show that conformal primaries with analytically continued conformal dimension can be understood as certain contour integrals on the principal series. This clarifies how conformally soft Goldstone modes fit in but do not augment this basis. Conformally soft gravitons of dimension two and zero which are related by a shadow transform are shown to generate superrotations and non-meromorphic diffeomorphisms of the celestial sphere which we refer to as shadow superrotations. This dovetails the Virasoro and Diff(S2) asymptotic symmetry proposals and puts on equal footing the discussion of their associated soft charges, which correspond to the stress tensor and its shadow in the two-dimensional celestial CFT.

Magnetic soft charges, dual supertranslations, and ’t Hooft line dressings

We construct the Faddeev-Kulish asymptotic states in a quantum field theory of electric and magnetic charges. We find that there are two kind of dressings: apart from the well known (electric) Wilson line dressing, there is a magnetic counterpart which can be written as a 't Hooft line operator. The 't Hooft line dressings are charged under the magnetic large gauge transformation (LGT), but are neutral under electric LGT. This is in contrast to the Faddeev-Kulish dressings of electrons, which can be written as a Wilson line operator and are charged under electric LGT but neutral under magnetic LGT. With these dressings and the corresponding construction of the coherent states, the infrared finiteness of the theory of electric and magnetic charges is guaranteed. Even in the absence of magnetic monopoles, the electric and magnetic soft modes exhibit the electromagnetic duality of vacuum Maxwell theory. Using only the asymptotic form of three-point interactions in a field theory of electric and magnetic charges, we show that the leading magnetic dressings, like the leading electric ones, are exact in the field theory of electric and magnetic charges, in accordance with a conjecture of Strominger. We then extend the construction to perturbative quantum gravity in asymptotically flat spacetime, and construct gravitational 't Hooft line dressings that are charged under dual supertranslations. The duality in the quantum theory between the electric and magnetic soft charges and their dressings is thus made manifest.

A note on electric-magnetic duality and soft charges

We derive the asymptotic symmetries of the manifestly duality invariant formulation of electromagnetism in Minkoswki space. We show that the action is invariant under two algebras of angle-dependent u(1) transformations, one electric and the other magnetic. As in the standard electric formulation, Lorentz invariance requires the addition of additional boundary degrees of freedom at infinity, found here to be of both electric and magnetic types. A notable feature of this duality symmetric formulation, which we comment upon, is that the on-shell values of the zero modes of the gauge generators are equal to only half of the electric and magnetic fluxes (the other half is brought in by Dirac- string type contributions). Another notable feature is the absence of central extension in the angle-dependent u(1)2-algebra.

Soft charges from the geometry of field space

Infinite sets of asymptotic soft-charges were recently shown to be related to new symmetries of the S-matrix, spurring a large amount of research on this and related questions. Notwithstanding, the raison-d’être of these soft-charges rests on less firm ground, insofar as their known derivations through generalized Noether procedures tend to rely on the fixing of (gauge-breaking) boundary conditions rather than on manifestly gauge- invariant computations. In this article, we show that a geometrical framework anchored in the space of field configurations singles out the known leading-order soft charges in gauge theories. Our framework unifies the treatment of finite and infinite regions, and thus it explains why the infinite enhancement of the symmetry group is a property of asymptotic null infinity and should not be expected to hold within finite regions, where at most a finite number of physical charges — corresponding to the reducibility parameters of the quasi-local field configuration — is singled out. As a bonus, our formalism also suggests a simple proposal for the origin of magnetic-type charges at asymptotic infinity based on spacetime (Lorentz) covariance rather than electromagnetic duality.

Gravitational dressing, soft charges, and perturbative gravitational splitting

In gauge theories and gravity, field variables are generally not gauge-invariant observables, but such observables may be constructed by "dressing" these or more general operators. Dressed operators create particles, together with their gauge or gravitational fields which typically extend to infinity. This raises an important question of how well quantum information can be localized; one version of this is the question of whether soft charges fully characterize a given localized charge or matter distribution. This paper finds expressions for the non-trivial soft charges of such dressed operators. However, a large amount of flexibility in the dressing indicates that the soft charges, and other asymptotic observables, are not inherently correlated with details of the charge or matter distribution. Instead, these asymptotic observables can be changed by adding a general radiative (source-free) field configuration to the original one. A dressing can be chosen, perturbatively, so that the asymptotic observables are independent of details of the distribution, besides its total electric or Poincar\'e charges. This provides an approach to describing localization of information in gauge theories or gravity, and thus subsystems, that avoids problems associated with nonlocality of operator subalgebras. Specifically, this construction suggests the notions of electromagnetic or gravitational splittings, which involve networks of Hilbert space embeddings in which the charges play an important role.

Source and response soft charges for Maxwell theory on AdSd

We study asymptotic symmetries and their associated charges for Maxwell theory on anti de Sitter (AdS) background in any dimension. This is obtained by con- structing a conserved symplectic structure for the bulk and a theory on the boundary, which we specify. We show that the boundary phase space is described by two scalars and two sets of “source” and “response” boundary gauge transformations. The bulk dynamics is invariant under these two sets of boundary transformations. We study the (soft) charges associated with these two sets and show that they form an infinite dimensional Heisenberg type algebra. Studying the large AdS radius flat space limit, we show only the source soft charges survive. We also analyze algebra of charges associated with SO(d − 1, 2) isometries of the background AdSd space and study how they act on our source and response charges. We briefly discuss implication of our results for the AdS/CFT.

Generalized asymptotics for gauge fields

An interesting question is to characterize the general class of allowed boundary conditions for gauge theories, including gravity, at spatial and null infinity. This has played a role in discussions of soft charges, where antipodal symmetry has typically been assumed. However, the existence of electric and gravitational line operators, arising from gauge­invariant dressed observables, for example associated to axial or Fefferman-Graham like gauges, indicates the existence of non-antipodally symmetric initial data. This note studies aspects of the solutions corresponding to such non-symmetric initial data. The explicit evolution can be found, via a Green function, and bounds can be given on the asymptotic behavior of such solutions, evading arguments for singular behavior. Likewise, objections to such solutions based on infinite symplectic form are also avoided, although these solutions may be superselected. Soft charge conservation laws, and their modification, are briefly examined for such solutions. This discussion strengthens (though is not necessary for) arguments that soft charges characterize gauge field degrees of freedom, but not necessarily the degrees of freedom associated to the matter sourcing the field.

String memory effect

In systems with local gauge symmetries, the memory effect corresponds to traces inscribed on a suitable probe when a pure gauge configuration at infinite past dynamically evolves to another pure gauge configuration at infinite future. In this work, we study the memory effect of 2-form gauge fields which is probed by strings. We discuss the “string memory effect” for closed and open strings at classical and quantum levels. The closed string memory is encoded in the internal excited modes of the string, and in the open string case, it is encoded in the relative position of the two endpoints and the non-commutativity parameter associated with the D-brane where the open string endpoints are attached. We also discuss 2-form memory with D-brane probes using boundary state formulation and, the relation between string memory and 2-form soft charges analyzed in [1].

Null infinity, the BMS group and infrared issues

There has been a recent resurgence of interest in the structure of the gravitational field at null infinity, sparked by new results on soft charges and infrared issues related to the S matrix theory in perturbative quantum gravity. We summarize these developments and put them in the broader context of research in the relativity community that dates back to several decades. In keeping with intent of this series, this overview is addressed to gravitational scientists who are not experts in this specific area.

Membrane paradigm from near-horizon soft hair

The membrane paradigm posits that black hole microstates are dynamical degrees of freedom associated with a physical membrane vanishingly close to the black hole’s event horizon. The soft hair paradigm postulates that black holes can be equipped with zero-energy charges associated with residual diffeomorphisms that label near-horizon degrees of freedom. In this paper we argue that the latter paradigm implies the former. More specifically, we exploit suitable near-horizon boundary conditions that lead to an algebra of “soft hair charges” containing infinite copies of the Heisenberg algebra, associated with area-preserving shear deformations of black hole horizons. We employ the near-horizon soft hair and its Heisenberg algebra to provide a formulation of the membrane paradigm and show how it accounts for black hole entropy.

Soft charges and electric-magnetic duality

The main focus of this work is to study magnetic soft charges of the four dimensional Maxwell theory. Imposing appropriate asymptotic falloff conditions, we compute the electric and magnetic soft charges and their algebra both at spatial and at null infinity. While the commutator of two electric or two magnetic soft charges vanish, the electric and magnetic soft charges satisfy a complex U(1) current algebra. This current algebra through Sugawara construction yields two U(1) Kac-Moody algebras. We repeat the charge analysis in the electric-magnetic duality-symmetric Maxwell theory and construct the duality-symmetric phase space where the electric and magnetic soft charges generate the respective boundary gauge transformations. We show that the generator of the electric-magnetic duality and the electric and magnetic soft charges form infinite copies of iso(2) algebra. Moreover, we study the algebra of charges associated with the global Poincaré symmetry of the background Minkowski spacetime and the soft charges. We discuss physical meaning and implication of our charges and their algebra.

Localisation of soft charges, and thermodynamics of softly hairy black holes

Large gauge transformations (LGT) in asymptotically flat space are generated by charges defined at asymptotic infinity. No method for unambiguously localising these charges into the interior of spacetime has previously been established. We determine what this method must be, and use it to find localised expressions for the LGT charges. Applying the same principle to the case of a charged black hole spacetime leads to angle-dependent generalisations of the Smarr formula and the first law of black hole mechanics, both of which have important thermodynamical implications. In particular, the presence of a heat current intrinsic to the event horizon is observed.

Observable supertranslations

We show that large gauge transformations in asymptotically flat spacetime can be implemented by sandwiching a shell containing the ingoing hard particles between two finite-width shells of soft gauge excitations. Integration of the graviton Dirac bracket implies that our observable soft degrees of freedom obey the algebra imposed by Strominger on unobservable boundary degrees of freedom. Thus, we provide both a derivation and an observable realization of this algebra. The conservation laws associated with asymptotic symmetries are seen to arise physically from free propagation of infrared modes. This explains in physical terms our recent result that soft charges fail to constrain the hard scattering problem, and so cannot be relevant to the black hole information paradox.

Strolling along gauge theory vacua

We consider classical, pure Yang-Mills theory in a box. We show how a set of static electric fields that solve the theory in an adiabatic limit correspond to geodesic motion on the space of vacua, equipped with a particular Riemannian metric that we identify. The vacua are generated by spontaneously broken global gauge symmetries, leading to an infinite number of conserved momenta of the geodesic motion. We show that these correspond to the soft multipole charges of Yang-Mills theory.

Log corrections to entropy of three dimensional black holes with soft hair

We calculate log corrections to the entropy of three-dimensional black holes with “soft hairy” boundary conditions. Their thermodynamics possesses some special features that preclude a naive direct evaluation of these corrections, so we follow two different approaches. The first one exploits that the BTZ black hole belongs to the spectrum of Brown-Henneaux as well as soft hairy boundary conditions, so that the respective log corrections are related through a suitable change of the thermodynamic ensemble. In the second approach the analogue of modular invariance is considered for dual theories with anisotropic scaling of Lifshitz type with dynamical exponent z at the boundary. On the gravity side such scalings arise for KdV-type boundary conditions, which provide a specific 1-parameter family of multi-trace deformations of the usual AdS3/CFT2 setup, with Brown-Henneaux corresponding to z = 1 and soft hairy boundary conditions to the limiting case z → 0+. Both approaches agree in the case of BTZ black holes for any non-negative z. Finally, for soft hairy boundary conditions we show that not only the leading term, but also the log corrections to the entropy of black flowers endowed with affine û (1) soft hair charges exclusively depend on the zero modes and hence coincide with the ones for BTZ black holes.