A
bstract
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.
A
bstract
We introduce a functional covariant differential as a tool for studying field space geometry in a manifestly covariant way. We then touch upon its role in gauge theories and general ...relativity over bounded regions, and in BRST symmetry. Due to the Gribov problem, we argue that our formalism — allowing for a non-vanishing functional curvature — is necessary for a global treatment of gauge-invariance in field space. We conclude by suggesting that the structures we introduce satisfactorily implement the notion of a (non-asymptotic) observer in gauge theories and general relativity.
Boundaries in gauge theories are a delicate issue. Arbitrary boundary choices enter the calculation of charges via Noether's second theorem, obstructing the assignment of unambiguous physical charges ...to local gauge symmetries. Replacing the arbitrary boundary choice with new degrees of freedom suggests itself. But, concretely, such boundary degrees of freedom are spurious—i.e. they are not part of the original field content of the theory—and have to disappear upon gluing. How should we fit them into what we know about field-theory? We resolve these issues in a unified and geometric manner, by introducing a connection 1-form, ϖ, in the field-space of Yang–Mills theory. Using this geometric tool, a modified version of symplectic geometry—here called ‘horizontal’—is possible. Independently of boundary conditions, this formalism bestows to each region a physical notion of charge: the horizontal Noether charge. The horizontal gauge charges always vanish, while global charges still arise for reducible configurations characterized by global symmetries. The field-content itself is used as a reference frame to distinguish ‘gauge’ and ‘physical’; no new degrees of freedom, such as group-valued edge modes, are required. Different choices of reference fields give different ϖ's, which are cousins of gauge-fixings like the Higgs-unitary and Coulomb gauges. But the formalism extends well beyond gauge-fixings, for instance by avoiding the Gribov problem. For one choice of ϖ, would-be Goldstone modes arising from the condensation of matter degrees of freedom play precisely the role of the known group-valued edge modes, but here they arise as preferred coordinates in field space, rather than new fields. For another choice, in the Abelian case, ϖ recovers the Dirac dressing of the electron.
Colored tensor models have been recently shown to admit a large
N expansion, whose leading order encodes a sum over a class of colored triangulations of the
D-sphere. The present paper investigates ...in details this leading order. We show that the relevant triangulations proliferate like a species of colored trees. The leading order is therefore summable and exhibits a critical behavior, independent of the dimension. A continuum limit is reached by tuning the coupling constant to its critical value while inserting an infinite number of pairs of
D-simplices glued together in a specific way. We argue that the dominant triangulations are branched polymers.
The eliminative view of gauge degrees of freedom—the view that they arise solely from descriptive redundancy and are therefore eliminable from the theory—is a lively topic of debate in the philosophy ...of physics. Recent work attempts to leverage properties of the QCD
θ
YM
-term to provide a novel argument against the eliminative view. The argument is based on the claim that the QCD
θ
YM
-term changes under “large” gauge transformations. Here we review geometrical propositions about fiber bundles that unequivocally falsify these claims: the
θ
YM
-term encodes topological features of the fiber bundle used to represent gauge degrees of freedom, but it is
fully
gauge-invariant. Nonetheless, within the essentially classical viewpoint pursued here, the physical role of the
θ
YM
-term shows the physical importance of bundle topology (or superpositions thereof) and thus counts against (a naive) eliminativism.
A
bstract
We present a systematic procedure to renormalize the symplectic potential of the electromagnetic field at null infinity in Minkowski space. We work in
D ≥
6 spacetime dimensions as a toy ...model of General Relativity in
D ≥
4 dimensions. Total variation counterterms as well as corner counterterms are both subtracted from the symplectic potential to make it finite. These counterterms affect respectively the action functional and the Hamiltonian symmetry generators. The counterterms are local and universal. We analyze the asymptotic equations of motion and identify the free data associated with the renormalized canonical structure along a null characteristic. This allows the construction of the asymptotic renormalized charges whose Ward identity gives the QED soft theorem, supporting the physical viability of the renormalization procedure. We touch upon how to extend our analysis to the presence of logarithmic anomalies, and upon how our procedure compares to holographic renormalization.
This is the first of a series of papers dedicated to the study of the partition function of three-dimensional quantum gravity on the twisted solid torus with the aim to deepen our understanding of ...holographic dualities from a non-perturbative quantum gravity perspective. Our aim is to compare the Ponzano–Regge model for non-perturbative three-dimensional quantum gravity with the previous perturbative calculations of this partition function. We begin by reviewing the results obtained in the past ten years via a wealth of different approaches, and then introduce the Ponzano–Regge model in a self-contained way. Thanks to the topological nature of three-dimensional quantum gravity we can solve exactly for the bulk degrees of freedom and identify dual boundary theories which depend on the choice of boundary states, that can also describe finite, non-asymptotic boundaries. This series of papers aims precisely at the investigation of the role played by the different quantum boundary conditions leading to different boundary theories. Here, we will describe the spin network boundary states for the Ponzano–Regge model on the twisted torus and derive the general expression for the corresponding partition functions. We identify a class of boundary states describing a tessellation with maximally fuzzy squares for which the partition function can be explicitly evaluated. In the limit case of a large, but finely discretized, boundary we find a dependence on the Dehn twist angle characteristic for the BMS3 character. We furthermore show how certain choices of boundary states lead to known statistical models as dual field theories – but with a twist.
We study the expectation value of a nonplanar Wilson graph operator in SL(2,C) Chern–Simons theory on S3. In particular we analyze its asymptotic behavior in the double-scaling limit in which both ...the representation labels and the Chern–Simons coupling are taken to be large, but with fixed ratio. When the Wilson graph operator has a specific form, motivated by loop quantum gravity, the critical point equations obtained in this double-scaling limit describe a very specific class of flat connection on the graph complement manifold. We find that flat connections in this class are in correspondence with the geometries of constant curvature 4-simplices. The result is fully non-perturbative from the perspective of the reconstructed geometry. We also show that the asymptotic behavior of the amplitude contains, at the leading order, an oscillatory part proportional to the Regge action for the single 4-simplex in the presence of a cosmological constant. In particular, the cosmological term contains the full-fledged curved volume of the 4-simplex. Interestingly, the volume term stems from the asymptotics of the Chern–Simons action. This can be understood as arising from the relation between Chern–Simons theory on the boundary of a region, and a theory defined by an F2 action in the bulk. Another peculiarity of our approach is that the sign of the curvature of the reconstructed geometry, and hence of the cosmological constant in the Regge action, is not fixed a priori, but rather emerges semiclassically and dynamically from the solution of the equations of motion. In other words, this work suggests a relation between 4-dimensional loop quantum gravity with a cosmological constant and SL(2,C) Chern–Simons theory in 3 dimensions with knotted graph defects.
We analyze the partition function of three-dimensional quantum gravity on the twisted solid torus and the ensuing dual field theory. The setting is that of a non-perturbative model of ...three-dimensional quantum gravity, the Ponzano–Regge model, which is here reviewed in a self-contained manner and then used to compute quasi-local amplitudes for its boundary states. In this second paper of the series, we choose a particular class of boundary spin network states which impose Gibbons–Hawking–York boundary conditions to the partition function. The peculiarity of these states is to encode a two-dimensional quantum geometry peaked around a classical quadrangulation of the finite toroidal boundary. Thanks to the topological properties of three-dimensional gravity, the theory easily projects onto the boundary while crucially still keeping track of the topological properties of the bulk. This produces, at the non-perturbative level, a specific non-linear sigma-model on the boundary, akin to a Wess–Zumino–Novikov–Witten model, whose classical equations of motion can be used to reconstruct different bulk geometries: the expected classical one is accompanied by other “quantum” solutions. The classical regime of the sigma-model becomes reliable in the limit of large boundary spins, which coincides with the semi-classical limit of the boundary geometry. In a 1-loop approximation around the solutions to the classical equations of motion, we recover (with corrections due to the non-classical bulk geometries) results obtained in the past via perturbative quantum General Relativity and through the study of characters of the BMS3 group. The exposition is meant to be completely self-contained.
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