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'\^etre of these soft-charges rests on less firm ground, insofar as their known derivations through generalized Noether procedures are not fully gauge invariant: these derivations rely on incomplete gauge fixations that leave only the new symmetries unfixed. An a priori reason for leaving out yet more general symmetries is missing, motivating further enlargements of the symmetries put forward in the literature. In this article, we propose that a geometrical framework anchored in the space of field configurations gives a rationale for the leading order soft charges in gauge theories. Our framework also explains why this infinite enhancement of the symmetry group is a property of asymptotic 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. Finally, our formalism suggests a new proposal for the origin of magnetic-type charges at asymptotic infinity.

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 are not fully gauge invariant: these derivations rely on incomplete gauge fixations that leave only the new symmetries unfixed. An a priori reason for leaving out yet more general symmetries is missing, motivating further enlargements of the symmetries put forward in the literature. In this article, we propose that a geometrical framework anchored in the space of field configurations gives a rationale for the leading order soft charges in gauge theories. Our framework also explains why this infinite enhancement of the symmetry group is a property of asymptotic 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. Finally, our formalism suggests a new proposal for the origin of magnetic-type charges at asymptotic infinity.
The work of Strominger and collaborators on the "softtriangle" ( [1] and references therein) unveiled fascinating and unexpected relations among (i) Weinberg's soft theorems and their generalizations [2][3][4], (ii) the so-called (nonlinear) memory effects [5][6][7], and (ii) an enlarged group of asymptotic symmetries for the S-matrix, in both gauge theories (and gravity). Strong evidence for these enlarged symmetries has been gathered by "reverseengineering" their charges from the soft theorems. However, while descriptions in the asymptotic phase space exist (e.g. [1,[8][9][10][11]), a derivation of these charges from first principles is still missing. In particular, the relationship between what is "gauge" and what is "symmetry" is most often clouded by the employment of gauge fixings that happen to leave only the sought after symmetry parameters unfixed. Moreover, to interpret the soft modes as Goldstone modes [1], the broken symmetries must be global and not gauge in nature. Hence the questions: can the distinction between gauge and global symmetries be made a priori? Does this distinction lead to the correct charges? 1 Early on, the suspicion emerged that these questions are related to the interplay of gauge symmetry with the (infinitely far) boundaries 2 of the spacelike Cauchy surfaces and of I ± . This intuition suggests in turn that new symmetries and charges should emerge even at finite (fiducial) boundaries ( [14] and e.g. [15,Sect.1]). Here, on the other hand, we employ a framework [16][17][18] that (i) is manifestly gauge invariant at boundaries, (ii) provides a criterion to distinguish gauge from symmetry, (iii) can be applied to infinite Cauchy surfaces as well as to finite bounded regions, (iv ) associates nonvanishing charges to symmetries but not to gauge, (v ) engenders an infinite dimensional group of symmetries only at I , and (vi) associates to these sym- * ariello [ at ] perimeterinstitute [ dot ] ca 1 Curiously, these issues are better understood in gravity [12]. 2 However, see [13], who implicitly suggests that the new symmetries have more to do with the null-ness of I than with gauge. metries charges that reproduce the leading (electric [10]) soft theorems. Items (iii), (v ) and (vi) are entirely new results. As a byproduct we will also uncover a quasi-local analogue of the radiative degrees of freedom [19] as well as a new proposal for the origin of magnetic-type charges at asymptotic boundaries. The techniques employed are geometric in nature, although the investigated geometry is that of field space. We will focus on Yang-Mills (YM) theories in 4 spacetime dimensions. In appendix A, we comment on generalizations in the presence of massive charged matter and to higher dimensions. The strategy we follow proceeds in two steps. First, we exploit the fiducial fibre bundle structure and the kinematic supermetric on the configuration space of the YM theory to construct geometrically a gauge-invariant quasi-local symplecitc potential. Through this gaugeinvariant symplectic potential, we can successfully parse physical symmetries from gauge transformations, and devise a gauge-invariant Noether procedure which assigns nonvanishing charges to the former only. On (portions of) the finite-time Cauchy surfaces, our gauge-invariant symplectic form carries only the quasi-local generalization of the radiative mode, and leaves aside the constrained Coloumbic components of the gauge field which are fully determined by the Gauss constraint. Only a finite number of gauge-invariant global symmetries and charges is nonvanishing. However, far into the future, close to future null infinity, our construction unveils a symmetry enhancement whose associated leading order charges coincide with the leading order (electric) soft charges. At asymptotic null infinity, two circumstances are fundamental: the induced metric on the hypersurface becomes (conformally) null and the gauge field is assumed to admit an expansion in inverse powers of the radial coordinate 3 . The latter circumstance, related to the peeling property, is a foundational component of the asymptotic-infinity formulation of isolated systems (see e.g. [20]). No gauge-fixing, or restriction on the gauge parameters, apart from their asymptotic expansion, is ever required. The origin of the new soft charges is not, as it is sometimes suggested (e.g. [15,Sect.1]), to be found in the generic breaking of gauge symmetry at the boundary of the Cauchy surface. On the contrary, ensuring manifest gauge-invariance both in the bulk and at the boundary is a cornerstone of the present approach.

I. CONFIGURATION SPACE GEOMETRY
Let {Σ t } t be a Cauchy foliation of a (3+1)-dimensional spacetime M ∼ = Σ t × R. For notational simplicity, we restrict our analysis to foliations of unit lapse N = 1 (cf. appendix B). However, we keep the shift β i nontrivial, since it will be relevant in the retarded-time coordinates needed for the asymptotic limit. Our focus is the gauge-variant configuration space of the YM theory with compact semi-simple charge group G associated to {Σ t } t . More precisely, we consider the (sub)configuration space Φ = {A ∈ Ω 1 (R, Lie(G))} associated to a finite bounded region R ⊂ Σ such that R ∼ = B 3 (the 3-dimensional ball) with ∂R ∼ = S 2 . Crucially, we will not demand gauge transformation to trivialize at ∂R. This is because we are interested in identifying the quasi-local gauge-invariant degrees of freedom, and this cannot depend on a choice of gauge-fixing at ∂R, especially when R is a fiducial subregion of Σ [16][17][18]21].
We will denote by n the future-pointing unit timelike normal to Σ in M , and by s the spacelike outgoing unit normal to ∂R in Σ.
The configuration space Φ has the structure of a fiducial, infinite-dimensional, principal fibre bundle π : Φ → Φ/G, A → [A]. Here, G ∼ = C ∞ (R, G), equipped with point-wise multiplication in G, is the group of gauge transformations, or "gauge group", and Φ/G is the space of "physical configurations" [22][23][24][25][26]. It is natural to provide Φ with two geometric structures. The first, associated to Φ's fibre bundle structure, is a functional connection-form ('Var-Pie'), The second is the ultralocal supermetric that can be read off the kinetic term of the YM Lagrangian, L = where g ij is the induced Riemannian metric on Σ t (henceforth, we will omit the measure dvol = √ gd D x).
a. The functional connection By construction, ̟ tells the pure-gauge variations of A from the physical ones, in a gauge-covariant manner. Let ξ ♯ ∈ X 1 (Φ) be the field-space vector associated to an infinitesimal gauge transformation ξ ∈ Lie(G). With a mild abuse of notation, 4 Then, the defining properties of ̟ can be expressed as where L is the field-space Lie-derivative and δ the (fiducial) field-space de Rham differential. To avoid confusion between field-space and spacetime quantities, we reserved double-struck symbols for field-space. In the second equation, the bracket [·, ·] is the Lie bracket in Lie(G). Hence, this equation states that ̟ transforms covariantly when transported along the gauge orbits in Φ (if ξ is a field-independent gauge transformation, then δξ ≡ 0, see [16][17][18]). The first condition states that ̟ is essentially a projector on the tangent space of the gauge orbit, V A = Span(ξ ♯ A ) ⊂ T A Φ, and therefore its kernel defines the physical, or "horizontal", directions H A . Given a generic variation δA, its horizontal component is Notice that although a (local) gauge fixing σ : Φ/G ֒→ Φ defines a unique ̟ such that ImTσ ⊂ H, the converse is not true: first, because the horizontal distribution H might not be Frobenious-integrable (if F = δ̟ + ̟ 2 = 0), and second, because ̟ is defined along the entire gauge orbit and thus cannot select any one section of Φ. At this point it is important to notice that ̟ is not uniquely defined, since the algebraic split TΦ = V ⊕ H ̟ is not canonical. This leads us to the next ingredient.
b. The Singer-DeWitt connection If Φ is equipped with a supermetric which is constant in the gauge directions L ξ ♯ G = 0, the orthogonal decomposition TΦ = V ⊥ H ⊥ defines a functional connection ̟ ⊥ [17,18,[22][23][24][25][26][27], through the condition that δ H A (5) is orthogonal to all vertical vectors ξ ♯ (3), This readily leads to the elliptic boundary value problem 4 More precisely: where the s subindex stands for the contraction with the spacelike normal s i and D i is the gauge covariant generalization of the Levi-Civita (LC) derivative on Σ. We will refer to the above connection as the Singer-DeWitt (SdW) connection. The boundary condition crucially follows from the fact that gauge transformations have not been trivialized at ∂R. In electromagnetism on a flat hypersurface, the above is a Poisson equation with Neumann boundary conditions. Its solutions are unique up to a constant. This remark is the seed of some fundamental considerations that we postpone to the next section on symmetry and charges. We conclude by stressing the fact that (7) imposes boundary conditions on ̟, and not on δA, which is free to take any value at ∂R or elsewhere. In particular, no boundary condition is imposed on the gauge field.
and the symbolȦ (recall that β i is the shift of the foliation {Σ t } t ). In this "3+1" decomposition, A t is a scalar on Σ t . Inspection of the YM action shows that A t is a Lagrange multiplier for the Gauss constraint, Also, from these equations, it follows that the symplectic potential of a YM theory, when written in configuration space variables (i.e. in terms of the velocity fields, instead of the momenta) and restricted to R, reads Now, the definition of the horizontal variations δ H A allows us to introduce the horizontal (pre)symplectic potential θ H by simply replacing δA with its horizontal counterpart: Notice that in these formulas, we are interpreting the electric field E as a vector field in Φ. This is possible because -forgetting D i A t for now -E i is essentially equal to ∂ t A i (albeit corrected for a nontrivial shift of {Σ t } t ), and ∂ t A i can be interpreted as the tangent "velocity vector" to the history of A i in configuration space. With this interpretation, Π = G(E, ·) ∈ T * Φ is the momentum of A i . Decomposing E, understood as a vector, into its horizontal and vertical components, we find for some ϕ ∈ C ∞ (Σ, Lie(G)) that transforms gaugecovariantly: δ ξ ϕ = [ϕ, ξ]. This decomposition corresponds to writing A t as which is particularly convenient, since ̟(Ȧ), under a gauge transformation ξ, can be checked to transform This uses (4) and the fact thatȦ is a vector on Φ and thus L ξ ♯Ȧ = ξ ♯ ,Ȧ TΦ is a Lie bracket between vectors. The formula holds also for field-dependent gauge transformations, provided ∂ t is interpreted as a total derivative d/dt = ∂/∂ t + (∂ t A)δ/δA (for details see [19, Appendix A]). Therefore, (14) automatically takes care of the nontrivial transformation properties that the Lagrange multipliers must satisfy to have full consistency under time-dependent gauge transformations, the inhomogeneous piece being taken care of by ̟(Ȧ), which depends only on the configuration variables. These considerations hold for any choice of ̟. But, for the SdW connection, ̟ = ̟ ⊥ and ϕ = ϕ ⊥ , there is more: this is the only functional connection such that (i) the Gauss constraint C G depends only on ϕ = ϕ ⊥ , and (ii) ϕ = ϕ ⊥ drops from the horizontal symplectic potential: the two facts being closely related: they mirror the two steps of symplectic reduction, where one first solves the constraint and then removes the "conjugate" degree of freedom. The first equation is easy to verify by recalling that SdW-horizontal vectors δA = δ ⊥ A are by definition in the kernel of ̟ ⊥ and therefore annihilate the rhs of (7); while the second equation follows from (12) and the definition of the SdW connection ̟ ⊥ as the one for which horizontality is given by G-orthogonality to the fibres π −1 ([A]) ⊂ Φ.
b. Quasi-local degrees of freedom To recapitulate, we used the kinetic supermetric G to define a preferred functional connection ̟ = ̟ ⊥ , named after Singer and DeWitt (SdW), according to the slogan "horizontality from fibre orthogonality". The preferred status of ̟ ⊥ results from the fact that the SdW connection is the only functional connection for which the non-dynamical component of E, i.e. the one fixed by the Gauss constraint, drops from θ H = θ ⊥ . In other words, ̟ ⊥ is the only choice of functional connection such that the corresponding gauge-invariant (horizontal) symplectic potential θ H = θ ⊥ contains only the quasi-local analogue of the radiative degrees of freedom, with no Coulombic contribution left. This is possible because of the role the kinetic supermetric G plays in the (usual) symplectic form θ (when expressed in configuration space).
Notice that the quasi-local radiative degrees of freedom, i.e. the SdW-horizontal perturbations δ ⊥ A are nonlocally built from a generic perturbation δA, since this requires solving (7). Thanks to the presence of boundary conditions in (7), the nonlocality is limited to the region R and no further information from the rest of Σ is required. Finally, none of these considerations is affected by the presence of matter.
E.g., for G = SU(N ), a scalar ψ ∈ C ∞ (Σ, V ), with V ∼ = R N the fundamental representation of G, can be incorporated as follows: where the charge density ρ is defined by Tr(ρξ) = 1 2 (ψξψ − ψξψ) for all ξ ∈ Lie(G), and where the matter horizontal vectors read δ ⊥ ψ = δψ + ̟ ⊥ ψ. Crucially, ̟ ⊥ is still defined purely in terms of δA according to (7). In electromagnetism, δ ⊥ ψ and δ ⊥ A i can be interpreted in terms of quasi-local generalization of the variations of a Dirac-dressed scalar field and of transverse photons, respectively. However, these interpretations are not fully satisfactory. They carry the same benefits and drawbacks of an interpretation of ̟ as a gauge-fixing: although it might provide an intuition, it captures only a limited aspect of the formalism and therefore is, in the end, inadequate [18].

III. SYMMETRIES AND CHARGES
As remarked above, in flat space electromagnetism, the defining equation of the SdW connection (7) admits unique solutions only up to constant offsets. This is an important observation, since χ = cte corresponds to elements in Lie(G) such that δ χ A = 0. This generalizes to non-Abelian YM theories as follows: at A ∈ Φ, the SdW connection is defined up to reducibility parameters of A, i.e. up to a χ ∈ Lie(G) such that δ χ A = 0. Reducibility parameters are the analogue of Killing vector fields in general relativity, and exist only at peculiar, reducible, configurations in Φ. The geometrical reason for a kernel in the above equations is that Φ fails to be a bona-fide principal fibre bundle, because certain fibres (i.e. gauge orbits) degenerate, making Φ/G a stratified manifold. The lower the stratum of Φ/G to which [A] belongs, the more symmetric [A] is [18].
On a compact and boundary-less hypersurface Σ, the Hamiltonian generator of the gauge transformation ξ is Q[ξ] = θ(ξ ♯ ) = θ(A, ψ; δ ξ A, δ ξ ψ) and vanishes on-shell of the Gauss constraint, Q[ξ] ≈ 0. This means that gauge transformations have trivial charges. In presence of boundaries, however, Q[ξ] ≈ q[ξ] for q[ξ] = ∂R Tr(ξE s ) = 0. This poses a puzzle: do gauge transformation at the boundary of a fiducial region R ⊂ Σ suddenly turn into physical symmetries carrying nontrivial charges? This sounds implausible, unless (possibly gauge breaking) boundary conditions are imposed at ∂R. This puzzle is resolved if one makes use of the "horizontal symplectic geometry" [16][17][18]. Indeed, the contraction of a vertical vector ξ ♯ with the horizontal form θ ⊥ identically vanishes, unless ξ = χ is a reducibility parameter of A, i.e. Dχ = 0, in which case the generator coincides with the physical charge density of the matter distribution [17,18]: In contrast to the other equations in this section, (20) depends on the use of the SdW-connection since, in this case, ̟ ⊥ (χ ♯ ) = 0. This means that a reducible vertical transformation χ ♯ is also horizontal, i.e. "physical", with respect to the SdW connection [16][17][18]. Therefore, we see that another property of the SdW connection is to select the physically meaningful charges associated to actual global symmetries of the gauge field configuration. Notice that, in YM theory, if a continuous family of reducibility parameters {χ(t)} t exists along the on-shell motion γ = {A i (t)} t ⊂ Φ, then χ(t) is a reducibility parameter of the (on-shell) spacetime connection A µ (t). To these spacetime reducibility parameters, one associates proper dynamically conserved charges, since then the spacetime divergence D µ (J µ χ) vanishes on-shell (see also [26,28]). Therefore, our configuration space reducibility parameters are natural candidates for dynamically conserved charges (and in the instantaneous configurations space Φ this is the best one can do). In the next section, we apply this construction to fields on Σ t=+∞ = I + . This construction will feature fundamentally new properties with respect to the finite-time hypersurface case.

IV. ASYMPTOTIC INFINITY
Consider Minkowski space in the retarded coordinates (t, u, y A ), with lapse N = 1 and shift β i dx i = −du, and consider a family of regions R t ∈ Σ t defined by u ∈ [u i , u f ]. In the late time limit, t → ∞: Σ t → I + , γ AB becomes the (conformal) metric on the celestial sphere S ∼ = S 2 , and the gauge field is assumed to approach the vacuum configuration A (0) at a certain rate.
In particular, the assumption is made that in a neigh-bourhood of I + the field admits an expansion 5 in powers of t (which at leading order, and at fixed u, coincides with Penrose's conformal factor): Hence, at late time, configuration space Φ has an extra structure that gauge transformations must also respect. Therefore, they also come layered in inverse powers of t: Notice that we are not gauge-fixing the vacuum A (0) , nor any of the subleading orders of A (at least to the extent to which they admit the above expansion, see footnote 3). Once again, we will leave it to the configuration space geometry to distinguish gauge from physical symmetries. Observe that, in the asymptotic limit, (7) becomes Thus, using the fact that physical symmetries χ ♯ are given by transformations which are simultaneously vertical and horizontal, i.e. in the kernel of (24), we immediately find that -thanks to the degenerate nature of the limit metric g ij (t) and to the layering of A ∈ Φ t→∞ and ξ ∈ Lie(G) t→∞ -the leading order χ (0) does not need to be a reducibility parameter of A (0) i , as was the case for finite-boundaries: it suffices that χ (0) is covariantly constant in retarded time: Since A (0) i is pure gauge (this is not needed in an Abelian theory), this equation gives us a celestial-sphere-worth of asymptotic symmetries for all configurations in Φ t→∞ , rather than just a finite number only at the reducible configurations. Physically, this is the consequence of points on the celestial sphere being infinitely separated from each other. To compute the asymptotic charge of the physical symmetries χ (0) through the first equality in (20), we must first define θ ⊥ on Φ t→∞ , verify its finiteness and hence evaluate it at δA = χ (0)♯ . Observe that even for t → ∞, θ ⊥ can be written as if A t is assumed to admit an expansion in t −1 . This follows from the general equality, G(E + Dφ, δ ⊥ A) = G(E, δ ⊥ A) for all φ admitting an expansion in t −1 , due to the defining relation of ̟ ⊥ to G. To proceed, we expanḋ 5 At least up to a certain order ko < ∞.
is the charge aspect, i.e. the asymptotic electric field generated by a charge in the spacetime's bulk. Notice that F where D A is the gauge-covariant LC derivative of γ AB on the celestial sphere S, and A (0) u is a scalar on S. Hence, combining this and (26), and usingȦ (0) where we abbreviated I + = u f ui du S (D−1) √ γd D−1 y. Finally, defining the charge as in the first equality of (20) and using (25), we obtain These charges are readily recognized as matching Strominger's "new asymptotic charges", or "soft charges" [1,29]. The term "soft" is due to the fact that, since χ (0) is constant in time (25), the charge only involves the zero-mode (in retarded time) of the momentum conjugate to the gauge-invariant photon field δ ⊥ A Using the Gauss law and performing the integration in du, the soft charge expression can also be written as Therefore, both the asymptotic symplectic potential (28) and the soft charges involve, from a spacetime perspective, Coulombic information for the electric field. This might be puzzling, since we stated that θ ⊥ does not contain this information on any Σ t . The explanation resides in the term β k F ki correcting the naive expression of E i for the presence of a nonvanishing shift β i : although at finite t this term simply adds some magnetic field to the canonical momentum, asymptotically it gets boosted so much that it captures the retarded-time dynamics of A (0) A as well as -via the Gauss law -the bulk's Coulombic information. Another surprising feature of this framework lies in the fact that at finite times nontrivial charges require a nonvanishing charge density ρ (20), while on I no matter charge density is required (29).

V. TOWARDS MAGNETIC SOFT CHARGES
In [10], the relevance of magnetic, rather than electric, soft charges was identified. The relation of these charges to gauge transformations raises some puzzles, since electro-magnetic duality is explicitly broken -even in vacuum -by the introduction of an electromagnetic gauge potential A µ subject to gauge transformations. This is of course related to the fact that since the magnetic charge density identically vanishes, one turns the would-be source equation for the magnetic field into an algebraic identity (Bianchi). In the present construction, this state of affairs is highlighted by the fact that only the (total) electric charge arises as a horizontal charge on a finite-time hypersurface Σ t . Therefore, the proposed field-space analysis is consistent on finite-time hypersurfaces. Again, the question arises, whether it is possible for it to recover magnetic charges asymptotically.
Here, although we will not attempt to perform a complete analysis that could fully recover the results of [9,10] (but see also [30] for a challenging example), we will still put forward the following observation: asymptotically magnetic charges can arise from considering foliations Σ t → I more general than the one considered in the previous section.
E.g. consider a rotating foliation in the axial direction, φ → φ + τ t −1 , that comes at leading order in t −1 with a shift β i dx i = −du − τ dφ + O(t −2 ) (recall that t −1 is our conformal parameter). This shift corresponds to a frame with a finite rotational velocity at infinity. This more general shift introduces in the expression of the electric field asymptotically conjugate to δA A , a new term, τ F φA , see (8,9).
This type of soft charge shares properties with the magnetic charges entering Low's subleading soft-theorem [9,10,31]. Once again, we did not prove that the present framework is capable of naturally recovering the subleading charges: this would require considerably more work and likely some new ideas. However, we did show that magnetic charges do not have to arise necessarily from an electromagnetic duality, but in certain circumstance can be a consequence of a "covariantization" of the choice of foliation approaching I .

ACKNOWLEDGEMENT
It is a pleasure to thank Henrique Gomes, with whom many of the ideas present in the first part of this paper have been developed. I would also like to thank Beatrice Bonga, for her valuable comments on an earlier draft. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Economic Development, Job Creation and Trade.