Boundary electromagnetic duality from homological edge modes
BBoundary electromagnetic duality from homological edge modes
Philippe Mathieu ∗ and Nicholas Teh † University of Notre Dame (Dated: February 16, 2021)Recent years have seen a renewed interest in using ‘edge modes’ to extend the pre-symplectic struc-ture of gauge theory on manifolds with boundaries. Here we further the investigation undertaken inFreidel and Pranzetti [1] by using the formalism of homotopy pullback and Deligne-Beilinson coho-mology to describe an electromagnetic (EM) duality on the boundary of M = B × R . Upon breakinga generalized global symmetry, the duality is implemented by a BF-like topological boundary term.We then introduce Wilson line singularities on ∂M and show that these induce the existence of dualedge modes, which we identify as connections over a ( − − I. INTRODUCTION
In recent years, there has been a lot of interest in theuse of edge modes to construct an ‘extended phase space’,which yields gauge-invariant symplectic structures andcharges for gauge theories in the presence of a finiteboundary. The central idea at the heart of this appli-cation of edge modes is that the boundary symmetryshould not be conceived of as stemming from the ‘break-ing of gauge symmetry’ at the boundary, but rather as atrue physical symmetry that is related to gauge-invariantobservables.The above series of investigations was initiated by Don-nelly and Freidel in [2], who used edge modes to definea boundary (from the perspective of the Cauchy surface)symplectic potential that compensates for the failure ofgauge invariance of the bulk symplectic potential underfield-dependent gauge transformations. More recently,Freidel and Pranzetti [1] attempted to extend the aboveanalysis to construct EM duality only on the boundaryof a bulk spacetime, i.e. in the physically realistic case inwhich the bulk itself does not contain magnetic charges.Although such an EM duality was expected to hold frominvestigations into soft theorems and large gauge trans-formations in the asymptotic limit [3, 4], Freidel andPranzetti were the first to argue that one can constructthis EM duality without demanding that the large gaugetransformations play the role of a physical symmetry gen-erated by the charges. Furthermore, their analysis yieldsan important albeit heuristic result: The resulting mag-netic and electric boundary charges do not commute, butinstead give rise to a central charge.The reason that we call this result ‘heuristic’ is thatthe analysis in [1] does not properly take into accountthe global structure of fields on the boundary, as wellas the role that ‘singularities’ play in the derivation of acentral charge. Thus, one goal of this paper is to justifythe results of [1] by adopting a novel and systematic ap-proach to the construction of edge modes for boundary ∗ [email protected] † [email protected] EM duality. However, our approach goes much furtherthan merely justifying this result; it also allows us toanswer some fundamental questions concerning the ex-tended phase space construction from edge modes. Forinstance, what is the relationship between the originalextended pre-symplectic structure derived in [2] and thedistinct pre-symplectic structure that is necessary for theresults of [1] to hold? Our framework provides an answerto this question by introducing a mechanism for spon-taneous symmetry breaking on the boundary. Further-more, we find interesting links between our constructionand other topics that have seen a lot of recent interest,such as generalized global symmetries [5, 6], duality walls[7], the dressing field method [8], and the appearance ofhigher structures (i.e. gerbes) [7, 9] on the boundary ofa field theory.In Section II, we review the construction of Frei-del and Pranzetti [1] and note the necessity for a sys-tematic explanation of the edge modes, as well as theconstruction of a pre-symplectic structure that reflectsthe non-commutation of boundary electric and magneticcharges (which is induced by singularities in the bound-ary sphere). In Section III, we lay the groundwork for asystematic construction of the relevant edge modes. Inparticular, we review our previous work [10], where weshowed that the edge modes in [2] can be constructedby means of two ingredients: at the kinematical level,they arise when one implements topological boundaryconditions by means of a homotopy pullback, and at thedynamical level, the relevant bulk-to-boundary matchingconditions and boundary symplectic structure arise froma particular choice of boundary action (on the latter, seealso [11]).[12]The subsequent sections then apply these tools to con-struct EM duality on the boundary and non-commutingelectric and magnetic charges in codimension 2. SectionIV recalls that edge modes are intimately connected withthe ‘boundary’ spontaneous symmetry breaking of global1-form symmetries, and Section V explains that the low-energy limit of the symmetry-breaking action yields aboundary action for EM duality. In order to motivate theintroduction of the EM duality action that involves notonly edge modes, but also dual edge modes, Section VI a r X i v : . [ m a t h - ph ] F e b provides a description of regularized Wilson lines. Sec-tion VII then introduces an edge mode version of the EMduality action that includes a dressed Wilson line on theboundary of the theory; it is the latter (codimension 2)structure that induces the dual edge modes. We give anexplicit calculation showing how this results in the chargenon-commutation relation produced in [1], and note thatthe relevant boundary data can be formalized as a ‘non-trivial ( − II. REVIEW OF THE FREIDEL-PRANZETTICONSTRUCTION
In [1], Freidel and Pranzetti (FP) work with theLorentzian geometry M = B × R whose boundary is ∂M = S × R . They then impose the action S = 12 e (cid:90) M F ∧ (cid:63)F + (cid:90) M A ∧ (cid:63)J − π (cid:90) ∂M A ∧ d ˜ a, (1)where J is assumed to be a gauge-invariant current, and˜ a := d ˜ φ − ˜ A , where ˜ φ is a scalar edge mode living on ∂M which transforms as ˜ φ (cid:55)→ ˜ φ + ˜ (cid:15) when ˜ A undergoes theusual gauge transformation ˜ A (cid:55)→ ˜ A + d ˜ (cid:15) . ˜ φ is also knownas the dressing field and the gauge-invariant quantity ˜ a is the corresponding dressed field.FP then introduce by hand another edge mode φ ∈ Ω ( ∂M ) that is related to the gauge field A and use a := dφ − A to write down the ‘dressed form’ of thepre-symplectic form that follows from the action (1):Ω = 1 e (cid:90) Σ δA ∧ (cid:63)δF + 12 π (cid:90) ∂ Σ δa ∧ δ ˜ a, (2)where Σ = B is a Cauchy surface at some point in time.Finally, FP introduce what they call a ‘physical bound-ary symmetry’ φ (cid:55)→ φ − α, ˜ φ (cid:55)→ ˜ φ − ˜ α (an on-shell sym-metry) which acts only on the edge modes, and use it toderive the (on-shell) charges Q E = 1 e (cid:90) ∂ Σ α ∗ F (3) Q M = 0 , (4)where Σ is a spacelike Cauchy surface whose boundary ∂ Σ has the topology of S . Evidently, the commutator ofthese charges vanishes, but FP claim that upon imposinga ‘singularity’ in the A field on the boundary S the on-shell magnetic charge should be modified to Q M = (cid:90) S ˜ αa, (5)where S is a circle around the singular point on S , uponwhich the charge algebra is now centrally extended by { Q E .Q M } = − π (cid:90) S αd ˜ α. (6) While suggestive, this argument raises several ques-tions. First, when and how can one introduce the edgemodes in a fundamental way, instead of simply dressing‘by hand’ at the level of the pre-symplectic form? Secondand more importantly, the central charge (6) should arisefrom a particular pre-symplectic structure: What is thisstructure and how can one derive it from an action? Inwhat follows, we will try to answer these questions. In or-der to do so, one must (i) be very careful to construct theright gauge-invariant action from a generalized boundarycondition, and (ii) be explicit about what one means by‘singularities’ in the boundary sphere. We now turn tothe question of how the imposition of such boundary con-ditions is related to the existence of edge modes on theboundary. III. THE ORIGIN OF EDGE MODES:HOMOTOPICAL BOUNDARY CONDITIONS
In [10], we introduced the following way of imposing‘relaxed’ boundary conditions on the field content of agauge theory. Consider a U(1) gauge theory on M (whichis contractible). The space of kinematic bulk fields canbe formalized as BU(1) con ( M ), the groupoid of prin-cipal U(1) bundles with connection on M : more con-cretely, since M is contractible, we can think of this asthe groupoid whose objects are globally defined gaugefields A and whose morphisms are gauge transformations A (cid:55)→ A + dχ . We want to introduce a way of saying thatthe bulk fields restrict to a particular principal bundle(or principal bundle with connection) on ∂M ; however,we must be careful here, because it would be too strictto require that the restricted bulk field equals to someparticular boundary field configuration; instead, the cor-rect notion of comparison for gauge fields and bundlesis not equality but isomorphism — as we are about tosee, this simple point leads directly to the constructionof edge modes on the boundary.In order to implement this relaxed notion of bound-ary condition, we use the following ‘homotopy pullback’diagram: F ( M ) (cid:15) (cid:15) (cid:47) (cid:47) BU(1) con ( M ) h res (cid:15) (cid:15) {∗} p (cid:47) (cid:47) BU(1) ( ∂M ) (7)where p is a functor picking out a particular principalbundle in BU(1) ( ∂M ). We note that in this particulardiagram, we will impose the boundary condition of beinga trivial bundle ; however, in general one can constructa diagram that includes non-trivial bundles and connec-tions as boundary data if one likes, and we will find itnecessary to include connection data in Section VI. Thefield content resulting from imposing this boundary con-dition is derived by completing the pullback square toobtain the ‘homotopy pullback’ F ( M ).In Appendix A of [10], we provide a toolkit to computethe homotopy pullback F ( M ) for general field content. Inthis section, on the other hand, our gauge fields are globalobjects so it suffices to adopt an elementary and hands-onapproach to constructing the groupoid F ( M ). (In SectionVII, we will have to be more careful about computing F ( M ) because the boundary data is a non-trivial gerbewith connection; to perform this computation in a waythat is accessible to physicists, we have chosen to use theDeligne-Beilinson presentation of differential cohomologyas detailed in [14].)We now proceed to our elementary description of F ( M )for (7). First, an object of this groupoid is the pair ( A, φ ),where φ ∈ Ω ( ∂M ) is a morphism that relates p ( ∗ ) tores( A ), which is the boundary restriction of the bulk field.In other words, the ‘edge mode’ φ witnesses the statementthat the restriction of the bulk bundle is ‘the same as’(but not equal to) the trivial boundary bundle. Second,a morphism of this groupoid is a map that satisfies thefollowing commuting diagram: p ( ∗ ) = ∗ ϕ (cid:15) (cid:15) id ∗ (cid:47) (cid:47) ∗ = p ( ∗ ) ϕ (cid:48) (cid:15) (cid:15) res ( A ) = ∗ ε (cid:47) (cid:47) ∗ = res ( A (cid:48) ) (8)Thus, a morphism in F ( M ) is given by ( A, ϕ ) (cid:15) −→ ( A + d(cid:15), ϕ + ε ), where (cid:15) ∈ Ω ( M ). This is exactly thetransformation law that [2] posit for edge modes whenacted upon by the gauge symmetry. On the other hand,notice that the trivial boundary bundle p ( ∗ ) itself car-ries automorphisms that we can think of as an ‘external’transformation of the boundary condition: under suchan automorphism, A is left invariant but φ (cid:55)→ φ − α ,where α ∈ Ω ( ∂M ). The latter is precisely the trans-formation law for φ under what [1, 2] call the physical boundary symmmetry (as opposed to the gauge symme-try) of the edge mode. In other words, our constructionshows that edge modes come from a particular boundarycondition and that once one understands that boundarystructure, one obtains the so-called ‘physical boundarysymmetries’ for free. Finally, we emphasize that on theboundary ∂M , the edge modes can be used to ‘dress’ thegauge field, yielding the definition of the dressed photon a := dφ − A . IV. A HIGGS MODEL FROM EDGE MODES
We recall that [2] introduces a boundary pre-symplectic form Ω ∂ Σ that compensates for the failureof gauge invariance of the standard pre-symplectic formΩ Σ under field-dependent gauge transformations in thepresence of a finite boundary. For instance, in the caseof electromagnetism, the standard pre-symplectic formis Ω Σ = (cid:82) Σ (cid:63)δF ∧ δA and the boundary pre-symplecticform given in [2] is (cid:82) ∂ Σ (cid:63)δF ∧ δφ . Essentially, their con- struction proceeds by dressing the standard presymplec-tic form (which ensures gauge invariance), and noticingthat this leads to a boundary contribution. They alsouse gauge invariance to motivate the bulk-to-boundary‘matching condition’ (cid:63)F | ∂M = (cid:63) ∂ a , where (cid:63) ∂ denotesthe Hodge star operator with respect to the boundary ∂M .In the previous section, we showed that at the kine-matic level, such a dressing field φ naturally arises on theboundary by implementing our homotopical boundaryconditions. However, according to the covariant phasespace formalism, we need dynamical input in order todefine the boundary pre-symplectic form correspondingto such a φ : it should arise from a well-defined variationalproblem on the boundary. In [10] (see also [11]) we derivethe boundary pre-symplectic form given in [2] as well astheir bulk-to-boundary matching condition (indeed, wederive the symplectic structure of the entire BRST ex-tension of the theory) from the following action: S = 12 e (cid:90) M F ∧ (cid:63)F + t (cid:90) ∂M a ∧ (cid:63) ∂ a, (9)where (cid:63) ∂ is the Hodge star on ∂M and a := dφ − A can either be interpreted as the dressing of A , or as anaffine covariant derivative d A φ . The equations of motionresulting from varying this action are d (cid:63) F = 0, d (cid:63) a =0, and (cid:63)F | ∂M = (cid:63) ∂ a ; and the presymplectic 2-form is (cid:82) Σ (cid:63)δF ∧ δA + (cid:82) ∂ Σ (cid:63) ∂ δa ∧ δφ . We note that these structuresare of course gauge-invariant.When we view a as d A φ , it is clear that t a ∧ (cid:63) ∂ a pro-vides a description of the Higgs phase of the theory, i.e.it is the kinetic term for a charged scalar coupled to ourU(1) gauge theory. In other words, on the boundary ∂M ,the U(1) gauge symmetry of the bulk Maxwell action hasbeen Higgsed to a Z gauge symmetry. On the other hand,if we expand the boundary action in terms of A and φ ,we can also think of it as the Proca action for a mas-sive vector field A , where a Stuckelberg field φ has beenintroduced to maintain the gauge-invariance.[15]It is instructive to ask what happens to the global M when we couple it to the boundary action. We recall [5]that in a free Maxwell theory, there are two global 1-formsymmetries, viz. the electric and magnetic ones. Theelectric charge operator is (cid:82) C (cid:63)F and the magnetic chargeoperator is (cid:82) C F , where C is a surface. As discussed in [5],if one then goes on to Higgs the pure Maxwell theory bycoupling it to a charge N scalar, we break the global 1-form electric symmetry down to a Z N symmetry (so ourcase is simply the N = 1 case). The difference betweenthe scenario discussed in [5] and ours is that for us thisHiggsing only happens on the boundary ∂M , where wedo not need to postulate the existence of the Higgs field φ , because it arises for free from the homotopical bound-ary condition (7). One expects [16] that the breaking ofa 1-form global symmetry gives rise to a 1-form Gold-stone mode, and indeed the boundary term (cid:82) ∂M a ∧ (cid:63) ∂ a is identified in [17] as the Goldstone action. In otherwords, the emergent gauge-invariant degree of freedom issimply the dressed photon a , which transforms under thephysical boundary symmetry as a − dα . V. EM DUALITY IN THE LOW-ENERGYLIMIT
In the previous section, we established that Higgsinga U(1) theory on the boundary leads to the extendedphase space (and bulk-to-boundary matching conditions)invoked in the seminal paper [2] connecting edge modes tothe symplectic structure of gauge theories. However, theEM duality version of this construction [1] uses a differentaction, i.e. action (1), from the action given in [2], i.e.action (9). What then is the connection between thesetwo models (i.e. action (9) and action (1))? We notethat the key property of (1) is that it has a BF boundaryterm [18], and thus the connection is given by a standardargument [6] showing that a BF term appears when wetake the low energy limit( t → ∞ ) of the boundary termin (9). For completeness, we briefly review this argumentin our context.Consider again the boundary term (cid:82) ∂M a ∧ (cid:63) ∂ a in (9).We now dualize the edge mode φ by introducing a 2-formLagrange multiplier H and writing the boundary actionas 12 t (cid:90) ∂M H ∧ (cid:63) ∂ H + (cid:90) ∂M H ∧ a, (10)where the equivalence is demonstrated by varying H toyield the equation of motion (cid:63) ∂ H = t a . If we thenintegrate our edge mode φ from this form of the boundaryaction and take the low-energy limit t → ∞ , we obtainthe low-energy boundary action ˜ A ∧ F , where ˜ A is a 1-form such that H = d ˜ A .This low-energy boundary action looks like a BF actionin 3D, but as we are about to see, it differs from a freeBF theory whose equations of motion yield the flatnessof A and ˜ A . The new total (bulk plus boundary) actionis 12 e (cid:90) M F ∧ (cid:63)F + k π (cid:90) ∂M ˜ A ∧ F, (11)which yields the equation of motion (cid:63)F | ∂M = ˜ F := d ˜ A upon varying A , as well as a gauge-invariant presym-plectic potential and presymplectic form. We stress twopoints about this model. First, the equation of motionthat matches bulk to boundary fields is precisely thesource of the boundary EM duality. It is amusing tonote that this is nothing other than an elementary im-plementation of the duality wall formalism discussed in[7]: there, the prescription for witnessing a duality isto introduce a ‘wall operator’ on codimension 1 surfaceseparating the dual theories, and in the case of the S-transformation of EM duality, that wall operator takesthe form of a BF term. Second, the gauge-invariance of the presymplecticstructure relies on the fact that ∂M is closed and thatthere is no higher codimension stratum that would spoilthe gauge-invariance via Stokes’ theorem (thus obviatingthe need to introduce edge modes). However, as we willsoon see, in order to recover a non-commutation relationbetween electric and magnetic charges along the lines of[1], we need to introduce not only the edge modes comingfrom placing a (homotopical) boundary condition on thebulk U(1) bundle with connection A , but also the dual edge modes ˜ φ coming from placing a boundary conditionon the boundary dual bundle with connection ˜ A . In otherwords, we expect the dual edge modes to arise on a codi-mension 2 stratum (with respect to M ) that lies in theboundary ∂M . In the next section, we discuss how sucha stratum emerges from inserting electric Wilson linesinto the boundary, and in Section VII we use these edgemodes to write down a dressed action that leads to thedesired non-commutation relation. We will see that thisdressed action is equivalent to supplementing the action(11) with some codimension two data. VI. WILSON LINES FOR THE DRESSEDGAUGE FIELD
We recall that according to [1], the source of the non-trivial boundary magnetic charge and the central chargecan be traced to the existence of singularities piercingthe boundary sphere S . From the perspective of ourboundary BF theory, such singularities can be thoughtof as probe electric Wilson lines for the dressed photon a that are extended in time, i.e. as (cid:82) γ a along a timelikecurve γ = (cid:8) p t ∈ S × { t } , t ∈ R (cid:9) ⊂ ∂M . The curve γ can be visualized as piercing the spatial boundary S ×{ t } at p t for any time t ∈ R .Following [7], we now introduce a regularized descrip-tion of such a Wilson line into our theory. Let Z εγ = (cid:8) B ε ( p t ) , t ∈ R (cid:9) be a tubular neighborhood of γ where B ε ( p t ) is the 2-ball of radius ε centered at p t ∈ S × { t } .Hence, ∂Z εγ = (cid:8) S ε ( p t ) , t ∈ R (cid:9) .We can then construct a closed and co-closed 1-formΩ n on ∆ εγ = ∂M \ ◦ Z εγ (thus ∆ εγ ∼ = B × R ), with integralperiods (up to a 2 π factor) such that, for a fixed integer n :i) (cid:63) ∂ Ω n | ∂Z εγ = 0,ii) (cid:82) S ε ( p t ) Ω n = 2 πn for any t ∈ R .The existence of such a 1-form Ω n is proven in [7]. Akey point of the proof is the fact that the linking numberbetween γ and S ε ( p t ) is 1 for any t ∈ R by definition.The regularized Wilson line can then be defined as W ε ( γ ) = (cid:90) ∆ εγ a ∧ Ω n . (12)This picture can be generalized to the case of multipleWilson lines. In what follows, we will drop the index n of Ω n and the superscripts and subscripts of ∆ εγ . VII. DUAL EDGE MODES AND THECENTRAL CHARGE
In this section, we want to start with our homo-topy pullback construction and produce the kinematicresources to write down an action that1. Accommodates dressed boundary Wilson lines thatare analogous to (12).2. Reproduces the EM duality on ∂M .3. Gives rise to non-commuting electric and magneticcharges defined in terms of edge modes as conjec-tured in [1].We will do so by initially working with dressed fields onthe boundary, since this guarantees the gauge invarianceof the action under non-trivial boundary gauge trans-formations; but as we will soon see, it also leads to aninteresting reformulation in terms of undressed variables.From Section VI, we know that in order to includeregularized Wilson lines of the form (cid:82) ∆ A ∧ Ω, we need toconsider a submanifold ∆ ⊂ ∂M whose boundary ∂ ∆ isisomorphic to S × R . We thus proceed to formulate ourboundary bundles and action terms on ∆ and ∂ ∆. It willbe convenient for us to abuse language by also using ∆ torefer to the smallest open set (with respect to inclusion)containing ∆ (and thus ∂ ∆) as it is defined above.To be clear, there are two kinds of edge modes thatwe wish to construct in this section. First, the standardedge mode φ on the codimension 1 submanifold ∂M isconstructed in the same way as (7), i.e by means of thefollowing homotopy pullback square: F ( M ) (cid:15) (cid:15) (cid:47) (cid:47) BU(1) con ( M ) h res (cid:15) (cid:15) {∗} p (cid:47) (cid:47) BU(1) ( ∂M ) (13)Second, as we argued at the end of Section V, the dual edge modes ˜ φ should come from placing a codimension2 boundary condition on the boundary dual bundle thatlives on ∆. Thus, in this scenario the existence of thedual edge modes is actually induced by the regularizedWilson line, and we will take the relevant codimension 2surface to be ∂ ∆. More precisely, we will understand ˜ φ as arising from the following homotopy pullback,:˜ F (∆) (cid:15) (cid:15) (cid:47) (cid:47) BU(1) con (∆) h res (cid:15) (cid:15) {∗} p (cid:47) (cid:47) BU(1) − con ( ∂ ∆) (14) where we stress that we are now imposing a new kind ofboundary condition by means of the groupoid BU(1) − con .We will describe this groupoid explicitly in just a mo-ment, but the important thing to note is that it restrictsthe field content to a particular principal bundle on ∂ ∆(thus also restricting the transformations which preservethat data) and in addition incorporates a connection onthat bundle (which is natural because the Wilson linedata that induces ∂ ∆ involves a connection).In order to compute ˜ F (∆), we will first need to prop-erly define the elements at the other corners of diagram(14). In what follows, we will assume that ∆ is providedwith a good cover U = { U i } i (i.e. the open sets of U and their intersections are contractible or empty) whichinduces a good cover V = { V i } i on ∂ ∆.First, BU(1) con (∆) is the groupoid of U(1) connectionsover ∆. We remind the reader that a U(1)-connection ˜ A is a collection of local 1-forms ˜ A i in the U i ’s, a collectionof ˜Λ ij in the double intersections U ij = U i ∩ U j (i.e. thearguments of the transition functions of the U(1)-bundleover which the connection is defined) and a collection ofelements ˜ n ijk ∈ π Z in the triple intersections U ijk = U i ∩ U j ∩ U k (which define a ˇCech 2-cocyle indicatingthe isomorphism class of the U(1)-bundle over which theconnection is defined) satisfying the following so-calleddescent equations:˜ A i − ˜ A j − d ˜Λ ij = 0 in U ij (15)˜Λ ij − ˜Λ ik + ˜Λ jk − ˜ n ijk = 0 in U ijk (16)˜ n ijk − ˜ n ijl + ˜ n ikl − ˜ n jkl = 0 (17)More concisely, we can write this as D [1 , ˜ A = 0 (18)by means of the Deligne-Beilinson differential D [1 , = ˇ δ − d
00 ˇ δ − d δ (19)where ˇ δ is the ˇCech coboundary operator and d is theusual de Rham differential operator (which injects Z intothe set of constant functions over U ijk ) and˜ A = ˜ A i ˜Λ ij ˜ n ijk . (20)We say that ˜ A is a Deligne-Beilinson (DB) cocycle ofdegree 1, refering to the form degree of the local fields˜ A i , or a differential cocycle of degree 2 (where the latterconvention makes it possible to have a cup product thatis truly graded commutative). The set of DB 1-cocyclesover ∆ will be denoted as Z (∆).A general ‘gauge transformation’ of ˜ A is given by thefollowing set of relations:˜ A i −→ ˜ A i + d ˜ q i in U i (21)˜Λ ij −→ ˜Λ ij + ˜ q i − ˜ q j + ˜ m ij in U ij (22)˜ n ijk −→ ˜ n ijk + ˜ m ij − ˜ m ik + ˜ m jk in U ijk (23)or more concisely, ˜ A −→ ˜ A + D [1 , ˜ q (24)with D [1 , = d δ d δ (25)(note that D [1 , ◦ D [1 , = 0) and˜ q = (cid:18) ˜ q i ˜ m ij (cid:19) (26)As one can see from equations (21–23), this type of gaugetransformation corresponds to simultaneously perform-ing a bundle isomorphism (thus changing the represen-tative of a bundle isomorphism class) and a change ofsection. We say that ˜ q is a DB 0-cochain (the set ofDB 0-cochains over ∂ ∆ being denoted C ( ∂ ∆)) and D [1 , ˜ q is a DB coboundary of degree 1 (the set of DB1-coboundaries over ∂ ∆ being denoted B (∆)).In our present language, the groupoid of U(1)-connections over ∆ can now be described in terms ofthe following objects and morphisms:BU (1) con (∆) = (cid:40) Obj : ˜ A ∈ Z (∆)Mor : ˜ A ˜ q −→ ˜ A + D [1 , ˜ q, ˜ q ∈ C (∆) (27)This specifies the data of the upper right corner in dia-gram (14).Actually, we would also like to consider a more re-stricted scenario in order to specify the data of the lowerright corner in (14). Notice that if D [0 , ˜ q = 0 (28)where D [0 , = (cid:18) ˇ δ d δ (cid:19) (29)then the gauge transformation ˜ A −→ ˜ A + D [1 , ˜ q corre-sponds to a change of section on the same bundle. In thiscase, we will say that ˜ q is a 0-cocycle and use Z (∆) todenote the set of 0-cocycles over ∆. In the mathematicsliterature, it is customary to refer to ˜ q as a connectionover a ( − gerbe over ∆, a concept that generalizes 0-gerbes, which are nothing but U(1)-bundles. The differ-ence between (25) and (29) is a matter of truncation ofthe ˇCech-de Rham complex for the construction of DB complex, see [14] and Appendix C for more details. Thedata for the lower right corner in (14) can now be con-cisely described as the groupoid BU(1) − con ( ∂ ∆):BU(1) − con ( ∂ ∆) = (cid:40) Obj : ˜ B ∈ Z ( ∂ ∆)Mor : ˜ B ˜ φ −→ ˜ B + D [1 , ˜ φ, ˜ φ ∈ Z ( ∂ ∆) (30)where the morphisms are connections over ( − F (∆) of the diagram (14). First recall that the func-tor p just selects an element of BU(1) − con ( ∂ ∆), while thefunctor res restricts to ∂ ∆ the element of BU(1) con (∆),which is defined over ∆. Then, by definition of thehomotopy pullback, the objects of ˜ F (∆) are triplets (cid:16) ∗ , ˜ A, ˜ φ (cid:17) ∈ {∗} × Z (∆) × Z ( ∂ ∆) (and we willomit the first component ∗ in the following). A mor-phism (cid:16) ˜ A, ˜ φ (cid:17) −→ (cid:16) ˜ A (cid:48) , ˜ φ (cid:48) (cid:17) in ˜ F (∆) is a pair of mor-phisms (cid:16) id ∗ : ∗ −→ ∗ , ε : A −→ ˜ A (cid:48) = ˜ A + D [1 , ˜ ε (cid:17) that iscompatible with ˜ φ and ˜ φ (cid:48) , i.e. such that the diagram ∗ ˜ φ (cid:15) (cid:15) id ∗ (cid:47) (cid:47) ∗ ˜ φ (cid:48) (cid:15) (cid:15) res (cid:16) ˜ A (cid:17) ˜ ε (cid:47) (cid:47) res (cid:16) ˜ A (cid:48) (cid:17) (31)commutes. Hence, a morphism in ˜ F (∆) is given by (cid:16) ˜ A, ˜ φ (cid:17) −→ (cid:16) ˜ A (cid:48) = ˜ A + D [1 , ˜ ε, ˜ φ (cid:48) = ˜ φ + ˜ ε (cid:17) , where ˜ ε ∈ Z (∆) (and not simply C (∆) since both ˜ φ and ˜ φ (cid:48) are 0-cocycles) and we can rewrite˜ F (∆) = Obj : (cid:16) ˜ A, ˜ φ (cid:17) ∈ Z (∆) × Z ( ∂ ∆)Mor : (cid:16) ˜ A, ˜ φ (cid:17) ˜ ε −→ (cid:16) ˜ A + D [1 , ˜ ε, ˜ φ + ˜ ε (cid:17) , ˜ ε ∈ Z (∆)(32)Now, we can introduce the covariant derivative of ˜ φ : D A φ = D [1 , ˜ φ − ˜ A (33)or, written differently: D A ˜ φ = d
00 00 0 (cid:18) ˜ φ i ˜ m ij (cid:19) − ˜ A i ˜Λ ij ˜ n ijk = d ˜ φ i − ˜ A i − ˜Λ ij − ˜ n ijk (34)since D [0 , acts by definition like the null operator on Z ( ∂ ∆) and can be regarded as a submatrix of D [1 , .At this point, several remarks are in order. First, recallthat the n -gerbes over a manifold Y are classified upto isomorphism by H n +2 ( Y ). So here, the ( − ∂ ∆ are classified up to isomorphism by H ( ∂ ∆) ∼ = H (cid:0) S (cid:1) ∼ = Z . Hence, there are nontrivial ( − φ oversuch ( − R -valued function (the exponent of the transition function)globally defined over ∂ ∆. In fact, once we introduce anappropriate action, the cohomology class in H ( ∂ ∆) thatindexes the ( − A is globally defined.Hence, the first component of D A φ is globally definedalso (recall that since D [0 , ˜ φ = 0, the functions ˜ φ i in the U i ’s differ by integers in the U ij ’s). From now on, we willdenote this ‘dressed’ quantity as ˜ a :˜ a := d ˜ φ i − ˜ A (35)This will be our main (gauge-invariant) building block inthe following.We have now completed the computation of the fieldcontent arising from the homotopy pullback, and areready to use it in the following manifestly gauge-invariantaction: 12 e (cid:90) M F ∧ (cid:63)F + k π (cid:90) ∆ a ∧ ˜ F − k π (cid:90) ∂ ∆ a ∧ ˜ a + p π (cid:90) ∂ ∆ a ∧ Ω , (36)where k and p are integers, and the last term is the Wil-son line term introduced in Section VI. We can straight-forwardly apply the covariant phase space formalism to(36) to compute the corresponding charges and symplec-tic structure: we do so in (45)–(49) and (58)–(65) belowand show that one obtains precisely the charges that FPfind in [1], as well as the desired central charge.Before proceeding to these results, however, it is in-structive to consider an alternative form of the actionthat yields the same equations of motion and symplecticstructure, but which yields a different magnetic charge.Integrating (36) by parts, we note that up to an exactterm, the action (36) is equivalent to the following com-bination of undressed fields:12 e (cid:90) M F ∧ (cid:63)F − k π (cid:90) ∆ ˜ A ∧ F + k π (cid:90) ∂ ∆ ˜ φ ∧ F + p π (cid:90) ∂ ∆ a ∧ Ω , (37)where we now see that we have transformed this actioninto the form of our earlier low-energy action (11) alongwith two additional terms that live on a codimension 2stratum. Evidently, the third term has to be understood in the following way: (cid:90) S × R ˜ φdA = n − (cid:88) i =1 (cid:90) > P i P i +1 × R ˜ φ i dA − n (cid:88) i =1 (cid:90) P i × R ˜ m i ( i +1) A (38)where the P i ’s are points on S and˜ φ i − ˜ φ i +1 = ˜ m i ( i +1) ∈ π Z (39)in a neighborhood of P i +1 (which is nothing but a conse-quence of the fact that D [0 , φ = 0), where the elements˜ φ i and ˜ m ij transform as˜ φ i −→ ˜ φ i + ˜ n i (40)˜ m ij −→ ˜ m ij + ˜ n i − ˜ n j (41)that is ˜ φ −→ ˜ φ + D [0 , − ˜ n (42)where D [0 , − = (cid:18) d ˇ δ (cid:19) (43)and ˜ n = (˜ n i ) (44)An important remark is in order here: Since ˜ φ is definedonly locally, different choices of local representatives andchoices of a geometric decomposition like the P i ’s in (38)would lead to integrals that differ by n ∈ π Z . Thus, onlythe complex exponential of those integrals (which are thenatural quantum objects) is well-defined. However, forwhat we are interested in, i.e. the symplectic chargesand bracket, this ambiguity does not matter because itis eliminated by the functional derivative.The fact that (37) yields the same equations of motionand pre-symplectic form as (36) but not the same chargescan be traced to part of the so-called JKM ambiguity[19]: under the transformation L (cid:55)→ L + dK , we have onthe boundary S ∂ (cid:55)→ S ∂ + (cid:82) ∂ K and the pre-symplecticpotential θ (cid:55)→ θ + δK while the symplectic structure Ω = δθ remains invariant.There are several reasons to be interested in the formof the action given in (37). First, we note that this isessentially the extended BF action discussed in [11], andthus on the basis of that analysis, we would expect a cen-tral charge to arise. Second, the term ˜ φ ∧ F that is neededto compensate for the failure of gauge invariance of ˜ A ∧ F (in the presence of a codimension 2 stratum) can be giventhe following interpretation: it is a 2-dimensional BF ac-tion term that can function as a duality wall operator.Thus, what we would expect is that such a wall operatorterm in the action serves to dualize the electric Wilsonloop (cid:82) γ a . We now see that this is confirmed by vary-ing the action (36) to obtain the following equations ofmotion: d (cid:63) dA = 0 on M , (45) (cid:63)dA = e k π d ˜ A on ∆ , (46) k π ˜ a = p π Ω on ∂ ∆ , (47) (cid:63)dA = 0 on ∂M \ ∆ , (48) F = 0 on ∆ (and ∂ ∆) . (49)Equation (45) is nothing but the Maxwell equation,whereas (46) is the duality equation that enforces thecodimension 1 EM duality. Equation (47) may look sur-prising. Indeed, ˜ a = d ˜ φ − ˜ A and ˜ A is globally defined on∆ which is contractible. Hence, by the Poincar´e lemma,˜ A has to be exact, which might give the impression thatΩ itself is exact, which would contradict the fact that Ωis a nontrivial de Rham cocycle. The reason this impres-sion is false is that ˜ φ is not globally defined, although d ˜ φ is. Hence, (cid:90) S ε ( p t ) ˜ a = (cid:90) S ε ( p t ) d ˜ φ − ˜ A = (cid:90) S ε ( p t ) d ˜ φ (50)by Stokes theorem, since ˜ A is truly exact. Remark thatthe last integral in equation (50) formally looks like afirst Chern number, except that in this case it is com-puted from a connection over a ( − , ∂ ∆),in terms of the Chern number associated with the rel-ative Chern class of the bundle over ∆ with a chosentrivialization over ∂ ∆. The computation of this remain-ing integral can be treated in a similar way as (38). Usingthe same notation as before, we have: (cid:90) S ε ( p t ) d ˜ φ = n − (cid:88) i =1 (cid:90) > P i P i +1 d ˜ φ i (51)= n − (cid:88) i =1 (cid:16) ˜ φ i ( P i +1 ) − ˜ φ i ( P i ) (cid:17) = n − (cid:88) i =1 (cid:16) ˜ φ i ( P i +1 ) − ˜ φ i +1 ( P i +1 ) (cid:17) = n − (cid:88) i =1 ˜ m i ( i +1) ∈ π Z . (52)Hence, we can truly write: k π (cid:90) S ε ( p t ) ˜ a = p π (cid:90) S ε ( p t ) Ω (53)Note that since the ˜ m i ’s define a class in H ( ∂ ∆) thatindexes the isomorphism class of the ( − ∂ ∆ over which the connection ˜ φ is defined, it is clear thatΩ determines this class on-shell through the equation ofmotion. It is thus essential to have a nontrivial gerbe,otherwise ˜ φ would be globally defined and (51) wouldsimply be zero, making the equation of motion (47) im-possible to satisfy in general.We now proceed to computing the presymplectic po-tential from the action (36): θ = 1 e (cid:90) Σ δA ∧ (cid:63)dA + k π (cid:90) ∆ ∩ Σ δφ ∧ d ˜ A − k π (cid:90) ∂ ∆ ∩ Σ δφ ∧ ˜ a + p π (cid:90) ∂ ∆ ∩ Σ δφ ∧ Ω − k π (cid:90) ∆ ∩ Σ a ∧ δ ˜ A + k π (cid:90) ∂ ∆ ∩ Σ a ∧ δ ˜ φ (54)where Σ is a Cauchy surface, that is, a spacelike sub-manifold of M corresponding to B × { t } where a tubu-lar/disk neighborhood of a puncture on ∂B = S hasbeen removed so that ∂ Σ ∼ = B × { t } . Note that δ ˜ φ isglobally defined. Indeed, if we apply δ (functional differ-ential) to (39), we obtain δ ˜ φ i − δ ˜ φ i +1 = δ ˜ m i ( i +1) (55)and we claim δ ˜ m i ( i +1) = 0 , (56)which is equivalent to saying that, when we vary a con-nection ˜ φ over a gerbe whose isomorphism class is in-dexed by ˜ m , we vary it among the connections over thegerbes of the same isomorphism class. The intuitive ideabehind this claim is the following: The space of (piece-wise) smooth Z -valued functions is made of disconnectedsheets (indexed by ˜ m ), so one cannot continuously distortan element out of its sheet; hence we have δ ˜ φ i = δ ˜ φ i +1 =: δφ. (57)On-shell, the presymplectic potential simplifies to: θ = 1 e (cid:90) Σ δA ∧ (cid:63)dA + k π (cid:90) ∆ ∩ Σ δφ ∧ d ˜ A − k π (cid:90) ∆ ∩ Σ a ∧ δ ˜ A + k π (cid:90) ∂ ∆ ∩ Σ a ∧ δ ˜ φ (58)We want to consider now a vector field that generatesthe (on-shell) physical symmetry of the edge modes. Forthat, we dualize δφ and δ ˜ φ by δδφ and δδ ˜ φ respectively,which are globally defined, since δφ and δ ˜ φ are. Thedualization works as: δδφ ( x ) ( δφ ( y )) = δ ( x − y ) (59)and identically for δ ˜ φ and δδ ˜ φ .From the homotopy pullback, the vector field that gen-erates the symmetry of φ is globally defined, so it can bewritten as δ α := (cid:90) ∂M α δδφ (60)(recall that α is of the same type as ε ), whereas the vectorfield that generates the symmetry of ˜ φ is only locallydefined because of the coefficients ˜ α i (which are of thesame type as ˜ ε ): δ ˜ α := ( δ ˜ α i ) := (cid:18)(cid:90) ∂M ˜ α i δδ ˜ φ (cid:19) . (61)Following [1], we can use these symmetries to calculatethe electric charge: Q E = θ ( δ α )= k π (cid:90) x ∈ ∆ ∩ Σ δφ ( x ) (cid:18)(cid:90) y ∈ ∂M α ( y ) δδφ ( y ) (cid:19) ∧ d ˜ A ( x )= k π (cid:90) x ∈ ∆ ∩ Σ (cid:18)(cid:90) y ∈ ∂M α ( y ) δφ ( x ) (cid:18) δδφ ( y ) (cid:19)(cid:19) ∧ d ˜ A ( x )= k π (cid:90) x ∈ ∆ ∩ Σ (cid:18)(cid:90) y ∈ ∂M α ( y ) δ ( x − y ) (cid:19) ∧ d ˜ A ( x )= k π (cid:90) x ∈ ∆ ∩ Σ α ( x ) ∧ d ˜ A ( x ) Q E = k π (cid:90) ∆ ∩ Σ α ∧ d ˜ A (62)and in the same manner the collection of magneticcharges in each open set U i : Q M = (cid:0) Q Mi (cid:1) = ( θ ( δ ˜ α i )) = (cid:18) k π (cid:90) ∂ ∆ ∩ Σ a ∧ ˜ α i (cid:19) . (63)The symplectic structure is calculated (on-shell) as:Ω = − δθ = 1 e (cid:90) Σ δA ∧ (cid:63)dδA + k π (cid:90) ∆ ∩ Σ δφ ∧ dδ ˜ A + k π (cid:90) ∆ ∩ Σ δa ∧ δ ˜ A − k π (cid:90) ∂ ∆ ∩ Σ δa ∧ δ ˜ φ (64)from which we finally obtain the bracket of charges: (cid:8) Q E , Q M (cid:9) = Ω ( δ α , δ ˜ α ) = − k π (cid:90) ∂ ∆ ∩ Σ α ∧ d ˜ α i . (65)This is precisely the central charge that was suggested in[1], for which we have now provided a systematic justi-fication. We note that, strictly speaking, what we havehere is a collection of local brackets (cid:8) Q E , Q Mi (cid:9) whichturn out to be globally well-defined since they involve d ˜ α i , which is a globally well-defined 1-form. VIII. THE SCALAR-2-FORM DUALITY CASE
The situation we studied in the previous section canbe generalized to the case where A is a global n -formover M (that can be interpreted as a connection over atrivial ( n − M ), φ is a global ( n − ∂M (that can be interpreted as a connection overa trivial ( n − ∂M ), ˜ A is a global m -formover ∆ (that can be interpreted as a connection over atrivial ( m − φ is a connection overa non-trivial ( m − ∂ ∆. In this case, M = B m + n +1 × R , so ∂M = S m + n × R , ∆ = B m + n × R ⊂ ∂M and ∂ ∆ = S m + n − × R .Indeed, one can generalize the scenario even furtherto manifolds M = N × R , N being any manifold withboundary, with nontrivial gerbes for A ( A being then amore general element in Z n DB ( M )), φ ( φ being then amore general element in Z n − ( ∂M )), ˜ A ( ˜ A being then amore general element in Z m DB (∆)) and ˜ φ ( ˜ φ being thena more general element in Z m − ( ∂M )). Recall that p -gerbes over a manifold X are classified up to isomorphismby H p +2 ( X ), a 0-gerbe being nothing but a usual U(1)-bundle.We now apply this framework to the much simpler caseconsidered in [13], where n = 2 and m = 0 on B × R .We have ∆ = ∂M = S × R , so ∂ ∆ = ∅ ; in other words,we have no (regularized) puncture, and therefore no edgemode ˜ φ , i.e. no connection over a ( − B . While the authors introduce magnetic monopoles for B , they note that another way of describing the scenariowould be to keep the field approach for massive particlesand only introduce the duality at the boundary. Thelatter (which is of course the approach of [1]) is the de-scription that we will now provide, albeit in the case ofa finite (as opposed to asymptotic) boundary.Using the notations of [13], the action we will nowconsider is: − (cid:90) M H ∧ (cid:63)H + (cid:90) ∂M b ∧ dψ (66)where b = dω − B ( a = dφ − A with the previous no-tations) is the dressed field B ( A with the previous no-tations) and H = dB ( F = dA with the previous nota-tions). The field B is assumed to be a globally defined2-form (more abstractly, this is a connection over a 1-gerbe, that is, in general, a collection of 2-forms definedin the open sets of a good cover satisfying some descentequations, but here the gerbe is trivial and we choosea representative of the connection that is globally de-fined) and the edge mode ω is a globally defined 1-form.The pair ( B, ω ) transforms as (
B, ω ) −→ ( B + dβ, ω + β )where β is a globally defined 1-form. This makes b gaugeinvariant. We note that the transformation ω −→ ω + γ (where γ is a globally defined 1-form) alone (i.e. no as-sociated transformation B −→ B + dγ ) is an on-shell0symmetry of the action. Finally, the scalar field ψ ( ˜ A with the previous notations) is also assumed to be glob-ally defined. It transforms as ψ −→ ψ + dn where n isan integer and d represents the operator that formallyinjects the set of integers into the set of functions. Wenote that ψ is here a connection over a ( − − H (∆) = 0,there is only one isomorphism class of ( − ∂M , so ∂ ∆ = ∅ , there is no dual edge mode ( ˜ φ with the previousnotations).The equations of motion are: d (cid:63) dB =0 on M, (67) (cid:63)dB = − dψ on ∂M, (68) dB =0 on ∂M. (69)The symplectic potential is then: θ = − (cid:90) Σ δB ∧ (cid:63)dB (70)+ (cid:90) ∂M ∩ Σ δω ∧ dψ + (cid:90) ∂M ∩ Σ b ∧ δψ (71)and we can compute the charge associated with the edgemode ω : Q = θ ( δ α ) = (cid:90) ∂M ∩ Σ α ∧ dψ (72)which is, on-shell, Q = θ ( δ α ) = − (cid:90) ∂M ∩ Σ α ∧ (cid:63)H (73)This form of the charge looks almost the same as the2-form charge computed in [13], but with one crucial dif-ference: the symmetry used in [13] to compute the chargeis a generalized global symmetry of B , whereas here it isa physical symmetry of the edge mode ω that comes fromthe homotopy pullback. ACKNOWLEDGMENTS
We thank Laurent Freidel, Daniele Pranzetti, Alexan-der Schenkel, Stephan Stolz, Pavel Mnev, KonstantinWernli and Donald Youmans for their helpful remarkson the present work and our fruitful discussions.
Appendix A: Detailed computations of the equationsof motion
We consider the following action S = 12 e (cid:90) M dA ∧ (cid:63)dA + k π (cid:90) ∆ a ∧ d ˜ A (A1) − k π (cid:90) ∂ ∆ a ∧ ˜ a + p π (cid:90) ∂ ∆ a ∧ Ω (A2) where a = dφ − A and ˜ a = d ˜ φ − ˜ A .Let’s vary S with respect to A : ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 S ( A + tδA ) (A3)= 1 e (cid:90) M dδA ∧ (cid:63)dA − k π (cid:90) ∆ δA ∧ d ˜ A (A4)+ k π (cid:90) ∂ ∆ δA ∧ ˜ a − p π (cid:90) ∂ ∆ a ∧ Ω (A5)= 1 e (cid:90) M δA ∧ d (cid:63) dA + 1 e (cid:90) M d ( δA ∧ (cid:63)dA ) (A6) − k π (cid:90) ∆ δA ∧ d ˜ A + k π (cid:90) ∂ ∆ δA ∧ ˜ a (A7) − p π (cid:90) ∂ ∆ a ∧ Ω (A8)leading to the four first equations of motion (45) – (48).Now, let’s vary S with respect to φ : ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 S ( φ + tδφ ) (A9)= k π (cid:90) ∆ dδφ ∧ d ˜ A − k π (cid:90) ∂ ∆ dδφ ∧ ˜ a (A10)+ p π (cid:90) ∂ ∆ dδφ ∧ Ω (A11)= k π (cid:90) ∆ d (cid:16) δφ ∧ d ˜ A (cid:17) − k π (cid:90) ∂ ∆ d ( δφ ∧ ˜ a ) (A12)+ k π (cid:90) ∂ ∆ δφ ∧ d ˜ A + p π (cid:90) ∂ ∆ d ( δφ ∧ Ω) (A13)leading to d ˜ A = 0 on ∂ ∆ (A14)which is actually weaker than (47), as it is obtained bydifferentiating (47), taking into account the fact that Ωis closed.Now, let’s vary S with respect to ˜ A : ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 S (cid:16) ˜ A + tδ ˜ A (cid:17) (A15)= k π (cid:90) ∆ a ∧ dδ ˜ A + k π (cid:90) ∂ ∆ a ∧ δ ˜ A (A16)= − k π (cid:90) ∆ dA ∧ δ ˜ A − k π (cid:90) ∆ d (cid:16) a ∧ δ ˜ A (cid:17) (A17)+ k π (cid:90) ∂ ∆ a ∧ δ ˜ A (A18)from which we get our last equation of motion (49).Finally, let’s vary S with respect to ˜ φ : ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 S (cid:16) ˜ φ + tδ ˜ φ (cid:17) (A19)= − k π (cid:90) ∂ ∆ a ∧ dδ ˜ φ (A20)= k π (cid:90) ∂ ∆ dA ∧ δ ˜ φ + k π (cid:90) ∂ ∆ d (cid:16) a ∧ δ ˜ φ (cid:17) (A21)1which leads to a weaker version of (49) (the same equa-tion but on ∂ ∆ only). Appendix B: Explicit verification of the gaugeinvariance of the presymplectic potential. θ ( A + dε, φ + ε ) − θ ( A, φ ) (B1)= 1 e (cid:90) Σ dδε ∧ (cid:63)dA + k π (cid:90) ∆ ∩ Σ δε ∧ d ˜ A (B2) − k π (cid:90) ∂ ∆ ∩ Σ δε ∧ ˜ a + p π (cid:90) ∂ ∆ ∩ Σ δε ∧ Ω (B3)= − e (cid:90) Σ dδε ∧ d (cid:63) dA (B4)+ 1 e (cid:90) Σ d ( δε ∧ (cid:63)dA ) + k π (cid:90) ∆ ∩ Σ δε ∧ d ˜ A (B5) − k π (cid:90) ∂ ∆ ∩ Σ δε ∧ ˜ a + p π (cid:90) ∂ ∆ ∩ Σ δε ∧ Ω (B6)=0 (B7)on-shell since ∂ Σ = − ∆ ∩ Σ (Indeed, both ∂ Σ and ∆ ∩ Σare homeomorphic to B ×{ t } but ∂ Σ is reached throughΣ and ∆ ∩ Σ is reached through ∆ so they have oppositeorientation.) θ (cid:16) ˜ A + d ˜ ε, ˜ φ + ˜ ε (cid:17) − θ (cid:16) ˜ A, ˜ φ (cid:17) (B8)= − k π (cid:90) ∆ ∩ Σ a ∧ dδ ˜ ε + k π (cid:90) ∂ ∆ ∩ Σ a ∧ δ ˜ ε (B9)= k π (cid:90) ∆ ∩ Σ dA ∧ δ ˜ ε (B10)+ k π (cid:90) ∆ ∩ Σ d ( a ∧ δ ˜ ε ) + k π (cid:90) ∂ ∆ ∩ Σ a ∧ δ ˜ ε (B11)=0 (B12)on-shell. Appendix C: The DB double complex
We are actually interested in two distinct complexesthat can be regarded as ˇCech-de Rham bicomplexes withspecial constraints: Z -valued elements on one side, anda truncation on the other side. The first DB complex is represented on Fig. C andthe second is represented on Fig. C. On the firstone, Ω − ( U i ...i n ) = Z , Ω ( U i ...i n ) = Ω ( U i ...i n ) = C ∞ ( U i ...i n , R ) and d : Ω − ( U i ...i n ) −→ Ω ( U i ...i n )is the canonical injection. Remark that this is not theˇCech-de Rham complex. It is truncated after the 1-forms,that is, d : Ω ( U i ...i n ) −→ R but with Z . The differential ofthis DB complex is D [1 ,k ] and maps the k -th diagonalonto the k + 1-th ( k ≥ − d : Ω ( U i ...i n ) −→ D [0 ,k ] with( k ≥ − FIG. 1. Complex truncated at form degree 1FIG. 2. Complex truncated at form degree 0[1] L. Freidel and D. Pranzetti, Electromagnetic duality andcentral charge, Physical Review D , 10.1103/phys-revd.98.116008 (2018). [2] W. Donnelly and L. Freidel, Local subsystems in gaugetheory and gravity, Journal of High Energy Physics ,10.1007/jhep09(2016)102 (2016). [3] A. Strominger, Magnetic corrections to the soft photontheorem, Physical Review Letters , 10.1103/phys-revlett.116.031602 (2016).[4] V. Hosseinzadeh, A. Seraj, and M. M. Sheikh-Jabbari,Soft charges and electric-magnetic duality, Journal ofHigh Energy Physics , 10.1007/jhep08(2018)102(2018).[5] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett,Generalized global symmetries, Journal of High EnergyPhysics , 10.1007/jhep02(2015)172 (2015).[6] T. Banks and N. Seiberg, Symmetries and strings in fieldtheory and gravity, Physical Review D , 10.1103/phys-revd.83.084019 (2011).[7] A. Kapustin and M. Tikhonov, Abelian duality, walls andboundary conditions in diverse dimensions, Journal ofHigh Energy Physics , 006–006 (2009).[8] J. Attard, J. Fran¸cois, S. Lazzarini, and T. Masson, Thedressing field method of gauge symmetry reduction, areview with examples, Foundations of Mathematics andPhysics One Century After Hilbert , 377–415 (2018).[9] J. Fuchs, T. Nikolaus, C. Schweigert, and K. Wal-dorf, Bundle gerbes and surface holonomy (2009),arXiv:0901.2085 [math.DG].[10] P. Mathieu, L. Murray, A. Schenkel, and N. J.Teh, Homological perspective on edge modes in linearyang–mills and chern–simons theory, Letters in Mathe-matical Physics , 1559–1584 (2020).[11] M. Geiller and P. Jai-akson, Extended actions, dynam-ics of edge modes, and entanglement entropy (2019), arXiv:1912.06025 [hep-th].[12] [10] also constructs the ghost and antifield phase spacefor the field content, but this will not be needed in thepresent work.[13] M. Campiglia, L. Freidel, F. Hopfmueller, and R. M.Soni, Scalar asymptotic charges and dual large gaugetransformations, Journal of High Energy Physics ,10.1007/jhep04(2019)003 (2019).[14] M. Bauer, G. Girardi, R. Stora, and F. Thuillier, AClass of topological actions, JHEP , 027, arXiv:hep-th/0406221.[15] For an attempt to explain why such Stuckelbergingshould generically occur at the boundary of a subsystem,we refer the reader to [20].[16] E. Lake, Higher-form symmetries and spontaneous sym-metry breaking (2018), arXiv:1802.07747 [hep-th].[17] D. Hofman and N. Iqbal, Goldstone modes and pho-tonization for higher form symmetries, SciPost Physics , 10.21468/scipostphys.6.1.006 (2019).[18] By “BF term” we mean here a term whose Lagrangian is B ∧ F = B ∧ dA where A and B are U(1) connections,contrary to the more common case where B is simply aglobal background differential form.[19] J. Kirklin, Unambiguous phase spaces for subregions,Journal of High Energy Physics (2019).[20] G. Dvali, C. Gomez, and N. Wintergerst, St¨uckelbergformulation of holography, Physical Review D94