p-form electrodynamics as edge modes of a topological field theory

p-form electrodynamics in d ≥ 2 dimensions is shown to emerge as the edge modes of a topological field theory with a precise set of boundary conditions, through the Hamiltonian reduction of its action. Electric and magnetic charges correspond to Noether ones in the topological field theory. For chiral p-forms, the topological action can be consistently truncated, so that the Henneaux-Teitelboim action is recovered from a pure Chern-Simons theory, with a manifestly covariant stress-energy tensor at the boundary. Topologically massive p-form electrodynamics as well as axion couplings are also shown to be described through this mechanism by considering suitable (self-)interaction terms in the topological theory.

One of the main aims of our work is showing that p-form electrodynamics in d ≥ 2 dimensions emerges as the edge modes of a topological field theory of BF-type once endowed with a very precise set of boundary conditions, defined in Section 2, that requires the presence of a metric only at the boundary.The form of the stress-energy tensor at the boundary naturally suggests the link with p-form electrodynamics, which is explicitly performed through the Hamiltonian reduction of the topological action in Section 3. It is worth stressing that electric and magnetic charges can then be seen as Noetherian ones in the BF theory.The case of even p-forms in d = 2p + 2 spacetime dimensions is discussed in Section 4, where it is shown that the topological action can be consistently truncated to a pure Chern-Simons theory devoid of boundary terms, whose Hamiltonian reduction precisely yields the Henneaux-Teitelboim action for chiral p-forms.We conclude in Section 5, where (self-)interactions of the topological field theory are considered, which allows to reproduce topologically massive p-form electrodynamics, extensions of it, as well as axion couplings as edge modes for the same set of boundary conditions.

Topological field theory of BF-type
Let us consider the action principle of an Abelian BF theory (see e.g., [19][20][21][22][23][24][25][26][27][28]) on a manifold Ω of d + 1 dimensions, given by where B and C correspond to p + 1 and (d − p − 1)-forms, respectively.The boundary term B is defined on M = ∂Ω, being generically required in order to have a well-defined variational principle once the boundary conditions are specified.Its precise form can be obtained as follows.The variation of (2.1) reads so that the bulk terms vanish when the field equations, dB = 0 and dC = 0, hold.Thus, the action attains an extremum provided that the variation of the boundary term is given by which requires a precise choice of boundary conditions to be integrated.

Boundary conditions
In order to specify our boundary conditions we assume the existence of a metric structure at the boundary, so that g µν is defined only at M = ∂Ω.The boundary conditions are then defined by choosing the C-field to be the Hodge dual the B-field at the boundary, i.e.,1 The boundary condition (2.4) then allows to integrate the variation of the boundary term δB in (2.3), so that it is given by In sum, the action principle becomes well-defined for our choice of boundary conditions in (2.4).

Stress-energy tensor at the boundary
Note that the topological field theory under discussion has no local notion of energy.Nevertheless, since the boundary conditions incorporate a metric, it is possible to define a stress-energy tensor at the boundary M = ∂Ω along the lines of Brown and York [29], given by which for the action in (2.6) reads The explicit form of the stress-energy tensor in (2.8) then naturally suggests a link with p-form electrodynamics at the boundary, which is discussed in the next section.

p-form electrodynamics from Hamiltonian reduction
The Hamiltonian reduction of the topological field theory described by (2.6) can be readily performed due to its simplicity.Assuming the topology of Ω to be of the form R×Σ, indices can be split in space and time so that the action reads with α = (p + 1)!(d − p − 1)!, and the boundary term is given by for Lagrange multipliers, and hence the corresponding constraints fulfilling are locally solved by Thus, replacing the solution of the constraints in (3.5) back into (3.1),after suitable integration by parts, the full action reduces to a boundary term that can be written as It is then useful to fix the gauge according to so that B = dA.Besides, the boundary condition (2.4) allows to trade the C-field by the Hodge dual of B, and hence, the action (3.6) becomes that of p-form electrodynamics, given by where the dynamical field turns out to be the p-form A, whose field strength is B = dA.
One interesting direct consequence of the equivalence of the topological action (2.6) with that of p-form electrodynamics at the boundary, is that electric and magnetic charges can both be seen to emerge from Noether ones in the topological theory.Indeed, the suitably normalized conserved charges associated to the gauge transformations of the topological action (δB = dλ B , δC = dλ C ) once evaluated at the boundary are given by corresponding to the electric and magnetic charges of p-form electrodynamics, respectively.
4 Consistent truncation: chiral p-forms from a pure Chern-Simons theory Let us consider the case of even p-forms for d = 2p+2, so that one can perform the following change of basis in the fields of the topological theory In terms of B ± , the boundary condition (2.4) then reads which amounts to require the fields to be (anti-)selfdual at the boundary, and the topological action (2.6) becomes just the difference of two pure Chern-Simons forms, given by Note that the simple change of basis (4.1) yields to the action (4.3) that is devoid of boundary terms, and leads to a well-defined variational principle for (anti-)chiral fields at the boundary by construction.
It is worth highlighting that the action (4.3) can be consistently truncated to describe single chiral fields at the boundary, either for vanishing B + or B − .The link between chiral p-forms and pure Chern-Simons theories has also been explored in [30].

Chiral p-form action from Hamiltonian reduction
The Hamiltonian reduction of the action for an odd (p + 1)-form B ± in 2p + 3 spacetime dimensions that is (anti-)chiral at the boundary, is performed along the lines of Section 3. Splitting the indices in space and time, the action (4.4) can be written as ) with a boundary term B± that reads The constraint associated to the Lagrange multiplier ) is then exactly solved as so that the the full action (4.5) reduces to a boundary term given by where latin indices a 1 , . . ., a 2p+1 stand for the spacelike ones at M = ∂Ω.The (anti-)selfduality condition (4.2) fixes the Lagrange multiplier in terms of the field strength in (4.7), according to and hence, the action (4.8), reduces to the Henneaux-Teitelboim action for (anti-)chiral p-forms [31,32], given by written in terms of the magnetic field Here, we have also made use of the ADM decomposition of the metric so that N and N a stand for the lapse and shift functions, respectively, while the energy and momentum densities explicitly read It is worth pointing out that although covariance is not manifest in the action (4.10), invariance under diffeomorphisms that preserve the background metric holds by virtue of the fact that the energy and momentum densities fulfill the Dirac-Schwinger algebra [31,32].

Manifestly covariant stress-energy tensor for chiral p-forms
One of the advantages of obtaining the Henneaux-Teitelboim action for chiral p-forms (4.10) as an edge mode of the pure Chern-Simons action in (4.4) is that a manifestly covariant stress-energy tensor can be readily obtained as in Section 2.2.Indeed, the Brown-York stress-energy tensor in (2.7) evaluated at the boundary M = ∂Ω is given by being traceless, manifestly covariant and clearly conserved by virtue of the Bianchi identity and the (anti-)self-dual boundary condition (4.2).Its components relate to the energy and momentum densities by projecting along the normal vector to the spacelike hypersurface n µ = (N, 0), so that where H ± and H ± a are given by (4.13) and (4.14), respectively.

Extensions and final remarks
Axion-like couplings between diverse p-forms as well as topologically massive extensions can also be seen to emerge as edge modes of topological field theories that include suitable (self-)interaction terms deforming the gauge symmetries without breaking them.
5.1 Topologically massive p-form electrodynamics from a BF theory with a "cosmological term" One possibility is to endow the topological theory in (2.6) with a "cosmological term", extending that in [21] for even (p + 1)-forms in higher dimensions, so that our action in (2.6) is deformed as being clearly well-defined for the same boundary conditions in (2.4), provided that B stands for an even (p + 1)-form in d + 1 = m(p + 1) dimensions.Note that the integer m ranges as 2 ≤ m ≤ (d + 1)/2, and the gauge symmetries now become The Hamiltonian reduction can then be carried out as in Section 3, so that once the indices are split in space and time, the action is given by (3.1) with a deformed constraint where Thus, the constraints can also be locally solved, and once the solution is replaced back into the action, fixing the gauge as in (3.7), it reduces to a boundary term given by where the p-form A is the dynamical field with field strength B = dA, precisely reproducing topologically massive p-form electrodynamics for m = 2, and extending it otherwise.In particular, for a standard U (1) gauge field (p = 1), the original topologically massive electrodynamics of Deser, Jackiw and Templeton [33] is recovered in d = 3 (m = 2), while the graviphoton of five-dimensional supergravity is obtained in d = 5 (m = 3) for a precise value of the deformation parameter µ.More possibilities arise in higher dimensions, as in the case of d = 11, where three different theories can be obtained, for p = 1, 3, 5 (with m = 6, 3, 2, respectively).Note that the eleven-dimensional supergravity 3-form field [34] is described by (5.4), where the value of µ becomes fixed by local supersymmetry.

Axion-like couplings from interacting topological theories
A class of couplings between p-form fields for diverse values of p can also be described through the edge modes of a topological theory of BF-type with suitable interaction terms.As a precise example let us consider a five-dimensional action of the form (5.5) which is clearly well-defined for boundary conditions as in (2.4), regardless the value of the coupling λ.Following the same lines as in the previous cases, the Hamiltonian reduction of (5.5) reduces to a four-dimensional boundary term describing the axion coupling of Maxwell electrodynamics with the massless Klein-Gordon field, given by where B [2] = dA and B [1] = dϕ.As a closing remark, it is certainly worth exploring how topological invariants as the Ray-Singer torsion and the generalized linking number, known to be deeply connected with topological field theories of BF-type [19][20][21][22][23][24][25][26][27][28], reflect themselves in the context of p-form electrodynamics.
Note added.This is a slightly updated version of our unpublished preprint [35] that was presented in XVIII Chilean Symposium of Physics during November 2012.Our results possess some overlap with those recently reported in [36].