An interacting conformal chiral 2-form electrodynamics in six dimensions

Abstract

The strong-field limit for the 2-form potential on an M5-brane yields a conformal chiral 2-form electrodynamics in six dimensions, with gauge-invariant self-interactions but no adjustable coupling constant; the stress tensor is that of a null fluid. Lorentz invariance can be made manifest via an interpretation as a tensionless ‘space-filling M5-brane’, or as a truncation of the infrared dynamics of an M5-brane in AdS₇ × S⁴.

1. Introduction

In 1983, on the occasion of the 60th birthday of Jan Łopuszański, Iwo Bialynicki-Birula wrote an article about Born-Infeld (BI) electrodynamics in which he showed, inter alia, that there is a strong-field limit that yields a conformal-invariant interacting electrodynamics [1]. It was shown recently by Luca Mezincescu and the author [2] that many of the properties of this Bialynicki–Birula electrodynamics (BBE) follow from an interpretation of it as a truncation of the low-energy dynamics of a tensionless D3-brane. As we also showed, it has an alternative interpretation as a truncation of the infrared (IR) dynamics of a D3-brane in the AdS₅ × S⁵ vacuum of IIB supergravity.

On the occasion of Michael Duff’s 70th year (70 is the new 60), it is my pleasure to offer an extension of the ideas in [1,2] that makes contact with two principal themes of Michael’s research over the last 40 years. One is Eleven Dimensions and the other is Branes; Michael’s recent review [3] reminds us that the two topics merged about 30 years ago, only to meet with ‘fierce resistance’.

The starting point here will be the M5-brane of M-theory, and more specifically the dynamics of its worldvolume 2-form potential with self-dual 3-form field-strength. This 2-form potential is part of a (2,0) supermultiplet in six dimensions (6D) that couples to the boundary of a membrane [4]. Membranes stretched between parallel M5-branes become tensionless strings in the coincidence limit [5] and the IR limit of the resulting dynamics is some interacting (2,0)-supersymmetric 6D conformal field theory (CFT) [6]. This (2,0) theory remains ‘mysterious’, in part because it has no coupling constant and hence cannot be approached perturbatively from some free-field limit; it has been argued that, away from the conformal limit, it becomes a free-field theory of a (2,0) supermultiplet coupled to tensile self-dual 6D strings [7].

The main result here is the construction of an interacting conformal chiral 2-form electrodynamics in six dimensions; this too has no coupling constant and (for reasons given at the end of this paper) is likely to have a (2,0)-supersymmetric extension. It is also related to M5-brane dynamics, in the same way that BBE is related to D3-brane dynamics. It is unlikely (in the author’s estimation) to be the (2,0) CFT of multiple M5-brane dynamics but it may provide some useful insights into that problem. Because of the close analogy to BBE, it will be useful to first consider the basic structure of BBE; its method of construction, some of its properties and its relation to the D3-brane will be reviewed later.

A feature of BBE emphasized in [1] is that it has no configuration-space action. However, there is a phase-space action; for standard Minkowski space–time coordinates {t, σ} it is

$$S_{\text{BBE}}=\int \text{d}t\int {\text{d}}^3\sigma \{\mathbf{\text{E}}\cdot \mathbf{\text{D}}-|\mathbf{\text{D}}\times \mathbf{\text{B}}|\}.$$ 1.1

The (3-covector) electric field E and the (3-pseudovector density) magnetic-induction field B are defined in terms of the components (A₀, A) of a potential 1-form in the usual way

$$\mathbf{\text{E}}=\mathbf{\nabla }{A}_0-\dot{\mathbf{\text{A}}}\text{and}\mathbf{\text{B}}=\mathbf{\nabla }\times \mathbf{\text{A}}.$$ 1.2

Note that

$$\mathbf{\text{E}}\cdot \mathbf{\text{D}}=-\dot{\mathbf{\text{A}}}\cdot \mathbf{\text{D}}-A_0\mathbf{\nabla }\cdot \mathbf{\text{D}}+\text{total derivative},$$ 1.3

which shows that A₀ is a Lagrange multiplier for the Gauss-law constraint on the electric displacement 3-vector density D, which is canonically conjugate to −A. The BBE action is not obviously conformal invariant, and even its Lorentz invariance is not obvious, but these properties were established in [1] and can be simply explained in terms of its D3-brane origin [2].

A comment on terminology may be useful here. Given a phase-space action of the form

$$S=\int \text{d}t\int {\text{d}}^3\sigma \{\mathbf{\text{E}}\cdot \mathbf{\text{D}}-\mathcal{H}(D,B)\},$$ 1.4

the magnetic field is defined as $\mathbf{\text{H}}=\mathrm{\partial }\mathcal{H}/\mathrm{\partial }\mathbf{\text{B}}$; it is a (pseudo) 3-covector whereas the magnetic-induction field B is a (pseudo) 3-vector density. The Hamiltonian of Maxwell electrodynamics in a vacuum is such that H = B in cartesian coordinates (for which the distinction between vectors, covectors and vector densities is lost) and there is then no need to distinguish between H and B. However, the distinction is essential to the nonlinear BI theory, and to BBE. Similarly, the electric 3-covector field E may be defined as $\mathbf{\text{E}}=\mathrm{\partial }\mathcal{H}/\mathrm{\partial }\mathbf{\text{D}}$, but this is just the field equation derivable from the phase-space action by variation with respect to D. For the Maxwell case, and in cartesian coordinates, this equation is E = D, but equality no longer holds in the nonlinear BI theory, or in BBE.

The 2-form analogue of BBE in six dimensions also has no configuration-space action but it too has a phase-space action. To present it we must first introduce analogues of the electric field and magnetic-induction field in terms of the components of the 3-form field-strength F = dA for a 2-form potential A

$$\begin{array}{rl}{E}_{ij}& =F_{ij0}\equiv {\dot{A}\hspace{0.17em}}_{ij}-2{\mathrm{\partial }}_{[i}{A}_{j]0}\\ \text{and}{B}^{ij}& =\frac{1}{6}{\epsilon }^{ijklm}{F}_{klm}\equiv \frac{1}{2}{\epsilon }^{ijklm}{\mathrm{\partial }}_k{A}_{mn}.\end{array}\}$$ 1.5

The phase-space action involves an analogue of the electric displacement field (now an antisymmetric tensor density) with components D^ij that are canonically conjugate to A_ij but there is also a chirality constraint that allows it to be eliminated; the resulting action is a functional only of A_ij (i.e. the space components of the 2-form potential A). This action is

$$S[A]=\int \text{d}t\int {\text{d}}^5\sigma \{-\frac{1}{2}{\dot{A}\hspace{0.17em}}_{ij}{B}^{ij}-|B\wedge B|\},$$ 1.6

where $B\wedge B$ is a 5-vector density with components

$${(B\wedge B)}_i=\frac{1}{4}{\epsilon }_{ijklm}{B}^{jk}{B}^{lm}.$$ 1.7

The main aim of the remainder of this paper will be to explain why this action is Lorentz invariant, and not only that but also conformal invariant. This will be achieved in two ways, both related to limits and truncations of the dynamics of an M5-brane, for which the manifestly Lorentz invariant, and reparametrization invariant, action was found in [8], although our starting point will be the phase-space action constructed from it in [9]. The simplest way is via a tensionless limit of the M5-brane in the 11-dimensional (11D) Minkowski vacuum, and this will be discussed first. Then we shall see how the same theory arises from an IR limit of an M5-brane in the AdS₇ × S₄ vacuum of 11D supergravity. Throughout the paper, except for some final comments and speculations, all fermions are set to zero.

As already mentioned, most of these M5-brane-derived results are generalizations of the D3-brane-derived results of [2], and of the idea of a strong-field limit first explored in [1], so it will be very useful to cover some of that ground first. It was brought to the author’s attention after submission to the arXiv of the first version of this paper that some of this material is also covered in a paper of Gibbons and West [10], who further consider a ‘strong-coupling limit’ of the M5-brane with results that overlap with those found here.

2. Born-Infeld preliminaries

The BI theory of nonlinear electrodymamics has an action of the form

$$S_{\text{BI}}[A]=-T\int {\text{d}}^{4}x\{\sqrt{-det(\eta +T^{-\frac{1}{2}}F)}-1\},$$ 2.1

where T is a constant with dimensions of energy density (in units for which $\hslash =1$), and F = dA is the Faraday 2-form for the electromagnetic 1-form potential A. We shall need the corresponding phase-space action, which takes the form (1.4) with Hamiltonian density

$${\mathcal{H}}_{\text{BI}}=T\{\sqrt{1+T^{-1}(|\mathbf{\text{D}}{|}^2+|\mathbf{\text{B}}{|}^2)+T^{-2}|\mathbf{\text{D}}\times \mathbf{\text{B}}{|}^2}-1\}.$$ 2.2

Note that H_BI has an SO(2) invariance under rotations of (D, B).

In the weak-field limit, we have

$${\mathcal{H}}_{\text{BI}}\to {\mathcal{H}}_{\text{Maxwell}}=\frac{1}{2}(|\mathbf{\text{D}}{|}^2+|\mathbf{\text{B}}{|}^2).$$ 2.3

Note that the constant T has dropped out; this is related, of course, to the conformal invariance of Maxwell’s equations.

To investigate the strong-field limit, it is convenient to rewrite the Hamiltonian density as

$${\mathcal{H}}_{\text{BI}}=\sqrt{|\mathbf{\text{D}}\times \mathbf{\text{B}}{|}^2+T(|\mathbf{\text{D}}{|}^2+|\mathbf{\text{B}}{|}^2)+T^2}-T,$$ 2.4

and then use the fact that the strong-field limit for fixed T is equivalent to the T → 0 limit at fixed field strengths. This is precisely how Bialynicki–Birula found the Hamiltonian density

$${\mathcal{H}}_{\text{BB}}=|\mathbf{\text{D}}\times \mathbf{\text{B}}|.$$ 2.5

The corresponding BBE field equations are

$$\dot{\mathbf{\text{D}}}=\mathbf{\nabla }\times (\mathbf{\text{n}}\times \mathbf{\text{D}})\text{and}\dot{\mathbf{\text{B}}}=\mathbf{\nabla }\times (\mathbf{\text{n}}\times \mathbf{\text{B}}),$$ 2.6

where n is the unit-vector field

$$\mathbf{\text{n}}=\mathbf{\text{D}}\times \frac{\mathbf{\text{B}}}{|\mathbf{\text{D}}\times \mathbf{\text{B}}|}.$$ 2.7

These equations should be taken together with the constraint $\mathbf{\nabla }\cdot \mathbf{\text{D}}=0$ and the identity $\mathbf{\nabla }\cdot \mathbf{\text{B}}=0$. As observed in [1], the equations are nonlinear because only solutions with the same n can be superposed. Note that the BBE Hamiltonian is independent of the dimensionful constant T, which suggests that BBE might be conformal invariant. In fact, it is conformal invariant, and it is also invariant under an enlarged $Sl(2;\mathbb{R})$ electromagnetic duality group [1].

(a). Poincaré and conformal invariance

The BI configuration-space action (2.1) is manifestly Poincaré invariant. This symmetry is not obvious from the phase-space action with BI Hamiltonian density (2.4) but the equivalence of this action to (2.1) can be established by a straightforward elimination of the momentum variable D. However, this step cannot be performed once the strong-coupling limit has been taken as the BBE phase-space action of (1.1) is linear in D, and this puts into doubt the Poincaré invariance of BBE. The issue was addressed and resolved in [1] but there is another way to exhibit the space–time symmetries [2]; we review the idea here but taking the BI action as our starting point, rather than its Bialynicki–Birula limit.

In writing down the BI action (2.1), we implicitly made the standard choice of Minkowski coordinates for the Minkowski space–time. Let us now rename these Minkowski coordinates as {X^μ;μ = 0, 1, 2, 3}; then, in any other coordinate system {ξ^μ;μ = 0, 1, 2, 3}, specified by the functions X^μ(ξ), the Minkowski space–time metric is

$$g_{\mu \nu }(\xi )={\mathrm{\partial }}_{\mu }{X}^{\rho }{\mathrm{\partial }}_{\nu }{X}^{\sigma }{\eta }_{\rho \sigma },$$ 2.8

where ∂_μ is a partial derivative with respect to ξ^μ. The BI action is now

$$S_{\text{BI}}[A,X]=-T\int {\text{d}}^4\xi \{\sqrt{-det(g+T^{-\frac{1}{2}}F)}-\sqrt{-detg}\}.$$ 2.9

This action is reparametrization invariant; it depends on the four functions X^μ in addition to A, and this dependence must be taken into account when passing to the phase-space form of the action. After writing ξ⁰ = t and ξⁱ = σⁱ (i = 1, 2, 3), this new phase-space action takes the form

$$S[\mathbf{\text{A}},X;\mathbf{\text{D}},P;A_0,u]=\int \text{d}t\int {\text{d}}^3\sigma \hspace{0.17em}\{{\dot{X}}^{\mu }{P}_{\mu }+\mathbf{\text{E}}\cdot \mathbf{\text{D}}-u^{\mu }{\mathcal{H}}_{\mu }\},$$ 2.10

where P_μ is the momentum density canonically conjugate to X^μ. The new Lagrange multipliers u^μ impose an additional four constraints (i.e. in addition to the Gauss-law constraint); the constraint functions are

$$\begin{array}{rl}\hspace{0.17em}& {\mathcal{H}}_0=\frac{1}{2}\{{(P-TC)}^2+T(D^i{D}^j+B^i{B}^j)h_{ij}+T^{2}deth\}\\ \text{and}& {\mathcal{H}}_i={\mathrm{\partial }}_i{X}^{\mu }{P}_{\mu }-{\epsilon }_{ijk}{D}^i{B}^k,\end{array}\}$$ 2.11

where h is the space metric¹; i.e. h_ij = g_ij, and the 4-vector-density C has components

$$C_{\mu }=\frac{1}{6}{\epsilon }_{\mu \nu \rho \sigma }{\epsilon }^{ijk}{\mathrm{\partial }}_i{X}^{\nu }{\mathrm{\partial }}_j{X}^{\rho }{\mathrm{\partial }}_k{X}^{\sigma }.$$ 2.12

Useful identities are

$$C^2\equiv -deth\text{and}{\mathrm{\partial }}_i{X}^{\mu }{C}_{\mu }\equiv 0.$$ 2.13

Before proceeding, we pause to explain how the BI Hamiltonian density of (2.2) is recovered from the new, reparametrization invariant, action. The key point is that the new constraints are first-class; more precisely, the constraint functions ${\mathcal{H}}_{\mu }$ form a first class set, which requires the Poisson-bracket of each element of the set with any other constraint function to be zero on the surface in phase-space defined by the set of all constraints. Here it should be recalled that in addition to the new constraints imposed by the new Lagrange multipliers u^μ there is also the Gauss-law constraint imposed by A₀; this is first-class too because its PBs with ${\mathcal{H}}_{\mu }$ are zero, as expected because it generates the gauge transformation of A and the functions ${\mathcal{H}}_{\mu }$ are gauge invariant. It follows that the new constraint functions ${\mathcal{H}}_{\mu }$ will form a first-class set if the matrix of their PBs is zero when all ${\mathcal{H}}_{\mu }$ are zero. This first-class property, which was verified in detail in [2], implies that the new constraints also generate gauge invariances. These new gauge invariances are (on-shell) equivalent to diffeomorphisms of the four-dimensional (4D) Minkowski space–time, and this allows us to impose the Monge gauge

$$X^{\mu }(\xi )={\xi }^{\mu }(\mu =0,1,2,3)\Rightarrow h_{\mu \nu }={\eta }_{\mu \nu }.$$ 2.14

In this gauge $h_{ij}={\delta }_{ij}\hspace{0.17em}(\Rightarrow deth=1)$ and P^μ C_μ = P⁰, so that

$$2{\mathcal{H}}_0=-{(P^0+T)}^2+|\mathbf{\text{P}}{|}^2+T(|\mathbf{\text{D}}{|}^2+|\mathbf{\text{B}}{|}^2)+T^2.$$ 2.15

The solution of the constraints for P_μ is then

$$\begin{array}{rl}{P}^0& =\pm \sqrt{|\mathbf{\text{D}}\times \mathbf{\text{B}}{|}^2+(|\mathbf{\text{D}}{|}^2+|\mathbf{\text{B}}{|}^2)+T^2}-T\\ \text{and}\mathbf{\text{P}}& =\mathbf{\text{D}}\times \mathbf{\text{B}}.\end{array}\}$$ 2.16

The Monge-gauge Hamiltonian is P⁰, which is ${\mathcal{H}}_{\text{BI}}$ if we assume P⁰ > 0.

Having now confirmed its equivalence to the BI action, we return to the reparametrization-invariant action (2.10). Its advantage is that the Poincaré group now acts linearly, and only on (X, P). The Noether charges corresponding to space–time translations and Lorentz transformation are simply

$${\mathcal{P}}_{\mu }=\int {\text{d}}^3\sigma P_{\mu }\text{and}{\mathcal{J}}^{\mu \nu }=2\int {\text{d}}^5\sigma X^{[\mu }{P}^{\nu ]}.$$ 2.17

It is straightforward to verify, using the (X, P) equations of motion and the constraints, that these charges are time-independent for appropriate boundary conditions, and that they span the algebra of the Poincaré group with respect to the Poisson bracket relations derived from the action (2.10). As Noether charges are gauge invariant, they are unchanged by the imposition of the Monge gauge, except that the previously independent momentum variables P_μ must be replaced by their Monge-gauge expressions (2.2), and all PB relations must now be computed using the canonical PB relations of the Monge-gauge action. The Noether charges are still time-independent (now as a consequence of the Monge-gauge field equations) and their PB algebra is also unchanged. What does change is that the Noether charges now generate transformations of the phase-space variables (A, D), which were initially inert!

All of this discussion continues to apply in the T → 0 limit. In this limit, the constraint functions (2.11) simplify to

$${\mathcal{H}}_0=\frac{1}{2}{P}^2\text{and}{\mathcal{H}}_i={\mathrm{\partial }}_i{X}^{\mu }{P}_{\mu }-{\epsilon }_{ijk}{D}^i{B}^k.$$ 2.18

These are still Poincaré invariant but the action (2.10) is now invariant under the larger group of conformal isometries of the 4D Minkowski metric. By imposing the Monge gauge and solving the constraints for P_μ one recovers the BBE action (1.1), which is therefore also Poincaré, and conformal, invariant. The Monge-gauge expressions for the associated Noether charges can be used to find the symmetry transformations; this step was carried out for Lorentz transformations in [2], and the result was used to perform a direct verification of the Lorentz invariance of the BBE action of (1.1).

(b). Relation to the D3-brane

We have now arrived at a point from which it is easy to see the relation of BBE to the D3-brane. The action (2.10) is a truncation of the phase-space action for the D3-brane of tension T in the Minkowski vacuum of 10-dimensional (10D) IIB supergravity [11]: the transverse fluctuations of a static planar D3-brane are ignored (as are all anticommuting spinor variables) thereby reducing the 10D Minkowski space–time to a 4D Minkowski space–time that can be identified with the D3-brane worldvolume. As the brane now fills the available space this truncation was referred to as ‘space-filling’. The only remaining dynamics is that of the 1-form potential on the 4D Minkowski worldvolume; in general this is equivalent to the BI theory of nonlinear electrodynamics, with T being its dimensionful constant. The tensionless limit is thus equivalent to the strong-field limit of [1].

As already emphasized, the advantage of the D3-brane interpretation of BBE is that Lorentz invariance is linearly realized, and hence manifest. Conformal invariance is still nonlinearly realized but it is now related to the (easily established) conformal invariance of null brane dynamics. A further advantage of the D3-brane perspective is that it relates the $Sl(2;\mathbb{R})$ electromagnetic duality invariance group of BBE found in [1] to the $Sl(2;\mathbb{R})$ duality group of IIB supergravity [2]. As this latter, $Sl(2;\mathbb{R})$ is broken to $Sl(2;\mathbb{Z})$ in IIB superstring theory, we may anticipate that quantum effects will break the $Sl(2;\mathbb{R})$ invariance of BBE to $Sl(2;\mathbb{Z})$.

Finally, by considering a D3-brane in the AdS₅ × S⁵ vacuum of IIB supergravity one can interpret BBE as a truncation of the IR limit of the dynamics of a planar D3-brane coincident with the AdS₅ Killing horizon [2]. This suggests that one might find a chiral 2-form electrodynamics in six dimensions by taking the IR limit of a planar M5-brane in the AdS₇ × S⁴ vacuum of 11D supergravity. As we shall see later, this is indeed one way of arriving at the chiral 2-form electrodynamics advertised in the Introduction. However, in complete analogy with the D3-brane case, it can also be found more simply as a truncation of the action for a tensionless M5-brane in the 11D Minkowski vacuum, and this will be our starting point for what follows.

3. The M5-brane

A reparametrisation-invariant phase-space action for the M5-brane, in a general bosonic background, was obtained in [9] from the configuration-space M5-brane action of [8]. Here we take the background to be the 11D Minkowski vacuum, with Minkowski coordinates {X^m;m = 0, 1, …, 10}, and we set to zero all fermions. The worldvolume coordinates {ξ^μ;μ = 0, 1, …, 5} are split into ξ⁰ = t and ξⁱ = σⁱ (i = 1, …, 5). The action then takes the form

$$S_{M5}=\int \text{d}t\int {\text{d}}^5\sigma \{{\dot{X}}^m{P}_m+\frac{1}{2}{\dot{A}\hspace{0.17em}}_{ij}{D}^{ij}-u^{\mu }{\mathcal{H}}_{\mu }-\frac{1}{2}{\sigma }_{ij}{\chi }^{ij}\},$$ 3.1

where the variables u^μ and σ_ij (=−σ_ji) are Lagrange multipliers for constraints. The constraint functions ${\mathcal{H}}_{\mu }$ and χ^ij are functions of canonical variables that will be given below, we may ignore the Gauss-law-type constraint ∂_j D^ij = 0 imposed by A_i0 because it is implied by the chirality constraint. The canonical PB relations that one may read off from this action are

$$\begin{array}{rl}\hspace{0.17em}& {\{X^m(\mathit{\sigma }),P_n({\mathit{\sigma }}^{\prime }\}}_{PB}={\delta }_n^m\hspace{0.17em}\delta (\mathit{\sigma }-{\mathit{\sigma }}^{\prime })\\ \text{and}& {\{A_{ij}(\mathit{\sigma }),D^{kl}({\mathit{\sigma }}^{\prime })\}}_{PB}=({\delta }_i^k{\delta }_j^l-{\delta }_i^l{\delta }_j^k)\hspace{0.17em}\delta (\mathit{\sigma }-{\mathit{\sigma }}^{\prime }).\end{array}\}$$ 3.2

There is considerable freedom in the choice of the constraint functions ${\mathcal{H}}_{\mu }$; here we shall give them in a different basis to that of [9] and with fields rescaled² so that the weak-field Hamiltonian density in Monge gauge is independent of the M5-brane tension

$$\begin{array}{rl}{\mathcal{H}}_0& =\frac{1}{2}\{{\eta }^{mn}{P}_m{P}_n+\frac{T}{2}(D_{ij}{D}^{ij}+B_{ij}{B}^{ij})+T^{2}deth\}\\ \text{and}{\mathcal{H}}_i& ={\mathrm{\partial }}_i{X}^m{P}_m-V_i,\end{array}\}$$ 3.3

where h is again the space part of the induced worldvolume metric, and

$$V_i=-\frac{1}{4}{\epsilon }_{ijklm}{D}^{ij}{B}^{lm}.$$ 3.4

It should be appreciated that we are using a standard shorthand for which D_ij = h_ikh_kl D^kl and that ε_ijklm is the worldspace alternating invariant tensor density of opposite weight to ε^ijklm. In addition, we have the chirality constraint functions

$${\chi }^{ij}=D^{ij}+B^{ij}.$$ 3.5

Note that parity flips the relative sign in this expression because B is a pseudo-tensor density of the O(5) rotation group whereas D is a tensor density; a parity flip is needed to recover the results of [9] after undoing the rescaling mentioned above.

Our next task will be to compute the PBs of the constraint functions in order to determine the subset of first-class constraints that are required to generate the gauge transformations that are on-shell equivalent to diffeomorphisms of the M5-brane worldvolume.

(a). Poisson bracket algebra of constraints

It is convenient to choose a functional basis for the constraint functions by defining

$$H_0[\beta ]=\int {\text{d}}^5\sigma \beta {\mathcal{H}}_0\text{and}H[\mathit{\alpha }]=\int {\text{d}}^5\sigma {\alpha }^i{\mathcal{H}}_i,$$ 3.6

where β is a scalar inverse-density and α is a 5-vector field, with components αⁱ. We assume that β and α are smooth and have compact support, which will allow us to freely integrate by parts without the need to keep surface terms; this is equivalent to imposing appropriate boundary conditions on the worldvolume fields at spatial infinity. Similarly, we define

$$\chi [\omega ]=\frac{1}{2}\int {\text{d}}^5\sigma {\omega }_{ij}{\chi }^{ij},$$ 3.7

where ω_ij are the components of a smooth 2-form ω with compact support. We shall see below that the set with functionals H₀ and H as elements is first-class, but let us consider first the complementary set with functionals χ as elements; a calculation using the canonical PBs of (3.2) yields

$${\{\chi [\omega ],\chi [{\omega }^{\prime }]\}}_{PB}=2\int \omega \hspace{0.17em}\text{d}{\omega }^{\prime },$$ 3.8

where the integral of the 5-form ω dω′ is taken over the Euclidean 5-space. The right-hand side is not zero in general, which shows that the constraint functions imposing the self-duality condition do not form a first-class set, but neither do they form a second-class set because the right-hand side is zero when either ω or ω′ is an exact form. This was to be expected because ∂_iχ^ij = ∂_i D^ij, which generates the gauge transformation of A.

To compute the remaining PBs of the constraint functionals, it is convenient to begin by establishing that

$$\begin{array}{rl}\hspace{0.17em}& {\{B^{ij},(H_0[\beta ]+H[\mathit{\alpha }])\}}_{PB}=\frac{T}{2}{\epsilon }^{ijklm}{\mathrm{\partial }}_k(\beta D_{lm})+3{\mathrm{\partial }}_k({\alpha }^{[k}{B}^{ij]})\\ \hspace{0.17em}& {\{D^{ij},(H_0[\beta ]+H[\mathit{\alpha }])\}}_{PB}=\frac{T}{2}{\epsilon }^{ijklm}{\mathrm{\partial }}_k(\beta B_{lm})+3{\mathrm{\partial }}_k({\alpha }^{[k}{D}^{ij]})\\ \hspace{0.17em}& {\{X^m,(H_0[\beta ]+H[\mathit{\alpha }])\}}_{PB}=\beta P^m+{\alpha }^i{\mathrm{\partial }}_i{X}^m\\ \text{and}& {\{P^m,(H_0[\beta ]+H[\mathit{\alpha }])\}}_{PB}={\mathrm{\partial }}_i(\beta \hspace{0.17em}{\mathcal{O}}^{ij}{\mathrm{\partial }}_j{X}^m+{\alpha }^i{P}^m)\end{array}\},$$ 3.9

where

$${\mathcal{O}}^{ij}=T^2(deth)h^{ij}+T(D^{ik}{D}^{j\ell }+B^{ik}{B}^{j\ell })h_{k\ell }.$$ 3.10

It is also convenient to use a different basis by defining

$${\tilde{\mathcal{H}}}_i={\mathcal{H}}_i-A_{ij}{\mathrm{\partial }}_k{D}^{kj}\text{and}\tilde{H}[\mathit{\alpha }]=\int {\text{d}}^5\sigma \hspace{0.17em}{\alpha }^i{\tilde{\mathcal{H}}}_i.$$ 3.11

As ∂_k D^kj generates the gauge transformations of the 5-space 2-form potential A, it has zero PBs with the (gauge-invariant) functionals (H₀, H, χ); this means that PB relations among $(H_0,\tilde{H},\chi )$ will be the same as PB relations among (H₀, H, χ) on the surface in phase space determined by the full set of constraints (in any basis). The advantage of this replacement of H by $\tilde{H}$ is that it leads to a simpler result for the PB relations off this surface. For example, it is not difficult to establish, using the intermediate results of (3.9), that

$${\{\tilde{H}[\mathit{\alpha }],\tilde{H}[{\mathit{\alpha }}^{\prime }]\}}_{PB}=\tilde{H}[[\mathit{\alpha },{\mathit{\alpha }}^{\prime }]],$$ 3.12

where [α, α′] is the commutator of vector fields; this shows that the constraint functions ${\tilde{\mathcal{H}}}_i$ generate 5-space diffeomorphisms. A similar calculation, using the identity

$${\epsilon }^{pijkl}{D}_{ij}{B}_{kl}\equiv deth\hspace{0.17em}{h}^{pq}{\epsilon }_{qijkl}{D}^{ij}{B}^{kl},$$ 3.13

yields (as should be expected in the light of the interpretation just established for $\tilde{H}$)

$${\{\tilde{H}[\mathit{\alpha }],H_0[\beta ]\}}_{PB}=H_0[{\mathcal{L}}_{\mathit{\alpha }}\beta ]\text{and}{\{\tilde{H}[\mathit{\alpha }],\chi [\omega ]\}}_{PB}=\chi [{\mathcal{L}}_{\mathit{\alpha }}\omega ],$$ 3.14

where ${\mathcal{L}}_{\mathit{\alpha }}$ is the Lie derivative with respect to α. As β is a scalar inverse density and ω a 2-form, we have

$$\begin{array}{rl}{\mathcal{L}}_{\mathit{\alpha }}\beta & ={\alpha }^k{\mathrm{\partial }}_k\beta -({\mathrm{\partial }}_i{\alpha }^i)\beta \\ \text{and}{({\mathcal{L}}_{\mathit{\alpha }}\omega )}_{ij}& ={\alpha }^k{\mathrm{\partial }}_k{\omega }_{ij}+2({\mathrm{\partial }}_{[i}{\alpha }^k){\omega }_{j]k}.\end{array}\}$$ 3.15

So far, these PB relations could have been anticipated from the fact that the Lagrangian is an integral over Euclidean 5-space of a scalar density that is constructed from 5-space tensors or tensor-densities. In addition, one sees easily from (3.9) that

$${\{H_0[\beta ],\chi [\omega ]\}}_{PB}=T\chi [\mathrm{\Omega }(\beta ,\omega )]\text{and}{\mathrm{\Omega }}_{ij}=\frac{1}{2}{h}_{ip}{h}_{jq}{\epsilon }^{ijkpq}\beta {\mathrm{\partial }}_k{\omega }_{ij}.$$ 3.16

This has the form required for consistency of the chirality constraint, even though the specific form of Ω would be hard to guess.

This leaves the PB relations of the H₀ functionals. Using again the intermediate results of (3.9), we find that

$${\{H_0[\beta ],H_0[{\beta }^{\prime }]\}}_{PB}=H[\mathit{\alpha }(\beta ,{\beta }^{\prime })]-T\int {\text{d}}^5\sigma (\beta {\mathrm{\partial }}_i{\beta }^{\prime }-{\beta }^{\prime }{\mathrm{\partial }}_i\beta ){(D^2+B^2)}^{ij}{V}_j,$$ 3.17

where

$${\alpha }^i(\beta ,{\beta }^{\prime })={\mathcal{O}}^{ij}(\beta {\mathrm{\partial }}_j{\beta }^{\prime }-{\beta }^{\prime }{\mathrm{\partial }}_j\beta ),$$ 3.18

and (D²)^ij = D^ikh_klD^lj (and similarly for B²). The first term on the right-hand side is expected from known results for branes without worldvolume gauge potentials [12], but the additional term proportional to T is both unexpected and not obviously zero on the surface defined by the constraints. However, it is zero on this surface; this can be seen by using the chirality constraint to replace D by −B, which results in

$${(D^2+B^2)}^{ij}{V}_j\to \frac{1}{2}{(B^2)}^{ij}{\epsilon }_{jklpq}{B}^{kl}{B}^{pq}\equiv 0,$$ 3.19

where the following identity (for free index i) has been used:

$${\epsilon }_{jklpq}{B}_i{\hspace{0.17em}}^j{B}^{kl}{B}^{pq}\equiv 0.$$ 3.20

Therefore,

$${\{H_0[\beta ],H_0[{\beta }^{\prime }]\}}_{PB}{|}_{\chi =0}=H[\mathit{\alpha }(\beta ,{\beta }^{\prime })].$$ 3.21

This concludes the proof that the constraints corresponding to the functionals $(H_0,\tilde{H})$, and hence (H₀, H), form a first-class set.

Let us recall here that the phase-space action (3.1) with constraints (3.3) is equivalent to the action given in [9], which in turn was derived from the worldvolume reparametrization invariant configuration-space action of [8]. In view of this, it should not be surprising that the constraints corresponding to the functionals (H₀, H) form a first-class set; the above results should be seen as a check that no error has crept in along the way. However, the analysis has thrown up one surprise.

We know that the chirality constraint is a necessary feature of the full M5-brane action, which includes anticommuting (fermionic) variables, but one might have expected it to be optional in the context of the bosonic truncation. This appears not to be the case; if the chirality constraint is replaced by the Gauss-law type constraint ∂_k D^kj = 0, then most of the PB relations of the constraint functions $(H_0,\tilde{H})$ are unchanged, but the extra term proportional to T in (3.17) is now non-zero (for non-zero T) on the surface defined by the constraints so the constraints corresponding to these functionals no longer form a first-class set. It seems that the bosonic M5-brane action ‘knows’ that its 3-form field-strength must be self-dual even in the absence of any worldvolume fermions!

4. Nonlinear 2-form electrodynamics

Consider now a planar static M5-brane; its worldvolume is a 6D Minkowski space–time. There are five transverse dimensions into which this Minkowski worldvolume may fluctuate, but we may consistently set to zero these fluctuations, effectively reducing the space–time dimension from 11 to 6; the M5-brane worldvolume is mapped to the 6D space–time by the functions $\{X^{\mu }(\xi );\hspace{0.17em}\mu =0,1,\dots ,5\}$. This is a ‘space-filling’ truncation as the M5-brane fills the 5-space; the only remaining physical fluctuations are those of the 2-form potential on its worldvolume. The action (3.1) becomes

$$S=\int \hspace{0.17em}\text{d}t\int {\text{d}}^5\sigma \{{\dot{X}}^{\mu }{P}_{\mu }+\frac{1}{2}{\dot{A}\hspace{0.17em}}_{ij}{D}^{ij}-u^{\mu }{\mathcal{H}}_{\mu }-\frac{1}{2}{\sigma }_{ij}{\chi }^{ij}\},$$ 4.1

with χ^ij as before but now

$$\begin{array}{rl}\hspace{0.17em}& {\mathcal{H}}_0=\frac{1}{2}\{{\eta }^{\mu \nu }{P}_{\mu }{P}_{\nu }+\frac{T}{2}(D_{ij}{D}^{ij}+B_{ij}{B}^{ij})+T^{2}deth\}\\ \text{and}& {\mathcal{H}}_i={\mathrm{\partial }}_i{X}^{\mu }{P}_{\nu }-V_i.\end{array}\}$$ 4.2

It remains true that D_ij = h_ikh_kl D^kl, but now

$$h_{ij}(\xi )={\mathrm{\partial }}_i{X}^{\mu }{\mathrm{\partial }}_j{X}^{\nu }{\eta }_{\mu \nu }.$$ 4.3

This truncated M5-brane action still has a manifest Poincaré-invariance, but now in 6D with corresponding Noether charges

$${\mathcal{P}}_{\mu }=\int {\text{d}}^5\sigma \hspace{0.17em}{P}_{\mu }\text{and}{\mathcal{J}}^{\mu \nu }=2\int {\text{d}}^5\sigma \hspace{0.17em}{X}^{[\mu }{P}^{\nu ]}.$$ 4.4

Using the canonical PB relations, which are now

$$\begin{array}{rl}\hspace{0.17em}& {\{X^{\mu }(\mathit{\sigma }),P_{\nu }({\mathit{\sigma }}^{\prime }\}}_{PB}={\delta }_{\nu }^{\mu }\hspace{0.17em}\delta (\mathit{\sigma }-{\mathit{\sigma }}^{\prime })\\ \text{and}& {\{A_{ij}(\mathit{\sigma }),D^{kl}({\mathit{\sigma }}^{\prime })\}}_{PB}=({\delta }_i^k{\delta }_j^l-{\delta }_i^l{\delta }_j^k)\hspace{0.17em}\delta (\mathit{\sigma }-{\mathit{\sigma }}^{\prime }),\end{array}\}$$ 4.5

one may verify that the PB algebra of Noether charges is the 6D Poincaré algebra. The canonical PB relations may also be use to compute the PBs of the constraint functions. This is essentially the same calculation that was detailed earlier, with essentially the same result: the functions ${\mathcal{H}}_{\mu }$ form a first-class set that generate 6D diffeomorphisms.

(a). Monge gauge

The diffeomorphism invariance of the action (4.1) allows us to impose the Monge gauge. In the current context, this is simply an identification of the space–time coordinates with the worldvolume coordinates: X^μ(ξ) = ξ^μ. Having made this choice of coordinates we may solve for P_μ

$$\begin{array}{rl}\hspace{0.17em}& P^0=\pm T\sqrt{1+\frac{1}{2}{T}^{-1}(D_{ij}{D}^{ij}+B_{ij}{B}^{ij})+T^{-2}|\mathbf{\text{V}}{|}^2}\\ \text{and}& \mathbf{\text{P}}=\mathbf{\text{V}},\end{array}\}$$ 4.6

where V is the 5-vector density with components V_i, as given in (3.4). We should recall here that the Monge-gauge Euclidean 5-space metric is just the standard Euclidean metric: h_ij = δ_ij, and that the Monge-gauge Hamiltonian density is P⁰. The phase-space action in Monge-gauge is therefore

$$S_{\text{Monge}}=\int \text{d}t\int {\text{d}}^5\sigma \{\frac{1}{2}{\dot{A}\hspace{0.17em}}_{ij}{D}^{ij}-P^0-\frac{1}{2}{\sigma }_{ij}{\chi }^{ij}\}.$$ 4.7

For weak fields, and assuming P⁰ > 0, we have

$$P^0=T+\frac{1}{4}(D_{ij}{D}^{ij}+B_{ij}{B}^{ij})+\mathcal{O}(T^{-1}).$$ 4.8

We could arrange to cancel the constant T term as we did in the BI case. When this is done the T → ∞ limit can be taken; this limit is equivalent to the weak-field limit.³ If we also ignore the chirality constraint then elimination of D^ij yields the standard, and manifestly Lorentz-invariant, free-field Lagrangian density for 2-form electrodynamics in 6D Minkowski space–time, with field equations that propagate three parity-doublets of massless modes.

The chirality constraint χ^ij = 0 reduces the number of propagated modes from 6 to 3. To see this, we may use this constraint to eliminate D^ij, after which the weak-field Lagrangian density becomes [13]

$${\mathcal{L}}_{\text{chiral}}=-\frac{1}{2}{\dot{A}\hspace{0.17em}}_{ij}{B}^{ij}-\frac{1}{2}{B}_{ij}{B}^{ij}.$$ 4.9

The corresponding weak-field equation is

$${\dot{B}}^{ij}+\frac{1}{2}{\epsilon }^{ijklm}{\mathrm{\partial }}_k{B}_{lm}=0,$$ 4.10

which implies that $◻B^{ij}=0$. There are 10 components of B_ij but only six are independent (because of the identity ∂_i B^ij = 0) and only three of these satisfy the first-order field equation (the other three satisfy this equation with the opposite relative sign).

(b). The strong-field/tensionless limit

Let us return to the reparametrization-invariant action of (4.1) for a ‘space-filling’ M5-brane. The strong-field limit at fixed tension T is equivalent to the tensionless limit T → 0 at fixed field-strengths, and in this limit

$${\mathcal{H}}_0=\frac{1}{2}{P}^2\text{and}{\mathcal{H}}_i={\mathrm{\partial }}_i{X}^{\mu }{P}_{\mu }-V_i.$$ 4.11

In the functional basis for the constraint functions used previously, the non-zero PB relations are now

$$\begin{array}{rl}\hspace{0.17em}& {\{\tilde{H}[\mathit{\alpha }],\tilde{H}[{\mathit{\alpha }}^{\prime }]\}}_{PB}=\tilde{H}[[\mathit{\alpha },{\mathit{\alpha }}^{\prime }]],\\ \hspace{0.17em}& {\{\tilde{H}[\mathit{\alpha }],H_0[\beta ]\}}_{PB}=H_0[{\mathcal{L}}_{\mathit{\alpha }}\beta ]\\ \text{and}& {\{\tilde{H}[\mathit{\alpha }],\chi [\omega ]\}}_{PB}=\chi [{\mathcal{L}}_{\mathit{\alpha }}\omega ],\end{array}\}$$ 4.12

together with the PB relation of (3.8) for the chirality constraint functionals. The subalgebra of constraint functions ${\mathcal{H}}_{\mu }$ is now a Lie algebra (as noted for the D3 case in [2]). It is also valid on the full phase space rather than only on the surface defined by the chirality constraint; this means that the chirality constraint is now optional, in the sense that it can be consistently replaced by the Gauss-law-type constraint imposed by A_0i.

Note that the action (4.1) is still Poincaré invariant, with the Noether charges of (4.4). In fact, it is conformal invariant. A vector field k is a conformal Killing vector field on 6D Minkowski space–time if there exists some space–time scalar function f_k for which

$${({\mathcal{L}}_k\eta )}_{\mu \nu }=f_k\hspace{0.17em}{\eta }_{\mu \nu },$$ 4.13

where ${\mathcal{L}}_k\eta$ is the Lie derivative of the Minkowski metric η with respect to the vector field k(X). For any such k, the first-order variation

$${\delta }_k{X}^m=k^m,{\delta }_k{P}_m=-({\mathrm{\partial }}_m{k}^n)P_n,\hspace{0.17em}{\delta }_{k}e=e{f}_k,$$ 4.14

is an invariance of the action (4.1) with the phase-space functions ${\mathcal{H}}_{\mu }$ of (4.11). The corresponding Noether charge is

$$Q[k]=\int {\text{d}}^5\sigma \hspace{0.17em}{k}^{\mu }{P}_{\mu }.$$ 4.15

(c). The Monge-gauge action and its symmetries

In the Monge gauge, X^μ(ξ) = ξ^μ, the solution of the constraints ${\mathcal{H}}_{\mu }=0$ at zero tension is

$$P^0=\pm |\mathbf{\text{V}}|\text{and}\mathbf{\text{P}}=\mathbf{\text{V}}.$$ 4.16

Choosing P⁰ > 0, the Monge-gauge action for the T → 0 (equivalently, strong-field) limit is seen to be

$$S=\int \hspace{0.17em}\text{d}t\int {\text{d}}^5\sigma \{\frac{1}{2}{\dot{A}\hspace{0.17em}}_{ij}{D}^{ij}-|\mathbf{\text{V}}|-\frac{1}{2}{\sigma }_{ij}{\chi }^{ij}\}.$$ 4.17

The field equations of this action are jointly equivalent to

$${\dot{A}\hspace{0.17em}}_{ij}=-B_{ijk}{n}^k+{\sigma }_{ij},$$ 4.18

where nⁱ are the components of the unit 5-vector field

$$\mathbf{\text{n}}=\frac{\mathbf{\text{V}}}{|\mathbf{\text{V}}|},$$ 4.19

and the constraints

$$D^{ij}+B^{ij}=0\text{and}{\mathrm{\partial }}_{[i}{\sigma }_{jk]}=0.$$ 4.20

Some constraint on the 5-space 2-form σ was to be expected because Lagrange multipliers for second-class constraints are determined by the equations of motion; the fact that only dσ is determined was also to be expected because the exact part of σ (in its Hodge decomposition) is what imposes the first-class constraint associated with the Abelian 2-form gauge invariance, and Lagrange multipliers for first-class constraints are not determined by the field equations [14]. The equation for A and the chirality constraint jointly imply the following gauge-invariant field equations

$${\dot{B}}^{ij}=-3{\mathrm{\partial }}_k(n^{[k}{B}^{ij]})\text{and}{\dot{D}}^{ij}=-3{\mathrm{\partial }}_k(n^{[k}{D}^{ij]}).$$ 4.21

Note that these equations imply

$${\dot{\chi }}^{ij}=3{\mathrm{\partial }}_k(n^{[i}{\chi }^{jk]}),$$ 4.22

which shows that the chirality constraint effects a consistent truncation of the equations that we would have without this constraint.

This action (4.17) is manifestly invariant under 5-space rotations but Lorentz invariance is no longer manifest. Nevertheless, it is Lorentz invariant. The Noether charges are unchanged by gauge fixing except that we must use the gauge-fixed expressions when evaluating them or taking Poisson brackets, which must now be computed using the canonical PBs of the gauge-fixed action

$${\{A_{ij}(\mathit{\sigma }),D^{kl}({\mathit{\sigma }}^{\prime })\}}_{PB}=({\delta }_i^k{\delta }_j^l-{\delta }_i^l{\delta }_j^k)\hspace{0.17em}\delta (\mathit{\sigma }-{\mathit{\sigma }}^{\prime }).$$ 4.23

For example, the Monge-gauge charges for time and space translations are

$${\mathcal{P}}^0=\int {\text{d}}^5\sigma \hspace{0.17em}|\mathbf{\text{V}}|\text{and}\mathcal{P}=\int {\text{d}}^5\sigma \mathbf{\text{V}}.$$ 4.24

It is instructive to verify that these quantities are conserved as a consequence of the equations of motion. This can be done even without the use of the chirality constraint but in this case one must use instead the Gauss-law-type constraint ∂_i D^ij = 0. Using the identity

$$\frac{1}{2}({\mathrm{\partial }}_p{n}^j){\epsilon }_{ijklm}(D^{pk}{B}^{lm}+B^{pk}{D}^{lm})\equiv ({\mathrm{\partial }}_i{n}^j)V_j-({\mathrm{\partial }}_j{n}^j)V_i$$ 4.25

and the fact that V = P, one finds that

$${\dot{P}}_i=-{\mathrm{\partial }}_j(n^j{P}_i)(\Rightarrow {\dot{P}}^0=-\mathbf{\nabla }\cdot \mathbf{\text{P}}),$$ 4.26

and hence ${\dot{\mathcal{P}}}_{\mu }=0$ for appropriate boundary conditions.

More generally, the Monge-gauge expression for the Noether charge (4.15) associated with any vector field k is

$$Q[k]=\int {\text{d}}^5\sigma \hspace{0.17em}{k}^{\mu }(t,\mathit{\sigma })P_{\mu },$$ 4.27

and so

$$\begin{array}{rl}\dot{Q}[k]& =\int {\text{d}}^5\sigma \{{\dot{k}}^0{P}_0+{\dot{k}}^i{P}_i+k^0{\dot{P}}_0+k^i{\dot{P}}_i\}\\ \hspace{0.17em}& =\int {\text{d}}^5\sigma \{{\dot{k}}^0{P}_0+(\dot{\mathbf{\text{k}}}-\mathbf{\nabla }{k}^0)\cdot \mathbf{\text{P}}+({\mathrm{\partial }}_j{k}^i)n^j{n}_i{P}^0\}+\int {\text{d}}^5\sigma \mathbf{\nabla }\cdot [(k^0-\mathbf{\text{n}}\cdot \mathbf{\text{k}})\mathbf{\text{P}}],\end{array}$$ 4.28

where the second equality involves the use of equations (4.26), an integration by parts, and use of the relation P = nP⁰. Next, we observe that k generates a conformal isometry of the 6D Minkowski metric η if ${\mathcal{L}}_k\eta =f_k\eta$ for some function f_k; this is equivalent to the equations

$${\dot{k}}^0=f_k,\dot{\mathbf{\text{k}}}-\mathbf{\nabla }{k}^0=0,\hspace{0.17em}{\mathrm{\partial }}_{(i}{k}_{j)}=f_k{\delta }_{ij},$$ 4.29

from which we deduce that $\dot{Q}[k]=0$ for appropriate boundary conditions. This establishes conformal invariance of the action (4.17).

(d). Lorentz invariance and the stress tensor

The Lorentz boost generator is

$$\mathbf{\text{L}}=t\mathcal{P}-\int {\text{d}}^5\sigma \hspace{0.17em}\mathit{\sigma }{P}^0.$$ 4.30

In Monge gauge, we find, for constant uniform 5-vector parameter w, that

$${\{A_{ij},\mathbf{\text{w}}\cdot \mathbf{\text{L}}\}}_{PB}=\frac{1}{2}{\epsilon }_{ijklm}{N}^k{B}^{lm},$$ 4.31

where

$$\mathbf{\text{N}}=t\mathbf{\text{w}}-(\mathbf{\text{w}}\cdot \mathit{\sigma })\mathbf{\text{n}}.$$ 4.32

This implies that

$${\{B^{ij},\mathbf{\text{w}}\cdot \mathbf{\text{L}}\}}_{PB}=3{\mathrm{\partial }}_k(N^{[k}{B}^{ij]}).$$ 4.33

A separate PB calculation yields

$${\{D^{ij},\mathbf{\text{w}}\cdot \mathbf{\text{L}}\}}_{PB}=3{\mathrm{\partial }}_k(N^{[k}{D}^{ij]}).$$ 4.34

A consequence of the above results is that

$${\{{\chi }^{ij},\mathbf{\text{w}}\cdot \mathbf{\text{L}}\}}_{PB}=3{\mathrm{\partial }}_k(N^{[k}{\chi }^{ij]}),$$ 4.35

which shows that the chirality constraint is Lorentz invariant. Another consequence is

$$\begin{array}{rl}\hspace{0.17em}& {\{\mathbf{\text{V}},\mathbf{\text{w}}\cdot \mathbf{\text{L}}\}}_{PB}=-\mathbf{\text{w}}|\mathbf{\text{V}}|+{\mathrm{\partial }}_k(N^k\mathbf{\text{V}})\\ \text{and}& {\{|\mathbf{\text{V}}|,\mathbf{\text{w}}\cdot \mathbf{\text{L}}\}}_{PB}=-(\mathbf{\text{w}}\cdot \mathbf{\text{n}})|\mathbf{\text{V}}|+{\mathrm{\partial }}_k(N^k|\mathbf{\text{V}}|).\end{array}\}$$ 4.36

Of course, the first of these equations implies the second.

Recall that P_μ in Monge gauge has components ( − |V|, V). However, these are not the components of a Lorentz co-vector field, despite the notation.⁴ Let us define a new set of components

$${\hat{P}}_{\mu }=\frac{P_{\mu }}{\sqrt{P^0}}.$$ 4.37

In Monge gauge,

$${\hat{P}}_0=-\sqrt{|\mathbf{\text{V}}|}\text{and}\widehat{\mathbf{\text{P}}}=\frac{\mathbf{\text{V}}}{\sqrt{|V|}}.$$ 4.38

These are the components of a Lorentz co-vector field. A direct calculation using (4.36) yields

$$\begin{array}{rl}\hspace{0.17em}& {\{\widehat{\mathbf{\text{P}}},\mathbf{\text{w}}\cdot \mathbf{\text{L}}\}}_{PB}=N^k{\mathrm{\partial }}_k\widehat{\mathbf{\text{P}}}+{\hat{P}}_0[\mathbf{\text{w}}+\frac{1}{2}\mathbf{\text{n}}(\mathbf{\text{w}}\cdot \mathit{\sigma })(\mathbf{\nabla }\cdot \mathbf{\text{n}})],\\ \text{and}& {\{{\hat{P}}_0,\mathbf{\text{w}}\cdot \mathbf{\text{L}}\}}_{PB}=N^k{\mathrm{\partial }}_k{\hat{P}}_0+{\hat{P}}_0[(\mathbf{\text{w}}\cdot \mathbf{\text{n}})+\frac{1}{2}(\mathbf{\text{w}}\cdot \mathit{\sigma })(\mathbf{\nabla }\cdot \mathbf{\text{n}})].\end{array}\}$$ 4.39

Again, the first of these equations implies the second.

This result for the Lorentz transformation of ${\hat{P}}_{\mu }$ may be compared with its Lie derivative with respect to the vector field ζ with components

$${\zeta }^0=\mathbf{\text{w}}\cdot \mathit{\sigma }\text{and}\mathit{\zeta }=t\mathbf{\text{w}}.$$ 4.40

Using equations (4.26) to replace time-derivatives by space derivatives, one may calculate this Lie derivative from its definition

$${\mathcal{L}}_{\zeta }{\hat{P}}_{\mu }:={\zeta }^{\nu }{\mathrm{\partial }}_{\nu }{\hat{P}}_{\mu }+({\mathrm{\partial }}_{\mu }{\zeta }^{\nu }){\hat{P}}_{\nu }.$$ 4.41

Comparing the result with (4.39), one sees that

$${\{{\hat{P}}_{\mu },\mathbf{\text{w}}\cdot \mathbf{\text{L}}\}}_{PB}={({\mathcal{L}}_{\zeta }\hat{P})}_{\mu }.$$ 4.42

To see that this is a Lorentz transformation, we observe (i) that to first order in the 5-vector parameter w,

$$t\to t^{\prime }=t+(\mathbf{\text{w}}\cdot \mathit{\sigma })\text{and}\mathit{\sigma }\to \mathit{\sigma }=\mathit{\sigma }+t\mathbf{\text{w}}$$ 4.43

is a Lorentz boost transformation of the space–time coordinates, and (ii) that for any scalar field Φ we have, again to first order in w,

$$\mathrm{\Phi }(t^{\prime },{\mathit{\sigma }}^{\prime })=\mathrm{\Phi }(t,\mathit{\sigma })+[{\mathcal{L}}_{\zeta }\mathrm{\Phi }](t,\mathit{\sigma }).$$ 4.44

Similarly, the first-order variation of any tensor or tensor-density field under a Lorentz boost with parameter w is given by its Lie derivative with respect to ζ.

Since ${\hat{P}}_{\mu }$ is a Lorentz 6-vector, it follows that

$$T_{\mu \nu }={\hat{P}}^{\mu }{\hat{P}}^{\nu }={(P^0)}^{-1}{P}_{\mu }{P}_{\nu }$$ 4.45

is a symmetric tensor of the Lorentz group. It is also ‘conserved’ in the sense that

$${\mathrm{\partial }}^{\mu }{T}_{\mu \nu }=0.$$ 4.46

To check this, we observe that we already know from (4.26) that ∂^μ P_μ = 0, so that

$${\mathrm{\partial }}^{\mu }{T}_{\mu \nu }=P^{\mu }{\mathrm{\partial }}_{\mu }\left(\frac{P_{\nu }}{P^0}\right).$$ 4.47

The right-hand side is trivially zero for ν = 0, so (taking into account that P/P⁰ = n) we see that what needs to be checked is that

$${\dot{n}}^i+\mathbf{\text{n}}\cdot \mathbf{\nabla }{n}^i=0,$$ 4.48

but this is a consequence of the equations of (4.26).

The expression (4.45) for the stress tensor may be rewritten in terms of the energy density P⁰ = |V| and a null 6-vector n as

$$T_{\mu \nu }=n_{\mu }{n}_{\nu }|\mathbf{\text{V}}|\text{and}{n}_{\mu }=(-1,\mathbf{\text{n}}).$$ 4.49

Since n is null this stress tensor is traceless, as it must be given the conformal invariance of the field equations. This is the stress tensor of a null fluid, as found for BBE in [15] and in agreement with [10].

(e). Simplifying the action

So far we have maintained the chirality constraint as one imposed by a Lagrange multiplier in the action. This has the advantage that the symplectic form defined by the phase-space action is in standard Darboux form, from which we can easily read off the PBs, which we have been using extensively in the above discussion of symmetries of the action. However, now that we have dealt with this topic it is convenient to use the chirality constraint to eliminate D from the action; we simply replace it by −B. We then have

$$V_i\to \frac{1}{4}{\epsilon }_{ijklm}{B}^{jk}{B}^{lm}\equiv {(B\wedge B)}_i,$$ 4.50

and the action (4.17) becomes a functional of the space 2-form A alone

$$S[A]=\int \text{d}t\int {\text{d}}^5\sigma \{-\frac{1}{2}{\dot{A}\hspace{0.17em}}_{ij}{B}^{ij}-|B\wedge B|\}.$$ 4.51

This is the action advertised in the Introduction. Its field equations are, as could be expected from (4.21),

$${\dot{B}}^{ij}=3{\mathrm{\partial }}_k(n^{[i}\hspace{0.17em}{B}^{jk]}),$$ 4.52

where the direction of the unit vector field n is now determined by $B\wedge B$. These are nonlinear equations, analogous to the BBE equations (2.6); as in that case, the nonlinearity is due to the dependence of the unit 5-vector field n on B.

It is instructive to verify the Lorentz invariance of this simplified action. From (4.31), we see that the first-order Lorentz transformation of A is

$${\delta }_{\mathbf{\text{v}}}{A}_{ij}=\frac{1}{2}{\epsilon }_{ijklm}{N}^k{B}^{lm},$$ 4.53

where N is given in (4.32). This implies⁵

$${\delta }_{\mathbf{\text{v}}}|B\wedge B|=-(\mathbf{\text{n}}\cdot \mathbf{\text{v}})|B\wedge B|+\text{total space derivative}.$$ 4.54

The Lorentz variation of the Hamiltonian is therefore not zero but we should not expect it to be zero because the transformation of A involves an explicit time-dependence through its dependence on N, and this will produce a variation of the geometric term in the action. Specifically, one finds that

$${\delta }_{\mathbf{\text{v}}}(-{\dot{A}\hspace{0.17em}}_{ij}{B}^{ij})=-(\mathbf{\text{n}}\cdot \mathbf{\text{v}})|B\wedge B|+\text{total time derivative},$$ 4.55

where the second line makes use of the identity $\mathbf{\text{n}}\cdot \dot{\mathbf{\text{n}}}\equiv 0$. The action S[A] is therefore Lorentz invariant, despite appearances.

We also learn something else from this check of Lorentz invariance. It might be thought that we could generalize the action S[A] to

$$S_{\lambda }[A]=\int \text{d}t\int {\text{d}}^5\sigma \{-\frac{1}{2}{\dot{A}\hspace{0.17em}}_{ij}{B}^{ij}-\lambda |B\wedge B|\},$$ 4.56

for arbitrary constant λ. However, this generalized action is Lorentz invariant only for λ = 1 (given our assumption of a positive Hamiltonian density). Also, it is not possible to introduce λ by rescaling A since both terms in the action (the geometrical term and the Hamiltonian) are homogeneous of second degree in A. Despite this fact, the Hamiltonian is not quadratic in A; if it were, S[A] would be a free-field action, with free-field equations, but the equations are not free-field equations.

5. IR limit of an M5-brane in AdS₇ × S⁴

So far we have considered the M5-brane, or some truncation of its dynamics, in the 11D Minkowski vacuum of 11D supergravity. Now we turn to the M5-brane in the AdS₇ × S⁴ vacuum. For an S⁴ of radius R, the AdS₇ × S⁴ metric is

$$\text{d}{s}_{11}^2=\left(\frac{r}{R}\right)\text{d}{x}^{\mu }\hspace{0.17em}\text{d}{x}^{\nu }{\eta }_{\mu \nu }+{\left(\frac{R}{r}\right)}^2(\text{d}{r}^2+r^2\hspace{0.17em}\text{d}{\mathrm{\Omega }}_4^2),$$ 5.1

where r is the radial coordinate for spherical polar coordinates on ${\mathbb{E}}^5$ and $\text{d}{\mathrm{\Omega }}_4^2$ is the SO(5)-invariant unit metric on S⁴. If we set

$$r=\frac{4R^3}{z^2},$$ 5.2

then the metric becomes

$${\left(\frac{2R}{z}\right)}^2[\text{d}{x}^{\mu }\hspace{0.17em}\text{d}{x}^{\nu }{\eta }_{\mu \nu }+\text{d}{z}^2+{\left(\frac{z}{2}\right)}^2\text{d}{\mathrm{\Omega }}_4^2],$$ 5.3

where z is now an inverse-square radial coordinate. The boundary of AdS₇ is at z = 0 whereas the Killing horizon of our Poincaré patch coordinates is at z = ∞.

We now consider a static planar M5-brane in this background, at fixed z and fixed position on the 4-sphere, so that its worldvolume is coincident with the 6D Minkowski space with coordinates x^μ. This is a solution of the M5-brane equations of motion and we wish to consider fluctuations about it. The induced metric on this fluctuating M5-brane is

$$h_{ij}={\left(\frac{2R}{z}\right)}^2[{\mathrm{\partial }}_i{x}^{\mu }{\mathrm{\partial }}_j{x}^{\nu }{\eta }_{\mu \nu }+{\mathrm{\partial }}_{i}z{\mathrm{\partial }}_{j}z+{\left(\frac{z}{2}\right)}^2{\mathrm{\partial }}_i{\psi }^I{\mathrm{\partial }}_j{\psi }^J{\overline{g}}_{IJ}],$$ 5.4

where ${\overline{g}}_{ij}$ is the metric on the unit 4-sphere in angular coordinates {ψ^I;I = 1, 2, 3, 4}. If we now use this result for the induced metric in the constraint functions (3.3) for the M5-brane phase-space action, and rescale $u^0\to {\tilde{u}}^0$ such that $u^0{\mathcal{H}}_0={\tilde{u}}^0{\tilde{\mathcal{H}}}_0$ for

$${\tilde{\mathcal{H}}}_0={\left(\frac{2R}{z}\right)}^2{\mathcal{H}}_0,$$ 5.5

then we find that

$$\begin{array}{rl}\hspace{0.17em}& {\tilde{\mathcal{H}}}_0=\frac{1}{2}\{{\eta }^{\mu \nu }{p}_{\mu }{p}_{\nu }+p_z^2+\frac{4L^2}{z^2}+\mathcal{O}\left(\frac{T{R}^6}{z^6}\right)\}\\ \text{and}& {\mathcal{H}}_i={\mathrm{\partial }}_i{x}^{\mu }{p}_{\mu }+{\mathrm{\partial }}_{i}z{p}_z+{\mathrm{\partial }}_i{\psi }^I{p}_I-{(B\wedge B)}_i,\end{array}\}$$ 5.6

where

$$L^2={\overline{g}}^{IJ}{p}_i{p}_J,$$ 5.7

which is the square of the S⁴ angular momentum, and we have used the chirality constraint to set D = −B in the equation for ${\mathcal{H}}_i$.

However, the constraint functions (3.3) require some T-dependent modifications when the 11D supergravity 3-form potential C⁽³⁾ is non-zero, as it is for the AdS₇ × S⁴ vacuum since dC⁽³⁾ is proportional to the volume 4-form on S⁴ [16]. In addition, there is a coupling, with coefficient T, to the dual 6-form C⁽⁶⁾ that is defined for solutions of the 11D supergravity field equations; it takes the following form for the AdS₇ × S⁴ solution [17]:

$$C^{(6)}\propto {\left(\frac{2R}{z}\right)}^6\hspace{0.17em}\text{d}{x}^0\hspace{0.17em}\text{d}{x}^1\cdots \hspace{0.17em}\text{d}{x}^5.$$ 5.8

The coupling to C⁽⁶⁾ therefore contributes a further $\mathcal{O}(T{R}^6/z^6)$ term.

As these modifications all come with a factor of T they are not relevant to the T → 0 limit, but now we can consider a different limit in which z → ∞. Recall that as z → 0 the M5-brane moves to the 6D Minkowski boundary of AdS₇, where its dynamics becomes that of a free field theory. In the context of the AdS/CFT correspondence, this is the UV limit of the M5-brane dynamics. The z → ∞ limit is an IR limit in which the brane moves to the null Killing horizon (which is the boundary of the Poincaré patch covered by the coordinates that we chose for the AdS₇ metric). All $\mathcal{O}(T{R}^6/z^6)$ terms can still be ignored, now because they involve inverse powers of z, but this still leaves modifications arising from couplings to C⁽³⁾, which takes the form

$$C^{(3)}\propto R^3{c}_{IJK}(\{\psi \})\hspace{0.17em}\text{d}{\psi }^I\hspace{0.17em}\text{d}{\psi }^J\hspace{0.17em}\text{d}{\psi }^K.$$ 5.9

All modifications due to the non-zero background form fields, in particular those due to C⁽³⁾, can be viewed (by field redefinitions) as modifications to the phase-space constraints only. One such modification is $B\to B-T{\mathcal{C}}^{(3)}$, where B = dA and ${\mathcal{C}}^{(3)}$ is the pullback of C³ to the M5-brane. Another is the replacement P_m → P_m − TC_m, where C_m is the 5-space Hodge-dual of the 5-form obtained by contraction with the vector field ∂_m of a 6-form consisting of a linear combination of $C^{(3)}\wedge C^{(3)}$ and $C^{(3)}\wedge B$ [9]. In each case⁶, there is a factor of ${\mathcal{C}}^{(3)}$. These terms are also irrelevant in the z → ∞ limit, provided that the limit is taken keeping the momentum variables p_I, and hence L², finite. The L²/z² term then drops out of ${\tilde{\mathcal{H}}}_0$ and the p_I variables in the action become Lagrange multipliers for the constraints

$${\dot{\psi }}^I-u^i{\mathrm{\partial }}_i{\psi }^I=0.$$ 5.10

These equations should hold for all possible uⁱ, which are generically non-zero and vary with solutions of the equations of motion. The position of the M5-brane worldvolume on S⁴ is therefore fixed (independent of worldvolume coordinates) in the z → ∞ limit.

To conclude, the z → ∞ limit leads to the action

$$S_{M5}^{(IR)}=\int \text{d}t\int {\text{d}}^5\sigma \{{\dot{x}}^{\mu }{p}_{\mu }+\dot{z}{p}_z-\frac{1}{2}{\dot{A}\hspace{0.17em}}_{ij}{B}^{ij}-{\tilde{u}}^0{\tilde{\mathcal{H}}}_0-u^i{\mathcal{H}}_i\},$$ 5.11

where

$${\tilde{\mathcal{H}}}_0={\eta }^{\mu \nu }{p}_{\mu }{p}_{\nu }+p_z^2\text{and}{\mathcal{H}}_i={\mathrm{\partial }}_i{x}^{\mu }{p}_{\mu }+{\mathrm{\partial }}_{i}z{p}_z-{(B\wedge B)}_i.$$ 5.12

Choosing the Monge gauge and solving the constraints yields the equivalent action

$$S=\int \hspace{0.17em}\text{d}t\int {\text{d}}^5\sigma \{\dot{z}{p}_z-\frac{1}{2}{\dot{A}\hspace{0.17em}}_{ij}{B}^{ij}-\mathcal{H}\},$$ 5.13

where

$$\mathcal{H}=\sqrt{{\delta }^{ij}{p}_i{p}_j+p_z^2}\text{and}{p}_i={(B\wedge B)}_i-{\mathrm{\partial }}_{i}z{p}_z.$$ 5.14

A consistent truncation of this action is to set dz = 0 and p_z = 0, in which case we recover the chiral 2-form electrodynamics with the action S[A] of (1.6). This derivation of it allows us to interpret it as the dynamics of a chiral 2-form on an M5-brane at the null Killing horizon of AdS₇.

6. Discussion

It has been shown here that the interacting conformal-invariant BBE [1] has an analogue in six dimensions but for a 2-form potential, which is subject to a chirality condition that halves the number of degrees of freedom. Just as BBE is a strong-field limit of the nonlinear (and non-conformal) Born-Infeld electrodynamics, which is a truncation of the Dirac-Born-Infeld action for a D3-brane, the new conformal chiral 2-form electrodynamics is a strong-field limit of the nonlinear (and non-conformal) 2-form electrodynamics that survives a similar truncation of the analogous action for an M5-brane. In both cases, this strong-field limit can also be taken prior to the truncation, in two different ways.

One way is to take the tensionless limit, either of a D3-brane in the 10D Minkowski vacuum of IIB supergravity (as done in [2], where it leads to a generalization of BBE to include scalar fields) or of an M5-brane in the 11D Minkowski vacuum of 11D supergravity, which yields a similar generalization (not explored here) of the 6D chiral 2-form electrodynamics. These limits must be taken within a phase-space formulation of the action, but these are known from earlier work in [11] (D-branes) and [9] (M5-brane); in the latter work the tensionless limit of the M5-brane action was also considered but with inconclusive results; the problematic features of this limit that were noted there have been resolved here. Essentially, one should choose definitions of the worldvolume fields for which the tension drops out in the weak-field limit; the tensionless limit then becomes possible and is equivalent to a strong-field limit.

The other way to arrive at both BBE and the new 6D chiral 2-form electrodynamics is to consider, respectively, a D3-brane in the AdS₅ × S⁵ vacuum of 10D IIB supergravity and an M5-brane in the AdS₇ × S⁴ vacuum of 11D supergravity. In Poincaré patch coordinates for AdS an infinite static planar brane, with Minkowski worldvolume, will solve the brane equations of motion if it is placed at a fixed radial distance from the Killing horizon in these coordinates. Moving the brane to larger distances corresponds to a renormalization group flow towards the ultraviolet limit of the brane dynamics (which is the free conformal theory of the weak-coupling limit) whereas moving it closer to the horizon corresponds to a flow towards the IR. Again, the limit cannot be taken in the configuration-space form of the brane action, but it can be taken in the phase-space form of the action.

As shown for the D3-brane in [2] and here for the M5-brane, this IR limit describes a brane at the AdS Killing horizon with all transverse fluctuations frozen except for fluctuations into the AdS bulk, determined by a single scalar. This is generic, and would apply to the M2-brane in AdS₄ × S⁷ but for the D3-brane and M5-brane there is the additional dynamics of the worldvolume albelian gauge potential, a 1-form for D3 and a 2-form for M5. A consistent truncation to this form-field dynamics yields precisely BBE in the D3 case and the new conformal chiral 2-form electrodynamics in the M5 case.

Whereas a tensionless limit of the D3-brane or M5-brane is a rather artificial one within the String/M-theory context for which these branes have physical relevance, the IR limit just desribed is a natural one to consider. The ‘AdS×S’ background is just a low-energy description of the effect of a large number, N + 1 say, of parallel coincident branes; from an AdS/CFT perspective the ‘AdS×S’ background is the AdS bulk description of the ground state of the lR dynamics of these branes. If one of the N + 1 branes is separated from the remaining N but remains parallel to them then it will appear as a probe brane with a worldvolume coincident with one of the 6D Minkowski slices of the Poincaré patch of the ADS space–time (with a slightly increased constant radius of curvature). The IR limit thus corresponds to the return to the fold of a lone and isolated brane; it is therefore plausible that its IR dynamics will contain some information about the IR dynamics of the collection of N + 1 branes. In the M5-case, this is the ‘mysterious’ (2,0)-supersymmetry 6D CFT.

This brings us to the question of whether the conformal chiral 2-form 6D electrodynamics described here can be incorporated into a supersymmetric extension, in particular, a (2, 0)-supersymmetric extension. One could attempt to answer this question directly but it could also be addressed by returning to the phase-space action for the M5-brane but now including all fermions; this action was given in [9]. The weak-field limit of the configuration space M5-brane action of [8] was worked out in detail in [17]; not surprisingly, this limit reduces the M5-brane dynamics to that of a free (2, 0) 6D supermultiplet. It is, therefore, plausible that the strong-field limit of the full phase-space action of the M5-brane will be a (2, 0)-supersymmetric extension of the simple chiral 2-form electrodynamics presented here, but any detailed verification of this is likely to require significantly more effort than the author has exerted to obtain the results reported here.

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Competing interests

We declare we have no competing interest.

Funding

This work was partially supported by STFCconsolidated grant no. ST/L000385/1.