A manifestly Lorentz invariant chiral boson action

The problem of finding a manifestly Lorentz invariant action for chiral n-form electrodynamics is addressed in the context of a recent solution for n = 2 based on a truncation of M5-brane dynamics; it is explained here why this solution requires interactions. In contrast, the field equation of the analogous action for “chiral 0-form electrodynamics” is the free chiral boson equation for a scalar field, but with a zero mode that is eliminated by a gauge invariance.

The problem of finding a manifestly Lorentz invariant action for chiral n-form electrodynamics is addressed in the context of a recent solution for n = 2 based on a truncation of M5-brane dynamics; it is explained here why this solution requires interactions. An analogous action for n = 0 based on a truncation of string dynamics is shown to reduce, upon gauge fixing, to the Floreanini-Jackiw action for a free chiral boson with its zero mode eliminated by a gauge invariance.

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Maxwell's conformal-invariant equations for electrodynamics have a natural generalization to n-form electrodynamics in a Minkowski spacetime of 2(n + 1) dimensions, and these equations are derivable from a Lorentzinvariant action quadratic in the gauge-invariant (n + 1)form field strength (see e.g. [1]). For even n one may consistently impose a self-duality condition on this (n + 1)form; the resulting parity-violating, but still Lorentz (and conformal) invariant, equations are those of "chiral n-form electrodynamics" (see e.g. [2]). The n = 2 case arises naturally in M-theory as a linearization of the non-linear 2-form electrodynamics on a planar static M5-brane [3]. One may also consider n = 0, for which the self-dual 'field-strength' 1-form is the differential of a scalar field called a "chiral boson".
A long-standing problem has been to find a manifestly Lorentz invariant action for these chiral n-form field theories. Many solutions have been proposed, each with some feature that could be considered undesirable; the most recent proposal is that of Sen [4] to whom we defer for references to earlier proposals. A feature of all proposed solutions to date is that they apply to the free field theory, in some cases as a free-field limit; e.g. for the non-linear n = 2 chiral electrodynamics on the M5-brane [5,6], which was found in non-covariant form in [7]. In a recent work, the author has shown that another solution is possible for n = 2 provided interactions implied by M5-brane dynamics are included [8]. One purpose of this paper is to explain why this solution cannot be used for the free-field theory, but the main purpose is to present an analogous chiral 0-form electrodynamics, which turns out to be a manifestly Lorentz invariant action for the free scalar field theory of Floreanini and Jackiw [9]. The field equation is the free "chiral boson" equation but with a gauge invariance that eliminates the 'zero mode'.
We shall begin with a brief summary of the 6D chiral 2-form electrodynamics arising as a truncation of M5brane dynamics. The phase-space action is a functional of the space components {A ij ; i, j = 1, 2, . . . , 5} of the 6D 2-form potential A, of the form [10] where B ij = 1 2 ε ijklm ∂ k A lm , and the overdot indicates a time-derivative; the Hamiltonian density H can be rewritten, after some rescaling of fields, in the form [8] where T is a constant with dimensions of energy density (the M5-brane tension) and Notice that The free-field equations for A ij in this limit are the equations of chiral 2-form electrodynamics obtained in [2]. As pointed out in [8], the M5-brane origin of (1) implies that it is equivalent to a manifestly Lorentz-invariant action obtained by truncation of the M5-brane phase-space action of [10]; this includes a parity-violating "chirality" constraint, which implies that the variables canonically conjugate to A ij are (up to a sign) B ij . After using this constraint, the truncated M5-brane action takes the form where {X µ ; µ = 0, 1 . . . 5} are 6D Minkowski coordinates as functions of the now arbitrary local coordinates (t, σ), with canonically conjugate momentum densities P µ , and u µ are Lagrange multipliers for the remaining constraints. The constraint functions are where h ij = ∂ i X µ ∂ j X ν η µν , the space components of the Minkowski metric g in the new local coordinates, and C µ is such thatẊ µ C µ = √ − det g. The constraint functions are "first-class" and hence generate gauge transformations, which allow us to impose the Monge gauge conditions (X 0 , X) = (t, σ). In this gauge, H = P 0 and the constraints may be solved for P µ ; the action (1) with Hamiltonian density (2) is then recovered if we choose H to be positive.
The advantage of the action (5), as compared with (1), is that its Lorentz invariance is now a linearly realized 'internal' symmetry that acts only on the variables (X µ , P µ ). The Monge gauge invariance breaks this symmetry but preserves a combination of it with a 'compensating' diffeomorphism, with respect to which the other fields transform. In this way, the non-linear Lorentz transformations of A ij that leave invariant the gaugefixed action emerge upon gauge fixing. Further discussion of this point may be found in [8] but it will also be discussed here in the context of the chiral 0-form electrodynamics to be introduced below.
Before moving on to 0-form electrodynamics, we pause to explain why interactions are necessary for manifest Lorentz invariance of the action for chiral 2-form electrodynamics. We have seen that the T → ∞ limit of (2) leads to a phase-space action for the free 6D chiral 2-form electrodynamics. One might think that an equivalent action with a linearly realized Lorentz invariance could be found by taking T → ∞ in (5). However, the cancellation of terms proportional to T in the T → ∞ limit of (2) corresponds to a cancellation in H 0 of the T 2 det h term with the T 2 C 2 term (since C 2 ≡ − det h), but this is now insufficient for a well-defined T → ∞ limit, and a rescaling of H 0 to make it possible results in a constraint that cannot be solved for P 0 .
We conclude that manifest Lorentz invariance requires T to be finite, but this is consistent with T = 0. The T → 0 limit was first considered in [11] as a generalization of a limit proposed in [12] for 4D Born-Infeld electrodynamics, and more recently in [8], where it was observed (following a similar observation in the context of Born-Infeld electrodynamics [13]) that the T → 0 limit can also be taken in the manifestly Lorentz invariant action (5). However, this limit yields a chiral 2-form electrodynamics that is still interacting; in Monge gauge the Hamiltonian density is H = |B ∧ B|, which is homogeneous of second degree in A, but not quadratic.
Consider now the following action for a scalar field ϕ in a 2D Minkowski spacetime with Minkowski coordinates {x µ ; µ = 0, 1}: where T is a constant with dimensions of energy density. There are interactions for finite T but Passing to the Hamiltonian formulation by introducing a momentum density π(x) canonically conjugate to ϕ(x), we arrive at a phase-space action of the form where t = x 0 and σ = x 1 , and the Hamiltonian density is where the prime indicates a derivative with respect to σ. Notice that in which case elimination of π by its field equation π = ϕ ′ yields the action (8).
We may also take the T → 0 limit: In this case the field equations depend on the sign of ϕ ′ π. We could resolve this ambiguity by imposing the constraint π = ±ϕ ′ on the field equations for some choice of sign; this choice is reversed by parity so we may choose π = ϕ ′ without loss of generality, in which case we are left with the single equation This is the standard "chiral boson" equation. Rather than impose the chirality condition π = ϕ ′ on the field equations we can instead impose it in the action (9). The Hamiltonian density of (10) then reduces to the T -independent expression H = 1 2 (ϕ ′ ) 2 , which yields the Floreanini-Jackiw (FJ) action: The corresponding scalar field equation is This is the chiral-boson equation again, but now for ϕ ′ rather than ϕ; the general solution for ϕ is The zero mode is an unphysical gauge degree of freedom because the action has a (restricted) gauge invariance: ϕ(ξ) → ϕ(ξ) + a(t) for arbitrary function a(t).
It is not obvious that the above steps leading to the FJ action preserve Lorentz invariance, but this action is known to be not only Lorentz invariant but also conformal invariant [9]. We now aim to find an alternative action that makes the Lorentz invariance manifest; we shall see that it also clarifies conformal invariance.
We begin by constructing a manifestly Lorentz invariant alternative to the phase space action (9). The action (7), although manifestly Lorentz invariant, will not serve our purpose as it does not allow the incorporation of a chirality condition. However, the remedy for this difficulty starts with a rewriting of (7) using arbitrary local coordinates {ξ µ ; µ = 0, 1} for the 2D Minkowski spacetime, where now ξ 0 = t and ξ 1 = σ. The previous Minkowski coordinates x µ are replaced by local functions X µ (ξ), and the Minkowski metric is now The action (7) now takes the form In passing to the Hamiltonian formulation we must take into account that we have new fields X µ (ξ), which will require their own conjugate momentum fields P µ (ξ). In addition, because the configuration space action is invariant under reparametrizations of the 2D Minkowski spacetime, the Hamiltonian density is replaced by a sum of constraints imposed by Lagrange multipliers. The final result for the phase-space action is where (e, u) are the Lagrange multiplier fields, and with C µ = ε µν (X ν ) ′ . Useful identities are The canonical Poisson bracket relations determined by the 'geometrical' part of the action are These may be used to compute the PB relations of the constraint functions; the computation is greatly simplified by choosing a functional basis for them: where α is a 1D vector field and β is a scalar inversedensity, equivalent to a vector field in 1D. Assuming that (α, β) are smooth, and have compact support, one finds that where [., .] indicates a commutator of (1D) vector fields. This result shows that the constraints are first-class, and hence that they generate gauge invariances; these are, of course, equivalent ("on-shell") to the invariance under 2D diffeomorphisms that we introduced initially. We may therefore impose the Monge gauge condition In this gauge the Hamiltonian is P 0 , and we may solve the constraints for P µ . Assuming that P 0 is positive, we find that where H is the Hamiltonian density of (10).
We may now incorporate the chirality constraint into the action (19) via a Lagrange multiplier field λ, to get To complete the functional basis for the constraints we define where γ(σ) is a 1D scalar. Further PB calculations show, as could be anticipated, that which tells us that the functionals (H 0 , H 1 ) still span a set of first-class constraints. We also have The right hand side is non-zero for non-zero γγ unless γ ′γ′ = 0. In fact χ(γ) has zero PBs with all other constraint functions if γ ′ = 0, so the set of first-class constraint includes where the second equality assumes boundary conditions such that dσϕ ′ = 0 (which allows periodic identification of σ, although this would break the 2D Lorentz invariance). The complementary set of functionals {χ[γ]; γ ′ = 0} is second class. These results show that inclusion of the chirality constraint preserves the gauge invariances that allow us to impose the Monge gauge in the Lorentz invariant action (27), which yields They also imply an invariance of this action under a gauge transformation generated by χ 0 , which is The FJ action (14) is now recovered on using the constraint χ = 0 to eliminate π from (32).
We have now seen that the FJ chiral scalar action is a gauge-fixed version of the action (27), which has 2D Lorentz invariance realized linearly as an 'internal' symmetry, with (Lorentz boost) Noether charge This charge is precisely what generates the Lorentz transformations of the canonical variables of (32) after the Monge gauge choice is made, even though these variables are inert under the internal Lorentz symmetry of (27). This is true for any (finite) choice of T because (as noted earlier) the Monge gauge action is T -independent, but we can simplify the computation of the Lorentz transformation of ϕ by taking the T → 0 limit. In this T → 0 limit we have The Monge gauge expression for L at T = 0 is therefore The Lorentz boost transformations of (ϕ, π) generated by this functional, with respect to the canonical PB relations of the action (32), are where w is the Lorentz boost parameter. These variations imply δ w χ = w(ξ + χ) ′ , which confirms the Lorentz invariance of the constraint χ = 0. We saw earlier that using χ = 0 in (32) to eliminate π yields the FJ action (14) for ϕ; we thereby confirm the result of [9] that this action is Lorentz invariant, with Lorentz transformation δ w ϕ = −w ξ + ϕ ′ to first order in the Lorentz boost parameter w. More generally, the conformal invariance of the FJ action is a consequence of the invariance of the manifestly Lorentz invariant action (27) under transformations generated by the Noether charge for any conformal Killing vector field k(X) of the 2D Minkowski spacetime. In the Monge gauge, this Noether charge reduces to which generates the transformation δ k ϕ = k + ϕ ′ . Ignoring surface terms, it may be verified directly that this induces a variation of the FJ action that is a surface term if, and only if, ∂ − k + = 0; in other words, for k + = k + (ξ + ). Using (16) we have δ k Φ = k + ∂ + Φ ≡ L k Φ, which is the first-order variation of a scalar field under a conformal transformation of the 2D Minkowski coordinates. To conclude this discussion of "chiral 0-form electrodynamics" we remark that the action (18), omitting the −T √ − det g term, is the Nambu-Goto action for a string of tension T in a 3D Minkowski spacetime with coordinate functions (X 0 , X 1 , X 2 ), and conjugate momentum densities (P 0 , P 1 , P 2 ), where In this notation the chirality constraint is If we suppose that ϕ takes values in S 1 , which breaks the 3D Lorentz invariance to 2D Lorentz invariance, then the assumption that dσϕ ′ = 0 becomes equivalent to the assumption that the winding number of the string around the compact space dimension is zero. When combined with the chirality constraint, this implies that the total momentum p 2 = dσP 2 in the compact direction is zero. The canonically conjugate would-be variable is the zero-mode of ϕ that is rendered unphysical by the gauge invariance generated by p 2 . Finally, we remark that the mechanism exploited here for constructing manifestly Lorentz invariant actions for chiral n-form theories has a precursor in the manifestly Lorentz invariant Green-Schwarz superstring action [14]. In that case the anticommuting variables are worldsheet scalars that become worldsheet spinors in the lightcone gauge. More generally, the anticommuting variables of Green-Schwarz-type superbrane actions are worldvolume scalars that become worldvolume spinors in the Monge gauge [15,16]. An analogous transmutation mechanism is at work in this paper, most obviously for the chiral 2-form electrodynamics because its 'covariant' action includes fields that are scalars with respect to a global 'internal' Lorentz symmetry but SO(5) tensors with respect to 5-space diffeomorphisms; they become SO(5) tensors of the rotation subgroup of a non-linearly realized spacetime Lorentz invariance in the Monge gauge, as shown in detail in [8].