Canonical analysis of E6(6)(R) invariant five dimensional (super-)gravity

Kreutzer, Lars T.

doi:10.1063/5.0037092

We investigate the canonical structure of the bosonic sector of the unique maximal supergravity theory in five dimensions that is manifestly invariant under the global action of $E_{6 (6)} (R)$ ⁠. Starting from the Lagrangian formulation of the theory, we construct the Hamiltonian formulation and the full set of canonical constraints. We determine all gauge transformations and compute the algebra formed by the canonical constraints under the Poisson bracket. We re-derive the number of physical degrees of freedom and construct the extended Hamiltonian, describing the most general time evolution of the theory, where the full gauge freedom is manifest.

I. INTRODUCTION

In this work, we investigate the canonical structure of the bosonic sector of the unique ungauged maximal supergravity theory in five dimensions that is manifestly invariant under the global action of $E_{6 (6)} (R)$ ⁠.

Toroidal compactification of 11 dimensional supergravity leads to symmetries described by the exceptional Lie groups $E_{n (n)} (R) (n = 6, 7, 8)$ in the lower dimensional theories in 11 − n dimensions. This fact was first discovered by Cremmer and Julia in 1979,¹ and Cremmer went on to describe the five dimensional case with $E_{6 (6)} (R)$ symmetry in detail in 1980.² The occurrence of exceptional symmetries in maximal supergravity remains one of the most remarkable features of these theories and remains to be fully understood at the quantum level.³

We choose to analyze the $E_{6 (6)} (R)$ invariant theory since it is one of the simplest examples of a theory with an exceptional symmetry. In five dimensions, no self-dual forms exist and the symmetries are symmetries of the Lagrangian itself, not just of the equations of motion. Furthermore, the group $E_{6 (6)} (R)$ itself is easier to work with than the larger exceptional groups. Moreover, we choose to analyze the bosonic sector of the theory since this greatly reduces the complexity of the analysis while still keeping many interesting features intact.

$E_{6 (6)} (R)$ exceptional field theory—first described in 2013⁴—is an $E_{6 (6)} (R)$ covariant extension of 11 dimensional supergravity on a 5 + 27 dimensional generalized space–time with external and internal coordinates (x^μ, Y^M).^4–7 The internal coordinates Y^M carry an index in the fundamental representation of $E_{6 (6)} (R)$ ⁠. The so-called section condition of the theory is a $E_{6 (6)} (R)$ covariant constraint on the coordinate derivatives ∂_M of the internal geometry, which requires that only a subset of the 27 internal coordinates is physical.

Exceptional field theory has since developed into a very active field with too many applications and directions to mention here in detail. Some examples are the construction of consistent truncations and duality covariant solutions of supergravity theories (including also less than maximal supersymmetry), a unifying treatment for brane solutions and exotic branes, non-geometric backgrounds, duality covariant graviton amplitudes, and many more—we refer the reader to the following review and publications.^8–10

The ungauged $E_{6 (6)} (R)$ invariant supergravity theory in five dimensions is related to the $E_{6 (6)} (R)$ exceptional field theory by taking the trivial solution to the section condition ∂_M = 0, thus removing all traces of the exceptional generalized geometry.⁵ This procedure is equivalent to the compactification of 11 dimensional supergravity on a 6 dimensional torus.⁵

So far the canonical formulation of exceptional field theory has not been investigated, at least in part due to the great complexity of the exceptional generalized geometry, the complicated topological term and the many covariantization terms involved. However, a comprehensive canonical analysis of exceptional field theory should cast light on the exact role of the section condition—in particular on its relevance to the closure of the gauge algebra. Furthermore, it would explicitly demonstrate the role of the external diffeomorphisms in interconnecting the terms of the bosonic exceptional field theory Lagrangian to fix all the relative coefficients.⁵ Moreover, one may see the canonical formulation as the starting point for the canonical quantization procedure and a potential way to identify a higher dimensional Ashtekar connection.^11–13 Likewise, one may be interested in the canonical formulation of exceptional field theory as a tool for finding the solution to the local initial value problem for the exceptional generalized geometry as well as for finding the correct definition of the Arnowitt–Deser–Misner (ADM) energy and momentum in this setting.

This work is intended as a preparation and first stepping stone toward the canonical analysis of the full $E_{6 (6)} (R)$ exceptional field theory. We aim to make this analysis as explicit, self-contained, and complete as possible in order to provide a useful reference for future work. By analyzing the $E_{6 (6)} (R)$ supergravity theory on its own, we expect to gain insight into and greatly facilitate the canonical analysis of the full $E_{6 (6)} (R)$ exceptional field theory—nonetheless, the $E_{6 (6)} (R)$ supergravity theory is a very interesting theory in its own right, and hence, we expect there to be applications of this work that are not related to exceptional field theory.

We use the canonical formalism for this analysis because it is an algorithmic framework that can be applied to virtually all gauge theories. The canonical analysis yields much of the relevant information about a theory, such as its symmetries, gauge transformations, gauge algebra, and physical degrees of freedom. Furthermore, it is the starting point for the canonical quantization of a theory, and in its extended form, the canonical formulation goes beyond the Lagrangian framework by making the full gauge freedom manifest.^14,15

The outline of this work is as follows: In Sec. I A, we clarify the notation and conventions used in this work. In Sec. I B, we review the necessary principles of the Arnowitt–Deser–Misner formulation of general relativity. In Sec. I C, we describe the Lagrangian formulation of the $E_{6 (6)} (R)$ invariant supergravity theory.

We begin the construction of the canonical theory in Sec. II. In Sec. II A, we compute the canonical momenta and perform some redefinitions. We then state the canonical Hamiltonian in Sec. II B. The primary constraints are discussed in Sec. II C. The complete set of canonical constraints is constructed in Sec. II D. The total Hamiltonian is stated in Sec. II E.

We begin the canonical analysis itself in Sec. III. We define the diffeomorphism weight in Sec. III A and compute all gauge transformations of all canonical coordinates in Sec. III B. The algebra of constraints is calculated in Sec. III C. The extended Hamiltonian is stated in Sec. III D. In Sec. III E, we calculate the number of physical degrees of freedom of the theory.

We conclude with a summary of the results in Sec. IV.

For a short review of the notation and basics of canonical analysis, see Appendix A. In Appendix B, we list Poisson bracket relations that are needed and useful in computations. In Appendix C, we discuss some other useful formulas needed in the computations. In Appendix D, we discuss the treatment of the scalar coset constraints for the example of the SL(n)/SO(n) coset.

A. Notation and conventions

We work in the 1 + 4 dimensional ADM decomposition (see Subsection I B) of the 5 dimensional space–time and use the signature (− + + + +). We assume to work with a smooth, metric, Lorentzian, and globally hyperbolic manifold. Indices from the middle of the Greek alphabet, e.g., μ, ν, ρ, σ, τ, …, denote five dimensional curved space–time indices. We use Greek indices from the beginning of the alphabet, e.g., α, β, …, to denote five dimensional flat indices. We use Latin letters from the middle of the alphabet, e.g., k, l, m, n, …, for four dimensional curved spatial indices and Latin letters from the beginning of the alphabet, e.g., a, b, c, d, e, …, for flat spatial indices. We exclude the letter t from this list, since we use it to denote the curved time index. The spatial flat indices are lowered and raised by the four dimensional Euclidean metric, and thus, their placement is irrelevant. For a simple reference of the types of space–time indices used, see Table I.

TABLE I.

The types of letters used for space–time indices with regard to their dimension and whether they are curved or flat.

	Flat	Curved
1 + 4 dimensional	α, β, γ, …	μ, ν, ρ, σ, τ, …
4 dimensional	a, b, c, d, e, …	k, l, m, n, …

In addition to the letters mentioned above, we will reserve the indices 0 and t to denote the flat and curved time directions, meaning α = (0, a) and μ = (t, m).

Furthermore, we use capitalized Latin letters, e.g., K, L, M, N, …, to denote indices of the fundamental representation of the $E_{6 (6)} (R)$ Lie group, and these indices run from 1, …, 27. These are taken from the middle of the Latin alphabet; however, in this work, there will be no other types of capitalized Latin indices.

The object ϵ^μνρστ is the totally antisymmetric Levi–Cività symbol and is independent of the metric and coordinates. We choose the convention ϵ⁰¹²³⁴ ≔ +1.

B. Arnowitt–Deser–Misner (ADM) formulation of general relativity

In this section, we state the main results concerning the ADM decomposition which we need for the canonical formulation of supergravity. For many more details on the topic, see Refs. 16–19.

If we assume that the manifold is globally hyperbolic, we can foliate the manifold into a set of space-like hypersurfaces and decompose the metric in the following way.²⁰

We write the metric in the vielbein formalism (also known as the tetrad-, local frame-, or Cartan formalism) as follows: ${E_{μ}}^{α}$ is the five dimensional frame field (or fünfbein), and η_αβ is the Minkowski metric with signature (− + + + +),²⁰

G_{μ ν} = {E_{μ}}^{α} {E_{ν}}^{β} η_{α β} .

(1.1)

The following equivalent identities define the inverse vielbein:

{E_{μ}}^{α} {E_{α}}^{ν} = δ_{μ}^{ν},

(1.2)

{E_{β}}^{μ} {E_{μ}}^{α} = δ_{β}^{α} .

(1.3)

By introducing the lapse function N, the shift vector N^a, and the spatial vierbein (four dimensional frame field) ${e_{m}}^{a}$ ⁠, we can parameterize the fünfbein (five dimensional frame field) ${E_{μ}}^{α}$ in the following way:

{E_{μ}}^{α} ≕ (\begin{matrix} N & N^{a} \\ 0 & {e_{m}}^{a} \end{matrix}) .

(1.4)

Using Eq. (1.2), we find that the parameterization of the inverse fünfbein is given by the following equation:

{E_{α}}^{μ} = (\begin{matrix} N^{- 1} & - N^{- 1} N^{m} \\ 0 & {e_{a}}^{m} \end{matrix}) .

(1.5)

The components of the metric that follow from these parameterizations are given in Eqs. (1.6)–(1.8). The four dimensional metric on the spatial hypersurfaces is given by g_mn,

G_{00} = N^{a} N_{a} - N^{2},

(1.6)

G_{0 n} = N_{n} ≔ N_{a} {e_{n}}^{a},

(1.7)

G_{m n} = g_{m n} ≔ {e_{m}}^{a} {e_{n}}^{b} δ_{a b} .

(1.8)

When decomposing the inverse metric, one should note that the components of the inverse metric are not the inverse of the components of the metric, in particular with respect to the spatial metric inverse [see Eq. (1.11)]. The inverse metric components are written out as follows:

G^{00} = - N^{- 2},

(1.9)

G^{0 n} = N^{- 2} N^{n},

(1.10)

G^{m n} = g^{m n} - N^{- 2} N^{m} N^{n} .

(1.11)

The relation between the determinant of the five dimensional frame field E and the determinant of the spatial four dimensional frame field e is given by a factor of the lapse function,

E = N \cdot e .

(1.12)

The determinant of the metric G can be expressed in the following form:

G = - E^{2} = - N^{2} e^{2} .

(1.13)

Other formulas relating to the geometry, which are useful in computations, can be found in Appendix C.

When matter is coupled to the Einstein–Hilbert action, we arrive at the ADM decomposition by inserting the decomposition of the metric or vielbein fields and then separate the matter field components by their space–time index structure. For example, a one form A_μ splits into A_t and A_m, a two form B_μν into B_tn and B_mn, and a scalar field remains unchanged.

C. $E_{6 (6)} (R)$ invariant five dimensional (super-)gravity

We are interested in the bosonic sector of the maximal supergravity in five dimensions that is manifestly $E_{6 (6)} (R)$ invariant. The bosonic field content of this theory is given by ${G_{μ ν}, A_{μ}^{M}, M_{M N}}$ or in the ADM decomposition by ${N, N^{a}, {e_{m}}^{a}, A_{t}^{M}, A_{m}^{M}, M_{M N}}$ ⁠.^2,21 The lapse function, the shift vector field, and the spatial vielbein were already introduced in Sec. I B.²²

The fields $A_{μ}^{M}$ are Abelian vector gauge fields with one Lorentz index and one fundamental $E_{6 (6)} (R)$ index.

The fields M_MN are Lorentz scalars and carry two fundamental $E_{6 (6)} (R)$ indices that are symmetric. They are elements of the $E_{6 (6)} (R) /$ USp(8) coset, which is 42 dimensional; hence, not all components of M_MN are independent.² Due to the coset structure, M_MN transforms covariantly under the global action of $E_{6 (6)} (R)$ and is invariant under the local action of USp(8). The scalar fields can be interpreted as an $E_{6 (6)} (R)$ metric, and if one considers also the fermionic sector of the theory, it is, in fact, necessary to also rewrite the scalar fields in terms of the 27 dimensional $E_{6 (6)} (R)$ vielbein (see Ref. 2). The $E_{6 (6)} (R)$ metric interpretation becomes much clearer when one considers the full $E_{6 (6)} (R)$ exceptional field theory (see Refs. 5 and 6).

We treat the $E_{6 (6)} (R) /$ USp(8) coset constraints as being implicit—meaning that in this formalism, M_MN is a priori treated as if it is a generic symmetric matrix with 378 components (M, N = 1, …, 27) until all Poisson brackets have been evaluated.

This allows us to sum over all components of M_MN without distinguishing whether they are truly independent coset parameters or not and simplifies the notation and canonical analysis significantly. However, there are implicit coset constraints on this M_MN that ultimately—after evaluating all Poisson brackets—guarantee that it is an E₆₍₆₎/USp(8) coset representative with 42 degrees of freedom. We want to treat the coset constraints on M_MN implicitly because of their great complexity and because we do not need to use them explicitly in the context of this work—one should think of them as being added to the Lagrangian with appropriate Lagrange multipliers. If we were to do this explicitly, they would appear as canonical constraints and their consistency conditions would generate further canonical constraints on the canonical momenta Π^MN(M). The constraints would relate various components of M_MN and their canonical momenta Π^MN(M) among each other—implying among other things that det(M) = 1. Because they are canonical constraints, we are not allowed to apply them before fully evaluating all Poisson brackets. Indeed, if we treated them explicitly, they would require the introduction of a Dirac bracket because the canonical coset constraints of the fields and the momenta will form a second class system of constraints. To clarify this implicit treatment of the coset constraints, we demonstrate in Appendix D the explicit and implicit formalism for the much simpler case of the coset SL(n)/SO(n)—which has as its only coset constraint det(M) = 1. The scope of the implicit formalism will be sufficient for all calculations done in this work, and we will find that it leads to the right gauge transformations and dynamics when comparing it to the Lagrangian formulation.

In the Lagrangian formalism, the theory is given by the action of Eq. (1.14) and the Lagrangian density can be written in the form of Eq. (1.15). The Lagrangian density was first described in Ref. 2. Another way of obtaining Eq. (1.15) is to apply the trivial (∂_M = 0) solution of the section condition to the full $E_{6 (6)} (R)$ exceptional field theory Lagrangian density, thus removing all terms related to the exceptional generalized geometry,⁵

S = \int d^{5} x L_{5 D},

(1.14)

\begin{align} L_{5 D} = & + E^{(5)} R \\ + \frac{1}{24} E G^{μ ν} \partial_{μ} M_{M N} \partial_{ν} M^{M N} \\ - \frac{1}{4} E M_{M N} F_{μ ν}^{M} F_{ρ σ}^{N} G^{μ ρ} G^{ν σ} \\ + κ ϵ^{μ ν ρ σ τ} d_{L M N} A_{μ}^{L} F_{ν ρ}^{M} F_{σ τ}^{N} . \end{align}

(1.15)

The gravitational part of the theory is described by the Einstein–Hilbert term and minimal coupling to the other fields. ⁽⁵⁾R = ⁽⁵⁾ ${R^{μ}}_{ν μ σ} G^{ν σ}$ is the Ricci scalar in five dimensions.

The kinetic term of the scalar fields is a non-linear sigma model of the $E_{6 (6)} (R) /$ USp(8) coset—with the coset constraints treated as explained above. The inverse scalar fields M^MN are defined by $M_{M P} M^{P N} = δ_{M}^{N}$ ⁠.

The vector field kinetic term is described by a Maxwell theory type term with the additional contraction of the $E_{6 (6)} (R)$ indices by the scalar fields. The Abelian field strength is given by $F_{μ ν}^{M} ≔ 2 \partial_{[) μ} A_{ν (]}^{M}$ ⁠. The Abelian gauge group is U(1)²⁷ due to the fact that 27 is the dimension of the fundamental representation of $E_{6 (6)} (R)$ ⁠, and we have 27 copies of the vector field. The Bianchi identity $\partial_{[) μ} F_{ν ρ (]}^{M} = 0$ is not a canonical constraint, since it does not put any constraints on the canonical variables. The Bianchi identity simply follows from the commutativity of partial derivatives.

In addition, there is a metric independent topological term of the form A ∧ F ∧ F—note that this term is also second order in derivatives. The coefficient $κ = + \frac{\sqrt{10}}{24}$ of the topological term (the sign is convention dependent) is needed for maximal supersymmetry in five dimensions.²² This value of κ also guarantees $E_{8 (8)} (R)$ invariance upon reduction to three dimensions (cf. Ref. 23 for the reduction from 11 dimensions). The precise value is not relevant for this paper, and we will keep the coefficient general.

The symbols d_LMN and d^LMN are fully symmetric and carry three fundamental $E_{6 (6)} (R)$ indices. They are the (up to factors) unique invariant symbols of the fundamental representation of $E_{6 (6)} (R)$ ⁠, and we use the normalization given by $d_{M N Q} d^{M N P} = δ_{Q}^{P}$ ⁠.^2,5 In this work, we only need the fact that they are fully symmetric. For more details and useful identities regarding these symbols, see Ref. 5.

All objects in the Lagrangian are in 5 dimensions, and all objects in Secs. II and III are written in the 1 + 4 dimensional ADM decomposition of the 5 dimensional fields, as described in Sec. I B.

II. CANONICAL FORMULATION: HAMILTONIAN AND CANONICAL CONSTRAINTS

In this section, we construct the Hamiltonian formulation of the theory, starting from the Lagrangian formulation described in Sec. I C. As a reminder or as a brief introduction, we summarize the main definitions and notations of the canonical formalism and canonical analysis in Appendix A.

A. Canonical momenta

We will employ the notation that a capital Π(X)—with appropriate indices—signifies the canonical momentum associated with the field X. In many cases, the specification of X can be skipped since it is usually clear from the index structure of Π alone which field is conjugate to it.

Calculating the canonical momenta [see Eq. (A2) for the definition] of the lapse function, shift vector, and the time component of the gauge field, we immediately find the following primary constraints of the form Π(X) = 0:

Π (N) = 0,

(2.1)

Π^{a} (N_{a}) = 0,

(2.2)

Π_{M} (A_{t}^{M}) = 0 .

(2.3)

They vanish due to the fact that the Lagrangian density in Eq. (1.15) does not depend on the time derivative of their conjugate fields. We refer to this type of constraint as a shift type constraint since the gauge transformations generated by these constraints are shifts of the conjugate fields (as we will see in Sec. III B).

The remaining canonical momenta do not vanish since the Lagrangian does contain time derivatives of their conjugate fields. A dot above a function—such as in Eq. (2.6)—indicates a ∂_t derivative operator,

{Π^{m}}_{a} (e) = + 2 e e^{b m} [Ω_{0 (a b)} - δ_{a b} Ω_{0 c c}],

(2.4)

Π_{T}^{l} (A) = + \frac{e}{N} g^{l n} M_{T N} [F_{t n}^{N} + N^{k} F_{n k}^{N}] + 4 κ ϵ^{l m n r} d_{M N T} A_{m}^{M} F_{n r}^{N},

(2.5)

Π^{R S} (M) = + \frac{1}{6} \frac{e}{N} [{\dot{M}}_{Q P} M^{Q R} M^{P S} + N^{n} \partial_{n} M^{R S}] .

(2.6)

The component of the coefficient of anholonomy $Ω_{0 b c} = N^{- 1} \cdot [{e_{b}}^{n} (\partial_{t} - N^{m} \partial_{m}) e_{n c} - {e_{b}}^{m} e_{n c} \partial_{m} N^{n}]$ in Eq. (2.4) follows from the ADM decomposition of the definition of the coefficient of anholonomy $Ω_{α β γ} ≔ 2 {E_{[) α}}^{μ} {E_{β (]}}^{ν} \partial_{μ} E_{ν γ}$ ⁠.¹⁹

It is useful to define the contractions Π_ab(e) and Π(e) of the vielbein with the vielbein momentum,

Π_{a b} (e) ≔ + e_{m () a} {Π^{m}}_{b ()} (e),

(2.7)

Π (e) ≔ + {e_{m}}^{a} {Π^{m}}_{a} (e) .

(2.8)

To compare with the metric formulation of canonical general relativity (see, e.g., Refs. 16, 17, and 24), one can use Eq. (2.9) to relate the canonical momenta of the spatial vielbein to the canonical momenta of the metric,

Π^{m n} (g) = \frac{1}{2} e_{a}^{() m} Π^{n () a} (e) .

(2.9)

Starting from Eq. (2.5), we can redefine $Π_{T}^{l} (A)$ to simplify the Hamiltonian and hence the gauge transformations. We define $P_{L}^{l} (A)$ as the canonical momentum $Π_{T}^{l} (A)$ subtracted by the topological term contribution. This way $P_{L}^{l} (A)$ takes the form expected of the theory without topological term [see Eq. (2.11)],

P_{L}^{l} (A) ≔ Π_{L}^{l} (A) - 4 κ ϵ^{l m n r} d_{M N L} A_{m}^{M} F_{n r}^{N}

(2.10)

= \frac{e}{N} g^{l n} M_{L N} [F_{t n}^{N} + N^{k} F_{n k}^{N}] .

(2.11)

One has to be careful with this redefinition, however, since it is not a canonical transformation. This can be seen from the fact that $P_{L}^{l} (A)$ has a non-vanishing Poisson bracket with itself ${P_{L}^{l} (A), P_{K}^{k} (A)} \neq 0$ ⁠. If we explicitly compute this Poisson bracket, we find Eq. (2.12). The upper letter at the derivatives indicates the coordinate of differentiation. We refrain from using this equation in the following calculations since it is rather cumbersome—it is nonetheless a valid identity. A more manageable approach to the calculations is to proceed order by order in the coefficient of the topological term κ,

\begin{align} {P_{L}^{l} (A) (x), P_{K}^{k} (A) (y)} = & + 8 κ ϵ^{k l r s} d_{L K M} \partial_{r}^{y} A_{s}^{M} (y) δ (x - y) \\ + 8 κ ϵ^{k l r s} d_{L K M} A_{r}^{M} (y) \partial_{s}^{x} δ (x - y) \\ + 8 κ ϵ^{k l r s} d_{L K M} \partial_{r}^{x} A_{s}^{M} (x) δ (x - y) \\ + 8 κ ϵ^{k l r s} d_{L K M} A_{r}^{M} (x) \partial_{s}^{x} δ (x - y) . \end{align}

(2.12)

Nonetheless, the use of the momentum $P_{L}^{l} (A)$ greatly simplifies the Hamiltonian, and it has nice transformation properties, as we will see in Sec. III B.

The scalar momentum Π^RS(M) from Eq. (2.6) is computed using the part of the variation of the Lagrangian given by Eq. (2.13). We furthermore assume Eq. (2.14) as the definition of the fundamental Poisson bracket relation of Π^RS(M) with the scalar fields. As explained in Sec. I C and Appendix D, we treat the scalar fields and their momenta as generic symmetric matrices until all the Poisson brackets have been fully evaluated, and hence, Eq. (2.14) is the fundamental Poisson bracket of a symmetric matrix. Equation (2.14) furthermore eliminates the need for any distinction between diagonal and off-diagonal elements of the scalar fields and momenta. We assume that sums run over the full index range,

δ L_{5 D} = \frac{1}{2} Π^{M N} (M) δ {\dot{M}}_{M N} + \dots,

(2.13)

{M_{M N} (x), Π^{P Q} (M) (y)} = (δ_{M}^{P} δ_{N}^{Q} + δ_{N}^{P} δ_{M}^{Q}) δ^{(4)} (x - y) .

(2.14)

B. Canonical Hamiltonian

Having found all the canonical momenta in Sec. II A, we can now calculate the Hamiltonian density associated with Eq. (1.15) by a Legendre transformation. To do so, we calculate the ADM decomposition of all terms of the Lagrangian density—as described in Sec. I B—and then perform the Legendre transformation with respect to all canonical momenta [see Eq. (A3)].

The canonical Hamiltonian density is given in Eq. (2.15). Here, we can factor out the Lagrange multipliers—meaning the fields whose momenta are primary constraints of shift type—thus already making the secondary constraints apparent,

\begin{align} H_{5 D} = & + N \cdot [+ \frac{1}{4 e} Π_{a b} (e) Π_{a b} (e) - \frac{1}{12 e} Π {(e)}^{2} - e R) \\ (+ \frac{3}{2 e} Π^{M N} (M) Π^{R S} (M) M_{M R} M_{N S} - \frac{e}{24} g^{k l} \partial_{k} M_{M N} \partial_{l} M^{M N}) \\ (+ \frac{e}{4} M_{M N} g^{r m} g^{s n} F_{r s}^{M} F_{m n}^{N} + \frac{1}{2 e} g_{l m} M^{K L} P_{L}^{l} P_{K}^{m}] \\ + N^{n} \cdot [+ 2 {Π^{m}}_{a} (e) \partial_{[) n} e_{m (] a} - e_{n a} \partial_{m} {Π^{m}}_{a} (e)) \\ (+ \frac{1}{2} Π^{M N} (M) \partial_{n} M_{M N}) (+ F_{n l}^{M} P_{M}^{l}] \\ + A_{t}^{M} \cdot [- \partial_{l} P_{M}^{l} - 3 κ ϵ^{l m n r} d_{M N P} F_{l m}^{N} F_{n r}^{P}] \\ + \dot{N} \cdot Π (N) + {\dot{N}}_{a} \cdot Π^{a} (N_{a}) + {\dot{A}}_{t}^{M} \cdot Π_{M} (A_{t}) . \end{align}

(2.15)

We can see that there are just three terms that contain the redefined momentum $P_{M}^{l} (A)$ and only one topological term—just like in the Lagrangian density. Thus, using $P_{M}^{l} (A)$ instead of $Π_{M}^{l} (A)$ gives a much simpler Hamiltonian—as can be seen from reinserting the definition of $P_{M}^{l} (A)$ ⁠.

The terms in the last line of the Hamiltonian stem from the Legendre transformation; since the Lagrangian does not depend on the time derivatives of the Lagrange multipliers, these terms stay as they are.

It is common to work on the surface of primary constraints, consequently, removing the last line from the Hamiltonian. However, we aim to keep the setting as general as possible and do not want to restrict the analysis to a subregion in phase space.

C. Primary constraints

Calculating the canonical momenta, we have already seen that some of them vanish to yield primary constraints of shift type (see Sec. II A). There are six more primary constraints, called the Lorentz constraints L_ab, with L_ab = L_[ab]. Since the Lorentz symmetry is manifest in the vielbein formalism, there are constraints associated with this symmetry. The constraints L_ab are not of shift type and are not obvious, but they do follow immediately from the momenta of the spatial vielbein [see Eq. (2.4)] and take the explicit form of Eq. (2.19).¹⁹ The complete list of all 38 primary constraints is as follows:

Π (N) = 0,

(2.16)

Π^{a} (N_{a}) = 0,

(2.17)

Π_{M} (A_{0}^{M}) = 0,

(2.18)

L_{a b} ≔ e_{m [) a} {Π^{m}}_{b (]} (e) = 0 .

(2.19)

D. Secondary constraints

In this section, we follow the procedure for finding the complete set of constraints and guaranteeing their consistency, as outlined in Appendix A.

In order for the primary constraints to be consistent, we require that their time evolution is constant, i.e., that their time derivative vanishes.¹⁵ We are free to add the primary constraints—with arbitrary phase space functions C₀, C₁, C₂, C₃ as parameters—to the Hamiltonian, since we can arbitrarily extend the Hamiltonian away from the primary constraint surface in phase space. We arrive at a preliminary total Hamiltonian given by the following equation:

{\tilde{H}}_{T} ≔ H_{5 D} + C_{0} \cdot Π (N) + {(C_{1})}_{a} \cdot Π^{a} (N_{a}) + {(C_{2})}^{M} \cdot Π_{M} (A_{t}) + {(C_{3})}^{a b} \cdot L_{a b} .

(2.20)

Using ${\tilde{H}}_{T}$ ⁠, we can now test the consistency of a primary constraint Φ using the following equation [cf. Eq. (A8)]:

\dot{Φ} = {Φ, {\tilde{H}}_{T}} \overset{!}{=} 0

(2.21)

It is important to note here that the primary constraints all Poisson-commute among each other—with the exception of the Lorentz constraints with themselves {L_ab, L_cd} ≠ 0, since they form the Lorentz subalgebra, as we will see in Sec. III C.

First, we consider the consistency of the primary constraints of shift type—meaning vanishing canonical momenta. Since the primary constraints of shift type appear as Lagrange multipliers in the Hamiltonian and Poisson-commute with all other primary constraints, we find that their consistency requires that each shift type of primary constraint yields one secondary constraint. We call Eq. (2.22) the Hamilton constraint $H_{Ham} ≔ {H_{5 D}, Π (N)}$ ⁠, Eq. (2.23) the diffeomorphism constraint ${(H_{Diff})}_{n} ≔ {H_{5 D}, Π_{a} (N_{a})} e_{n}^{a}$ and Eq. (2.24) the Gauss constraint ${(H_{Gauss})}_{M} ≔ {H_{5 D}, Π_{M} (A_{0}^{M})}$ ⁠. The Gauss constraint is named due to the similarity to the constraint in Maxwell theory and Gauss’s law. Note that we have chosen to include a vielbein in the definition of the diffeomorphism constraint in order for the constraint to have a curved index. We will see in Sec. III B that these names are indeed justified,

\begin{align} H_{Ham} = & + \frac{1}{4 e} Π_{a b} (e) Π_{a b} (e) - \frac{1}{12 e} Π {(e)}^{2} - e R \\ + \frac{3}{2 e} Π^{M N} (M) Π^{R S} (M) M_{M R} M_{N S} - \frac{e}{24} g^{k l} \partial_{k} M_{M N} \partial_{l} M^{M N} \\ + \frac{e}{4} M_{M N} g^{r m} g^{s n} F_{r s}^{M} F_{m n}^{N} + \frac{1}{2 e} g_{l m} M^{K L} P_{L}^{l} P_{K}^{m}, \end{align}

(2.22)

\begin{align} {(H_{Diff})}_{n} = & + 2 {Π^{m}}_{a} (e) \partial_{[) n} e_{m (] a} - e_{n a} \partial_{m} {Π^{m}}_{a} (e) \\ + \frac{1}{2} Π^{M N} (M) \partial_{n} M_{M N} + F_{n l}^{M} P_{M}^{l}, \end{align}

(2.23)

{(H_{Gauss})}_{M} = - \partial_{l} P_{M}^{l} - 3 κ ϵ^{l m n r} d_{M N P} F_{l m}^{N} F_{n r}^{P} .

(2.24)

The secondary constraints restrict the canonical coordinates dynamically since we have made use of the equations of motion to find them. The secondary constraints are equivalent to the time components of the Lagrangian equations of motion. The Hamilton constraint is the time–time-component of the Einstein equation, the diffeomorphism constraints are the time-spatial-components of the Einstein equation. Similarly, the Gauss constraints are the time-components of the analog of the Maxwell equation.

It is worth noting that the Gauss constraints H_Gauss are independent of the metric and scalar degrees of freedom.

The gravitational part of the diffeomorphism constraint H_Diff can be rewritten using Eq. (2.25) to reveal a term containing the Lorentz constraint with a field dependent coefficient given by the spin connection ${ω_{n}}^{a b} = e^{a k} \nabla_{n} {e_{k}}^{b}$ ⁠. The covariant derivative ∇_n contains the Levi–Cività connection, and D_m contains the spin connection,

+ 2 {Π^{m}}_{a} (e) \partial_{[) n} e_{m (] a} - e_{n a} \partial_{m} {Π^{m}}_{a} (e) = - {e_{n}}^{a} D_{m} {Π^{m}}_{a} (e) + {ω_{n}}^{a b} L_{a b} .

(2.25)

One can redefine the diffeomorphism constraint as in Eq. (2.26), which makes the new diffeomorphism constraint ${\tilde{H}}_{Diff}$ and the Lorentz constraint Poisson-commute. However, we will see that the constraint of Eq. (2.23) gives the nicer and expected gauge transformations [see Eq. (3.22)]. We will thus continue to work with the diffeomorphism constraint of Eq. (2.23). The redefinition would simply represent a different choice of basis of the constraint algebra,

{({\tilde{H}}_{Diff})}_{n} ≔ {(H_{Diff})}_{n} - {ω_{n}}^{a b} L_{a b} .

(2.26)

The only primary constraints not of shift type are the Lorentz constraints. If we insert the Lorentz constraints in Eq. (2.21), we find that no new constraints are being generated. The consistency requirement does however restrict the coefficient ${(C_{3})}^{a b}$ of the Lorentz constraint term ${(C_{3})}^{a b} L_{a b}$ in the total Hamiltonian. The coefficient has to be ${(C_{3})}^{a b} = - N^{n} {ω_{n}}^{a b}$ for the constraints to be consistent. This is precisely the term found in Eq. (2.26). This means that the total Hamiltonian does not contain an intrinsic Lorentz constraint term. We will see in Sec. III D that there is a Lorentz constraint term in the extended Hamiltonian.

To verify the consistency of the secondary constraints, we also take Eq. (2.21) and now insert the secondary constraints as Φ. We find no tertiary constraints implying that the set of constraints we have found so far is complete and consistent. There is no constraint associated with the exceptional $E_{6 (6)} (R)$ symmetry of the theory since it is a global symmetry of the theory and not a gauge symmetry.

It is beneficial to make use of the concept of integrated or smeared constraints in order to avoid expressions that contain derivatives of the delta distribution. We define the smeared constraints by contracting all indices of the constraints with a tensor of test functions with the same symmetries and then integrating this expression over the entire spatial hypersurface. We denote the integrated constraint by adjoining brackets with the name of the test function to the constraint.

If we take the diffeomorphism constraint as an example, the smeared constraint is defined by Eq. (2.27), where λⁿ(x) is a vector of test functions on the spatial hypersurface,

H_{Diff} [λ] ≔ \int {(H_{Diff})}_{n} (x) λ^{n} (x) d^{4} x .

(2.27)

Note that no symmetry factors are inserted. The smeared version of the Lorentz constraints is given by Eq. (2.28), where γ^ab = γ^[ab],

L [γ] ≔ \int L_{a b} (x) γ^{a b} (x) d^{4} x .

(2.28)

From Eq. (2.15), we can now moreover see that the Hamiltonian is weakly vanishing $H_{5 D} \approx 0$ ⁠, demonstrating that the theory is generally covariant.¹⁵

E. Total Hamiltonian

The total Hamiltonian is given by Eq. (2.29). To arrive at the total Hamiltonian, we start with the canonical Hamiltonian of Eq. (2.15) and add all primary constraints with coefficients that satisfy the consistency conditions derived in Sec. II D,

\begin{align} H_{T} = & + N \cdot H_{Ham} + N^{n} \cdot {(H_{Diff})}_{n} + A_{t}^{M} \cdot {(H_{Gauss})}_{M} \\ + C_{0} \cdot Π (N) + {(C_{1})}_{a} \cdot Π^{a} (N_{a}) + {(C_{2})}^{M} \cdot Π_{M} (A_{t}) \\ - N^{n} {ω_{n}}^{a b} L_{a b} . \end{align}

(2.29)

The coefficients $C_{0}, {(C_{1})}_{a}$ and ${(C_{2})}^{M}$ are arbitrary phase space functions. The coefficient of the Lorentz constraint term is restricted to this particular form containing the spin connection, as explained in Subsection II D.

The time evolution of a phase space function F, generated by the total Hamiltonian, via Eq. (2.30) is equivalent to the original Lagrangian time evolution.¹⁵ Note that the equality in Eq. (2.30) is weak—this means that after evaluating the Poisson bracket, the equation holds only on the phase space surface where the constraints are satisfied (see Appendix A),

\dot{F} \approx {F, H_{T}} .

(2.30)

In Sec. III D, we construct the extended Hamiltonian and find that the exact coefficients in the total Hamiltonian are irrelevant to the most general time evolution since they are all replaced by arbitrary coefficient functions in the extended Hamiltonian.

In practice, it is often easiest to determine the time evolution by first computing all the gauge transformations. We compute all gauge transformations of all fields and momenta in Sec. III B.

III. CANONICAL ANALYSIS: GAUGE TRANSFORMATIONS AND GAUGE ALGEBRA

In this section, we analyze the constraints we found in Sec. II. We calculate the gauge transformations generated by the constraints, compute the algebra that is formed by the constraints under the Poisson bracket, determine the number of physical degrees of freedom, and discuss the extended Hamiltonian.

A. Diffeomorphism weight and the Lie derivative

We briefly define the diffeomorphism weight in this section, as the concept is required for Secs. III B–III E.

The Lie derivative of a tensor T with parameter ξ is denoted by $L_{ξ} T$ ⁠. The diffeomorphism weight Λ(T) is defined as the coefficient of the weight term in the Lie derivative. For example, if T is a vector, we write Eq. (3.1) for the components of the Lie derivative,

{(L_{ξ} T)}^{ν} = \underset{transport term}{\underset{︸}{ξ^{μ} \partial_{μ} T^{ν}}} - \underset{rotation term}{\underset{︸}{\partial_{μ} ξ^{ν} T^{μ}}} + \underset{weight term}{\underset{︸}{Λ \cdot \partial_{μ} ξ^{μ}}} .

(3.1)

Table II lists the diffeomorphism weights of all the relevant fields and momenta. The vielbein determinant is a tensor density, and since the Lagrangian includes a vielbein determinant, the canonical momenta have diffeomorphism weight one too.

TABLE II.

The diffeomorphism weights of the most important objects.

Object	Weight Λ
${e_{m}}^{a}$ ⁠, $A_{μ}^{M}$ ⁠, M_MN	0
${Π^{m}}_{a} (e)$ ⁠, $P_{M}^{m} (A)$ ⁠, Π^RS(M)	1
e	1

When Lie derivatives are used in Secs. III B–III E, they always include the weight term appropriate for the object that the derivative is applied to.

B. Gauge transformations

In this section, we compute the explicit form of all infinitesimal gauge transformations. To do so, we calculate the Poisson brackets of all the constraints Φ[λ] with all the canonical coordinates X via δX = {X, Φ[λ]} [see Eq. (A12)]. We use the notion of smeared constraints in Secs. III B–III E, as explained in Sec. II D. We only state the non-vanishing gauge transformations.

To compute Poisson brackets of the canonical coordinates, we define the fundamental Poisson brackets as follows:

{N (x), Π (N) (y)} = δ^{(4)} (x - y),

(3.2)

{N^{n} (x), Π_{m} (N^{k}) (y)} = δ_{m}^{n} δ^{(4)} (x - y),

(3.3)

{{e_{n}}^{a} (x), {Π^{m}}_{b} (e) (y)} = δ_{n}^{m} δ_{b}^{a} δ^{(4)} (x - y),

(3.4)

{A_{t}^{M} (x), Π_{N} (A_{t}^{K}) (y)} = δ_{N}^{M} δ^{(4)} (x - y),

(3.5)

{A_{m}^{M} (x), Π_{N}^{n} (A_{k}^{K}) (y)} = {A_{m}^{M} (x), P_{N}^{n} (y)} = δ_{N}^{M} δ_{m}^{n} δ^{(4)} (x - y),

(3.6)

{M_{M N} (x), Π^{P Q} (M) (y)} = (δ_{M}^{P} δ_{N}^{Q} + δ_{N}^{P} δ_{M}^{Q}) δ^{(4)} (x - y) .

(3.7)

Due to the implicit treatment of the coset constraints—as explained in Sec. I C and Appendix D—Eq. (3.7) is the fundamental Poisson bracket relation for a generic symmetric scalar matrix, and thus, there is no coset projector term.

Note that the redefinition of the momentum of $A_{m}^{M}$ from Eq. (2.10) does not affect the fundamental Poisson bracket, since the term that is subtracted in the redefinition only depends on the field $A_{m}^{M}$ ⁠.

In Appendix B, we list many other useful Poisson bracket identities that are needed in the computation of the gauge transformations.

To keep the expressions as simple as possible, we omit the notation of the coordinate dependence in the following. It is understood that the gauge transformation only depends on the coordinate of the field that the constraint acts upon.

We begin with the primary constraints of shift type. The only gauge transformations that one can generate using these constraints are shift transformations on the fields canonically conjugate to the vanishing momenta,

{N, Π (N) [λ_{1}]} = λ_{1},

(3.8)

{N_{a}, Π (N_{b}) [λ_{2}]} = {(λ_{2})}_{a},

(3.9)

{A_{t}^{N}, Π (A_{t}^{M}) [λ_{3}]} = {(λ_{3})}^{N} .

(3.10)

The Lorentz constraints generate Lorentz transformations on the spatial vielbein and its canonical momentum. The vielbein determinant is Lorentz invariant—as are all quantities that can be expressed solely through the metric tensor. The transformations take the form of a rotation of the flat index by the smearing tensor,

{{e_{n}}^{a}, L [γ]} = + e_{n b} γ^{b a},

(3.11)

{{Π^{n}}_{a} (e), L [γ]} = + {Π^{n}}_{c} (e) γ^{c b} δ_{b a} .

(3.12)

The Hamilton constraint generates time evolution, which we can interpret as a gauge transformation.¹⁵ The time evolution generated by just the Hamilton constraint—unlike the total Hamiltonian—does not capture any other gauge freedom. However, without the Hamilton constraint, time evolution is not possible at all. The time evolution of the fields generated by the Hamilton constraint is essentially given by the canonically conjugate momenta. The time evolution of the canonical momenta is more complicated and captures most of the dynamics. Since the vielbein—or equivalently the spatial metric—contracts all terms in the Lagrangian—with the exception of the topological term—its conjugate canonical momentum ${Π^{n}}_{a} (e)$ has a particularly complicated time evolution [see Eq. (3.14)]. In the third line of Eq. (3.14), we see the spatial Einstein equation in the vielbein form. It is useful to first compute the Poisson brackets of Eq. (B2) and the following equations.

The time evolution of ${Π^{n}}_{a} (e)$ can be compared to the time evolution of the canonical momentum of the metric in pure general relativity (see Refs. 16 and 17) using Eq. (2.9),

{e_{n a}, H_{Ham} [ϕ]} = + \frac{ϕ}{2 e} g_{m n} {Π^{m}}_{a} (e) - \frac{ϕ}{6 e} Π (e) e_{n a},

(3.13)

{{Π^{n}}_{a} (e), H_{Ham} [ϕ]} = + \frac{ϕ}{4 e} Π_{b c} (e) Π_{b c} (e) {e_{a}}^{n} - \frac{ϕ}{2 e} {Π^{k}}_{b} (e) {Π^{n}}_{b} (e) e_{k a}

(3.14)

\begin{align} - \frac{ϕ}{12 e} Π^{2} (e) {e_{a}}^{n} + \frac{ϕ}{6 e} Π (e) {Π^{n}}_{a} (e) \\ - 2 ϕ e (R^{n k} e_{k a} - \frac{1}{2} R {e_{a}}^{n}) \\ + 2 e (\nabla_{a} \nabla^{n} ϕ - \nabla^{k} \nabla_{k} ϕ {e_{a}}^{n}) \\ + \frac{3 ϕ}{2 e} Π^{M N} (M) Π^{R S} (M) M_{M R} M_{N S} {e_{a}}^{n} \\ + \frac{ϕ e}{24} \partial_{k} M_{M N} \partial_{l} M^{M N} g^{k l} {e_{a}}^{n} - \frac{ϕ e}{12} \partial_{k} M_{M N} \partial_{l} M^{M N} g^{l n} {e_{a}}^{k} \\ - \frac{ϕ e}{4} M_{M N} F_{r s}^{M} F_{k m}^{N} g^{r k} g^{s m} {e_{a}}^{n} + ϕ e M_{M N} F_{r s}^{M} F_{k m}^{N} g^{r k} g^{m n} {e_{a}}^{s} \\ + \frac{ϕ}{2 e} M^{K L} P_{L}^{l} (A) P_{K}^{k} (A) g_{l k} {e_{a}}^{n} - \frac{ϕ}{e} M^{K L} P_{K}^{n} (A) P_{L}^{l} (A) e_{l a}, \end{align}

{M_{M N}, H_{Ham} [ϕ]} = + \frac{6}{e} ϕ Π^{Q P} (x) M_{M Q} M_{N P},

(3.15)

\begin{align} {Π^{M N} (M), H_{Ham} [ϕ]} = & - \partial_{l} (\frac{ϕ e}{6} g^{k l} \partial_{k} M^{M N}) \\ - \frac{ϕ e}{6} g^{k l} \partial_{k} M_{K L} \partial_{l} M^{K M} M^{L N} \\ - \frac{6 ϕ}{e} Π^{P M} (M) Π^{N R} (M) M_{P R} \\ - \frac{ϕ e}{2} g^{r m} g^{s n} F_{r s}^{M} F_{m n}^{N} \\ + \frac{ϕ}{e} g_{l m} P_{L}^{l} P_{K}^{m} M^{K M} M^{L N}, \end{align}

(3.16)

{A_{n}^{N}, H_{Ham} [ϕ]} = + \frac{ϕ}{e} g_{n l} M^{N L} P_{L}^{l},

(3.17)

{P_{S}^{l}, H_{Ham} [ϕ]} = + \partial_{m} (e ϕ M_{N S} g^{r m} g^{l s} F_{r s}^{N})

(3.18)

- \frac{12 κ ϕ}{e} g_{m k} M^{K L} d_{S L M} ϵ^{l m r s} F_{r s}^{M} P_{K}^{k} .

(3.19)

The Gauss constraint generates Abelian U(1)²⁷ gauge transformations on the one form gauge field $A_{n}^{N}$ ⁠. The conjugate momentum $P_{N}^{n}$ is invariant under these transformations—like it is in the free theory—which is a nice property of the redefinition from Eq. (2.10). Since the constraint is independent of the metric and scalar fields, they do not transform,

{A_{n}^{N}, H_{Gauss} [ζ]} = + \partial_{n} ζ^{N},

(3.20)

{P_{N}^{n}, H_{Gauss} [ζ]} = 0 .

(3.21)

The diffeomorphism constraint generates diffeomorphisms on the spatial hypersurface via the Lie derivative (including the appropriate weight terms). If we were to use the redefined diffeomorphism constraint from Eq. (2.26), we would see additional terms in the transformation of the vielbein and its conjugate momentum,

{{e_{n}}^{a}, H_{Diff} [ξ]} = + L_{ξ} {e_{n}}^{a},

(3.22)

{{Π^{n}}_{a} (e), H_{Diff} [ξ]} = + L_{ξ} {Π^{n}}_{a} (e),

(3.23)

{M_{M N}, H_{Diff} [ξ]} = + L_{ξ} M_{M N},

(3.24)

{Π^{M N} (M), H_{Diff} [ξ]} = + L_{ξ} Π^{M N} (M),

(3.25)

{A_{n}^{N}, H_{Diff} [ξ]} = - ξ^{m} F_{n m}^{N}

(3.26)

\begin{align} = & + L_{ξ} A_{n}^{N} + δ_{(ξ^{m} A_{m})} A_{n}^{N}, \\ {P_{S}^{l}, H_{Diff} [ξ]} = & + L_{ξ} P_{S}^{l} + ξ^{l} {(H_{Gauss})}_{S} \end{align}

(3.27)

\approx + L_{ξ} P_{S}^{l} .

Due to the parameterization of the Lagrangian, the transformation of the one form gauge field is a Lie derivative only up to a U(1)²³ gauge transformation generated by the Gauss constraint H_Gauss. The notation is taken to mean $δ_{(ξ^{m} A_{m})} A_{n}^{N} = {A_{n}^{N}, H_{Gauss} [ξ^{m} A_{m}^{M}]} = + \partial_{n} (ξ^{m} A_{m}^{N})$ ⁠. In the case of its conjugate canonical momentum $P_{S}^{l}$ ⁠, the transformation is a Lie derivative up to the Gauss constraint H_Gauss and thus weakly equal to the Lie derivative.

The Schouten identity from Appendix C has to be used repeatedly to compute the transformation of the one form field and its momentum.

C. Algebra of constraints

The algebra that is spanned by the canonical constraints under the Poisson bracket is equivalent to the algebra of gauge transformations. We can interpret a Poisson bracket of two canonical constraints as the commutator of two infinitesimal gauge transformations.

In general, it is easiest to explicitly write out the simpler looking constraint and then make use of the gauge transformations from Sec. III B to compute the algebra. For relations that involve the diffeomorphism constraint, it is easiest to make use of the fact that all fields—except the one from gauge field and its conjugate momentum—transform as Lie derivatives under the diffeomorphism constraint.

The primary constraints of shift type Poisson-commute with all the other constraints and are therefore not listed below. The full constraint algebra can be written as follows:

\begin{align} {H_{Ham} [θ], H_{Ham} [τ]} & = H_{Diff} [(θ \nabla_{m} τ - τ \nabla_{m} θ) g^{m n}] \\ - L [(θ \nabla_{m} τ - τ \nabla_{m} θ) g^{m n} ω_{n a b}], \end{align}

(3.28)

{H_{Diff} [λ], H_{Ham} [θ]} = H_{Ham} [L_{λ} θ] + H_{Gauss} [\frac{θ}{e} λ^{p} g_{p k} P_{L}^{k} M^{L M}],

(3.29)

{H_{Ham} [θ], H_{Gauss} [ξ]} = 0,

(3.30)

{H_{Diff} [λ], H_{Diff} [κ]} = H_{Diff} [{[λ, κ]}^{n}] = H_{Diff} [L_{λ} κ^{n}],

(3.31)

{H_{Diff} [λ], H_{Gauss} [ξ]} = 0,

(3.32)

{H_{Gauss} [ξ], H_{Gauss} [ζ]} = 0,

(3.33)

{L [γ], L [κ]} = L [- 2 γ^{c [) a} κ^{b (] c}],

(3.34)

{H_{Ham} [ϕ], L [γ]} = 0,

(3.35)

{H_{Diff} [λ], L [γ]} = L [L_{λ} (γ^{a b})],

(3.36)

{H_{Gauss} [ξ], L [γ]} = 0 .

(3.37)

From Eq. (3.28), we see that two different orderings of time evolutions (as generated by the Hamilton constraint) can only differ by a diffeomorphism and a Lorentz transformation. This means that the time evolution with the Hamilton constraint is unique up to these gauge transformations.

Due to the use of the vielbein formalism and the choice of diffeomorphism constraint (see the discussion in Sec. II D), Eq. (3.28) contains a Lorentz constraint term whose smearing function depends on the spin connection. Nonetheless, Lorentz invariance is preserved since both the spin connection and the Lorentz constraint itself transform under a Lorentz transformation to cancel out the transformation of the diffeomorphism constraint term. From Eq. (2.26), one can moreover see that this additional term disappears when using the redefined constraint. However, the redefinition has many other consequences, which are not nice, including the introduction of additional terms in the algebra and in the gauge transformations as we have already mentioned.

In Eq. (3.29), we see that the Poisson bracket of a diffeomorphism and time evolution is the time evolution with a smearing function given by the Lie derivative of the parameter. There is furthermore a Gauss constraint term, which is due to the transformation property of the one form gauge field and its conjugate momentum under the diffeomorphism constraint.

Equations (3.28) and (3.29) moreover show that the algebra is actually an open- or pseudo-algebra, since the smearing functions on the right-hand side depend on canonical coordinates—see Refs. 15 and 21 for more information. Nonetheless, the term algebra is commonly used in such cases.

Note that even in pure general relativity, the smearing function of the diffeomorphism constraint in Eq. (3.28) contains the inverse metric. The constraint algebra of pure general relativity has been discussed in Refs. 24, 25, and 26. In addition, the canonical formulation and constraint algebra of 11 dimensional supergravity have been discussed in Refs. 27 and 28.

Equations (3.31), (3.33), and (3.34) show that diffeomorphisms, U(1)²⁷ gauge transformations, and Lorentz transformations each form a proper subalgebra of their own. Equation (3.33) thus also prove that the gauge field is indeed Abelian.

Equation (3.36) tells us that the Poisson bracket of the diffeomorphism constraint and the Lorentz constraint is given by a Lorentz transformation with the Lie derivative of the parameter. Using the redefinition of Eq. (2.26), one can make this term vanish, however, with the same unintended consequences as mentioned before.

We now know the full algebra of constraints and see that it does indeed close under the Poisson bracket. This fact implies that all canonical constraints Poisson-commute weakly—i.e., up to constraint terms the Poisson brackets of any two constraints vanish—or equivalently that all constraints are first class. This is no longer true, once one considers the full theory including fermions.

D. Extended Hamiltonian

The extended Hamiltonian is constructed from the total Hamiltonian [Eq. (2.29)] by adding all first class constraints with arbitrary coefficients. Since all the constraints are first class, the extended Hamiltonian is the linear sum over all constraints with arbitrary coefficient functions [see Eq. (3.38)]. The difference between the total and the extended Hamiltonian is in this case that the parameters of the secondary constraints and the Lorentz constraints become completely arbitrary, hence allowing for more general gauge transformations than the Lagrangian time evolution,

\begin{align} H_{E} = & + C_{Ham} \cdot H_{Ham} + {(C_{Diff})}^{n} \cdot {(H_{Diff})}_{n} + {(C_{Gauss})}^{M} \cdot {(H_{Gauss})}_{M} \\ + C_{0} \cdot Π (N) + {(C_{1})}_{a} \cdot Π^{a} (N_{a}) + {(C_{2})}^{M} \cdot Π_{M} (A_{t}) \\ + {(C_{3})}^{a b} L_{a b} . \end{align}

(3.38)

The most general time evolution of a phase space function F is then given by Eq. (3.39). Since we already know all the gauge transformations, we can use this to efficiently compute the time evolution of any function using Eq. (3.38) and the linearity of the Poisson bracket,

\dot{F} (q, p) \approx {F, H_{E}} .

(3.39)

The time evolution described by the extended Hamiltonian and the canonical Hamiltonian is equivalent for observables, since they are by definition gauge invariant.

E. Counting the degrees of freedom

With the full set of constraints as well as the constraint algebra known, we can re-derive the number of physical degrees of freedom of the theory. In Table III, we list all the field degrees of freedom and the primary and secondary constraints.

TABLE III.

A counting of the number of fields and the number of primary and secondary canonical constraints in the bosonic sector of $E_{6 (6)} (R)$ invariant five dimensional supergravity. The distinction between primary and secondary constraints is irrelevant to the number of physical degrees of freedom but illustrates where the constraints came from.

Fields	No.	Primary constraints	No.	Secondary constraints	No.
N	1	Π(N)	1	Hamilton constraint	1
N_a	4	Π(N_a)	4	Diffeomorphism constraints	4
e_ma	16	Lorentz constraints	6	…	0
M_(MN)	42	…	0	…	0
$A_{0}^{T}$	27	$Π (A_{0}^{T})$	27	Gauss constraints	27
$A_{l}^{T}$	108	…	0	…	0
Total	198	Total	38	Total	32

Fields	No.	Primary constraints	No.	Secondary constraints	No.
N	1	Π(N)	1	Hamilton constraint	1
N_a	4	Π(N_a)	4	Diffeomorphism constraints	4
e_ma	16	Lorentz constraints	6	…	0
M_(MN)	42	…	0	…	0
$A_{0}^{T}$	27	$Π (A_{0}^{T})$	27	Gauss constraints	27
$A_{l}^{T}$	108	…	0	…	0
Total	198	Total	38	Total	32

We count a total of 198 field variables—or 396 = 2 · 198 canonical coordinates in phase space. To arrive at this result, we have applied the implicit coset constraints to the scalar fields—as outlined in Appendix D—to reduce the number of scalar fields to the 42 independent variables, we need to describe the $E_{6 (6)} (R)$ /USp(8) coset. We also count a total of 70 = 38 + 32 canonical constraints, all of which are first class constraints and hence have to be counted twice.¹⁵ Thus, the theory has 396 − 2 · 70 = 256 physical dimensions in phase space or equivalently 128 physical (bosonic) degrees of freedom. This is in agreement with the well-known result that—at each point in space—maximal supergravity has 128 bosonic degrees of freedom.^2,21,29,30

IV. SUMMARY AND OUTLOOK

Starting from the Lagrangian formulation of the bosonic sector of the $E_{6 (6)} (R)$ invariant supergravity theory in five dimensions, we have constructed the Hamiltonian formulation of that theory. We then constructed the complete set of 70 canonical constraints. We calculated the gauge transformations generated by the canonical constraints and found that they generate time evolution, diffeomorphisms, Lorentz transformations, U(1)²⁷ gauge transformations, and shift transformations. As is to be expected, the $E_{6 (6)} (R)$ symmetry is not generated by canonical constraints since it is a global symmetry. We found that the algebra of gauge transformations closes and that all constraints of the bosonic theory are first class. Hence, the extended Hamiltonian—describing the most general time evolution of the theory, where the full gauge freedom is manifest—was constructed by summing over the complete set of constraints with fully arbitrary coefficient functions. We finally confirmed the well-known result that the number of physical degrees of freedom of the theory is 128.

This work is intended to lay the foundation for the canonical analysis of the full $E_{6 (6)} (R)$ exceptional field theory. Due to the complexity of the exceptional generalized geometry, the complicated topological term—so far also lacking an explicit non-integral expression—and the many covariantization terms involved, the complete treatment of the exceptional generalized geometry is beyond the scope of this preparatory work. As a result of this preparation, we can now focus on the more difficult tasks of understanding the internal exceptional geometry and working out its canonical structure. As was explained in the Introduction, there are many open questions one can ask about canonical exceptional field theory and the exceptional generalized geometry. Perhaps one of the most intriguing questions concerns the exact role of the section condition—notably its relevance to the closure of the gauge algebra. To cast light on this and other topics, we plan to proceed with the canonical analysis of the full $E_{6 (6)} (R)$ exceptional field theory in a future publication.

ACKNOWLEDGMENTS

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (“Exceptional Quantum Gravity,” Grant Agreement No. 740209).

L.T.K. was supported by the International Max Planck Research School for Mathematical and Physical Aspects of Gravitation, Cosmology and Quantum Field Theory.

L.T.K. is particularly grateful to Axel Kleinschmidt for his support, help, and many useful discussions. Furthermore, L.T.K. would like to thank Matteo Broccoli, Franz Ciceri, Jan Gerken, Emanuel Malek, Hermann Nicolai, and Stefan Theisen for comments and useful discussions.

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

APPENDIX A: BASICS OF CANONICAL ANALYSIS

In this section, we briefly review the basics of how to analyze a theory using the canonical formalism. Canonical analysis is a synonym for Hamiltonian analysis. A full introduction to this subject is far beyond the scope of this work, and we omit many details in this section. For an in-depth treatment, see Refs. 14 and 15. This section is mainly based on the first chapters in Ref. 15.

The starting point of the canonical analysis is most often the Lagrangian formulation of a theory. Starting from an action functional S, given by the d-dimensional space–time integral over some Lagrangian density $L (q, \dot{q})$ ⁠, which, in turn, depends only on some fields qⁿ(x) and their time derivatives ${\dot{q}}^{n} (x)$ [see Eq. (A1)]. The space–time coordinates are x^μ with indices μ = 0, …, d − 1. Note that in this section, the Latin indices are used to label fields and constraints of the theory.

The canonical formalism aims to treat the fields and their conjugate canonical momenta on an equal footing. We introduce the canonical momenta p_n as defined by Eq. (A2). In the case of a field theory, the derivative is to be understood as a functional derivative. We define the Hamiltonian density $H (q, p)$ as the Legendre transformation of the Lagrangian density with respect to the time derivatives of the fields [see Eq. (A3)]. One can then rewrite the action functional as in Eq. (A4),

S = \int d^{d} x L (q, \dot{q}),

(A1)

p_{n} ≔ \frac{δ L}{δ {\dot{q}}^{n}},

(A2)

H (q, p) ≔ {\dot{q}}^{n} p_{n} - L (q, \dot{q}),

(A3)

S = \int d^{d} x ({\dot{q}}^{n} p_{n} - H (q, p)) .

(A4)

Due to the Legendre transformation, we have now found a function $H (q, p)$ that depends on two sets of variables, which we call the canonical variablesqⁿ(x) and the canonical momentap_n(x). We call the tuple (q, p) the canonical coordinates or phase space coordinates and the space that they describe phase space.

We can think of gauge theories as theories where, at any given time, the dynamical content of the theory is relative to an arbitrary reference frame.¹⁵ Hence, the general solution of a gauge theory necessarily contains arbitrary functions of time, since it is always permitted to transform the reference frame. This arbitrariness leads to the canonical coordinates not being completely independent, and therefore, it is equivalent to the existence of constraints on the phase space. In a gauge theory, we will always find constraints {Φ_m(q, p) = 0, m = 1, …, M} that only depend on the canonical coordinates and not on their time derivatives.

The first type of constraint follows directly from the definition of the canonical momenta [Eq. (A2)], which is why we call them primary constraints. The simplest primary constraint is given by a vanishing canonical momentum Φ(q, p) = p_n = 0; however, more complicated types are also common. Since we have not made use of the time evolution, the primary constraints do not restrict the kinematics and are true identities.

A consequence of the existence of primary constraints is that the Hamiltonian becomes non-unique in phase space since we are free to add primary constraints with arbitrary coefficients u^m(q, p) to the Hamiltonian. We call this function the total Hamiltonian [see Eq. (A5)]. The phase space hypersurface described by {Φ_m = 0, ∀m} is called the primary constraint surface, and on this surface, the Hamiltonian is still uniquely defined,

H_{T} ≔ H + u^{m} (q, p) Φ_{m} .

(A5)

For two phase space functions F, G, we define the Poisson bracket by the following equation:

{F, G} ≔ \frac{δ F}{δ q^{n}} \frac{δ G}{δ p_{n}} - \frac{δ G}{δ q^{n}} \frac{δ F}{δ p_{n}} .

(A6)

Using the Poisson bracket, we can write the time evolution of a phase space function F that follows from the total Hamiltonian of Eq. (A5) as Eq. (A7). At this point, this is the most general time evolution we can write down

\dot{F} = {F, H} + u^{m} {F, Φ_{m}} .

(A7)

Since we need the primary constraints to hold for all times—in order for the formalism to be consistent—we arrive at Eq. (A8). If Eq. (A8) yields a relation that is independent of the arbitrary parameters u^m and independent of the primary constraints, then we call it a secondary constraint. Since we need to ensure that it is also conserved in time, we iterate this procedure for all secondary constraints and potentially end up with more constraints. The consistency requirements that stem from secondary constraints are sometimes also referred to as tertiary, quaternary, etc., constraints. For reasonable physical theories, the termination of this procedure is guaranteed,

{\dot{Φ}}_{m} = {Φ_{m}, H} + u^{m^{'}} {Φ_{m}, Φ_{m^{'}}} = 0 .

(A8)

The only difference between primary and secondary constraints is that secondary constraints do restrict the kinematics, since we have made use of the time evolution in order to find them. Note that this means that we are not allowed to add the secondary constraints to the Hamiltonian in the manner that we have done with the primary constraints in Eq. (A5). Since further distinction between primary and secondary constraint is not needed, we denote the complete set of constraints by {Φ_j, j = 1, …, J}.

If we apply the time evolution equation (A7) to the full set of constraints, we will not find any new constraints. However, the set of time evolution equations can be seen as a set of differential equations for the—a priori—arbitrary phase space functions u^m(q, p). Solving this system, we can write the general solution using the homogeneous solution $V_{a}^{m}$ and a particular solution U^m as in Eq. (A9). The coefficients of the homogeneous solution v^a are then truly arbitrary,

u^{m} = U^{m} + v^{a} V_{a}^{m} .

(A9)

We write the ≈ sign to indicate an equality that holds on the constraint surface, e.g., $H_{T} \approx H$ ⁠, we call this relation weak equality. The time evolution generated by the total Hamiltonian via $\dot{F} \approx {F, H_{T}}$ is equivalent to the Lagrangian time evolution.

An important property of the constraints is whether they are first class [see Eq. (A10)] or second class constraints [see Eq. (A11)]. Second class constraints require further treatment, since they do not occur in the theory being analyzed in this work and we again refer the reader to Ref. 15. Note that the definition of the class relies on the weak equality, meaning that first class constraints Poisson-commute with all other constraints up to terms proportional to constraints. This also implies that if all the constraints are first class, the Poisson bracket algebra of the constraints closes automatically,

Φ_{k} first class constraint \Leftrightarrow \forall j : {Φ_{k}, Φ_{j}} \approx 0,

(A10)

Φ_{k} second class constraint \Leftrightarrow \exists j : {Φ_{k}, Φ_{j}} ≉ 0 .

(A11)

First class constraints can be interpreted as generators of gauge transformations. Mathematically, this is only guaranteed for first class primary constraints; however, for reasonable physical theories, this is also true for first class secondary constraints. This is also known as the Dirac conjecture. Counterexamples exist and are known.¹⁵ Gauge transformations δF on a phase space function F are generated as in Eq. (A12), where $Φ_{a} ≔ Φ_{m} V_{a}^{m}$ ⁠,

δ F = {F, Φ_{a}} v^{a} .

(A12)

We can add all first class constraints to the total Hamiltonian to describe the time evolution with the full gauge freedom accounted for. We call this the extended Hamiltonian [see Eq. (A13)], where {γ_a} is the set of all first class constraints,

H_{E} ≔ H_{T} + u^{a} γ_{a} .

(A13)

The extended Hamiltonian generates time evolution via Eq. (A14). This can be seen as an extension of the Lagrangian framework since the full gauge freedom is now manifest in the time evolution,

\dot{F} (q, p) \approx {F, H_{E}}, Φ_{j} \approx 0 .

(A14)

This only concerns gauge variant quantities, since for gauge invariant quantities (observables), all the Hamiltonian time evolutions are equivalent $H_{E} \Leftrightarrow H_{T} \Leftrightarrow H$ ⁠.

Furthermore, it is a general feature of generally covariant theories that the Hamiltonian vanishes weakly $H \approx 0$ —implying that the Hamiltonian consists of a linear combination of constraints. In particular, general relativity and, thus, also all supergravity theories are generally covariant theories. In this form, the time evolution is interpreted as a yet another gauge transformation.

APPENDIX B: POISSON BRACKET RELATIONS

In this appendix, we list some Poisson bracket relations that are intermediate results or otherwise useful for computations,

{M^{M N}, Π^{P Q} (M)} = (- M^{M P} M^{Q N} - M^{M Q} M^{P N}) δ^{(4)} (x - y),

(B1)

\begin{align} \int d^{4} x ϕ (x) {- e R (x), {Π^{n}}_{a} (e) (y)} = & + 2 e ϕ ({R^{n}}_{a} - \frac{1}{2} R {e_{a}}^{n}) \\ - 2 e (\nabla_{a} \nabla^{n} ϕ - {e_{a}}^{n} \nabla^{k} \nabla_{k} ϕ), \end{align}

(B2)

{{e_{b}}^{k}, {Π^{n}}_{a} (e)} = - {e_{a}}^{k} {e_{b}}^{n},

(B3)

{e, {Π^{n}}_{a} (e)} = + e {e_{a}}^{n},

(B4)

\{\frac{1}{e}, {Π^{n}}_{a} (e)\} = - \frac{{e_{a}}^{n}}{e},

(B5)

\{g_{k l}, {Π^{n}}_{a} (e)\} = + 2 δ_{() k}^{n} e_{l () a},

(B6)

\{g^{k l}, {Π^{n}}_{a} (e)\} = - 2 g^{n () k} {e_{a}}^{l ()},

(B7)

{H_{Ham} [ϕ], e} = + \frac{ϕ Π (e)}{6},

(B8)

\{H_{Ham} [ϕ], \frac{1}{e}\} = - \frac{ϕ Π (e)}{6 e^{2}},

(B9)

{H_{Ham} [ϕ], {e_{b}}^{k}} = + \frac{ϕ}{2 e} Π_{a b} (e) {e_{a}}^{k} - \frac{ϕ}{6 e} Π (e) {e_{b}}^{k},

(B10)

{H_{Ham} [ϕ], g^{r m}} = + \frac{ϕ}{e} e^{a () r} {Π^{m ()}}_{a} (e) - \frac{ϕ}{3 e} Π (e) g^{r m},

(B11)

{H_{Ham} [ϕ], M^{M N}} = + \frac{6}{e} ϕ Π^{M N} (M),

(B12)

{L [γ], {e_{a}}^{k}} = + γ_{a b} e^{b k} .

(B13)

APPENDIX C: OTHER USEFUL FORMULAS

In this appendix, we discuss some general formulas that are needed for the computations of the main sections.

From the definition of the (vielbein) determinant and the fact that its variation can be expressed as $δ e = e {e_{n}}^{a} δ {e_{a}}^{n}$ ⁠, we find the following identity:

{e_{a}}^{n} \partial_{p} {e_{n}}^{a} = e^{- 1} \partial_{p} e = - e \partial_{p} e^{- 1} .

(C1)

Furthermore, we can exploit the fact that we can over-antisymmetrize a tensor to make it vanish—e.g., we take an object with D + 1 indices in D dimensions and antisymmetrize them to get zero—this is sometimes called the Schouten identity. One can think of this identity as the fact that there cannot be D + 1 linearly independent vectors in D dimensions. Equation (C2) states the identity in four dimensions, where ϵ is the Levi–Cività symbol in four dimensions and v^k are vector components,

ϵ^{[) l m n r} v^{k (]} = 0 .

(C2)

If we expand this expression, we arrive at the following equation:

ϵ^{l m n r} v^{k} = - 4 ϵ^{k [) l m n} v^{r (]} .

(C3)

APPENDIX D: TREATMENT OF COSET CONSTRAINT EXAMPLE SL(n)/SO(n)

As explained in Sec. I C, we want to treat the $E_{6 (6)} (R) /$ USp(8) coset constraints on M_MN as being implicit. To explain this formalism, we consider the much simpler coset SL(n)/SO(n) as an example to demonstrate how one would explicitly or implicitly treat the coset constraints.

We start with the sigma model Lagrangian of a generic symmetric matrix M_MN with M, N = 1, …, n and just the time derivative for simplicity. The inverse matrix is defined via $M_{M N} M^{N K} = δ_{M}^{K}$ ⁠. We then explicitly add the SL(n)/SO(n) coset constraint c ≔ det(M) − 1 with the Lagrange parameter ϕ to the Lagrangian arriving at Eq. (D1). This is the starting point for the explicit treatment of the coset constraints,

L = - \frac{1}{2} {\dot{M}}_{M N} {\dot{M}}^{M N} + c ϕ .

(D1)

We can now calculate the canonical momenta and find the primary constraint (D3),

Π^{M N} (M) = - 2 {\dot{M}}^{M N},

(D2)

Π (ϕ) = 0 .

(D3)

The canonical Hamiltonian of this theory is given by Eq. (D4), where Π_MN(M) ≔ −Π^RS(M) M_RMM_SN. The total Hamiltonian is given by $H_{T} ≔ H + u Π (ϕ)$ ⁠,

H = + \frac{1}{8} Π_{M N} (M) Π^{M N} (M) + \dot{ϕ} Π (ϕ) - c ϕ .

(D4)

As explained in Appendix A, we now need to check the consistency of the primary constraint (D3) using Eq. (A7) and find that there is one secondary constraint given by Eq. (D5) as intended,

c = \det (M) - 1 = 0 .

(D5)

Considering Eq. (A7) now for the consistency of the secondary constraint, we find that there is one tertiary constraint given by Eq. (D6). In this calculation, we have used Eq. (D7), which follows directly from the fundamental Poisson bracket of a symmetric matrix with its canonical momentum (D8)—which is in form identical to the fundamental Poisson bracket in Eq. (3.7),

\det (M) M_{M N} Π^{M N} (M) ≕ p = 0,

(D6)

{\det (M), Π^{M N} (M)} = \det (M) M^{M N},

(D7)

{M_{M N} (x), Π^{P Q} (M) (y)} = (δ_{M}^{P} δ_{N}^{Q} + δ_{N}^{P} δ_{M}^{Q}) δ (x - y) .

(D8)

The tertiary constraint p—induced by the consistency of the coset constraint det(M) = 1—can be interpreted as a coset constraint on the momenta implying the tracelessness of the sl(n) algebra element M_RN Π^NS(M).

Applying Eq. (A7) to the tertiary constraint, we do not find any new constraints and the constraint system is consistent.

We immediately find that the primary constraint Π(ϕ) is a first class constraint. The secondary and tertiary constraints however form a second class system of constraints since the right-hand side of Eq. (D11) necessarily contains a constant term and cannot be rewritten solely in terms of the secondary constraint—the factor n here comes from $M_{M N} M^{M N} = δ_{M}^{M} = n$ ⁠,

{c, c} = 0,

(D9)

{p, p} = 0,

(D10)

{c, p} = n \det {(M)}^{2} .

(D11)

The presence of these two second class constraints tells us that we should define a Dirac bracket {g, f}_DB as defined in Ref. 15 given by Eq. (D12), where f, g are two functions on phase space. We find by construction that all constraints commute in the Dirac bracket,

{f, g}_{D B} ≔ {f, g} - \frac{1}{n \det {(M)}^{2}} ({f, p} {c, g} - {f, c} {p, g}) .

(D12)

One may have noted that Eq. (D7)—which is a general result for any symmetric matrix—seems incompatible with the secondary constraint det(M) = 1, which defines the SL(n)/SO(n) coset. This seeming inconsistency is resolved by the fact that we are not allowed to apply the canonical constraints before evaluating all Poisson brackets fully.

For the Dirac bracket, this is no longer true, however,¹⁵ and we are free to apply the canonical constraints inside the Dirac bracket yielding the following more expected equation:

{\det (M), Π^{M N} (M)}_{D B} = 0 .

(D13)

We can furthermore compute the number of degrees of freedom. We have the canonical variables M_MN, Π^KL(M), ϕ, Π(ϕ), which have $\frac{n (n + 1)}{2} + \frac{n (n + 1)}{2} + 1 + 1$ components—a priori the variables M_MN, Π^KL(M) are still generic symmetric matrices. We have one first class primary, one second class secondary, and one second class tertiary constraints, which means we should count them as 2 · 1 + 1 + 1.¹⁵ This leaves us with n² + n − 2 physical degrees of freedom in phase space or $\frac{n^{2}}{2} + \frac{n}{2} - 1$ physical degrees of freedom in field space.

This is the full explicit treatment of the coset constraints in the SL(n)/SO(n) case.

The implicit treatment of the coset constraints uses the Lagrangian without adding the coset constraints explicitly $L = - \frac{1}{2} {\dot{M}}_{M N} {\dot{M}}^{M N}$ ⁠. We can nonetheless apply them after all Poisson brackets have been computed, thus considering them implicitly. The price one pays for the implicit treatment of the constraints is that we are not able to apply canonical constraints inside the brackets. In return, we are able to skip the canonical analysis of the coset constraints—which are much more complicated for the $E_{6 (6)} (R) /$ USp(8) coset—if we are not interested in applying them inside brackets. We can compute the canonical degrees of freedom by looking at the dimension of the coset, in this case dim(SL(n)/SO(n)) $= \frac{n^{2}}{2} + \frac{n}{2} - 1$ ⁠. We thus arrive at the same result as in the explicit treatment.

For an alternative canonical treatment of coset space sigma models using the vielbein formalism, see Ref. 31.

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Author(s)

Canonical analysis of $E_{6 (6)} (R)$ invariant five dimensional (super-)gravity

I. INTRODUCTION

A. Notation and conventions

B. Arnowitt–Deser–Misner (ADM) formulation of general relativity

C. $E_{6 (6)} (R)$ invariant five dimensional (super-)gravity

II. CANONICAL FORMULATION: HAMILTONIAN AND CANONICAL CONSTRAINTS

A. Canonical momenta

B. Canonical Hamiltonian

C. Primary constraints

D. Secondary constraints

E. Total Hamiltonian

III. CANONICAL ANALYSIS: GAUGE TRANSFORMATIONS AND GAUGE ALGEBRA

A. Diffeomorphism weight and the Lie derivative

B. Gauge transformations

C. Algebra of constraints

D. Extended Hamiltonian

E. Counting the degrees of freedom

IV. SUMMARY AND OUTLOOK

ACKNOWLEDGMENTS

DATA AVAILABILITY

APPENDIX A: BASICS OF CANONICAL ANALYSIS

APPENDIX B: POISSON BRACKET RELATIONS

APPENDIX C: OTHER USEFUL FORMULAS

APPENDIX D: TREATMENT OF COSET CONSTRAINT EXAMPLE SL(n)/SO(n)

REFERENCES

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Canonical analysis of E6(6)(R) invariant five dimensional (super-)gravity

I. INTRODUCTION

A. Notation and conventions

B. Arnowitt–Deser–Misner (ADM) formulation of general relativity

C. E6(6)(R) invariant five dimensional (super-)gravity

II. CANONICAL FORMULATION: HAMILTONIAN AND CANONICAL CONSTRAINTS

A. Canonical momenta

B. Canonical Hamiltonian

C. Primary constraints

D. Secondary constraints

E. Total Hamiltonian

III. CANONICAL ANALYSIS: GAUGE TRANSFORMATIONS AND GAUGE ALGEBRA

A. Diffeomorphism weight and the Lie derivative

B. Gauge transformations

C. Algebra of constraints

D. Extended Hamiltonian

E. Counting the degrees of freedom

IV. SUMMARY AND OUTLOOK

ACKNOWLEDGMENTS

DATA AVAILABILITY

APPENDIX A: BASICS OF CANONICAL ANALYSIS

APPENDIX B: POISSON BRACKET RELATIONS

APPENDIX C: OTHER USEFUL FORMULAS

APPENDIX D: TREATMENT OF COSET CONSTRAINT EXAMPLE SL(n)/SO(n)

REFERENCES

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Canonical analysis of $E_{6 (6)} (R)$ invariant five dimensional (super-)gravity

C. $E_{6 (6)} (R)$ invariant five dimensional (super-)gravity