Noncommutative hamiltonian formalism

for noncommutative gravity

Leonardo Castellani

Dipartimento di Scienze e Innovazione Tecnologica
Università del Piemonte Orientale, viale T. Michel 11, 15121 Alessandria,
Italy
INFN, Sezione di Torino, via P. Giuria 1, 10125 Torino, Italy
Regge Center for Algebra, Geometry and Theoretical Physics, via P. Giuria 1,
10125 Torino, Italy

We present a covariant canonical formalism for noncommutative (NC) gravity,
and in general for NC geometric theories defined via a twisted $\star$-wedge
product between forms. Noether theorems are generalized to the NC setting, and
gauge generators are constructed in a twisted phase space with
$\star$-deformed Poisson bracket. This formalism is applied to NC $d=4$
vierbein gravity, and allows to find the canonical generators of the tangent
space $\star$-gauge group.

In memoriam of Alessandro D’ Adda

<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 A summary of the covariant hamiltonian formalism
    1. 2.1 Geometric action and field equations
    2. 2.2 Form Hamiltonian
    3. 2.3 Form Poisson bracket
  3. 3 Twisted hamiltonian formalism
    1. 3.1 $\star$-Hamiltonian
    2. 3.2 $\star$-Poisson bracket and infinitesimal canonical transformations
    3. 3.3 $\star$-Symmetries and $\star$-Noether theorems
  4. 4 Twisted noncommutative gravity in $d=4$
    1. 4.1 Classical action and symmetries
    2. 4.2 NC Action and NC symmetries
      1. 4.2.1 NC gravity action for the component fields
      2. 4.2.2 $\star$-gauge variations
  5. 5 NC Hamiltonian formulation of NC gravity
    1. 5.1 $\star$-Legendre transformation and $\star$-Hamiltonian
    2. 5.2 Construction of $\star$-gauge generators
  6. 6 Conclusions
  7. A Twist differential geometry
  8. B Gamma matrices in $d=4$
    1. B.1 Useful identities

##  1 Introduction

In this paper we present a hamiltonian formalism tailored for noncommutative
(NC) geometric field theories. Both hamiltonian techniques and noncommutative
geometry have a long history in theoretical physics, and have been brought
together already many decades ago when phase space, the geometrical arena of
canonical formalism, was studied by Dirac [1, 2] as a primordial example of
noncommutative space. This inspired the idea of spacetime coordinates as non-
commuting operators [3, 4], with commutation relations of the type
$[x^{\mu},x^{\nu}]=i\theta^{\mu\nu}$. Such relations lead to an uncertainty
principle that smears spacetime at distances shorter than $\sqrt{\theta}$.
Since quantum theory prevents us to “measure” geometry at distances smaller
than the Planck length $L_{P}$ (at this scale the curvature radius of
spacetime becomes comparable to the wavelength of a probe particle), the above
commutation relations seem to make good physical sense when
$\sqrt{\theta}\approx L_{P}$. A quantum theory of gravity containing or
predicting noncommutativity of spacetime coordinates at small distances would
therefore stand a good chance to be intrinsically regulated.

Geometric theories, as for example (super)gravity, can be formulated on a NC
spacetime, by deforming the usual wedge product between forms to a twisted
wedge product, or $\wedge_{\star}$ product. This program has been applied for
instance to Yang-Mills theories coupled to fermions (for a NC version of the
standard model see for ex. [5, 6]), to gravity coupled to fermions [7], and
also to supergravity, see [8].

Canonical quantization of these NC field theories can be achieved by applying
the traditional hamiltonian methods to the classical $\star$-deformed
theories, expanded in the noncommutativity parameter $\theta$. Another way,
which we advocate in the present paper, is to design a NC hamiltonian
formalism that can be applied directly to the NC theory, before expressing it
as a classical deformed field theory.

Here we generalize to a NC setting the covariant hamiltonian formalism,
introduced in the eighties in ref.s [9]-[13], and further developed more
recently in [14, 15]. This formalism is well adapted to geometric theories,
governed by Lagrangian $d$-forms.

By extending the covariant Legendre transformation of [14, 15] to a NC
covariant Legendre transformation, we can define NC momenta and Hamiltonian.
This we do explicitly on the example of the twisted noncommutative (pure)
gravity in $d=4$ of ref. [7].

The power of hamiltonian methods is particularly useful when investigating
symmetries. Starting from the Hamiltonian, an algorithm [16] exists to
construct all the canonical symmetry generators, based on the Dirac treatment
of constrained hamiltonian systems [17, 18, 19]. The algorithm has been
generalized to the covariant hamiltonian framework in [14, 15], and in the
present paper further generalized to NC theories, thus providing a NC analogue
of Noether theorems.

The paper is organized as follows. Section 2 summarizes the covariant
hamiltonian method for geometric theories. Section 3 generalizes this method
to NC twisted geometric theories. In Sections 4 and 5 we give a detailed
account of $d=4$ NC gravity in this formalism: NC momenta, Hamiltonian and
Poisson brackets are found, and the canonical generator of the NC symmetries
of the theory is explicitly constructed via the NC generalization of the
algorithm of [14, 15]. The Appendices contain a micro-review of twisted NC
geometry (for a more detailed treatment see for ex. [20]), and some gamma
matrix conventions.

##  2 A summary of the covariant hamiltonian formalism

###  2.1 Geometric action and field equations

Consider the action :

$S=\int_{{\cal M}^{d}}L(\phi_{i},d\phi_{i})$ (2.1)

where the Lagrangian $L$ depends on a collection of $p_{i}$-form fields
$\phi_{i}$ and their exterior derivatives, and is integrated on a
$d$-dimensional manifold ${\cal M}^{d}$.

The variational principle reads

$\delta S=\int_{{\cal
M}^{d}}\delta\phi_{i}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}L\over\partial\phi_{i}}+d(\delta\phi_{i}){\stackrel{{\scriptstyle\rightarrow}}{{\partial}}L\over\partial(d\phi_{i})}=0$
(2.2)

All products are exterior products between forms, satisfying

$AB=(-)^{ab+\eta_{a}\eta_{b}}BA$ (2.3)

with $a,b$ and $\eta_{a},\eta_{b}$ the degrees and the fermionic gradings of
the forms $A$ and $B$ respectively ($\eta=0$ for bosons and $\eta=1$ for
fermions). In the following we consider for simplicity only bosonic fields.
The symbol
${\stackrel{{\scriptstyle\rightarrow}}{{\partial}}L\over\partial\phi_{i}}$
indicates the right derivative of $L$ with respect to a $p_{i}$-form
$\phi_{i}$, defined by first bringing $\phi_{i}$ to the left in $L$ (taking
into account the sign changes due to the form degrees) and then canceling it
against the derivative.

After integrating by parts the second term in the action variation (2.2), one
obtains the Euler-Lagrange equations

$d~{}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}L\over\partial(d\phi_{i})}-(-)^{p_{i}}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}L\over\partial\phi_{i}}=0$
(2.4)

These are $(d-p_{i})$-form equations.

###  2.2 Form Hamiltonian

We define the $d$-form Hamiltonian as:

$H\equiv d\phi_{i}~{}\pi^{i}-L$ (2.5)

where the ($d-p_{i}-1$)-form momenta are given by:

$\pi^{i}\equiv{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}L\over\partial(d\phi_{i})}$
(2.6)

The Hamiltonian does not depend on the “velocities” $d\phi_{i}$, the proof
being the same as in the usual case, unless primary constraints are present
(see later).

The form-analogue of the Hamilton equations reads:

$d\phi_{i}=(-)^{(d+1)(p_{i}+1)}~{}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}H\over\partial\pi^{i}},~{}~{}~{}d\pi^{i}=(-)^{p_{i}+1}~{}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}H\over\partial\phi_{i}}$
(2.7)

The first equation is equivalent to the momentum definition, and is obtained
by taking the right derivative of $H$ as given in (2.5) with respect to
$\pi^{i}$, and then using (2.6), and
$(d-p_{i}-1)(p_{i}+1)=(d+1)(p_{i}+1)(mod~{}2)$. The second is equivalent to
the Euler-Lagrange form equations, and is obtained by taking the right
derivative of $H$ with respect to $\phi_{i}$. Both can be deduced as
variational equations for the action (2.1) rewritten with canonical variables:

$S=\int_{{\cal M}^{d}}d\phi_{i}~{}\pi^{i}-H(\phi_{i},\pi^{i})$ (2.8)

###  2.3 Form Poisson bracket

The form Hamilton equations allow to express the (on shell) exterior
differential of any $p$-form $F(\phi_{i},\pi^{i})$ as

$dF=d\phi_{i}~{}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}F\over\partial\phi_{i}}+d\pi^{i}~{}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}F\over\partial\pi^{i}}=(-)^{(d+1)(p_{i}+1)}~{}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}H\over\partial\pi^{i}}~{}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}F\over\partial\phi_{i}}+(-)^{p_{i}+1}~{}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}H\over\partial\phi_{i}}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}F\over\partial\pi^{i}}$
(2.9)

Using left derivatives this expression simplifies:

$dF={\stackrel{{\scriptstyle\leftarrow}}{{\partial}}H\over\partial\pi^{i}}~{}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}F\over\partial\phi_{i}}-(-)^{p_{i}d}~{}{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}H\over\partial\phi_{i}}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}F\over\partial\pi^{i}}$
(2.10)

Note: left derivatives are defined as “acting on the left” and for example
${\stackrel{{\scriptstyle\leftarrow}}{{\partial}}H\over\partial\phi_{i}}$
really means
$H{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}\over\partial\phi_{i}}$. It
is easy to verify that the left and right derivatives of an $f$-form $F$ with
respect to an $a$-form $A$ satisfy

${\stackrel{{\scriptstyle\leftarrow}}{{\partial}}F\over\partial
A}=(-)^{a(f+1)}~{}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}F\over\partial
A}$ (2.11)

and this relation is used to prove eq. (2.10).

The expression for the differential (2.10) suggests the definition of the form
Poisson bracket (FPB):

$\\{A,B\\}\equiv{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}B\over\partial\pi^{i}}~{}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}A\over\partial\phi_{i}}-(-)^{p_{i}d}~{}{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}B\over\partial\phi_{i}}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}A\over\partial\pi^{i}}$
(2.12)

so that

$dF=\\{F,H\\}$ (2.13)

The form Poisson bracket between the $a$-form $A$ and the $b$-form $B$ is a
($a+b-d+1$)-form, and canonically conjugated forms satisy:

$\\{\phi_{i},\pi^{j}\\}=\delta_{i}^{j}$ (2.14)

Using the definition (2.12), the following relations for the FPB of (2.12) can
be shown to hold [14]:

$\displaystyle\\{B,A\\}=-(-)^{(a+d+1)(b+d+1)}\\{A,B\\}$ (2.15)
$\displaystyle\\{A,BC\\}=B\\{A,C\\}+(-)^{c(a+d+1)}\\{A,B\\}C$ (2.16)
$\displaystyle\\{AB,C\\}=\\{A,C\\}B+(-)^{a(c+d+1)}A\\{B,C\\}$ (2.17)
$\displaystyle(-)^{(a+d+1)(c+d+1)}\\{A,\\{B,C\\}\\}+cyclic~{}=0$ (2.18)
$\displaystyle(-)^{(a+d+1)(b+d+1)}\\{\\{B,C\\},A\\}+cyclic~{}=0$ (2.19)

i.e. graded antisymmetry, derivation property, and form-Jacobi identities.

As discussed in [14, 15], canonical transformations, symmetry generators and
Noether theorems have their covariant counterpart. We will treat them directly
in the twisted case.

##  3 Twisted hamiltonian formalism

In this Section we extend the covariant hamiltonian formalism to geometric
theories on noncommutative spaces, and more precisely to twisted geometric
theories with a noncommutative $\star$-product. Many formulae of the preceding
Section continue to hold, with the usual exterior product replaced by the
$\star$-exterior product between forms.

###  3.1 $\star$-Hamiltonian

Starting from a geometric action

$S=\int_{{\cal M}^{d}}L(\phi_{i},d\phi_{i},\wedge_{\star})$ (3.1)

where all the fields are multiplied with $\wedge_{\star}$ products, the Euler-
Lagrange equations are obtained by varying with respect to $\phi_{i}$. This
variation takes the same form as in the classical (undeformed) case (2.2),
with two modifications:

$\bullet$ exterior products $\wedge$ are replaced by $\wedge_{\star}$.

$\bullet$ the right derivative is defined via cyclic permutation. This means
that the field to be varied is first brought to the left via cyclic
permutation (this we can do since integration is cyclic), and then is varied.

The outcome is that the Euler-Lagrange equations take the same form as the
classical ones, with the right derivative being defined with the cyclic
procedure.

The $\star$-Hamiltonian $d$-form is defined as

$H\equiv d\phi_{i}\wedge_{\star}\pi^{i}-L$ (3.2)

where the ($d-p_{i}-1$)-form $\star$-momenta are given by:

$\pi^{i}\equiv{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}L\over\partial(d\phi_{i})}$
(3.3)

with cyclic right derivative. Note that here only the integral of $H$ is
independent of $d\phi_{i}$, since cyclicity is essential for the proof, and
only integrated $\wedge_{\star}$ products are cyclic.

Varying the action

$S=\int_{{\cal M}^{d}}d\phi_{i}~{}\wedge_{\star}\pi^{i}-H$ (3.4)

yields the $\star$–Hamilton equations: these are formally the same as the
classical ones in (2.7), with the proviso that the right derivative is the
cyclic one.

In the following right and left derivatives are always defined to be cyclic,
and $\wedge_{\star}$ products between forms are understood when omitted.

###  3.2 $\star$-Poisson bracket and infinitesimal canonical transformations

Formula (2.9), with cyclic derivatives and $\wedge_{\star}$ products, holds
now only under integration, so that $F$ must be a ($d-1$)-form. Still, the
relation (2.11) continues to hold, with cyclic left and right derivatives, and
we can define $\star$-Poisson brackets formally as in (2.12):

$\\{A,B\\}_{\star}\equiv{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}B\over\partial\pi^{i}}\wedge_{\star}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}A\over\partial\phi_{i}}-(-)^{p_{i}d}~{}{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}B\over\partial\phi_{i}}\wedge_{\star}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}A\over\partial\pi^{i}}$
(3.5)

As in the classical case, we can also define infinitesimal canonical
transformations on the basic fields and momenta as follows:

$\displaystyle\delta\phi_{i}=\\{\phi_{i},G\\}_{\star}={\stackrel{{\scriptstyle\leftarrow}}{{\partial}}G\over\partial\pi^{i}}$
$\displaystyle\delta\pi^{i}=\\{\pi^{i},G\\}_{\star}=-(-)^{p_{i}d}{\stackrel{{\scriptstyle\leftarrow}}{{\partial}}G\over\partial\phi_{i}}$
(3.6)

where $G=G(\phi_{i},\pi^{i})$ is a ($d-1$)–form, the generator of the
canonical transformation. The generator $G$ can contain parameters
$\varepsilon(x)$ depending only on spacetime, for example as
$G=\varepsilon(x)G^{\prime}$. For a $d$-form $A$, the following relation holds
under integration:

$\delta
A=\delta\phi_{i}~{}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}A\over\partial\phi_{i}}+\delta\pi^{i}~{}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}A\over\partial\pi^{i}}=\\{A,G\\}_{\star}$
(3.7)

The second equality follows immediatedly from eq.s (3.6). Note that
$\\{A,G\\}_{\star}$ is a $d$-form like $A$.

In the following we always consider ($d$-1)–form generators of the type
$\varepsilon(x)~{}\pi\phi$ (“point canonical transformations“ ) or of the more
general form $d\varepsilon(x)~{}\pi+\varepsilon(x)~{}\pi\phi$, and it is easy
to check that the transformations (3.6) preserve the canonical FPB relations
(2.14). Indeed

$\displaystyle\\{\phi^{\prime}_{i},\pi^{\prime
j}\\}_{\star}=\\{\phi_{i}+\\{\phi_{i},G\\}_{\star},\pi^{j}+\\{\pi^{j},G\\}_{\star}\\}_{\star}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\\{\phi_{i},\pi^{j}\\}_{\star}+\\{\phi_{i},\\{\pi^{j},G\\}_{\star}\\}_{\star}+\\{\\{\phi_{i},G\\}_{\star},\pi^{j}\\}_{\star}+O(\varepsilon^{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\\{\phi_{i},\pi^{j}\\}_{\star}+O(\varepsilon^{2})$
(3.8)

since the two central terms of the second line cancel each other, as can be
verified using the definition of $\star$-Poisson bracket (3.5). Therefore the
variations (3.6) deserve to be called $\star$-canonical (infinitesimal)
transformations.

###  3.3 $\star$-Symmetries and $\star$-Noether theorems

Consider now the generic variation of the action (3.4)

$\displaystyle\delta S=\int_{{\cal
M}^{d}}d(\delta\phi_{i})\pi^{i}+d\phi_{i}~{}\delta\pi^{i}-\delta H=\int_{{\cal
M}^{d}}d(\delta\phi_{i}\pi^{i})-(-)^{p_{i}}\delta\phi_{i}~{}d\pi^{i}+d\phi_{i}~{}\delta\pi^{i}-\delta
H$ $\displaystyle~{}~{}~{}~{}=\int_{{\cal
M}^{d}}d(\delta\phi_{i}\pi^{i})-(-)^{p_{i}}(-)^{(d-p_{i})p_{i}}d\pi^{i}~{}\delta\phi_{i}+d\phi_{i}~{}\delta\pi^{i}-\delta
H$ (3.9)

where $\wedge_{\star}$ products are omitted. Recalling the variations (3.6),
and the relation (2.11) between right and left derivatives, the variation of
the action becomes

$\displaystyle\delta S=\int_{{\cal
M}^{d}}d({\stackrel{{\scriptstyle\leftarrow}}{{\partial}}G\over\partial\pi^{i}}\pi^{i})-d\pi^{i}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}G\over\partial\pi^{i}}-d\phi_{i}{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}G\over\partial\phi_{i}}-\\{H,G\\}_{\star}=$
$\displaystyle~{}~{}~{}~{}=\int_{{\cal
M}^{d}}d({\stackrel{{\scriptstyle\leftarrow}}{{\partial}}G\over\partial\pi^{i}}\pi^{i}-G)-\\{H,G\\}_{\star}$
(3.10)

where we have used (3.7) for the variation of $H$, since $H$ is a $d$–form. In
the final expression for $\delta S$ we have equated $dG$ to the two terms with
right derivatives: this is correct as long as $G$ contains only $\phi$ and
$\pi$ fields, plus possibly constant (infinitesimal) parameters $\varepsilon$.
If it contains arbitrary functions $\varepsilon(x)$, for example
$\mathbb{G}=\varepsilon(x)G$ (we denote in this case the generator with
$\mathbb{G}$, while $G$ is still a function of only $\phi$ and $\pi$), the two
terms with right derivatives become $-d(\varepsilon(x)G)+d\varepsilon(x)~{}G$
under integration. With appropriate conditions on the boundary of ${{\cal
M}^{d}}$, the total derivative can be ignored and we are left with:

$\delta S=\int_{{\cal M}^{d}}d\varepsilon~{}G-\\{H,\varepsilon G\\}_{\star}$
(3.11)

For constant $\varepsilon$, we recover the analogue of Noether’s theorem for
global symmetries: the symmetry generator $\mathbb{G}=\varepsilon G$ has
vanishing $\star$–Poisson bracket with the Hamiltonian. For constant
$\varepsilon$ this also implies $\\{H,G\\}_{\star}=0$. As in the classical
case, this leads to a conservation law, since the on-shell differential of $G$
can be expressed (under integration) as
$dG=\\{G,H\\}_{\star}=-\\{H,G\\}_{\star}$. Then on shell $\int_{{\cal
M}^{d}}dG=0$ and the integral of $G$ on the boundary ${\partial{\cal M}^{d}}$
vanishes. Taking vanishing fields at spatial infinity, the integral of $G$ on
a spacelike surface is conserved in time.

When $\varepsilon$ is nonconstant, the variation (3.11) vanishes iff $G=0$ and
$\\{H,\varepsilon G\\}_{\star}=0$ (a nonconstant $\varepsilon$ cannot be taken
outside the $\star$–Poisson bracket, since it $\star$–multiplies $G$). Thus
generators of local symmetries have to vanish, together with their
$\star$–Poisson bracket with the Hamiltonian, a result analogous to Noether’s
theorem for local symmetries.

In gauge and gravity theories the symmetry transformations on the basic fields
can include derivatives of the parameter. In our geometric formalism these can
only be exterior derivatives $d\varepsilon$. Therefore we must consider
generators $\mathbb{G}$ of the type

$\mathbb{G}=\varepsilon G+d\varepsilon~{}F$ (3.12)

where $F$ is a ($d-2$)–form, and the action variation becomes

$\delta S=\int_{{\cal M}^{d}}d\varepsilon~{}G-\\{H,\varepsilon
G\\}_{\star}-\\{H,d\varepsilon F\\}_{\star}$ (3.13)

Thus $\star$-symmetry generators of this type must satisfy separately

$\\{H,\varepsilon G\\}_{\star}=0,~{}~{}~{}d\varepsilon~{}G-\\{H,d\varepsilon
F\\}_{\star}=0$ (3.14)

since $\varepsilon$ is an arbitrary function. To check whether these
conditions (3.14) hold we can use cyclic reorderings, since they originate
from the varied integral (3.13). Note that in presence of hamiltonian
constraints $\Phi=0$ (as is the case when the action is invariant under gauge-
type transformations) one must also require the variation of the constraints
to vanish weakly, i.e. modulo constraints:

$\delta\Phi\approx 0$ (3.15)

with $\approx$ meaning weak equality. In the commutative case, these
variations can be expressed with the Poisson bracket $\\{\Phi,\mathbb{G}\\}$,
whereas in the NC setting $\delta\Phi=\\{\Phi,\mathbb{G}\\}_{\star}$ holds
only under integration, with $\Phi$ a $d$–form.

This generalizes to the noncommutative setting the algorithm of [16] (adapted
to the covariant hamiltonian formalism in [14]) for the construction of
canonical gauge symmetry generators. The steps of the algorithm are:

1) look for first class $(d-1)$–forms $d\varepsilon F(\phi,\pi)$, first class
meaning that they have weakly vanishing $\star$-Poisson bracket with all the
constraints.

2) compute the $\star$–Poisson bracket $\\{H,d\varepsilon F\\}_{\star}$. The
result leads to a candidate for $G$.

3) Check whether this candidate, possibly with the addition of weakly
vanishing pieces, satisfies $\\{H,\varepsilon G\\}_{\star}=0$. In fact this
request usually fixes the weakly vanishing pieces (combinations of
constraints) that need to be added.

4) Check whether the variations generated by $\mathbb{G}=\varepsilon
G+d\varepsilon~{}F$ preserve (weakly) the constraints $\Phi$.

Then $\mathbb{G}$ generates a symmetry of the action (3.4).

In the next Sections we apply this formalism to noncommutative (twisted) $d=4$
vierbein gravity.

##  4 Twisted noncommutative gravity in $d=4$

###  4.1 Classical action and symmetries

We start by rewriting the classical Einstein-Hilbert action in a compact form,
using an index-free notation:

$S=\int Tr\left(iR\wedge V\wedge V\gamma_{5}\right)$ (4.1)

The fundamental fields are the 1–forms $\Omega$ (spin connection) and $V$
(vierbein):

$\Omega={1\over
4}\omega^{ab}\gamma_{ab},~{}~{}~{}~{}~{}V=V^{a}\gamma_{a}~{}~{}~{}~{}~{}~{}$
(4.2)

The curvature 2–form $R$ is defined as

$R=d\Omega-\Omega\wedge\Omega,$ (4.3)

Thus all fields are $4\times 4$ matrices in the spinor representation (see
Appendix A for $d=4$ gamma matrix conventions). The trace $Tr$ is taken on
this representation. Using the $d=4$ gamma matrix identities

$\gamma_{abc}=i\varepsilon_{abcd}\gamma^{d}\gamma_{5},~{}~{}~{}~{}~{}~{}~{}Tr(\gamma_{ab}\gamma_{c}\gamma_{d}\gamma_{5})=-4i\varepsilon_{abcd}$
(4.4)

we recover the usual action

$S=\int R^{ab}\wedge V^{c}\wedge V^{d}~{}\varepsilon_{abcd}$ (4.5)

The action is invariant under local diffeomorphisms (it is the integral of a
4-form on a 4-manifold) and under the local Lorentz rotations:

$\delta_{\epsilon}V=-[V,\epsilon],~{}~{}~{}\delta_{\epsilon}\Omega=d\epsilon-[\Omega,\epsilon],$
(4.6)

with gauge parameter (0-form)

$\epsilon={1\over 4}\epsilon^{ab}\gamma_{ab}$ (4.7)

The invariance can be directly checked on the action (4.1) noting that

$\delta_{\epsilon}R=-[{R},\epsilon],$ (4.8)

and using the cyclicity of the trace $Tr$, and the fact that $\epsilon$
commutes with $\gamma_{5}$. The Lorentz rotations close on the Lie algebra:

$[\delta_{\epsilon_{1}},\delta_{\epsilon_{2}}]=-\delta_{[\epsilon_{1},\epsilon_{2}]}$
(4.9)

After substituting (4.2) and (4.7) into (4.6), simple gamma algebra yields the
gauge variations of the component fields in (4.2):

$\delta_{\epsilon}V^{a}=\epsilon^{a}_{~{}b}V^{b},~{}~{}~{}\delta_{\epsilon}\omega^{ab}=d\epsilon^{ab}-\omega^{a}_{~{}c}\epsilon^{cb}+\omega^{b}_{~{}c}\epsilon^{ca}\equiv{\cal
D}\epsilon^{ab}$ (4.10)

Similarly, the variation of the curvature components is found to be

$\delta R^{ab}=\epsilon^{a}_{~{}c}R^{cb}-\epsilon^{b}_{~{}c}R^{ca}$ (4.11)

Thus all quantities in the action (4.5) transform homogeneously under Lorentz
local rotations, and since $\varepsilon_{abcd}$ is an invariant tensor of
$SO(1,3)$, the action is likewise invariant. Here the proof of invariance
looks simple both in the index-free and in the component formulation. Note
however that in general the index-free proof is much simpler.

###  4.2 NC Action and NC symmetries

In this subsection we recall the noncommutative generalization of the action
(4.1), the bosonic part of the NC action studied in ref. [7]. It is obtained
by replacing exterior products by deformed exterior products:

$S=\int Tr\left(i{R}\wedge_{\star}V\wedge_{\star}V\gamma_{5}\right)$ (4.12)

with

$R=d\Omega-\Omega\wedge_{\star}\Omega,$ (4.13)

Almost all formulae in Section 4.1 continue to hold, with $\star$–products and
$\star$–exterior products. However, the expansion of the fundamental fields on
the Dirac basis of gamma matrices must now include new contributions:

$\Omega={1\over 4}\omega^{ab}\gamma_{ab}+i\omega
1+\widetilde{\omega}\gamma_{5},~{}~{}~{}~{}~{}V=V^{a}\gamma_{a}+\widetilde{V}^{a}\gamma_{a}\gamma_{5}~{}~{}~{}~{}~{}~{}$
(4.14)

Similarly for the curvature :

$R={1\over 4}R^{ab}\gamma_{ab}+ir1+\tilde{r}\gamma_{5}$ (4.15)

and for the gauge parameter:

$\epsilon={1\over 4}\varepsilon^{ab}\gamma_{ab}+i\varepsilon
1+\widetilde{\varepsilon}\gamma_{5}$ (4.16)

Indeed now the $\star$–gauge variations read:

$\delta_{\epsilon}V=-V\star\epsilon+\epsilon\star
V,~{}~{}~{}\delta_{\epsilon}\Omega=d\epsilon-\Omega\star\epsilon+\epsilon\star\Omega,$
(4.17)

and in the variations for $V$ and $\Omega$ also anticommutators of gamma
matrices appear, due to the noncommutativity of the $\star$–product. Since for
example the anticommutator $\\{\gamma_{ab},\gamma_{cd}\\}$ contains $1$ and
$\gamma_{5}$, we see that the corresponding fields must be included in the
expansion of $\Omega$. Similarly, $V$ must contain a $\gamma_{a}\gamma_{5}$
term due to $\\{\gamma_{ab},\gamma_{c}\\}$. Finally, the composition law for
gauge parameters becomes:

$[\delta_{\epsilon_{1}},\delta_{\epsilon_{2}}]=\delta_{\epsilon_{2}\star\epsilon_{1}-\epsilon_{1}\star\epsilon_{2}}$
(4.18)

so that $\epsilon$ must contain the $1$ and $\gamma_{5}$ terms, since they
appear in the composite parameter
$\epsilon_{2}\star\epsilon_{1}-\epsilon_{1}\star\epsilon_{2}$.

The invariance of the noncommutative action (4.12) under the
$\star$-variations is proved in exactly the same way as for the commutative
case, noting that

$\delta_{\epsilon}R=-R\star\epsilon+\epsilon\star R,$ (4.19)

and using now, besides the cyclicity of the trace $Tr$ and the fact that
$\epsilon$ still commutes with $\gamma_{5}$, also the graded cyclicity of the
integral. The local $\star$-symmetry satisfies the Lie algebra of $GL(2,C)$,
and centrally extends the $SO(1,3)$ Lie algebra of the commutative theory.

Invariance under diffeomorphisms $x\rightarrow x^{\prime}(x)$ holds since the
action is a 4-form integrated on 4-dimensional spacetime. More precisely, the
action (4.12) is invariant under diffeomorphisms generated by the Lie
derivative ${\ell}_{\varepsilon}$, where
$\varepsilon=\varepsilon^{\mu}\partial_{\mu}$ is an infinitesimal tangent
vector. Indeed using the Cartan formula
${\ell}_{\varepsilon}=d\iota_{\varepsilon}+\iota_{\varepsilon}d$ we find that
the variation of the action is a total derivative, since the Lagrangian is a
top form. The (infinitesimal) variation obeys the Leibniz rule, amounting to
vary in turn all the fields (forms and tangent vectors) present in the action.
Form fields $\tau$ vary with the Lie derivative, i.e.
$\delta_{\varepsilon}\tau=\ell_{\varepsilon}\tau$, and the tangent vectors
$X_{A}=X_{A}^{\mu}\partial_{\mu}$ defining the $\wedge_{\star}$ product (see
Appendix A) transform as $\delta_{\varepsilon}X_{A}=[\varepsilon,X_{A}]$. With
these variations the action varies into a total derivative111In the present
setting, the tangent fields $X_{A}$ are background fields. Their promotion to
dynamical fields is discussed in ref. [21]..

Finally, in the next two subsections we give the NC action in terms of the
component fields ($V^{a}$, $\omega^{ab}$, $\widetilde{V}^{a}$, $\omega$,
$\widetilde{\omega}$), and the gauge variations of these fields.

####  4.2.1 NC gravity action for the component fields

Using the expansions (4.14) and (4.15), the action (4.12) takes the form:

$\displaystyle S=$ $\displaystyle\int
R^{ab}\wedge_{\star}(V^{c}\wedge_{\star}V^{d}-\widetilde{V}^{c}\wedge_{\star}\widetilde{V}^{d})\epsilon_{abcd}-2i~{}R^{ab}\wedge_{\star}(V_{a}\wedge_{\star}\widetilde{V}_{b}-\widetilde{V}_{a}\wedge_{\star}V_{b})$
$\displaystyle-4~{}r\wedge_{\star}(V^{a}\wedge_{\star}\widetilde{V}_{a}-\widetilde{V}^{a}\wedge_{\star}V_{a})+4i~{}\tilde{r}\wedge_{\star}(V^{a}\wedge_{\star}V_{a}-\widetilde{V}^{a}\wedge_{\star}\widetilde{V}_{a})$

with

$\displaystyle R^{ab}=d\omega^{ab}-{1\over
2}\omega^{a}_{~{}c}\wedge_{\star}\omega^{cb}+{1\over
2}\omega^{b}_{~{}c}\wedge_{\star}\omega^{ca}-i(\omega^{ab}\wedge_{\star}\omega+\omega\wedge_{\star}\omega^{ab})-$
$\displaystyle~{}~{}~{}~{}~{}~{}-{i\over
2}\epsilon^{ab}_{~{}~{}cd}(\omega^{cd}\wedge_{\star}\widetilde{\omega}+\widetilde{\omega}\wedge_{\star}\omega^{cd})$
(4.21) $\displaystyle r=d\omega-{i\over
8}\omega^{ab}\wedge_{\star}\omega_{ab}-i(\omega\wedge_{\star}\omega-\widetilde{\omega}\wedge_{\star}\widetilde{\omega})$
$\displaystyle\tilde{r}=d\widetilde{\omega}+{i\over
16}\varepsilon_{abcd}\omega^{ab}\wedge_{\star}\omega^{cd}-i(\omega\wedge_{\star}\widetilde{\omega}+\widetilde{\omega}\wedge_{\star}\omega)$
(4.22)

####  4.2.2 $\star$-gauge variations

The action (LABEL:action2NC) is invariant under the $\star$-gauge
transformations

$\displaystyle\delta_{\varepsilon}V^{a}={1\over 2}(\varepsilon^{a}_{~{}b}\star
V^{b}+V^{b}\star\varepsilon^{a}_{~{}b})+{i\over
4}\epsilon^{a}_{~{}bcd}(\widetilde{V}^{b}\star\varepsilon^{cd}-\varepsilon^{cd}\star\widetilde{V}^{b})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+i(\varepsilon\star
V^{a}-V^{a}\star\varepsilon)-\widetilde{\varepsilon}\star\widetilde{V}^{a}-\widetilde{V}^{a}\star\widetilde{\varepsilon}$
(4.23) $\displaystyle\delta_{\varepsilon}\widetilde{V}^{a}={1\over
2}(\varepsilon^{a}_{~{}b}\star\widetilde{V}^{b}+\widetilde{V}^{b}\star\varepsilon^{a}_{~{}b})+{i\over
4}\epsilon^{a}_{~{}bcd}(V^{b}\star\varepsilon^{cd}-\varepsilon^{cd}\star
V^{b})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+i(\varepsilon\star\widetilde{V}^{a}-\widetilde{V}^{a}\star\varepsilon)-\widetilde{\varepsilon}\star
V^{a}-V^{a}\star\widetilde{\varepsilon}$ (4.24)
$\displaystyle\delta_{\varepsilon}\omega^{ab}=d\varepsilon^{ab}+{1\over
2}(\varepsilon^{a}_{~{}c}\star\omega^{cb}-\varepsilon^{b}_{~{}c}\star\omega^{ca}+\omega^{cb}\star\varepsilon^{a}_{~{}c}-\omega^{ca}\star\varepsilon^{b}_{~{}c})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+i(\varepsilon^{ab}\star\omega-\omega\star\varepsilon^{ab})+{i\over
2}\epsilon^{ab}_{~{}~{}cd}(\varepsilon^{cd}\star\widetilde{\omega}-\widetilde{\omega}\star\varepsilon^{cd})$
(4.25)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+i(\varepsilon\star\omega^{ab}-\omega^{ab}\star\varepsilon)+{i\over
2}\epsilon^{ab}_{~{}~{}cd}(\widetilde{\varepsilon}\star\omega^{cd}-\omega^{cd}\star\widetilde{\varepsilon})$
(4.26) $\displaystyle\delta_{\varepsilon}\omega=d\epsilon-{i\over
8}(\omega^{ab}\star\varepsilon_{ab}-\varepsilon_{ab}\star\omega^{ab})+i(\varepsilon\star\omega-\omega\star\varepsilon-\widetilde{\varepsilon}\star\widetilde{\omega}+\widetilde{\omega}\star\widetilde{\varepsilon})$
(4.27)
$\displaystyle\delta_{\varepsilon}\widetilde{\omega}=d\widetilde{\varepsilon}+{i\over
16}\epsilon_{abcd}(\omega^{ab}\star\varepsilon^{cd}-\varepsilon^{cd}\star\omega^{ab})+i(\varepsilon\star\widetilde{\omega}-\widetilde{\omega}\star\varepsilon+\widetilde{\varepsilon}\star\omega-\omega\star\widetilde{\varepsilon})$
(4.28)

obtained by inserting the expansions (4.14), (4.15) and (4.16) into the
variations (4.17).

##  5 NC Hamiltonian formulation of NC gravity

In this Section we apply the $\star$-Hamiltonian formalism to the NC gravity
action (LABEL:action2NC).

###  5.1 $\star$-Legendre transformation and $\star$-Hamiltonian

We begin by giving the definition of momenta:

$\displaystyle\pi_{a}={\stackrel{{\scriptstyle\rightarrow}}{{\partial}}L\over\partial(dV^{a})}=0$
(5.1)
$\displaystyle{\widetilde{\pi}}_{a}={\stackrel{{\scriptstyle\rightarrow}}{{\partial}}L\over\partial(d\widetilde{V}^{a})}=0$
(5.2)
$\displaystyle\pi_{ab}={\stackrel{{\scriptstyle\rightarrow}}{{\partial}}L\over\partial(d\omega^{ab})}=\epsilon_{abcd}(V^{c}V^{d}-\widetilde{V}^{c}\widetilde{V}^{d})+2i(\widetilde{V}^{[a}V^{b]}-V^{[a}\widetilde{V}^{b]})$
(5.3)
$\displaystyle\pi={\stackrel{{\scriptstyle\rightarrow}}{{\partial}}L\over\partial(d\omega)}=4(\widetilde{V}^{c}V^{c}-V^{c}\widetilde{V}^{c})$
(5.4)
$\displaystyle{\widetilde{\pi}}={\stackrel{{\scriptstyle\rightarrow}}{{\partial}}L\over\partial(d\widetilde{\omega})}=4i(V^{c}V^{c}-\widetilde{V}^{c}\widetilde{V}^{c})$
(5.5)

The $\star$-Hamiltonian is therefore

$\displaystyle
H=dV^{a}\pi_{a}+d\widetilde{V}^{a}{\widetilde{\pi}}_{a}+d\omega^{ab}\pi_{ab}+d\omega~{}\pi+d\widetilde{\omega}~{}{\widetilde{\pi}}-L=$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}dV^{a}\Phi_{a}+d\widetilde{V}^{a}{\widetilde{\Phi}}_{a}+d\omega^{ab}\Phi_{ab}+d\omega~{}\Phi+d\widetilde{\omega}~{}{\widetilde{\Phi}}+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+[\omega^{c[a}\omega^{b]}_{~{}~{}c}+i(\omega^{ab}\omega+\omega\omega^{ab})+{i\over
2}(\omega_{ef}\widetilde{\omega}+\widetilde{\omega}\omega_{ef})\epsilon^{abef}]\cdot$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}\cdot(V^{c}V^{d}-\widetilde{V}^{c}\widetilde{V}^{d})\epsilon_{abcd}+2i(\widetilde{V}_{a}V_{b}-V_{a}\widetilde{V}_{b})]+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+4({i\over
8}\omega^{ab}\omega_{ab}-i\widetilde{\omega}\widetilde{\omega}+i\omega\omega)(\widetilde{V}^{c}V_{c}-V^{c}\widetilde{V}_{c})-$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}-4i({i\over
16}\omega^{ab}\omega^{cd}\epsilon_{abcd}-i\omega\widetilde{\omega}-i\widetilde{\omega}\omega)(V^{c}V_{c}-\widetilde{V}^{c}\widetilde{V}_{c})$
(5.6)

where the constraints $\Phi$ are defined by subtracting the right-hand sides
from the left-hand sides in (5.1) - (5.5).

###  5.2 Construction of $\star$-gauge generators

We apply now the algorithm for the search of symmetry generators. We start
from the first class 2–forms $\pi_{ab}$, $\pi$, ${\widetilde{\pi}}$. They
commute with all the constraints $\Phi$, and are therefore good candidates for
the $F$ part of the gauge generator $\mathbb{G}$ in (3.12). Their Poisson
bracket with $H$ is given by:

$\displaystyle{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}H\over\partial\omega^{ab}}=\pi_{c[a}\omega^{c}_{~{}~{}b]}+\omega_{[b}^{~{}~{}c}\pi_{a]c}+i(\omega\pi_{ab}-\pi_{ab}\omega)+{i\over
2}(\widetilde{\omega}\pi^{cd}-\pi^{cd}\widetilde{\omega})\epsilon_{abcd}+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+{i\over
8}(\omega_{ab}\pi-\pi\omega_{ab})+{i\over
16}({\widetilde{\pi}}\omega^{cd}-\omega^{cd}{\widetilde{\pi}})\epsilon_{abcd}$
(5.7)
$\displaystyle{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}H\over\partial\omega}=i(\omega^{ab}\pi_{ab}-\pi_{ab}\omega^{ab}+\omega\pi-\pi\omega+\widetilde{\omega}{\widetilde{\pi}}-{\widetilde{\pi}}\widetilde{\omega})$
(5.8)
$\displaystyle{\stackrel{{\scriptstyle\rightarrow}}{{\partial}}H\over\partial\widetilde{\omega}}={i\over
2}(\omega_{ab}\pi_{cd}-\pi_{cd}\omega_{ab})\epsilon^{abcd}+i(\omega{\widetilde{\pi}}-{\widetilde{\pi}}\omega-\widetilde{\omega}\pi+\pi\widetilde{\omega})$
(5.9)

Thus, if we take
$d\varepsilon_{ab}\pi^{ab}+d\varepsilon~{}\pi+d\widetilde{\varepsilon}~{}{\widetilde{\pi}}$
as the $F$ part of the symmetry generator $\mathbb{G}$, the second condition
in (3.14) is satisfied when the $G$ part is the sum of the right-hand sides in
the above equations. However we have still to satisfy the first condition in
(3.14). This we achieve by adding to the $G$ part a combination of
constraints. The result is the $\star$–$GL(2,C)$ gauge generator:

$\displaystyle\mathbb{G}=d\varepsilon_{ab}\pi^{ab}+d\varepsilon~{}\pi+d\widetilde{\varepsilon}~{}{\widetilde{\pi}}+$
$\displaystyle~{}~{}~{}+{1\over
2}\varepsilon_{ab}(V^{a}\pi^{b}+\pi^{b}V^{a})+{i\over
4}\varepsilon^{cd}(\widetilde{V}^{b}\pi^{a}-\pi^{a}\widetilde{V}^{b})\epsilon_{abcd}+$
$\displaystyle~{}~{}~{}+{1\over
2}\varepsilon_{ab}(\widetilde{V}^{a}{\widetilde{\pi}}^{b}+{\widetilde{\pi}}^{b}\widetilde{V}^{a})+{i\over
4}\varepsilon^{cd}(V^{b}{\widetilde{\pi}}^{a}-{\widetilde{\pi}}^{a}V^{b})\epsilon_{abcd}-$
$\displaystyle~{}~{}~{}-\varepsilon^{ca}(\omega^{b}_{~{}~{}c}\pi_{ab}+\pi_{ab}~{}\omega^{b}_{~{}~{}c})-i\varepsilon^{ab}(\omega\pi_{ab}-\pi_{ab}\omega)-{i\over
2}\varepsilon_{cd}(\widetilde{\omega}\pi_{ab}-\pi_{ab}\widetilde{\omega})\epsilon^{abcd}-$
$\displaystyle~{}~{}~{}-{i\over
8}\varepsilon_{ab}(\omega^{ab}\pi-\pi\omega^{ab})+{i\over
16}\varepsilon^{cd}(\omega^{ab}{\widetilde{\pi}}-{\widetilde{\pi}}\omega^{ab})\epsilon_{abcd}-$
$\displaystyle~{}~{}~{}-i\varepsilon(V^{a}\pi_{a}-\pi_{a}V^{a})+\widetilde{\varepsilon}(\widetilde{V}^{a}\pi_{a}+\pi_{a}\widetilde{V}^{a})-i\varepsilon(\widetilde{V}^{a}{\widetilde{\pi}}_{a}-{\widetilde{\pi}}_{a}\widetilde{V}^{a})+\widetilde{\varepsilon}(V^{a}{\widetilde{\pi}}_{a}+{\widetilde{\pi}}_{a}V^{a})-$
$\displaystyle~{}~{}~{}-i\varepsilon(\omega^{ab}\pi_{ab}-\pi_{ab}\omega^{ab})-{i\over
2}\widetilde{\varepsilon}(\omega_{ab}\pi_{cd}-\pi_{cd}\omega_{ab})\epsilon^{abcd}$
$\displaystyle~{}~{}~{}-i\varepsilon(\omega\pi-\pi\omega+\widetilde{\omega}{\widetilde{\pi}}-{\widetilde{\pi}}\widetilde{\omega})+i\widetilde{\varepsilon}(\widetilde{\omega}\pi-\pi\widetilde{\omega}-\omega{\widetilde{\pi}}+{\widetilde{\pi}}\omega)$
(5.10)

By simple inspection one can verify that indeed it generates the
$\star$-$GL(2,C)$ symmetry variations listed in (4.23)-(4.28). Moreover, we
can now obtain also the $\star$-$GL(2,C)$ symmetry variations of the momenta:

$\displaystyle\delta_{\varepsilon}\pi_{a}={\stackrel{{\scriptstyle\rightarrow}}{{\partial}}\mathbb{G}\over\partial
V^{a}}={1\over 2}(\pi^{b}\varepsilon_{ab}+\varepsilon_{ab}\pi^{b})-{i\over
4}({\widetilde{\pi}}^{b}\varepsilon^{cd}-\varepsilon^{cd}{\widetilde{\pi}}^{b})\epsilon_{abcd}-$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-i(\pi_{a}\varepsilon-\varepsilon\pi_{a})+{\widetilde{\pi}}_{a}\widetilde{\varepsilon}+\widetilde{\varepsilon}{\widetilde{\pi}}_{a}$
(5.11)
$\displaystyle\delta_{\varepsilon}{\widetilde{\pi}}_{a}={\stackrel{{\scriptstyle\rightarrow}}{{\partial}}\mathbb{G}\over\partial\widetilde{V}^{a}}={1\over
2}({\widetilde{\pi}}^{b}\varepsilon_{ab}+\varepsilon_{ab}{\widetilde{\pi}}^{b})-{i\over
4}(\pi^{b}\varepsilon^{cd}-\varepsilon^{cd}\pi^{b})\epsilon_{abcd}+$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\pi_{a}\widetilde{\varepsilon}+\widetilde{\varepsilon}\pi_{a}-i({\widetilde{\pi}}_{a}\varepsilon-\varepsilon{\widetilde{\pi}}_{a})$
(5.12)
$\displaystyle\delta_{\varepsilon}\pi_{ab}={\stackrel{{\scriptstyle\rightarrow}}{{\partial}}\mathbb{G}\over\partial\omega^{ab}}=-\pi_{c[a}\varepsilon_{b]}^{~{}~{}c}-\varepsilon_{c[b}\pi_{a]}^{~{}~{}c}-{i\over
8}(\pi\varepsilon_{ab}-\varepsilon_{ab}\pi)+{i\over
16}({\widetilde{\pi}}\varepsilon^{cd}-\varepsilon^{cd}{\widetilde{\pi}})\epsilon_{abcd}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-i(\pi_{ab}\varepsilon-\varepsilon\pi_{ab})-{i\over
2}(\pi^{cd}\widetilde{\varepsilon}-\widetilde{\varepsilon}\pi^{cd})\epsilon_{abcd}$
(5.13)
$\displaystyle\delta_{\varepsilon}\pi={\stackrel{{\scriptstyle\rightarrow}}{{\partial}}\mathbb{G}\over\partial\omega}=-i(\pi_{ab}\varepsilon^{ab}-\varepsilon^{ab}\pi_{ab})-i(\pi\varepsilon-\varepsilon\pi+{\widetilde{\pi}}\widetilde{\varepsilon}-\widetilde{\varepsilon}{\widetilde{\pi}})$
(5.14)
$\displaystyle\delta_{\varepsilon}{\widetilde{\pi}}={\stackrel{{\scriptstyle\rightarrow}}{{\partial}}\mathbb{G}\over\partial\widetilde{\omega}}=-{i\over
2}(\pi^{ab}\varepsilon^{cd}-\varepsilon^{cd}\pi^{ab})\epsilon_{abcd}+i(\pi\widetilde{\varepsilon}-\widetilde{\varepsilon}\pi-{\widetilde{\pi}}\varepsilon+\varepsilon{\widetilde{\pi}})$
(5.15)

Finally, using the above variations of the momenta and the variations of the
fields given in subsection 4.2.2, it is straightforward to check that the
$\Phi$ constraints defined in (5.6) are weakly preserved.

##  6 Conclusions

In this paper we have developed a covariant hamiltonian formalism for
noncommutative geometric theories, where noncommutativity is implemented via
an abelian twist. In this canonical framework we have introduced
$\star$-Poisson brackets, and the notion of $\star$-canonical gauge generator,
providing also an algorithm for its construction. This is of relevance when
studying the (local or global) symmetries of a geometric theory like gravity
or supergravity, and allows to generalize Noether theorems to a NC setting.

The NC hamiltonian formalism presented here can be directly applied to
supergravity theories, whose covariant symmetry generators have been
constructed in [15]. The use of noncanonical symplectic structures (see for
ex. [22]) could provide another interesting generalization of the covariant
hamiltonian setting, and of its NC version.

## Acknowledgements

We thank Paolo Aschieri and Chiara Pagani for useful discussions. We
acknowledge partial support from INFN, CSN4, Iniziativa Specifica GSS. This
research has a financial support from Università del Piemonte Orientale.

##  Appendix A Twist differential geometry

The noncommutative deformation of gravity considered in this paper (and
introduced in ref.s [7, 8]) relies on the existence of an associative
$\star$-product between functions and more generally an associative
$\wedge_{\star}$ exterior product between forms, satisfying the following
properties:

$\bullet~{}~{}$ Compatibility with the exterior differential:

$d(\tau\wedge_{\star}\tau^{\prime})=d(\tau)\wedge_{\star}\tau^{\prime}=\tau\wedge_{\star}d\tau^{\prime}$
(A.1)

$\bullet~{}~{}$ Graded cyclicity of the integral:

$\int\tau\wedge_{\star}\tau^{\prime}=(-1)^{deg(\tau)deg(\tau^{\prime})}\int\tau^{\prime}\wedge_{\star}\tau$
(A.2)

with $deg(\tau)+deg(\tau^{\prime})$ = $d$ = dimension of the spacetime
manifold, and where $\tau$ and $\tau^{\prime}$ have compact support (i.e. we
require (A.2) to hold up to boundary terms).

$\bullet~{}~{}$ Compatibility with complex conjugation:

$(\tau\wedge_{\star}\tau^{\prime})^{*}=(-1)^{deg(\tau)deg(\tau^{\prime})}\tau^{\prime*}\wedge_{\star}\tau^{*}$
(A.3)

Following [7] we describe here a class of twists whose associated
$\star$-products have all these properties. This class includes the
Groenewold-Moyal $\star$-product

$f\star
g=\mu\big{\\{}e^{\frac{i}{2}\theta^{\rho\sigma}\partial_{\rho}\otimes\partial_{\sigma}}f\otimes
g\big{\\}},$ (A.4)

where the map $\mu$ is the usual pointwise multiplication: $\mu(f\otimes
g)=fg$, and $\theta^{\rho\sigma}$ is a constant antisymmetric matrix.

Abelian Twists

Let $\Xi$ be the linear space of smooth vector fields on a smooth manifold
$M$, and $U\Xi$ its universal enveloping algebra. A twist ${\cal F}\in
U\Xi\otimes U\Xi$ defines the associative $\star$-product

$\displaystyle f\star g$ $\displaystyle=$ $\displaystyle\mu\big{\\{}{\cal
F}^{-1}f\otimes g\big{\\}}$ (A.5)

where the map $\mu$ is the usual pointwise multiplication: $\mu(f\otimes
g)=fg$. The product associativity relies on the defining properties of the
twist, see for ex. [20].

Explicit examples of twist are provided by the so-called abelian twists:

${\cal F}^{-1}=e^{\frac{i}{2}\theta^{AB}X_{A}\otimes X_{B}}$ (A.6)

where $\\{X_{A}\\}$ is a set of mutually commuting vector fields globally
defined on the manifold, and $\theta^{AB}$ is a constant antisymmetric matrix.
The corresponding $\star$-product is in general position dependent because the
vector fields $X_{A}$ are in general $x$-dependent. In the special case that
there exists a global coordinate system on the manifold we can consider the
vector fields $X_{A}={\partial\over\partial x^{A}}$, corresponding to the
Moyal twist, cf. (A.4):

${\cal
F}^{-1}=e^{\frac{i}{2}\theta^{\rho\sigma}\partial_{\rho}\otimes\partial_{\sigma}}$
(A.7)

Deformed exterior product

For abelian twists (A.6), the deformed exterior product between forms is
defined as

$\displaystyle\tau\wedge_{\star}\tau^{\prime}\equiv\sum_{n=0}^{\infty}\left({i\over
2}\right)^{n}\theta^{A_{1}B_{1}}\cdots\theta^{A_{n}B_{n}}(\ell_{X_{A_{1}}}\cdots\ell_{X_{A_{n}}}\tau)\wedge(\ell_{X_{B_{1}}}\cdots\ell_{X_{B_{n}}}\tau^{\prime})$
$\displaystyle~{}~{}=\tau\wedge\tau^{\prime}+{i\over
2}\theta^{AB}(\ell_{X_{A}}\tau)\wedge(\ell_{X_{B}}\tau^{\prime})+{1\over
2!}{\left(i\over
2\right)^{2}}\theta^{A_{1}B_{1}}\theta^{A_{2}B_{2}}(\ell_{X_{A_{1}}}\ell_{X_{A_{2}}}\tau)\wedge(\ell_{X_{B_{1}}}\ell_{X_{B_{2}}}\tau^{\prime})+\cdots$

where the commuting tangent vectors $X_{A}$ act on forms via the Lie
derivatives ${\ell}_{X_{A}}$. This product is associative, and the above
formula holds also for $\tau$ or $\tau^{\prime}$ being a $0$-form (i.e. a
function).

Exterior derivative

The exterior derivative satisfies the usual (graded) Leibniz rule, since it
commutes with the Lie derivative:

$\displaystyle d(f\star g)=df\star g+f\star dg$ (A.8) $\displaystyle
d(\tau\wedge_{\star}\tau^{\prime})=d\tau\wedge_{\star}\tau^{\prime}+(-1)^{deg(\tau)}~{}\tau\wedge_{\star}d\tau^{\prime}$
(A.9)

Integration: graded cyclicity

If we consider an abelian twist (A.6) given by globally defined commuting
vector fields $X_{A}$, then the usual integral is cyclic under the
$\star$-exterior products of forms, i.e. equation (A.2) holds up to boundary
terms. Indeed:

$\int\tau\wedge_{\star}\tau^{\prime}=\int\tau\wedge\tau^{\prime}=(-1)^{deg(\tau)deg(\tau^{\prime})}\int\tau^{\prime}\wedge\tau=(-1)^{deg(\tau)deg(\tau^{\prime})}\int\tau^{\prime}\wedge_{\star}\tau$
(A.10)

up to boundary terms. For example at first order in $\theta$,

$\int\tau\wedge_{\star}\tau^{\prime}=\int\tau\wedge\tau^{\prime}-{i\over
2}\theta^{AB}\int{\cal L}_{X_{A}}(\tau\wedge{\cal
L}_{X_{B}}\tau^{\prime})=\int\tau\wedge\tau^{\prime}-{i\over 2}\theta^{AB}\int
d{i}_{X_{A}}(\tau\wedge{\cal L}_{X_{B}}\tau^{\prime})$ (A.11)

where we used the Cartan formula ${\cal L}_{X_{A}}=di_{X_{A}}+i_{X_{A}}d$.

Complex conjugation

If we choose real fields $X_{A}$ in the definition of the twist (A.6), it is
immediate to verify that:

$(f\star g)^{*}=g^{*}\star f^{*}$ (A.12)
$(\tau\wedge_{\star}\tau^{\prime})^{*}=(-1)^{deg(\tau)deg(\tau^{\prime})}\tau^{\prime*}\wedge_{\star}\tau^{*}$
(A.13)

since sending $i$ into $-i$ in the twist (A.7) amounts to send $\theta^{AB}$
into $-\theta^{AB}=\theta^{BA}$, i.e. to exchange the order of the factors in
the $\star$-product.

##  Appendix B Gamma matrices in $d=4$

We summarize in this Appendix our gamma matrix conventions in $d=4$.

$\displaystyle\eta_{ab}=(1,-1,-1,-1),~{}~{}~{}\\{\gamma_{a},\gamma_{b}\\}=2\eta_{ab},~{}~{}~{}[\gamma_{a},\gamma_{b}]=2\gamma_{ab},$
(B.1) $\displaystyle\gamma_{5}\equiv
i\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3},~{}~{}~{}\gamma_{5}\gamma_{5}=1,~{}~{}~{}\varepsilon_{0123}=-\varepsilon^{0123}=1,$
(B.2)
$\displaystyle\gamma_{a}^{\dagger}=\gamma_{0}\gamma_{a}\gamma_{0},~{}~{}~{}\gamma_{5}^{\dagger}=\gamma_{5}$
(B.3)
$\displaystyle\gamma_{a}^{T}=-C\gamma_{a}C^{-1},~{}~{}~{}\gamma_{5}^{T}=C\gamma_{5}C^{-1},~{}~{}~{}C^{2}=-1,~{}~{}~{}C^{\dagger}=C^{T}=-C$
(B.4)

###  B.1 Useful identities

$\displaystyle\gamma_{a}\gamma_{b}=\gamma_{ab}+\eta_{ab}$ (B.5)
$\displaystyle\gamma_{ab}\gamma_{5}={i\over 2}\epsilon_{abcd}\gamma^{cd}$
(B.6)
$\displaystyle\gamma_{ab}\gamma_{c}=\eta_{bc}\gamma_{a}-\eta_{ac}\gamma_{b}-i\varepsilon_{abcd}\gamma_{5}\gamma^{d}$
(B.7)
$\displaystyle\gamma_{c}\gamma_{ab}=\eta_{ac}\gamma_{b}-\eta_{bc}\gamma_{a}-i\varepsilon_{abcd}\gamma_{5}\gamma^{d}$
(B.8)
$\displaystyle\gamma_{a}\gamma_{b}\gamma_{c}=\eta_{ab}\gamma_{c}+\eta_{bc}\gamma_{a}-\eta_{ac}\gamma_{b}-i\varepsilon_{abcd}\gamma_{5}\gamma^{d}$
(B.9)
$\displaystyle\gamma^{ab}\gamma_{cd}=-i\varepsilon^{ab}_{~{}~{}cd}\gamma_{5}-4\delta^{[a}_{[c}\gamma^{b]}_{~{}~{}d]}-2\delta^{ab}_{cd}$
(B.10) $\displaystyle
Tr(\gamma_{a}\gamma^{bc}\gamma_{d})=8~{}\delta^{bc}_{ad}$ (B.11)
$\displaystyle
Tr(\gamma_{5}\gamma_{a}\gamma_{bc}\gamma_{d})=-4~{}\varepsilon_{abcd}$ (B.12)
$\displaystyle
Tr(\gamma^{rs}\gamma_{a}\gamma_{bc}\gamma_{d})=4(-2\delta^{rs}_{cd}\eta_{ab}+2\delta^{rs}_{bd}\eta_{ac}-3!\delta^{rse}_{abc}\eta_{ed})$
(B.13) $\displaystyle
Tr(\gamma_{5}\gamma^{rs}\gamma_{a}\gamma_{bc}\gamma_{d})=4(-i\eta_{ab}\varepsilon^{rs}_{~{}~{}cd}+i\eta_{ac}\varepsilon^{rs}_{~{}~{}bd}+2i\varepsilon_{abc}^{~{}~{}~{}e}\delta^{rs}_{ed})$
(B.14)

where
$\delta^{ab}_{cd}\equiv\frac{1}{2}(\delta^{a}_{c}\delta^{b}_{d}-\delta^{b}_{c}\delta^{a}_{d})$,
$\delta^{rse}_{abc}\equiv{1\over
3!}(\delta^{r}_{a}\delta^{s}_{b}\delta^{e}_{c}$ \+ 5 terms), and indices
antisymmetrization in square brackets has total weight $1$.

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