Since in the renormalization of the restricted functional integral no higher derivative terms will be
generated, it suffices to consider 4D gravity actions with second derivatives only. Specifically we consider 4D
Einstein gravity coupled to Abelian gauge fields and
scalars in a way they typically arise from
higher-dimensional (super-)gravity theories. We largely follow the treatment in [45
, 46
]. The
higher-dimensional origin of their 3D reductions is explored in [56
]. The 4D action is of the form
In brief this is done as follows. The field equation for the gauge fields can be
interpreted as the Bianchi identity for a field strength
derived from
dual potentials
. For later convenience one chooses
with some
constant orthogonal matrix
. In view of
they satisfy the linear relation
Concerning the reduction, we consider here only the case when both Killing vector fields are
spacelike everywhere and commuting. The other signature (one timelike and one spacelike Killing vector
field) is most efficiently treated by relating it to the spacelike case via an Abelian T-duality transformation
(see [154
]). Alternatively one can perform the reduction in two steps and perform a suitable Hodge
dualization in-between (see [45
, 56
]).
Thus, from here on we take ,
, to be Killing vector fields on the Lorentzian
manifold
that are spacelike everywhere and commuting:
and
. Their
Lorentzian norms and inner product form a symmetric
matrix
. For the
resulting three scalar fields on the 4D spacetime it is convenient to adopt a lapse-shift type
parameterization. This gives
We deliberately refrained from picking coordinates so far to emphasize the geometric nature of the
reduction. As usual however the choice of adapted coordinates is advantageous. We now pick
(“Killing”) coordinates in which acts as
, for
. In these coordinates
the components of
and
are independent of
and
and thus are functions of
the remaining (nonunique “non-Killing”) coordinates
and
only. We write
,
, for the components of
in such a coordinate system. Since both Killing vectors are
spacelike,
has eigenvalues
and can be brought into the form
, by
a change of the non-Killing coordinates, where
is the metric of flat
-dimensional
Minkowski space. This can be taken to define
. Alternatively one can introduce
by
The matter content in Equation (3.1) consists of the
Abelian gauge fields and the sigma-model
scalars
,
. For the scalars the reduction is trivial, and simply amounts to
considering configurations constant in the Killing coordinates. For the gauge fields it turns out that
the
components of
,
, give rise to
fields
,
,
which transform as scalars under a change of the non-Killing coordinates
. In brief this
comes about as follows. The field equation
for the gauge fields in
Equation (3.1
) can be interpreted as the Bianchi identity for a field strength
derived from dual potentials
,
. We can take one of the Killing vectors, say
, and built
spacetime scalars by contraction
and
.
Reduction with respect to the other Killing vector
just requires that these scalars are
constant in the corresponding Killing coordinate
. The dependence on
is constrained by
gauge invariance. If
and
, the scalars change by a term
and
, respectively, and hence are invariant under
independent gauge
transformations. Thus, if the 4D gauge potentials and their duals, together with the corresponding
transformations are taken to be independent of
, a set of gauge invariant scalars
and
arises. As a matter of fact a constant remnant of the gauge transformations remains
and gives rise to a symmetry of the reduced system (see the discussion after Equation (3.11
)
below). We arrange the
fields
,
in a column vector
,
. For
convenience we summarize the field content of the 2-Killing vector subsector of Equation (3.1
) in
Table 1.
|
We combine all but and
into an
components scalar field
We briefly digress on the isometries of Equation (3.11). By virtue of the
invariance of the action
has
Killing vectors of which
are algebraically independent. Interestingly, the
action (3.10
) is also invariant under
,
, with a constant
column
. These symmetries can be viewed as residual gauge transformations; note however that a
compensating transformation of the gravitational potential
is needed. Finally constant
translations in
and scale transformations
, are obvious
symmetries of the action. The associated Killing vectors
of
generate a Borel subalgebra
of
, i.e.
. Together the metric (3.11
) always has
Killing
vectors.
In contrast the last generator
is only a Killing vector of
under certain conditions on
. If these are satisfied a remarkable ‘symmetry enhancement’ takes place in that
is the metric of
a much larger symmetric space
, where
is a non-compact real form of a simple Lie group with
. The point is that if
exists as Killing vector its commutator with
the gauge transformations is nontrivial and yields
additional symmetries (generalized “Harrison
transformations”). Since
always has
Killing vectors, the additional
then
match the dimension of
. For the number of dependent Killing vectors, i.e. the dimension of the
putative maximal subgroup
one expects
. Indeed under the
conditions stated the symmetric space
exists and is uniquely determined by
(and
the signature of the Killing vectors). See [45
] for a complete list. Evidently the gauge fields
are crucial for the symmetry enhancement. Among the systems in [45] only pure gravity has
.
From now on we restrict attention to the cases where such a symmetry enhancement takes place. The
scalars can then be arranged into a coset nonlinear sigma-model whose
-dimensional
target space is of the form
. Here
is always a simple noncompact Lie group and
a maximal
subgroup; the coset is a Riemannian space with metric
. Being the metric of a symmetric space
enjoys the properties
The reduced classical field theories (3.10) have some remarkable properties which we discuss
now:
Taken together these properties make the 2-Killing vector reductions a compelling laboratory to study the quantum aspect of the gravitational field.
For later reference we also briefly outline the Hamiltonian formulation of the system. In a Hamiltonian
formulation of this two-dimensional diffeomorphism invariant system one fixes at the expense
of a Hamiltonian constraint
and a diffeomorphism constraint
. The properly normalized
constraints come out of a lapse and shift decomposition of the form
Performing a standard Hamiltonian (“ADM type”) analysis based on Equation (3.10) and
Equation (3.14
), using the formulas of Section 3.2.3 one finds
A shortcut to arrive at the constraints (3.15) is to start from the Lagrangian (3.10
) in conformal gauge
(
,
, using
),
For the computation of the Poisson algebra it is convenient to put on-shell throughout (as its
equation of motion
is trivially solved) and to interpret the improvement terms in
Equation (3.15
) (here, those linear in the canonical variables) such that second time derivatives are
eliminated. As expected, the
generates infinitesimal spatial reparameterizations and the covariance of
the fields is a merely kinematical property. Explicitly a spatial density
of weight
transforms as
, the right-hand-side being the infinitesimal version of
, under
. The canonical momenta
,
,
are spatial densities of weight
, while
,
, and
are densities of weight 2.
The Hamiltonian constraint on the other hand resumes its usual kinematical-dynamical double
role.
The advantage of having the constraints defined with respect to the densitized lapse and shift
functions (3.14) is that the Poisson algebra generated by
and
is a Lie algebra on-shell, and
equivalent to the algebra of 2D conformal transformations. (Otherwise it yields the algebra of “surface
deformations”; cf. [214
]). To illustrate the difference let us note that with only the
equation of motion
imposed one computes
Here we collect some useful formulas for 2D gravity theories in a lapse/shift parameterization of the metric,
taken from [156]. As a byproduct we obtain a closed expression for the current
of the Euler
density
expressed in terms of the metric only. See [62
] for a discussion.
Our curvature conventions are the ones used throughout, the metric
has eigenvalues
.
In 2D a ‘densitized’ lapse-shift parameterization is convenient (see e.g. [197]),
This provides an explicit though noncovariant expression for the current in terms of the metric.
Related formulas either invoke the zweibein or use an explicit parameterization. The one given in [62
] is
based on an
type parameterization of
and is equivalent to Equation (3.22
).
Compared to Equation (2.11) in [62] a curl term
has been added which allows one
to express
solely in terms of the metric. Another advantage of Equation (3.22
) is that
the separation in dynamical and nondynamical variables is manifest:
is a function of
and the combination
only; the former is the dynamical variable, the latter
can be parameterized in terms of the lapse and shift functions. They can be anticipated to be
nondynamical in that no time derivatives of lapse and shift appear in
, as is manifest from
Equation (3.21
).
In the actions considered the term always multiplies a scalar field
, say. Using
Equation (3.20
) the Hamiltonian associated with an Lagrangian of the form
The Poisson algebra of the constraints ,
is the algebra of surface deformations, as required.
Alternatively the form (3.22
) can be used beforehand to get
Later on we aim at a Dirac quantization of the 2-Killing subsector of the theories (3.1). The
functional measure is then defined with the reduced Lagrangian (3.16
) in conformal gauge, and a
quantum version of the constraints
is imposed subsequently. For the renormalization
the symmetries of the Lagrangian (3.16
) are crucial. The
invariance of course gives rise
to a set of Lie algebra valued Noether currents
. Let
,
, denote a
basis of the Lie algebra with Killing form
. Let further
denote the
Killing vectors of
. Then we define
through its projection onto the basis
, via
More interestingly there are two ‘conformal currents’ which are not conserved on-shell but whose
divergence reproduces the Lagrangian (3.16) up to a multiple
Finally there exists an infinite set of nonlocal conserved currents whose charges are Dirac observables
and which be constructed explicitly(!) in terms of the dynamical fields, that is, without having to solve the
field equations. These currents can be found by different techniques similar to those used in nonlinear
sigma-models [150, 124]. For illustration we present the lowest one which is (for all on-shell configurations)
defined in terms of the dual potentials and
, with
. Then
It may be worthwhile to point out what is trivial and what is nontrivial about the relations (3.29). Once
the expression for the current
is known it is trivial to verify its conservation using the definition of the
potentials
and
. Since
generates time translations on the basis fields
the associated
conserved charge Poisson commutes with
(and trivially with
) and thus qualifies as a genuine
Dirac observable. What is nontrivial about Equation (3.29
) is that a Dirac observable can be constructed
explicitly in a way that does not require a solution of the Cauchy problem. The potentials
,
are only
defined on-shell but one does not need to know how they are parameterized by initial data. In stark contrast
the known abstract construction principles for Dirac observables in full general relativity always
refer to a solution of the Cauchy problem (see [70] for a recent account). The bonus feature of
the 2-Killing vector reduction that allows for this feat is the existence of a solution generating
group [90, 91] (“Geroch group”) and, related to it, the existence of a Lax pair. The latter allows one to
convert the Cauchy problem into a linear singular integral equation [4, 103] (which is still
nontrivial to solve) and at the same time it underlies the techniques used to find an infinite set of
nonlocal conserved currents of which the one in Equation (3.29
) is the lowest (least nonlinear)
one.
In the quantum theory, a construction of observables from first principles has not yet been achieved.
Existence of a quantum counterpart of the first charge (3.29) would already be a very nontrivial indication
for the quantum integrability of the systems. For its construction the procedure of Lüscher [139] could be
adopted. Independent of this, the ‘exact’ bootstrap formulation of [158] shows that the existence of a
‘complete’ set of quantum obeservables is compatible with the quantum integrability of the
system.
![]() |
http://www.livingreviews.org/lrr-2006-5 |
© Max Planck Society and the author(s)
Problems/comments to |