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 . 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) one sees that these conditions are satisfied if and , , where is the involution whose set of fixed points defines . Clearly this restricts the allowed cosets . For the admissible ones then determines the couplings , as functions of the scalars. Since is faithful the determination is unique for a given choice of section . Since the result does not dependent on the choice of section, i.e. for some -valued function determines the same and . We refrain from presenting more details here, which can be found in [45, 46], since the result of the 2-Killing vector reduction can be understood without them.
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 ). 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[90, 91] that there exists a one-to-one correspondence between tensor fields on the “orbit space” and tensor fields on with vanishing Lie derivative along and which are “completely orthogonal” to . This will be used for the matter fields below. Given subject to Equation (3.5) and one can reconstruct the original metric tensor as
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[45, 56]) gives the form of the action (3.1) on the field configurations compatible with the two Killing vectors. The result is
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  for a complete list. Evidently the gauge fields are crucial for the symmetry enhancement. Among the systems in  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). Note also that a constant field trivializes the system in that implies , in which case the only solutions of the constraints are , . Then is harmonic function and can be set to unity by a conformal transformation. In other words, for the only solutions of the classical field equations is Minkowski space with constant matter fields.
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. ). 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 . As a byproduct we obtain a closed expression for the current of the Euler density expressed in terms of the metric only. See  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. ),[214, 197]). Using
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  is based on an type parameterization of and is equivalent to Equation (3.22). Compared to Equation (2.11) in  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  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  could be adopted. Independent of this, the ‘exact’ bootstrap formulation of  shows that the existence of a ‘complete’ set of quantum obeservables is compatible with the quantum integrability of the system.
© Max Planck Society and the author(s)