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3.1 2 + 2 truncation of Einstein gravity + matter

In accordance with the general picture the renormalization flow will also dictate here to a certain extent the form of the renormalized actions. As mentioned the truncated 2 + 2 functional integral turns out to inherit the lack of perturbative renormalizability (with finitely many couplings) from the gravitational part of the full functional integral. However the restricted functional integral is more benign insofar as it is possible to preserve the conformal geometry in field space and insofar as no higher derivative terms are required for the absorption of cutoff dependencies. The strategy is similar as in the perturbative treatment of the full theory advocated by Gomis and Weinberg [94]: One works with a propagator free of unphysical poles and takes into account all counter terms enforced, but only those. (For the reasons explained in Section 2.3.2 we deliberatly avoid using the ‘relevant/irrelevant’ terminology here.) To emphasize the fact that no higher derivative terms are needed we shall refer to the quantum theory defined that way as the symmetry truncation of Quantum Einstein Gravity. We anticipate this fact in the following by taking a classical gravity + matter action as a starting point which is quadratic in the derivatives only (see Equation (3.1View Equation) below).

3.1.1 Gravity theories

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 k Abelian gauge fields and ¯n scalars in a way they typically arise from higher-dimensional (super-)gravity theories. We largely follow the treatment in [45Jump To The Next Citation Point46Jump To The Next Citation Point]. The higher-dimensional origin of their 3D reductions is explored in [56Jump To The Next Citation Point]. The 4D action is of the form

∫ [ ] 4 √--- 1--α -- q- T αβ ∗ αβ S4 = d x − g R (g) − 2⟨J ,J α⟩¯g − 4F αβ(μF − ν F ) . (3.1 )
Here gαβ,1 ≤ α,β ≤ 4, is the spacetime metric with eigenvalues (− ,+, +, + ), R (g) is its scalar curvature, and indices are raised with gαβ. There are k real Abelian vector fields arranged in a column ˆı B α = (Bα ), ˆı = 1,...,k, with field strength Fαβ = ∂αB β − ∂βB α and dual field strength ∗Fαβ = 21√−gεαβγδFγδ. The scalars -- ϕi, -- i = 1,...,n, parameterize a non-compact Riemannian symmetric space G-∕H-- with metric 𝔪- (ϕ-) ij. Its dim G- Killing vectors give rise to a Lie algebra valued Noether current -- J α. In terms of them the sigma-model Lagrangian for the scalars can be written as --α -- ⟨J ,Jα⟩¯g, where ⟨ ⋅, ⋅⟩¯g is an invariant scalar product on the Lie algebra ¯g. Finally the coupling matrices -- μ = μ(ϕ ) and -- ν = ν (ϕ ) are symmetric k × k matrices that depend on the scalars; the constant q > 0 has been extracted for normalization purposes. The vector fields are supposed to contribute positively to the energy density which requires that μ is a positive definite matrix. As such it has a unique positive square root 1∕2 μ to be used later. The coupling matrices μ and ν are now chosen in a way that renders the field equations derived from S4 – though in general not the action itself – -- G-invariant.

In brief this is done as follows. The field equation for the gauge fields ∇ (μF αβ − ν ∗F αβ) = 0 α can be interpreted as the Bianchi identity for a field strength G αβ = ∂αC β − ∂βC α derived from dual potentials Cα. For later convenience one chooses ∗G αβ = η(μF αβ − ν∗Fαβ) with some constant orthogonal matrix η ∈ O (k ). In view of ∗∗F = − F they satisfy the linear relation

(F ) ( ∗F ) ( μ1∕2 νμ− 1∕2) ( 0 ηT ) = Υ 𝒱c𝒱Tc ∗ with 𝒱c = − 1∕2 , Υ = , (3.2 ) G G 0 ημ − η 0
where the subscript c is mnemonic for ‘coupling’. If one now assumes that the column (F) G transforms linearly under a faithful 2k-dimensional real matrix representation c of -- G, i.e. (F) −1 T(F) G ↦→ c(¯g ) G, -- ¯g ∈ G, one finds that Equation (3.2View Equation) transforms covariantly if 𝒱c ↦→ c(¯g)𝒱chc, with an orthogonal matrix hc and c(¯g−1)T = Υc (¯g)Υ −1. Comparing this with the transformation law of the -- G-valued coset representatives 𝒱 ∗ (see e.g. [237Jump To The Next Citation Point]) one sees that these conditions are satisfied if c(𝒱 ) = 𝒱 ∗ c and −1 T c(¯τ(¯g)) = c(¯g ), -- ¯g ∈ G, where ¯τ is the involution whose set of fixed points defines --- H. Clearly this restricts the allowed cosets -- --- G∕ H. For the admissible ones c(𝒱∗) = 𝒱c then determines the couplings -- μ (ϕ), -- ν(ϕ) as functions of the scalars. Since c is faithful the determination is unique for a given choice of section 𝒱∗. Since 𝒱c𝒱Tc = c(𝒱∗¯τ(𝒱−∗ 1)) the result does not dependent on the choice of section, i.e. ^ 𝒱∗ = 𝒱∗h for some H-valued function h determines the same μ(ϕ-) and ν(ϕ). We refrain from presenting more details here, which can be found in [45Jump To The Next Citation Point46], since the result of the 2-Killing vector reduction can be understood without them.

3.1.2 2-Killing vector reduction

Concerning the reduction, we consider here only the case when both Killing vector fields K1,K2 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 [154Jump To The Next Citation Point]). Alternatively one can perform the reduction in two steps and perform a suitable Hodge dualization in-between (see [45Jump To The Next Citation Point56Jump To The Next Citation Point]).

Thus, from here on we take Ka = K α∂α a, a = 1, 2, to be Killing vector fields on the Lorentzian manifold (ℳ, g) that are spacelike everywhere and commuting: ℒKag αβ = 0 and [K1, K2 ] = 0. Their Lorentzian norms and inner product form a symmetric 2 × 2 matrix M = (Mab )1≤a,b≤2. For the resulting three scalar fields on the 4D spacetime it is convenient to adopt a lapse-shift type parameterization. This gives

( ) α β ρ Δ2 + ψ2 ψ Mab := gαβK a K b = Ka ⋅ Kb, M =: Δ- ψ 1 . (3.3 )
In the general relativity literature the fields Δ > 0 and ψ are known as the real and imaginary parts of the “Ernst potential”. The parameterization (3.3View Equation) is chosen such that the (non-negative) “area element” detM is the square of one of the fields. Taking the positive square root one has
ρ := ∘K----⋅ K-K--⋅ K--−-(K--⋅ K-)2 ≥ 0. (3.4 ) 1 1 2 2 1 2
By definition the metric g is left unchanged along the flow lines of the Killing vector fields. We denote the space of orbits by Σ. A projection operator onto the (co-)tangent space to each point in Σ is given by γ β := δ β − M abK K β α α aα b, where M ab are the components of M −1 and β Ka α := gαβK a. The associated Lorentzian metric α′ β′ ab γαβ := γ α γ β gα′β′ = g αβ − M Ka αKb β satisfies
α ℒKa γαβ = 0, K aγαβ = 0. (3.5 )
Since γ β α is a projector of rank two, the metric γ αβ has three independent components (not accounting for diffeomorphism redundancies). Generally one can show [90Jump To The Next Citation Point91Jump To The Next Citation Point] that there exists a one-to-one correspondence between tensor fields on the “orbit space” (Σ, γ) and tensor fields on (ℳ, g) with vanishing Lie derivative along Ka α and which are “completely orthogonal” to Kaα. This will be used for the matter fields below. Given γ subject to Equation (3.5View Equation) and M one can reconstruct the original metric tensor as
ab gαβ = γαβ + M Ka αKb β. (3.6 )
The 10 components are parameterized by the 3 + 3 independent functions in γ and M. Each of these functions is constant along the flow lines of the Killing vector fields but may vary arbitrarily within Σ.

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 Ka acts as a ∂∕ ∂y, for a = 1,2. In these coordinates the components of γ and M are independent of y1 and y2 and thus are functions of the remaining (nonunique “non-Killing”) coordinates x0 and x1 only. We write γμν(x), μ = 0,1, for the components of γ in such a coordinate system. Since both Killing vectors are spacelike, γμν has eigenvalues (− ,+) and can be brought into the form σ(x) γμν(x) = e ημν, by a change of the non-Killing coordinates, where η is the metric of flat 1 + 1-dimensional Minkowski space. This can be taken to define σ. Alternatively one can introduce σ by

∂ρ ⋅ ∂ρ = Δ(x )γ μν(x)∂μρ ∂νρ = Δ (x)e−σ(x)∂μρ ∂μρ. (3.7 )
On the left-hand-side is the (coordinate-independent) Lorentzian norm of the gradient of the 4D scalar field ρ; on the right-hand-side we evaluated this norm in the Killing coordinates where it must be proportional to γμν(x) ∂μρ∂ νρ. Adjusting also the non-Killing coordinates then gives the rightmost expression in Equation (3.7View Equation), which could also be taken to define σ. The upshot is that the most general 4D metric with two commuting Killing vectors is parameterized by four scalar fields, ρ,σ and Δ, ψ. In the adapted coordinates the 4D metric then reads
dS2 = eσ[− (dx0 )2 + (dx1)2] + ρ-(dy1 + ψdy2 )2 + ρΔ (dy2)2. (3.8 ) Δ
As already mentioned, the fields Δ and ψ are known as the real and the imaginary part of the “Ernst potential”; we shall refer to ρ as the “area radius” associated with the two Killing vectors, and to σ as the conformal factor. To motivate the latter note that a Weyl transformation gα β → eωgαβ of the 4D metric compatible with the 2-Killing vectors amounts to the simultaneous rescalings γ (x) → eω(x)γ (x) μν μν and ω(x) ρ(x) → e ρ(x).

The matter content in Equation (3.1View Equation) consists of the k Abelian gauge fields and the sigma-model scalars ϕi, i = 1,...,¯n. 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 4k components of ˆı Bα, ˆı = 1,...,k, give rise to 2k fields I A, I = 1,...,2k, which transform as scalars under a change of the non-Killing coordinates 0 1 (x ,x ). In brief this comes about as follows. The field equation ∇ α(μF αβ − ν ∗F αβ) = 0 for the gauge fields in Equation (3.1View Equation) can be interpreted as the Bianchi identity for a field strength G αβ = ∂αC β − ∂βC α derived from dual potentials C ˆı α, ˆı = 1,...,k. We can take one of the Killing vectors, say K = K1, and built 2k spacetime scalars by contraction ˆı : ˆı α B = B αK and ˆı : ˆı α C = C αK. Reduction with respect to the other Killing vector K2 just requires that these scalars are constant in the corresponding Killing coordinate y2. The dependence on y1 is constrained by gauge invariance. If B ˆıα ↦→ Bˆıα + ∂αbˆı and Ciα ↦→ Ciα + ∂αci, the scalars change by a term K α∂ bˆı α and K α∂ cˆı α, respectively, and hence are invariant under y1 independent gauge transformations. Thus, if the 4D gauge potentials and their duals, together with the corresponding transformations are taken to be independent of 1 2 y ,y, a set of gauge invariant scalars ˆı B (x) and C ˆı(x) 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 (3.11View Equation) below). We arrange the 2k fields B ˆı, C ˆı in a column vector AI, I = 1,..., 2k. For convenience we summarize the field content of the 2-Killing vector subsector of Equation (3.1View Equation) in Table 1.






4D fields fields in 2-Killing subsector


gαβ metric Δ, ψ,ρ, σ
B ˆıα Abelian gauge fields AI, I = 1,...,2k
--i ϕ KK scalars -i ϕ , i = 1,...¯n





Table 1: Field content of the 2-Killing vector subsector of the gravity theories 3.1View Equation.

We combine all but ρ and σ into an n := 2 + n¯+ 2k components scalar field

-- -- ϕ = (Δ,ψ, ϕ1,...,ϕ ¯n,A1,...,A2k ) (3.9 )
on the 2D orbit space with metric γμν and coordinates 0 1 (x ,x ). A lengthy computation (which is best done in a two step procedure; see [45Jump To The Next Citation Point56]) gives the form of the action (3.1View Equation) on the field configurations compatible with the two Killing vectors. The result is
∫ -1- 2 √ -- μν −2 μν i j S = 2λ d x ρ γ[2R(γ ) + γ ρ ∂μρ ∂νρ − γ 𝔪ij(ϕ )∂μϕ ∂νϕ ]. (3.10 )
Here λ is the gravitational constant per unit volume of the internal space. One sees that the reduced action has the form of a 2D nonlinear sigma-model non-minimally coupled to 2D gravity via the area radius ρ of the two Killing vectors. The target space of the sigma-model has dimension n = 2 + n¯+ 2k, we take Equation (3.9View Equation), viewed as a column, as field coordinates. With the normalization -- -- -- -- -i --j ⟨J ˆα,Jˆβ⟩¯g = 𝔪ij(ϕ )∂ ˆαϕ ∂ˆβϕ the metric then comes out as
( | | ) -1- 0 | | 0 | Δ2 | | q | || 0 Δ12- | | − 2Δ2 AT Υ || 𝔪 (ϕ) = | ---------------|--------|--------------------------| . (3.11 ) |( ---------------|-𝔪-(ϕ-)--|--------------------------|) 0 -q-ΥA | | q𝒱 𝒱T − -q2ΥA ⊗ AT Υ 2Δ2 Δ c c 4Δ2
Here Υ and 𝒱c are as in Equation (3.2View Equation) and q > 0 is a parameter used to adjust normalizations. The metric (3.11View Equation) has Riemannian signature (if the reduction was performed with respect to one spacelike and one timelike Killing vector it had 2k negative eigenvalues).

We briefly digress on the isometries of Equation (3.11View Equation). By virtue of the -- G invariance of the action 𝔪 has -- dim G Killing vectors of which -- --- n-= dim G ∕H are algebraically independent. Interestingly, the action (3.10View Equation) is also invariant under A ↦→ A + a, ψ ↦→ ψ − qAT Υa 2, with a constant 2k column a. These symmetries can be viewed as residual gauge transformations; note however that a compensating transformation of the gravitational potential ψ is needed. Finally constant translations in ψ and scale transformations -- -- (Δ, ψ, ϕ,A ) ↦→ (sΔ, sψ, ϕ,s1∕2A),s > 0, are obvious symmetries of the action. The associated Killing vectors e,h of 𝔪 generate a Borel subalgebra of sl 2, i.e. [h,e] = − 2e. Together the metric (3.11View Equation) always has dim G-+ 2k + 2 Killing vectors.

In contrast the last sl2 generator f is only a Killing vector of 𝔪 under certain conditions on -- --- G ∕H. If these are satisfied a remarkable ‘symmetry enhancement’ takes place in that 𝔪 is the metric of a much larger symmetric space G ∕H, where G is a non-compact real form of a simple Lie group with -- dim G = dim G + 4k + dim SL (2). The point is that if f exists as Killing vector its commutator with the gauge transformations is nontrivial and yields 2k additional symmetries (generalized “Harrison transformations”). Since 𝔪 always has -- dim G + 2k + 2 Killing vectors, the additional 1 + 2k then match the dimension of G. For the number of dependent Killing vectors, i.e. the dimension of the putative maximal subgroup H ⊂ G one expects dim H = dim H--+ 1 + 2k. Indeed under the conditions stated the symmetric space G ∕H exists and is uniquely determined by -- --- G ∕H (and the signature of the Killing vectors). See [45Jump To The Next Citation Point] for a complete list. Evidently the gauge fields are crucial for the symmetry enhancement. Among the systems in [45] only pure gravity has k = 0.

From now on we restrict attention to the cases where such a symmetry enhancement takes place. The scalars ϕi can then be arranged into a coset nonlinear sigma-model whose 2k + ¯n + 2-dimensional target space is of the form G∕H. Here G is always a simple noncompact Lie group and H a maximal subgroup; the coset is a Riemannian space with metric 𝔪ij (ϕ ). Being the metric of a symmetric space 𝔪 enjoys the properties

∇mRijkl (𝔪 ) = 0, Rij(𝔪) = ζ1𝔪ij, (3.12 )
which will be important later on. By construction G ∕H contains as subcosets the space SL (2,ℝ )∕SO (2) on which the gravitational potentials are coordinates and the original coset ----- G∕H of the scalars -- ϕ:
SL (2,ℝ)∕SO (2) ⊂ G ∕H ⊃ G-∕H.- (3.13 )
A brief list of examples of 4D theories (3.1View Equation) and the cosets G ∕H they give rise to in Equation (3.10View Equation) is: Pure gravity in 4D corresponds to SL(2,ℝ )∕SO (2), Einstein–Maxwell theory gives rise to SU (2,1)∕S [U (2) × U(1)], a 4D Einstein–dilaton-axion theory gives a coset SO (3,2 )∕SO (3 ) × SO (2), and the reduction of N = 8 supergravity leads to a bosonic sector with E8(+8)∕SO (16 ) coset.

The reduced classical field theories (3.10View Equation) have some remarkable properties which we discuss now:

  1. First, the field equations and the symplectic structure derived from Equation (3.11View Equation) coincide with the restriction of the field equations and the symplectic structure derived from Equation (3.1View Equation). In fact, this is a general feature of this type of symmetry reductions, and can be understood in terms of the “principle of symmetric criticality” [76].
  2. The field equations are classical integrable in the sense that they can be written as the compatibility condition of a pair of first order matrix-valued differential operators, depending on a free complex parameter. As a consequence large classes of solutions can be constructed analytically (see the books [2612198] and [103Jump To The Next Citation Point4Jump To The Next Citation Point] for detailed expositions).
  3. The classical integrability also entails that an infinite number of nonlocal conserved charges can be constructed explicitly. These are generalizations of the Lüscher–Pohlmayer charges for the O (N ) model. Moreover, these charges Poisson commute with the Hamiltonian and the diffeomorphism constraint that arise in a Hamiltonian (“Arnowitt–Deser–Misner-type”) analysis of the covariant system (3.11View Equation). In other words, an infinite system of Dirac observables can be constructed explicitly as functionals of the metric and the matter fields! Given the fact that (apart from mass and angular momentum) not a single Dirac observable is known explicitly in full general relativity, this is a most remarkable feature.
  4. The system captures the crucial “spin-two” aspect of gravity. For example without matter the classical solutions comprise various types of (nonlinear) gravitational waves with two independent polarizations (per spacetime point).
  5. In conformal gauge, γ (x) = eσ(x)η μν μν, the curvature term in Equation (3.11View Equation) is proportional to μ ∂ ρ∂μσ. Upon diagonalization, μ μ ∂ (ρ + σ)∂ μ(ρ + σ ) − ∂ (ρ − σ)∂μ(ρ − σ), this is proportional to a sum of two standard kinetic terms, one of which invariably has the ‘wrong sign’. Since a Weyl rescaling gαβ ↦→ eω(x)gαβ of the metrics (3.3View Equation) amounts to ρ ↦→ eωρ, σ ↦→ σ + ω this appears to reflect a conformal factor instability of the 4D gravitational action(s). Upon closer inspection it signals the absence of a genuine instability (see Section 3.2).

Taken together these properties make the 2-Killing vector reductions a compelling laboratory to study the quantum aspect of the gravitational field.

3.1.3 Hamiltonian formulation

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 γμν = eσημν at the expense of a Hamiltonian constraint ℋ 0 and a diffeomorphism constraint ℋ 1. The properly normalized constraints come out of a lapse and shift decomposition of the form

( − n2 + s2 s) γμν = eσ . (3.14 ) s 1
The lapse n and the shift s here are spatial densities of weight − 1, while eσ carries weight 2. In Section 3.1.4 we collect some useful formulas which allows one to streamline the Hamiltonian analysis of 2D gravity theories.

Performing a standard Hamiltonian (“ADM type”) analysis based on Equation (3.10View Equation) and Equation (3.14View Equation), using the formulas of Section 3.2.3 one finds

ℋ = T = 1(T + T ) 0 00 2 ++ −− λ ij ρ i j 1 2 2 = --𝔪 (ϕ )πi(ϕ)πj(ϕ) + ---𝔪ij(ϕ)∂1ϕ ∂1 ϕ + λπρπσ + --∂1σ∂1ρ − -∂1ρ, 2ρ 2λ λ λ (3.15 ) 1- ℋ1 = T10 = 2(T++ − T−− ) i = πi(ϕ)∂1ϕ + πσ∂1σ + π ρ∂1ρ − 2∂1πσ,
with a self-explanatory notation for the canonical momenta. The linear combinations T±± are introduced for later use. The action (3.10View Equation) can then be recast in Hamiltonian form, where the Hamiltonian is, up to possible boundary terms, given by ∫ dx (nℋ0 + sℋ1 ). If boundary terms are present they are separately conserved local charges (see [197Jump To The Next Citation Point]). Note also that a constant ρ field trivializes the system in that ρ = const implies πσ = 0, in which case the only solutions of the constraints are ϕi = const, πi(ϕ ) = 0. Then σ is harmonic function and eσ can be set to unity by a conformal transformation. In other words, for ρ = const the only solutions of the classical field equations is Minkowski space with constant matter fields.

A shortcut to arrive at the constraints (3.15View Equation) is to start from the Lagrangian (3.10View Equation) in conformal gauge (γμν = eσημν, η = diag(− 1,1), using R(eση) = − e−σ ∂2σ),

-1- [ μ i j μ − 1 μ ] L (ϕ,ρ,σ ) = − 2 λ ρ 𝔪ij(ϕ)∂ ϕ ∂μϕ − 2∂ ρ∂μσ − ρ ∂ ρ∂μρ , (3.16 )
and to work out the ‘would be’ energy momentum tensor Tμν of the Poincaré invariant Lagrangian (3.16View Equation). The relation between the velocities and the momenta then is of course different from that based on the lapse and shift analysis. However the Hamiltonian constraint ℋ0 and the 1D diffeomorphism constraint ℋ1, regarded as functions of the momenta, coincide with T00 and T01, respectively, as derived from Equation (3.16View Equation) as the components of the energy momentum tensor. Note that the trace of the ‘would-be’ energy momentum tensor μ μ λT μ = − 2∂ ∂μρ vanishes if the equation of motion ∂ μ∂μρ = 0 of ρ is imposed. We shall freely switch back and forth between both interpretations of the constraints.

For the computation of the Poisson algebra it is convenient to put ρ on-shell throughout (as its equation of motion μ ∂ ∂μρ = 0 is trivially solved) and to interpret the improvement terms in Equation (3.15View Equation) (here, those linear in the canonical variables) such that second time derivatives are eliminated. As expected, the ℋ1 generates infinitesimal spatial reparameterizations and the covariance of the fields is a merely kinematical property. Explicitly a spatial density d(x) of weight s transforms as {ℋ (x),d(y)} = ∂ dδ(x − y) − sd (y )δ ′(x − y) 1 1, the right-hand-side being the infinitesimal version of ^ ′ − s d(x ) → d(˜x) = [f (˜x)] d (f (˜x)), under −1 x → x˜= f (x). The canonical momenta πρ, πσ, πi are spatial densities of weight s = 1, while ℋ0, ℋ1, and eσ 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 (3.14View Equation) is that the Poisson algebra generated by ℋ0 and ℋ1 is a Lie algebra on-shell, and equivalent to the algebra of 2D conformal transformations. (Otherwise it yields the algebra of “surface deformations”; cf. [214Jump To The Next Citation Point]). To illustrate the difference let us note that with only the ρ equation of motion imposed one computes

{T± ±(x),T± ±(y)} = ∓2 [T±± (x) + T±±(y)]δ′(x − y), (3.17 ) {T++ (x),T− − (y)} = 0.
If also ∂∓T ±± = 0 is imposed the first equation can be rewritten as
′ {T ±±(x),T ±±(y)} = ∂ ±T±± (y)δ(x − y) ∓ 4T±± (y)δ(x − y). (3.18 )
As expected, the T ±± generate infinitesimal conformal transformations on the basis fields ρ, σ,ϕj: {T ±±(x),d(y )} = ∂±d(y)δ(x − y).

3.1.4 Lapse and shift in 2D gravity theories

Here we collect some useful formulas for 2D gravity theories in a lapse/shift parameterization of the metric, taken from [156Jump To The Next Citation Point]. As a byproduct we obtain a closed expression for the current K μ of the Euler density √ γR (2)(γ) = − ∂μK μ expressed in terms of the metric only. See [62Jump To The Next Citation Point] for a discussion. Our curvature conventions are the ones used throughout, the metric γ μν has eigenvalues (− ,+ ).

In 2D a ‘densitized’ lapse-shift parameterization is convenient (see e.g. [197Jump To The Next Citation Point]),

( ) σ − n2 + s2 s γμν = e s 1 . (3.19 )
The lapse n and the shift s here are spatial densities of weight − 1, while eσ carries weight 2. The rationale for this densitized lapse and shift parameterization is that the associated constraints automatically generate the proper Lie algebra of surface deformations (see [214197]). Using
( 1 s ) √ ---- − -- -- − γγμν = |( n n 2|) , (3.20 ) s- n − s- n n
one checks that the Hamiltonian associated with a free field Lagrangian − 1√ − γ-γμν∂μφ∂ νφ 2 is n [π2 + (∂ φ )2] + s∂ φπ 2 φ 1 1 φ, as it should be. Evaluating √ −-γR (2)(γ) in this parameterization gives the following expression (which we were unable to locate in the literature):
[ ] [ ] √ ----(2) -1 -1( 2 2 2 2) − γR (γ ) = ∂0 n (∂0σ − s∂1σ − 2 ∂1s) + ∂1 n (s − n )∂1σ − s∂0 σ + ∂1(s − n ) .(3.21 )
It can be rewritten as
√ ---- − γR (2)(γ ) = − ∂μK μ, √ ---- √ ---- √ ---- √ ---- √ ---- (3.22 ) K μ := − γγ μν∂ν ln − γ + ∂ν [ − γγ μν] + εμν − γγ01∂ν ln(− − γγ00),
where 01 10 ε01 = − ε10 = 1 = − ε = ε.

This provides an explicit though noncovariant expression for the current K μ in terms of the metric. Related formulas either invoke the zweibein or use an explicit parameterization. The one given in [62Jump To The Next Citation Point] is based on an SL(2,ℝ ) type parameterization of √ ---- γμν∕ − γ and is equivalent to Equation (3.22View Equation). Compared to Equation (2.11) in [62] a curl term εμν∂νφ has been added which allows one to express K μ solely in terms of the metric. Another advantage of Equation (3.22View Equation) is that the separation in dynamical and nondynamical variables is manifest: K μ is a function of det γ and the combination √ ----μν − γγ 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 K μ, as is manifest from Equation (3.21View Equation).

In the actions considered the term √ −-γR (2)(γ) always multiplies a scalar field ρ, say. Using Equation (3.20View Equation) the Hamiltonian associated with an Lagrangian of the form

√ ---[ 1 ] L = ρ − γ R (2)(γ) − -γ μν ∂μφ ∂νφ (3.23 ) 2
is readily worked out. As a function of the momenta π = − 1(∂ σ − s∂ σ − 2∂ s) ρ n 0 1 1 and 1 π σ = n(− ∂0ρ + s∂1ρ), one has √---- (2) 2 ρ − γR (γ ) ≃ n[− π ρπ σ + ∂1ρ∂1σ − 2∂1ρ ], where ‘≃’ denotes ‘modulo total derivatives’. For the Hamiltonian this gives
H = nℋ0 + sℋ1, 1 2 ρ 2 ℋ0 = ---πφ + -(∂1φ ) − (πρπ σ + ∂1 ρ∂1σ − 2∂1ρ), (3.24 ) 2ρ 2 ℋ1 = πφ∂1φ + πρ∂1ρ + πσ∂1σ − 2∂1 πσ.

The Poisson algebra of the constraints ℋ 0, ℋ 1 is the algebra of surface deformations, as required. Alternatively the form (3.22View Equation) can be used beforehand to get

√ ---- √ ---- √---- [√---- ] √---- [ √ ---- ] ρ − γR (2)(γ) ≃ − γγμν∂ μρ∂ν ln − γ + ∂μ ρ∂ν − γ γμν + − γγ01εμν∂ μρ∂ν ln − − γ γ00(,3.25 &#x0029
where ≃ denotes again ‘modulo total derivatives’.

3.1.5 Symmetries and currents

Later on we aim at a Dirac quantization of the 2-Killing subsector of the theories (3.1View Equation). The functional measure is then defined with the reduced Lagrangian (3.16View Equation) in conformal gauge, and a quantum version of the constraints ℋ0 ± ℋ1 is imposed subsequently. For the renormalization the symmetries of the Lagrangian (3.16View Equation) are crucial. The G invariance of course gives rise to a set of Lie algebra valued Noether currents Jμ. Let ta, a = 1,...,dim G, denote a basis of the Lie algebra with Killing form ⟨ , ⟩g. Let further Ya = Yia(ϕ)∂∕ ∂ϕi denote the Killing vectors of 𝔪. Then we define Jμ through its projection onto the basis ta, via

i j ⟨ta,Jμ⟩g := 2Y a(ϕ)mij(ϕ )∂μϕ . (3.26 )
From the equation of motion for ϕ and the Killing vector equation one readily checks that ∂μ (ρJ μ) = 0. Further, the invariance of Equation (3.16View Equation) under constant shifts in σ gives the trivial current ∂μ ρ.

More interestingly there are two ‘conformal currents’ which are not conserved on-shell but whose divergence reproduces the Lagrangian (3.16View Equation) up to a multiple

( ) μ 1- ∂ Cμ = L, λC μ = ρ∂μ σ + 2 lnρ , (3.27 ) μ ∂ D μ = − ln ρL, λD μ = ρ(σ∂ μlnρ − lnρ ∂μσ).
In fact, C μ and D μ are the only currents with that property. Their origin are the following scaling relations,
2 2 2 L ↦→ ω L if (ρ, σ) ↦→ (ω ρ,σ + clnω ), c ∈ ℝ, [ ( ) ] ω− 1 1 − 1 1- (3.28 ) L ↦→ ρ L if (ln ρ,σ + 2 ln ρ) ↦→ ωln ρ,ω σ + 2 ln ρ ,
where L is the Lagrangian (3.16View Equation). The parameter c is arbitrary, c = − 1∕2 matches the algebra of the conformal Killing vectors in Section 3.3, and c = 1 corresponds to scale gαβ ↦→ ω2gαβ transformations of the original 4D metric (3.6View Equation). Since for the action (3.10View Equation) the response under such a rescaling vanishes on-shell (see Equation (1.1View Equation)), the inessential nature of Newton’s constant is preserved by the reduction and remains visible through the conformal current Cμ in the gauge fixed Lagrangian (3.16View Equation).

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 [150124]. For illustration we present the lowest one which is (for all on-shell configurations) defined in terms of the dual potentials ∂ χ := − ε ρJν μ μν and ∂ ˜ρ := − ε ρν μ μν, with ε = − ε = 1 01 10. Then

{ ∫ } dx 𝒥0, ℋμ = 0, μ = 0,1, (3.29 ) 1- 2 ν 𝒥μ = 2 [ρJ μ,χ] + 2^ρρJμ − ρ εμνJ .

It may be worthwhile to point out what is trivial and what is nontrivial about the relations (3.29View Equation). Once the expression for the current 𝒥μ is known it is trivial to verify its conservation using the definition of the potentials χ and ^ρ. Since ℋ0 generates time translations on the basis fields ρ,σ,ϕi the associated conserved charge Poisson commutes with ℋ0 (and trivially with ℋ1) and thus qualifies as a genuine Dirac observable. What is nontrivial about Equation (3.29View Equation) is that a Dirac observable can be constructed explicitly in a way that does not require a solution of the Cauchy problem. The potentials χ, ^ρ 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 [9091] (“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 [4103] (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 (3.29View Equation) 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 (3.29View Equation) 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.


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