Gaurav Narain
email: narain@sissa.it

Roberto Percacci
email: rpercacci@perimeterinstitute.ca

Renormalization Group Flow in Scalar-Tensor Theories. I

We study the renormalization group flow in a class of scalar-tensor theories involving at most two derivatives of the fields. We show in general that minimal coupling is self consistent, in the sense that when the scalar self couplings are switched off, their beta functions also vanish. Complete, explicit beta functions that could be applied to a variety of cosmological models are given in a five parameter truncation of the theory in d = 4. In any dimension d > 2 we find that the flow has only a "Gaussian Matter" fixed point, where all scalar self interactions vanish but Newton's constant and the cosmological constant are nontrivial. The properties of these fixed points can be studied algebraically to some extent. In d = 3 we also find a gravitationally dressed version of the Wilson-Fisher fixed point, but it seems to have unphysical properties. These findings are in accordance with the hypothesis that these theories are asymptotically safe.

I. INTRODUCTION

Fundamental scalar fields have not yet been observed, but they play a crucial role in the standard model and in grand unified theories, as the order parameters whose VEV is used to distinguish between otherwise undifferentiated gauge interactions. Whether such scalar order parameters are elementary fields, as in the standard model, or composites, as in technicolor theories, is still an open question. Known examples of the Higgs phenomenon (superconductivity, the chiral condensate in QCD) point to the latter possibility, but even if this was the case it might still be possible to use scalar theory as an effective description (á la Landau-Ginzburg) at sufficiently low energy.

Scalar fields also play an important role in theories of gravity. Due to their simplicity they are very often used as models for matter. Also, because of the ease by which one can generate a nontrivial VEV, with an energy momentum tensor that resembles a cosmological constant, a scalar field is the most popular option as a driver of inflation. Furthermore, scalar fields easily mingle with the metric: by means of Weyl transformations it is possible to rewrite the dynamics in different ways [1], sometimes leading to new insight or to simplifications. Theories of gravity in which a scalar is present are often called scalar-tensor theories. In this paper we will discuss the quantum properties of a class of theories of this type.

The original motivation for this work comes from the progress that has been made in recent years towards understanding the UV behaviour of gravity. It seems that pure gravity possesses a Fixed Point (FP) with the right properties to make it asymptotically safe, or in other words nonperturbatively renormalizable [START_REF] Weinberg | General Relativity: An Einstein centenary survey[END_REF][START_REF] Reuter | [END_REF][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]39] (see also [20] for reviews). Let us assume for a moment that this ambitious goal can be achieved, and that pure gravity can be shown to be asymptotically safe. Still, from the point of view of phenomenology, we could not be satisfied because the real world contains also dozens of matter fields that interact in other ways than gravitationally, and a their presence affects also the quantum properties of the gravitational field, as is known since long [START_REF] Hooft | [END_REF]. Indeed, in a first investigation along these lines, it was shown in [22] that the presence of minimally coupled (i.e. non self interacting) matter fields shifts the position of the gravitational FP and the corresponding critical exponents. In some cases the FP ceases to exist, so it was suggested that this could be used to put bounds on the number of matter fields of each spin. More generally the asymptotic safety program requires that the fully interacting theory of gravity and matter has a FP with the right properties. Given the bewildering number of possibilities, in the search for such a theory one needs some guiding principle. One possibility that naturally suggests itself is that all matter self-interactions are asymptotically free [33]. Then, asymptotic safety requires the existence of a FP where the matter couplings approach zero in the UV, while the gravitational sector remains interacting. We will call such a FP a "Gaussian Matter FP" or GMFP. Following a time honored tradition, as a first step in this direction, scalar self interactions have been studied in [34,35]. Here we pursue that study further.

The tool that we use is the Wetterich equation, an exact renormalization group (RG) flow equation for a type of Wilsonian effective action Γ k , called the "effective average action". This functional, depending on an external energy scale k, can be formally defined by introducing an IR suppression in the functional integral for the modes with momenta lower than k. This amounts to modifying the propagator of all fields, leaving the interactions untouched. Then one can obtain a simple functional RG equation (FRGE) for the dependence of Γ k on k [25][26][27][28][29][30]. Insofar as the effective average action contains information about all the couplings in the theory, this functional RG equation contains all the beta functions of the theory. In certain approximations one can use this equation to reproduce the one loop beta functions, but in principle the information one can extract from it is nonperturbative, in the sense that is does not depend on the couplings being small.

The most common way of approximating the FRGE is to do derivative expansion of effective average action and truncate it at some order. In the case of a scalar theory the lowest order of this expansion is the local potential approximation (LPA), where one retains a standard kinetic term plus a generic potential [25,26,31,32]. In the case of pure gravity, the derivative expansion involves operators that are powers of curvatures and derivatives thereof. This has been studied systematically up to terms with four derivatives [11,15,16] and for a limited class of operators (namely powers of the scalar curvature) up to sixteen derivatives of the metric [13,14]. In the case of scalar tensor theories of gravity, one will have to expand both in derivatives of the metric and of the scalar field.

In this paper we study (Euclidean) effective average actions of the form

Γ k [g, φ] = d d x √ g V (φ 2 ) -F (φ 2 )R + 1 2 g µν ∂ µ φ∂ ν φ + S GF + S gh . ( 1
)
This can be seen as a generalization of the LPA, where one also includes terms with two derivatives of the metric.

In [35] it was shown that in d = 4, and assuming that V and F are polynomials in φ 2 , this theory admits a GMFP where only the lowest (φ 2 -independent) coefficient in V and F are nonzero. In this paper we extend and generalize this result in various ways. First of all, using so called "optimized" cutoff types [36] it is possible to write the beta functions in closed form, whereas in [35] they could only be studied numerically. This makes the subsequent analyses much more transparent. Unlike in [35], we will not assume from the outset that V and F are polynomials. Then, using the optimized cutoff it is possible to write explicit beta functionals for V and F . Exploiting general properties of these functionals we will be able to prove certain properties of the linearized flow in the neighborhood of the FP which had only been numerically observed previously. Namely we show that the matrix describing the linearized flow only has nonzero entries on the diagonal and on three lines next to it, and furthermore it has a block structure such that knowledge of the first two 2 × 2 blocks determines the whole matrix.

The discussion in this paper is also more general than that of [35] in two ways: we keep the dimension of spacetime arbitrary and we allow for a more general gauge fixing, depending on two arbitrary parameters. Keeping the dimension general is useful in view of possible applications to popular "large extra dimensions" theories [9], and also to higher dimensional dilatation-symmetric models which lead to vanishing cosmological constant in four dimensions [24]. Furthermore, with the closed form beta functions we can also perform a better search for other FP's where the scalar interactions are not all turned off. In [35] a numerical search was conducted on a grid of points in the neighborhood of the Gaussian FP, and no nontrivial FPs were found. Here, having the closed form of the beta functions, we can look for FPs by different methods. In d = 4, 5 and 6, some such points are found, but they appear to be spurious. On the other hand in d = 3 there is a FP which seems to be a genuine generalization of the Wilson-Fischer FP [40], but it has unphysical properties.

In a companion paper [START_REF] Narain | Renormalization Group Flow in Scalar-Tensor Theories[END_REF] we will extend the discussion to a more general class of effective actions,

Γ[g, φ] = d d x √ g L(φ 2 , R) + 1 2 g µν ∂ µ φ∂ ν φ + S GF + S gh . ( 2
)
where L is in general a function of the curvature scalar and of the scalar field.

Even though the original motivation of our work was to study the UV properties of the theory, it is important to stress that the beta functions that we obtain are completely general: they hold for any energy range. Depending on the ratio between the parameters of the theory (the cosmological constant, Newton's constant, the scalar mass and all the dimensionful higher couplings) different terms in the beta functions will come to dominate. However, this is something that has not been put in a priori. Thus the beta functions can be used also to study IR or mesoscopic problems, provided the system can be accurately modelled by a scalar tensor theory.

There is clearly much scope for applications to cosmology. Early work in this direction has been done in [23], using the beta functions of pure gravity. Along a different line, given the role played by scalar fields in inflation it seems likely that the RG running of couplings could have significant effects. This seems particularly true of recent attempts to use the standard model Higgs field as an inflaton, which use a special case of the action (1)

√ g 1 2 g µν ∂ µ φ∂ ν φ + 1 2 m 2 φ 2 + λφ 4 + 1 16π 2 (2Λ -R) + 1 2 ξφ 2
with a large value for the nonminimal coupling ξ [37]. The beta functions given in Appendix A, contain the full dependence on Λ, G, m 2 , λ and ξ, including threshold effects and a resummation of infinitely many perturbative contributions.

One can imagine also applications in the IR, for example along the lines of [24]. We mention that the appearance of a scalar field in the low energy description of gravity has been also stressed in [38]. For a FRGE-based approach to that issue see also [39].

This paper is organized as follows. In section 2 we will derive the "beta functionals" for V and F . In section 3 we discuss the general properties of the GMFP, in any gauge and dimension. In section 4 we discuss numerical solutions for the GMFP. In section 5 we will discuss other FP's with nontrivial potentials and we conclude in section VI with some additional remarks. Apendix A contains some lengthy formulae for the beta functions of five couplings in four dimensions.

II. THE BETA FUNCTIONS

In this paper we will obtain beta functionals for the functions V and F defined in (1). To achieve this, we use Wetterich's functional renormalization group equation (FRGE) [25]

∂ t Γ k = 1 2 STr δ 2 Γ k δΦ δΦ + R k -1 ∂ t R k , ( 3
)
where Φ are all the fields present in the theory and STr is the generalized functional trace including a minus sign for fermionic variables and a factor 2 for complex variables, and R k is a suitable tensorial cutoff.

A. Second variations

In order to evaluate the r.h.s. of (3) we start from the second functional derivatives of the functional (2). These can be obtained by expanding the action to second order in the quantum fields around classical backgrounds: g µν = ḡµν + h µν and φ = φ + δφ, where φ is constant. The gauge fixing action is given by

S GF = 1 2α d d x √ ḡ F (φ 2 )ḡ µν χ µ χ ν , ( 4
)
χ µ = ∇ν h νµ - β + 1 d ∇µ h .
and S gh is the corresponding ghost action given by

S GH = -d d x √ ḡ Cµ δ ρ µ ¯ + 1 - 2(1 + β) d ∇µ ∇ρ + Rρ µ C ρ . ( 5
)
These terms are already quadratic in the quantum fields. The second variation of eq. ( 1) is, Γ

= 1 2 d d x √ g 1 4 h 2 - 1 2 h µν h µν V (φ 2 ) -F (φ 2 ) R + F (φ 2 ) h h µν R µν + 1 2 F (φ 2 ) h h +F (φ 2 ) h µν ∇ µ ∇ ρ h ρ ν -F (φ 2 ) h α ν h µα R µν -F (φ 2 ) h µν R ρµσν h ρσ - 1 2 F (φ 2 )h µν h µν -F (φ 2 ) h∇ µ ∇ ν h µν + d d x √ g h • φ V (φ 2 ) -F (φ 2 ) R δφ + 2 φF (φ 2 ) h µν R µν δφ -2 φ F (φ 2 ) δφ (∇ µ ∇ ν h µν -h) + 1 2 d d x √ g δφ -+ 2 V (φ 2 ) + 4 φ 2 V (φ 2 ) -R 2 F (φ 2 ) + 4 φ 2 F (φ 2 ) + S GF + S gh . ( (2) k
) 6
Since we will never have to deal with the original metric g µν and scalar field φ, in order to simplify the notation, in the preceding formula and everywhere else from now on we will remove the bars from the backgrounds. As explained in detail in [START_REF] Reuter | [END_REF], the functional that obeys the FRGE (3) depends separately on the background field ḡµν and on a "classical field" (g cl ) µν = ḡµν + (h cl ) µν , where (h cl ) µν is Legendre conjugate to the sources that couple linearly to h µν . The same applies to the scalar field. In this paper, like in most of the literature on the subject, we will restrict ourselves to studying the effective average action in the case when (g cl ) µν = ḡµν and φ cl = φ. From now on the notation g µν and φ will be used to denote equivalently the "classical fields" or the background fields. For a discussion of the effective average action of pure gravity in the more general case when (g cl ) µν = ḡµν we refer to [START_REF] Manrique | [END_REF].

B. Decomposition

In order to partially diagonalize the kinetic operator, we use the decomposition of h µν into irreducible components

h µν = h T µν + ∇ µ ξ ν + ∇ ν ξ µ + ∇ µ ∇ ν σ - 1 d g µν σ + 1 d g µν h , ( 7
)
where h T µν is the (spin 2) transverse and traceless tensor, ξ µ is the (spin 1) transverse vector component, σ and h are (spin 0) scalars. In some cases this decomposition allows an exact inversion of the propagator. This happens for example in the case of maximally symmetric background metric. Thus with that in mind we will work on a d-dimensional sphere. This change of variables in the functional integral gives rise to Jacobian determinants, which however can be absorbed by further field re- 6,13]. Then the inverse propagators for various components of the field are easily read from the second variation of the effective action. Thus for the spin-2 component h T µν we get the following inverse propagator:

definitions ξµ = --R d ξ µ and σ = √ - --R d-1 σ [4,
1 2 F (φ 2 ) -+ d 2 -3d + 4 d(d -1) R - 1 2 V (φ 2 ) . ( 8
)
For the spin-1 component ξ we have the following inverse propagator:

1 α F (φ 2 ) -- R d -V (φ 2 ) + d -2 d F (φ 2 ) R . ( 9
)
The two spin-0 components of the metric, σ and h, mix with the fluctuation of φ resulting in an inverse propagator given by a symmetric 3 × 3 matrix S, with the following entries:

S σσ = 1 - 1 d 1 2 -1 - 1 α 1 - 1 d F (φ 2 ) (-) - 1 2 V (φ 2 ) + d -4 2d F (φ 2 ) R + 1 - 1 α F (φ 2 ) R d , S σh = S hσ = 1 2 1 - 1 d 2 d β α + 1 -1 F (φ 2 ) - -- R d -1 , S σφ = S φσ = -2 φ 1 - 1 d F (φ 2 ) - -- R d -1 , S hh = d -2 4d -1 - 1 d + 2β 2 αd(d -2) 2 F (φ 2 ) (-) + V (φ 2 ) - d -4 d F (φ 2 ) R , S hφ = S φh = -2 1 - 1 d φ F (φ 2 ) (-) + φ V (φ 2 ) -1 - 2 d φ F (φ 2 ) R , S φφ = -+ 2V (φ 2 ) + 4φ 2 V (φ 2 ) -(2F (φ 2 ) + 4φ 2 F (φ 2 ))R . ( 10
)
In order to diagonalize the kinetic operator occuring in the ghost action eq. ( 5), we perform a similar decomposition of the ghost field into transverse and longitudinal parts in the following manner:

Cµ = CµT + ∇ µ C , C µ = C T µ + ∇ µ C . ( 11
)
where CµT and C T µ satisfy the following constraints,

∇ µ CµT = 0 , ∇ µ C T µ = 0. ( 12
)
Again this decomposition would give rise to a non trivial Jacobian in the path-integral, which is cancelled by the further redefinition Ĉ = √ -C. For spin-1 component of the ghost, the inverse propagator is

-- R d , ( 13
)
while for spin-0 component we have the following inverse propagator

2 - 2(1 + β) d (-) - 2R d , (14)
Now we have to specify the cutoff R k occuring in FRGE eq. ( 3). We define R k by the rule that Γ

(2)

k + R k has the same form as Γ (2)
k except for the replacement ofby P k (-), where P k (z) = z + R k (z). R k (z) is a profile function which tends to k 2 for z → 0 and it approaches zero rapidly for z > k 2 . The quantity Γ

(2) k + R k is the "modified inverse propagator". This procedure applies both to the bosonic degrees of freedom and to the ghosts. The cutoff R k occuring in the FRGE depends on k not only through the profile function R k (z), but also through k dependent couplings present in the function F (φ 2 ) and F (φ 2 ). Thus the derivative k d dk = d dt acts not only on the profile function R k (z), but also on the k-dependent couplings present in F (φ 2 ) and F (φ 2 ). When this is neglected one recovers the one loop results. The presence of the beta functions on the RHS of the FRGE, produces a coupled system of linear equations, which has to be solved algebraically to yield the beta functions.

C. The β-functionals in d = 4

To read off the beta functions we have to compare the r.h.s. of the FRGE with the t-derivative of eq. ( 1), namely

∂ t Γ[g, φ] = d d x √ g ∂ t V (φ 2 , R) -∂ t F R . ( 15
)
(the gauge fixing and the kinetic term are not allowed to run in our approximations). Since the background R and φ are constant, the space-time integral produces just a volume factor, which eventually cancels with the same factor appearing on the RHS of the FRGE. Thus the running of V and F can be calculated using,

∂ t V (φ 2 ) = 1 V ol ∂ t Γ k R=0 , ∂ t F (φ 2 ) = - 1 V ol ∂(∂ t Γ k ) ∂R R=0 , ( 16
)
where V ol is the space-time volume. In order to exhibit the explicit form of these beta functionals we go to d = 4, where V ol = 384π 2 R 2 , and set α = 0 and β = 1 (De-Donder gauge). Furthermore, we choose the optimized cutoff [36], which allows to perform the integrations occuring in FRGE trace in closed form (see appendix A in [13]). From the FRGE we then get

R k (z) = (k 2 -z)θ(k 2 -z)
∂ t V = k 4 192π 2 6 + 30 V Ψ + 6(k 2 Ψ + 24 φ 2 k 2 F Ψ + k 2 F Σ 1 ) ∆ + 4 F + 5 k 2 Ψ + k 2 Σ 1 ∆ ∂ t F + 24 φ 2 k 2 Ψ ∆ ∂ t F , ( 17
)
∂ t F = k 2 2304π 2 150 + 120 k 2 F (3 k 2 F -V ) Ψ 2 - 24 ∆ 24 φ 2 k 2 F Ψ + k 2 Ψ + k 2 F Σ 1 - 36 ∆ 2 -4 φ 2 (6 k 4 F 2 + Ψ 2 ) ∆ +4 φ 2 Ψ Ψ (7 k 2 F -V ) (Σ 1 -k 2 ) + 4 φ 2 Σ 1 (7 k 2 F -V ) (2 Ψ V -V Ψ ) + 2 k 4 Ψ 2 Σ 2 + 48 k 4 F φ 2 Ψ Ψ Σ 2 -24 k 4 F φ 2 Ψ 2 Σ 2 - ∂ t F F 30 - 10 k 2 F (7 Ψ + 4 V ) Ψ 2 + 6 ∆ 2 k 2 F Σ 1 ∆ + 4 φ 2 V Ψ ∆ -24 k 4 F φ 2 Ψ 2 Σ 2 -4 φ 2 k 2 F Ψ Σ 1 (7 k 2 F -V ) + ∂ t F 24 k 2 φ 2 ∆ 2 (k 2 F + 5 V )∆ -12 k 2 Ψ Ψ Σ 2 -2 (7 k 2 F -V ) Ψ Σ 1 ( 18
)
where we have defined the shorthands:

Ψ = k 2 F -V ; Σ 1 = k 2 + 2 V + 4 φ 2 V ; Σ 2 = 2 F + 4 φ 2 F ; ∆ = 12 φ 2 Ψ 2 + Ψ Σ 1 .
Let us define the dimensionless fields φ = k

2-d 2 φ, R = k -2 R and the dimensionless functions Ṽ ( φ2 ) = k -d V (φ 2 ) and F ( φ2 ) = k 2-d F (φ 2
). The beta functionals of the dimensionless and dimensionful functions are related as follows:

(∂ t Ṽ )[ φ2 ] = -d Ṽ ( φ2 ) + (d -2) φ2 Ṽ ( φ2 ) + k -d (∂ t V )[φ 2 ] , ( 19
) (∂ t F )[ φ2 ] = -(d -2) F ( φ2 ) + (d -2) φ2 F ( φ2 ) + k -(d-2) (∂ t F )[φ 2 ] . ( 20
)
Some comments are in order. From the expressions of ∂ t V and ∂ t F given in eq. ( 17) and ( 18) respectively, we note that where ever there is occurence of φ 2 , it occurs in combinations like

φ 2 V V , φ 2 V F , φ 2 F F , φ 2 V ∂ t F , φ 2 F ∂ t F , φ 2 V
and φ 2 F . Occurence of such combinations are crucial, as they help us (as is demostrated in [START_REF] Narain | Renormalization Group Flow in Scalar-Tensor Theories[END_REF]) in proving that minimal coupling is self consitent. Because of the occurrence of ∂ t F in the r.h.s. of (17), the system of equations cannot be solved algebraically for ∂ t F . It may be possible to solve it as a differential equation, but here we shall not pursue this. Rather, we observe that if F and Ṽ are assumed to be finite polynomials in φ2 of the form

Ṽ ( φ2 ) = a n=0 λ2n φ2n ; F ( φ2 ) = b n=0 ξ2n φ2n . ( 21
)
with finite a and b, then ∂ t F is also a finite polynomial in the beta functions and it becomes possible to solve for the beta functions algebraically. As an explicit example, in the appendix we give these equations in the de-Donder gauge (α = 0 and β = 1) in d = 4 with five couplings truncation (a = 2, b = 1).

III. THE GAUSSIAN MATTER FIXED POINT A. Minimal coupling is self consistent

We assume that V and F are real analytic so that they can be Taylor expanded around φ 2 = 0. A given V and F define a FP if the corresponding dimensionless potentials satisfy ∂ t Ṽ = 0 and ∂ t F = 0. Because of analyticity, this is equivalent to requiring that all the derivatives of ∂ t Ṽ and ∂ t F with respect to φ2 , evaluated at φ2 = 0 are zero. Taking n derivatives of eq. ( 19) and eq. ( 20) with respect to φ2 we get

0 = (∂ t Ṽ ) (n) (0) = ((d -2)n -d) Ṽ (n) (0) + (k -d ∂ t V ) (n) (0) ; (22) 0 = (∂ t F ) (n) (0) = (n -1)(d -2) F (n) (0) + (k -(d-2) ∂ t F ) (n) (0) . ( 23
)
where in the last two terms the expressions in brackets can be thought of as functions of φ2 . We can rewrite them as

∂ n ∂( φ2 ) n (k -d ∂ t V ) = k (d-2)n-d ∂ n ∂(φ 2 ) n (∂ t V ) ; ∂ n ∂( φ2 ) n (k -(d-2) ∂ t F ) = k (d-2)(n-1) ∂ n ∂(φ 2 ) n (∂ t F ) .
We now make the following Ansatz:

V = k d λ0 , F = k d-2 ξ0 , ( 24
)
where λ0 and ξ0 are numbers to be determined. This corresponds to putting a = b = 0 in [START_REF] Hooft | [END_REF], or in other words to seeting to zero all scalar self couplings. We are assuming here that all the derivatives of V and F at φ 2 = 0 vanish, so that V and F are just constants. If a FP of this type exists, we call it a Gaussian Matter Fixed Point (GMFP).

In order to check that this ansatz defines a FP we need to show that eq. ( 22) and ( 23) are identically satisfied for all n ≥ 1, while for n = 0 they determine the numbers λ0 and ξ0 .

For n ≥ 1 the first term on the r.h.s. of eq. ( 22) and ( 23) vanishes because of the ansatz. There remains to show that

∂ n ∂(φ 2 ) n (∂ t V ) and ∂ n ∂(φ 2 ) n (∂ t F
) are zero at φ 2 = 0. In d = 4 one can check this explicitly by inspecting eq. ( 17) and eq. ( 18). The crucial point to observe is that in ∂ t V and ∂ t F , whenever φ 2 appears explicitly, it is multiplied by some derivative of V or F . So when the derivative removes φ 2 , what remains is zero because of the ansatz, and otherwise it is zero because there remains some positive power of φ 2 .

In other dimensions this crucial property remains valid. In other dimensions this crucial property remains valid, because it is true either for the second variations (in the case of the transverse traceless tensor and transverse vector components) or for the matrix trace of the second variations, in the case of the scalars. Since the beta functionals are obtained by taking functional traces of these expressions, this property will go through for them as well i.e. for n ≥ 1 the eq. ( 22) and ( 23) are identically satisfied. For a detailed proof see [START_REF] Narain | Renormalization Group Flow in Scalar-Tensor Theories[END_REF].

Thus in any dimension the ansatz works for all n ≥ 1. There remains to solve the equations for the constant terms in V and F , which are given by λ0 and ξ0 . We are going to do this numerically in section IV. In the meanwhile we assume that such a solution exists, and we study the properties of the linearized flow around it.

B. Linearized Flow around GMFP

To study the linearized flow around GMFP it will be convenient to Taylor expand V and F as follows:

V (φ 2 ) = ∞ n=0 λ 2n φ 2n ; F (φ 2 ) = ∞ n=0 ξ 2n φ 2n . ( 25
)
We define dimensionless couplings λ2n = k -d+(d-2)n λ 2n and ξ2n = k -(d-2)(1-n) ξ 2n , in such a way that the dimensionless potentials can be expanded as:

Ṽ ( φ2 ) = ∞ n=0 λ2n φ2n ; F ( φ2 ) = ∞ n=0 ξ2n φ2n . ( 26
)
To obtain the running of dimensionless couplings we take derivatives of eq. ( 19) and eq. ( 20) with respect to φ2 and use eq. ( 16)

∂ t λ2n = 1 n! δ n ∂ t Ṽ δ( φ2 ) n φ2 =0 ; ∂ t ξ2n = 1 n! δ n ∂ t F δ( φ2 ) n φ2 =0 . ( 27
)
Because of the presence of t-derivative on the RHS of FRGE, we do not obtain the beta functions of dimensionless couplings directly, rather we get algebraic equations for them, solving which one get the full beta functions.

Having defined the dimensionless couplings, we now define the stability matrix to be the matrix of derivatives of the dimensionless beta functions with respect to the dimensionless couplings at the FP. By definition it is a tensor quantity in the theory space. It will be convenient to write V 0 = V and V 1 = -F . One can then define the corresponding dimensionless potentials as Ṽa = k d-2a V a , where a is either 0 or 1. Then the stability matrix is given by,

(M ij ) ab = δ 1 i! ∂ t Ṽ (i) a (0) δ 1 j! Ṽ (j) b (0) F P . ( 28
)
From the above definition of the stability matrix we note that the couplings get arranged in the following order: λ 0 , ξ 0 , λ 2 , ξ 2 , λ 4 , ξ 4 . . .. Then the matrix M at the GMFP has the following form:

     M 00 M 01 0 0 • • • 0 M 11 M 12 0 • • • 0 0 M 22 M 23 • • • 0 0 0 M 33 • • • • • • • • • • • • • • • • • •      , ( 29
)
where each entry is a 2 × 2 matrix of the form

M ij =    ∂β λ (2i) ∂ λ(2j) ∂β λ (2i) ∂ ξ(2j) ∂β ξ (2i) ∂ λ(2j) ∂β ξ (2i) ∂ ξ(2j)    . ( 30
)
Moreover the various non zero entries of M are related to each other by the following recursion relations (in ddimensions):

M ii = (d -2) i + M 00 ; M i,i+1 = (i + 1) (2i + 1) M 01 , ( 31
)
where

M 00 = -d 0 0 -(d -2) + δM λ0 λ0 δM λ0 ξ0 δM ξ0 λ0 δM ξ0 ξ0 ; M 01 = δM λ0 λ2 δM λ0 ξ2 δM ξ0 λ2 δM ξ0 ξ2 . ( 32
)
We can prove these facts for the one loop beta functions, i.e. neglecting the t-derivatives of the couplings on the r.h.s. of FRGE. Using this we note that the running of dimensionless potentials can be written as follows:

∂ t Ṽa = -(d -2a) Ṽa + (d -2) φ2 Ṽ a + Ha Ṽb , φ2 Ṽ b Ṽ c , 2 Ṽ b + 4 φ2 Ṽ b . ( 33
)
We have indicated that the one loop beta functional depends on φ2 only through the three types of combinations indicated as the arguments for Ha . This can be verified in d = 4 by inspection of eq. ( 17) and eq. ( 18), when one drops the terms proportional to ∂ t F and ∂ t F in the r.h.s. The properties of the stability matrix given above follow by taking successive derivatives of ∂ t Ṽa with respect to φ2 at φ2 = 0. The i = 0 entries of eq. ( 28) can be calculated by setting φ2 = 0 in eq. ( 33):

∂ t Ṽa (0) = -(d -2a) Ṽa (0) + Ha ( Ṽa (0), 2 Ṽ a (0)) . ( 34
)
Since ∂ t Ṽa (0) depends only on Ṽa (0) and Ṽ a (0), in eq. ( 28) for i = 0, only j = 0, 1 will be non zero. Thus M 00 and M 01 are given by,

(M 00 ) ab = -(d -2a) δ ab + δ Ha ( Ṽc (0), 2 Ṽ c (0)) δ Ṽb (0) GM F P ; (M 01 ) ab = δ Ha ( Ṽc (0), 2 Ṽ c (0)) δ Ṽ b (0) GM F P . ( 35
)
Now we take first derivative of ∂ t Ṽa with respect to φ2 . This gives,

∂ t Ṽ a = -(d -2a) Ṽ a + (d -2) Ṽ a + (d -2) φ2 Ṽ a + δ Ha δ Ṽc Ṽ c + δ Ha δ( φ2 Ṽ c Ṽ d ) ( Ṽ c Ṽ d + φ2 Ṽ c Ṽ d + φ2 Ṽ d Ṽ c ) + δ Ha δ(2 Ṽ c + 4 φ2 Ṽ c ) (2 Ṽ c + 4 Ṽ c + 4 φ2 Ṽ c ) . ( 36
)
When we set φ2 = 0, we note from the above equation that ∂ t Ṽ a (0) depends only on Ṽa (0), Ṽ a (0) and Ṽ a (0). We use this in eq. ( 28) to calculate the i = 1 entries of the stability matrix. We note that M 1j = 0 for all j ≥ 3. Now we find the remaining possible non zero entries. For j = 0, we note that the dependence on Ṽa (0) is present only in δ Ha

δ Ṽc φ2 =0 , δ Ha δ( φ2 Ṽ c Ṽ d ) φ2 =0 and δ Ha δ(2 Ṽ c +4 φ2 Ṽ c ) φ2 =0
. But each of these terms are multiplied either with Ṽ a or Ṽ a , so when we calculate the stability matrix, these terms will not contribute due to GMFP conditions ( Ṽ (i) a = 0 for all i ≥ 1). Thus we conclude that M 10 = 0.

For j = 1, we take the derivative of ∂ t Ṽ a (0) with respect to Ṽ b . Thus using the condition of GMFP and eq. ( 35) we find,

(M 11 ) ab = -(d -2a)δ ab + (d -2)δ ab + δ Ha δ Ṽb φ2 =0 = (d -2)δ ab + (M 00 ) ab , ( 37
)
while for j = 2 we take derivatve of ∂ t Ṽ a (0) with respect to Ṽ b /2 and use eq. ( 35). Thus we get

(M 12 ) ab = 2 δ Ha δ(2 Ṽ c ) φ2 =0 • 6δ bc = 6 (M 01 ) ab . ( 38
)
Thus we see that for i = 1 we have,

M 10 = 0 ; M 11 = (d -2) • 1 + M 00 ; M 12 = 2 • 3 M 01 ; M 1j = 0 , ∀j ≥ 3 . ( 39
)
In order to understand the structure of the lines i ≥ 2 we will proceed by induction. We assume that the i-th derivative has the following structure,

(∂ t Ṽa ) (i) = -(d -2a) Ṽ (i) a + (d -2) φ2 Ṽ (i+1) a + i Ṽ (i) a + • • • + δ Ha δ Ṽc Ṽ (i) c + δ Ha δ( φ2 Ṽ c Ṽ d ) φ2 Ṽ c Ṽ d (i) + δ Ha δ(2 Ṽ c + 4 φ2 Ṽ c ) 2 Ṽ c + 4 φ2 Ṽ c (i) , ( 40
)
where the (• • • ) denote expressions having at least two factors of derivatives of potentials, which are irrelevant when calculating the entries of stability matrix. Clearly this property is true for i = 1. We show that if it holds for a given value of i, then it also holds for i + 1. Thus we take one more derivative eq.( 40) and we find

(∂ t Ṽa ) (i+1) = -(d -2a) Ṽ (i+1) a + (d -2) φ2 Ṽ (i+2) a + (i + 1) Ṽ (i+1) a + • • • + δ Ha δ Ṽc Ṽ (i+1) c + δ 2 Ha δ Ṽc δ Ṽd Ṽ (i) c Ṽ d + δ 2 Ha δ Ṽe δ( φ2 Ṽ c Ṽ d ) Ṽ (i) e φ2 Ṽ c Ṽ d + δ 2 Ha δ Ṽe δ(2 Ṽ c + 4 φ2 Ṽ c ) Ṽ (i) e 2 Ṽ c + 4 φ2 Ṽ c + δ Ha δ( φ2 Ṽ c Ṽ d ) φ2 Ṽ c Ṽ d (i+1) + δ 2 Ha δ Ṽe δ( φ2 Ṽ c Ṽ d ) Ṽ e φ2 Ṽ c Ṽ d (i) + δ 2 Ha δ( φ2 Ṽ e Ṽ f ) δ( φ2 Ṽ c Ṽ d ) ( φ2 Ṽ e Ṽ f ) φ2 Ṽ c Ṽ d (i) + δ 2 Ha δ(2 Ṽ e + 4 φ2 Ṽ e ) δ( φ2 Ṽ c Ṽ d ) (2 Ṽ e + 4 φ2 Ṽ e ) φ2 Ṽ c Ṽ d (i) + δ Ha δ(2 Ṽ c + 4 φ2 Ṽ c ) 2 Ṽ c + 4 φ2 Ṽ c (i+1) + δ 2 Ha δ Ṽe δ(2 Ṽ c + 4 φ2 Ṽ c ) Ṽ e 2 Ṽ c + 4 φ2 Ṽ c (i) + δ 2 Ha δ(2 Ṽ e + 4 φ2 Ṽ e ) δ( φ2 Ṽ c Ṽ d ) (2 Ṽ e + 4 φ2 Ṽ e ) (i) φ2 Ṽ c Ṽ d + δ 2 Ha δ(2 Ṽ c + 4 φ2 Ṽ c ) δ(2 Ṽ d + 4 φ2 Ṽ d ) (2 Ṽ c + 4 φ2 Ṽ c ) (i) (2 Ṽ d + 4 φ2 Ṽ d ) . ( 41
)
Aside from the new terms containing two factors of derivatives of the potentials, which can be neglected for our purposes, the remaining terms have the same structure as eq. ( 40). Thus by induction eq. ( 40) holds for all i.

We can now use this result to calculate the entries of the stability matrix in the i-th row. Using

2 Ṽ a + 4 φ2 Ṽ a (i) = 2 (2 i + 1) Ṽ (i+1) a + 4 φ2 Ṽ (i+2) a , ( 42
)
we note that at φ2 = 0 and using the condition of GMFP for calculating the stability matrix we have,

M ij = 0 , ∀j ≤ (i-1) ; M ii = (d-2) i+M 00 ; M i,i+1 = (i+1) (2 i+1)M 01 ; M ij = 0 , ∀j ≥ (i+2) . ( 43
)
This completes the proof of our statements in the one loop approximation. It is difficult to extend this proof to the exact equation, but we see in finite truncations that the previous properties of the stability matrix remain true.

Having established the properties of stability matrix we would like to compute its eigenvalues. The good feature of the block structure of stability matrix indicated in eq. ( 29) is that the eigenvalues are given just by the diagonal blocks. Since the consecutive diagonal blocks just differ by d -2, the eigenvalues of the consecutive diagonal blocks of M also differ by d -2. This is a very strong result, because it implies that, at a GMFP, the eigenvalues of M are all determined by the eigenvalues of M 00 . Furthermore, the off diagonal blocks of M are all determined by M 01 , so knowing M 00 and M 01 one can also determine all the eigenvectors. This is useful to understand the mixing among various operators at the FP. The smallest truncation that is required to calculate both M 00 and M 01 is when we retain terms up to φ 2 in each potential.

IV. NUMERICAL RESULTS

A. The GMFP in d = 4.

We now look for GMFP in various dimensions and calculate the critical exponents of the system, which are defined to be the opposites of the eigenvalues of M , i.e. θ i = -λ i , where λ i is the eigenvalue. As explained in the previous section, it is enough to calculate the eigenvalues of M 00 . We do this task first in d = 4.

In d = 4 for De-donder gauge we get the following FP equation,

24 λ0 16π 2 + 1 λ0 -ξ0 = 19, ( 44
) (265 -2304 π 2 ξ0 ) ξ0 + (127 + 2304 π 2 ξ0 ) λ0 = 160 λ2 0 λ0 -ξ0 . ( 45
)
On solving these, the only real solution that we get is λ * 0 = 0.00862 ; ξ * 0 = 0.02375 . [START_REF] Buchbinder | Effective action in quantum gravity[END_REF] We now compute the critical exponents θ of the stability matrix in this gauge. The relations given in eq. ( 31) between the various nonzero entries of the stability matrix are independent of the gauge. However the entires of M 00 and M 01 are gauge dependent. For d = 4 eq. ( 31) reduces to,

M ii = 2 i + M 00 ; M i,i+1 = (i + 1) (2 i + 1) M 01 . ( 47
)
In De-Donder gauge for d = 4, the entries of M 00 are

M λ0 λ0 = 1 32π 2 1 Θ ξ0 ξ2 0 9667 + 3456π 2 ξ0 (169 + 2304π 2 ξ0 ) + λ2 0 3279 + 1152π 2 ξ0 (275 + 6912π 2 ξ0 ) -18 λ0 ξ0 551 + 128π 2 ξ0 (331 + 6912π 2 ξ0 ) , ( 48
)
M λ0 ξ0 = 1 32π 2 1 Θ -48384 λ4 0 -443520π 2 ξ4 0 + λ0 ξ2 0 -9667 + 2304π 2 ξ0 (161 -3456π 2 ξ0 ) -3 λ3 0 1093 + 4608π 2 ξ0 (1 + 576π 2 ξ0 ) + 18 λ2 0 ξ0 551 + 192π 2 ξ0 (-1 + 4608π 2 ξ0 ) , ( 49
)
M ξ0 λ0 = 12 Θ ξ2 0 -252 λ0 ξ0 (3 + 128π 2 ξ0 ) + 3 λ2 0 (41 + 1536π 2 ξ0 ) + ξ2 0 (593 + 27648π 2 ξ0 ) , ( 50
)
M ξ0 ξ0 = -3 Θ -135 λ4 0 + 1309 ξ4 0 + 12 λ3 0 ξ0 (145 + 1536π 2 ξ0 ) + 36 λ0 ξ3 0 (77 + 3072π 2 ξ0 ) -6 λ2 0 ξ2 0 (811 + 23504π 2 ξ0 ) . ( 51
)
While the entries of M 01 are

M λ0 λ2 = 1 16π 2 1 Θ ξ2 0 (37 -1152π 2 ξ0 ) + 2 λ0 ξ0 (-5 + 1152π 2 ξ0 ) -λ2 0 (7 + 1152π 2 ξ0 ) , ( 52
)
M λ0 ξ2 = - 3 4π 2 (2 λ0 -5 ξ0 )( λ0 -ξ0 ) Θ , ( 53
)
M ξ0 λ2 = 24 ξ0 ( λ0 -ξ0 ) 2 Θ , ( 54
)
M ξ0 ξ2 = - 72 ξ0 ( λ0 -ξ0 ) 2 Θ , ( 55
)
where

Θ = -18 λ0 ξ0 (1 + 128π 2 ξ0 ) + 3 λ2 0 (5 + 384π 2 ξ0 ) + ξ2 0 (-17 + 1152π 2 ξ0 ) 2 . ( 56
)
The relations eq. ( 47) tells that the critical exponents of consecutive diagonal blocks will differ by 2. In the truncation where we keep terms till φ 2 in each potential, the critical exponents are,

2.143 ± 2.879i , 0.143 ± 2.879i (57)
The critical exponents 2.143 ± 2.879i correspond to eigenvalues of M 00 , while the critical exponents 0.143 ± 2. where the first complex conjugate pair of eigenvector correspond to critical exponents 2.143 ± 2.879i, while the second pair correspond to critical exponents 0.143 ± 2.879i.

We then looked for GMFP in other gauges. In d = 4 we consider various values of the gauge parameters α and β. To study the gauge dependence we considered 50 different values of α in the range 0 to 1.225 at step of 0.025, and 25 different values of β in the range -1 to 1.4 at interval of 0.1. For each combination of α and β we solved the FP equation obtained for λ0 and ξ0 . In general, this produces a set of FPs. In order to choose the correct GMFP from that set, we plot all the real FPs to see which one is continously followed in other gauge values and which ones are spurious. For example one can take any value of β, and plot all the real FPs for various values of α. Some FPs don't exits for all values of α, and are assumed to be truncation artifacts. Only one GMFP exists for all values, and is continuous. This observation of continuity in α and β is useful to write a code for selecting the right GMFP for various gauges. After calculating the GMFP we calculate the critical exponents of M 00 . We then plot the GMFP and critical exponents against the various gauge values and generate 3D graphs. In d = 4 we obtain the graphs shown in Fig. (1). We note that the existence of the FP has been actually verified in a much larger range of values of α and β.

B. The GMFP in other Dimensions

We now look for the GMFP in other dimensions. For any d > 2, in De-donder gauge, the FP equation for λ0 and ξ0 is given by,

2d λ0 + (3d -2) λ0 + (d -2)(d 2 + d -1) ξ0 (4π) d/2 Γ 2 + d 2 ( λ0 -ξ0 ) = 0 , ( 59
)
A λ2 0 + B λ0 ξ0 + C ξ2 0 ( λ0 -ξ0 ) + D ( λ0 -ξ0 ) = 0 , (60)
where

A = -(d -1)(d 3 + 2d 2 + 36d + 24) , B = -(d -1)(d 5 -17d 3 -38d 2 -96d -48) , C = (d 6 -13d 5 + 32d 4 -104d 3 + 72d 2 + 36d + 24) , D = 24d(d -1)(d -2)(4π) d/2 Γ 2 + d 2 . ( 61
)
Solving these equations we find that in other dimensions, it is possible to have more than one real solution. But when we plot all the real the solutions against various d in a graph, we notice that not all solutions exist in all dimensions.

Only one solution exists in all dimensions, and is continuous in d. Besides, the ones which don't exist in all dimensions, have large critical exponents and are probably unphysical. In Fig. 2 we plot the position of the GMFP for 2 < d ≤ 11.

After having found the GMFP in various dimensions, we set to calculate their critical exponents. In arbitrary dimensions, the various blocks of the stability matrix obey eq. (31). We plot the critical exponents of M 00 for various dimensions. From the graph Fig. (3) we note that around d = 2.8 there is bifurcation. Below d < 2.8 the critical exponents are no more complex.

A summary of the properties of the GMFP in various dimensions is given in table (I). Notice that for all the dimensions considered, the real part of the critical exponents is greater than d -2 and less than 2(d -2). As a result, in all these cases there are exactly two pairs of complex conjugate critical exponents with positive real part, i.e. four relevant directions. Finally we studied the gauge dependence in different dimensions in the same wasy as we did in d = 4, for example in d = 6 we obtain the graphs shown in Fig. (4

V. OTHER NON TRIVIAL FIXED POINTS

Having discussed the existence and properties of the GMFP, we can ask ourselves whether there exist other FP where the scalar field has nontrivial self-interactions. We look for (truncated) polynomial FP potentials of the form

Ṽ ( φ2 ) = a n=0 λ2n φ2n ; F ( φ2 ) = b n=0 ξ2n φ2n . ( 62
)
with finite a ≥ 1, b ≥ 0. Such potentials are known not to exist in a pure scalar theory in four dimensions [START_REF] Halpern | [END_REF], so we consider it unlikely that they exist in the presence of gravity, In fact the outcome of our numerical searches is that no such FP's appear to exist in dimensions 4, 5 and 6. (Some FP do appear in certain truncations but not in others, so they are likely to be just truncation artifacts.) The situation is somewhat different in three dimensions. We know that pure scalar theory in d = 3 has the Wilson-Fisher FP [44]. This FP can be seen in our calculations by taking the limit G → 0 (where Newton's constant G is related to ξ 0 = 1/16πG) and λ0 → 0, in which case gravity decouples. Solving the FP equations of the scalar field in the LPA, truncated to order φ 4 , one gets λ * 2 = -0.0385 and λ * 4 = 0.3234, with critical exponents θ 1 = 1.843 and θ 2 = -1.176. (These are not very good values, but we quote them here for the sake of comparison with what we find in the presence of gravity.) The FP persists when one goes to higher truncations.

One wonders whether there exists a "gravitationally dressed" Wilson-Fisher FP, with nonvanishing G, namely a FP where gravity and the scalar simultaneously have nontrivial interactions. Again in certain truncations one finds various FPs which turn out to be truncation artifacts. There seems however to exist one genuine FP: we find it in all truncations where a ≥ b, and it has very similar properties in all truncations. To explore its properties we have looked in two directions: increasing simultaneously a and b, or keeping b = 0 and increasing a. Tables (II) and (III) give the position and critical exponents of this FP for a ≥ b and b ≤ 4. One notices that λ * 2n < 0 for all n > 0 in the table. It is computationally demanding to continue in this direction, so to have some indication on the sign of λ * 2n for higher n we considered a simple truncation where F is constant, i.e. ξn = 0 for n > 0. In this case we could push the truncation up to a = 8. The results are given in tables (IV) and (V). One sees that the coefficients of the potential are indeed all negative. Furthermore, the coefficients grow in absolute value, so the series for V has a very small radius of convergence. This is similar to the situation discussed in [START_REF] Halpern | [END_REF], making the FP unphysical. So we conclude that also in three dimensions there is probably no physically viable FP besides the GMFP.

VI. CONCLUSIONS

The results of this paper confirm and extend the findings of [35]. The GMFP is found to exist also in other dimensions and in other gauges, and (with the possible exception of d = 3) there does not seem to be other FP's with nontrivial scalar self-interactions. In four dimensions this agrees with the findings of [START_REF] Halpern | [END_REF].

These results may be applied in various settings. The beta functions given in Appendix A contain the full dependence on the dimensionless parameters λ0 , ξ0 , λ2 , ξ2 , λ4 , without making any assumption on the value of these couplings (which in the case of the first three means the ratio between the dimensionful couplings λ 0 , ξ 0 , λ 2 and the RG scale k). In particular, threshold effects are taken into account by the denominators 1 + 2 λ2 and ξ0 -λ0 . One can easily recognize among various terms the ones that are obtained in perturbative approximations, but we emphasize that the derivation of these beta functions using the FRGE does not require that the couplings be small.

The most natural application of these results seems to be in the context of early cosmology, where a scalar field is used to drive inflation. In an asymptotic safety context, it would be attractive to obtain inflation as a result of FP behavior along the lines of [23]. In fact the energy scale involved is sufficiently high that one may expect quantum gravity effects to play some role. Alternatively, it would also be of interest to apply the flow equations derived here to the scalar tensor theory, e.g. to improve the results of [37].

According to various speculations, quantum effects may play a role also on very large scales, and then again the RG flow of scalar-tensor theory could become relevant. In this connection we recall that scalar-tensor theories of a different type also arise in the conformal reduction of pure gravity, and have been studied from the FRGE point of view in [10,39].

We have mentioned in the Introduction that scalar-tensor theories can be reformulated classically also as pure gravity theories with f (R) type actions, and one may wonder whether there is a relation also between their RG flows. In particular one could ask whether the FP that was found in [13,14] has a counterpart in the equivalent scalar-tensor theory. At first sight one would think that this is not the case, because the choice of cutoff breaks the classical equivalence between these theories. Still, this point deserves a more detailed investigation.

Another direction for research is the inclusion of other matter fields. As discussed in the introduction, if asymptotic safety is indeed the answer to the UV issues of quantum field theory, then it will not be enough to establish asymptotic safety of gravity: one will have to establish asymptotic safety for a theory including gravity as well as all the fields that occur in the standard model, and perhaps even other ones that have not yet been discovered. Ideally one would like to have a unified theory of all interactions including gravity, perhaps a GraviGUT along the lines of [45]. More humbly one could start by studying the effect of gravity on the interactions of the standard model or GUTs. Fortunately, for some important parts of the standard model it is already known that an UV Gaussian FP exists, so the question is whether the coupling to gravity, or some other mechanism, can cure the bad behavior of QED and of the Higgs sector. That this might happen had been speculated long ago [33]; see also [START_REF] Buchbinder | Effective action in quantum gravity[END_REF] for some detailed calculations. It seem that the existence of a GMFP for all matter interactions would be the simplest solution to this issue. In this picture of asymptotic safety, gravity would be the only effective interaction at sufficiently high scale. The possibility of asymptotic safety in a nonlinearly realized scalar sector has been discussed in [START_REF] Codello | [END_REF]. Aside from scalar tensor theories, the effect of gravity has been studied in [48] for gauge couplings and [49] for Yukawa couplings.

FIG. 1 :

 1 FIG. 1: The gauge dependence of GMFP and critical exponents in d = 4. For convenience of depicting the values in plot, we have rescaled λ * 0 and ξ * 0 by 1000 and 100 respectively.

GFIG. 2 :

 2 FIG. 2:In the first graph we plot the GMFP λ * 0 and ξ * 0 in various dimensions. In the second plot we calculate the corresponding FP values of the cosmological constant Λ * and Newton's constant G * in various dimensions.

2 FIG. 3 :

 23 FIG.3: Critical exponents at the GMFP in various dimensions. The left panel shows the real part of the critical exponents, the right panel shows the imaginary part of the critical exponents. We note that below d = 2.8 the critical exponents becomes real.

FIG. 4 :

 4 FIG.4: The gauge dependence of GMFP and critical exponents in d = 6. For convenience of depicting the values in plot, we have rescaled λ * 0 and ξ * 0 by 10 4 and 10 3 respectively.

Figure 2

 2 Figure 2 (narain1/xi_gd4.eps)

Figure 3 2 Figure 9

 329 Figure 3 (narain1/th_gd4.eps)

Figure 10

 10 Figure 10 (narain1/th_gd6.eps)

TABLE I :

 I ). ×10 -5 3.738 ×10 -5 2.853 ×10 -1 5.322 ×10 2 8.945 + 10.904 i 8.945 -10.904 i 9 3.380 ×10 -6 5.747 ×10 -6 2.941 ×10 -1 3.462 ×10 3 11.396 + 12.748 i 11.396 -12.748 i 10 4.960 ×10 -7 8.228 ×10 -7 3.014 ×10 -1 2.418 ×10 4 14.089 + 14.537 i 14.089 -14.537 i 11 6.817 ×10 -8 1.107 ×10 -7 3.079 ×10 -1 1.797 ×10 5 17.025 + 16.261 i 17.025 -16.261 i Position of GMFP and critical exponents for various dimensions.

	d	λ * 0	ξ * 0	Λ *	G *	θ1	θ2
	2.001 4.968 ×10 -5 2.386 ×10 2 1.041 ×10 -7 8.339 ×10 -5	2.001	0.001
	3 1.605 ×10 -2 1.047 ×10 -1 7.666 ×10 -2 1.900 ×10 -1 1.627 + 0.754 i	1.627 -0.754 i
	4 8.620 ×10 -3 2.375 ×10 -2 1.814 ×10 -1 8.375 ×10 -1 2.143 + 2.879 i	2.143 -2.879 i
	5 2.669 ×10 -3 5.744 ×10 -3 2.323 ×10 -1	3.463	3.236 + 4.996 i	3.236 -4.996 i
	6 6.230 ×10 -4 1.207 ×10 -3 2.581 ×10 -1 1.648 ×10	4.818 + 7.039 i	4.818 -7.039 i
	7 1.225 ×10 -4 2.235 ×10 -4 2.740 ×10 -1 8.900 ×10	6.744 + 9.004 i	6.744 -9.004 i
	8 2.133

TABLE II :

 II Position of Nontrivial FP in d = 3 for various truncations.

		8
	21 0.0196 -0.1646 -0.1595	0.1088 -0.03108
	31 0.01994 -0.1758 -0.1958 -0.2796	0.1096 -0.03810
	41 0.02002 -0.1783 -0.2041 -0.3466 -0.5579 0.1098 -0.03969
	22 0.01894 -0.1408 -0.1241	0.1071 -0.01122 0.04297
	32 0.01971 -0.1680 -0.1848 -0.2879	0.1089 -0.03131 0.01731
	42 0.01988 -0.1735 -0.1975 -0.3544 -0.5687 0.1093 -0.03542 0.01121
	33 0.01911 -0.1469 -0.1469 -0.1935	0.1074 -0.01420 0.05017 0.1617
	43 0.01953 -0.1618 -0.1768 -0.3083 -0.6569 0.1084 -0.02571 0.03197 0.1102
	44 0.01923 -0.1512 -0.1572 -0.2496 -0.4911 0.1077 -0.01728 0.04732 0.1765 0.3868

TABLE III :

 III Critical exponents at Non trivial FP in d = 3 for various truncations. When the critical exponents are complex we write them in the form θ ± iθ

	(a, b) θ 1	θ 1	θ3	θ4	θ5	θ6	θ7
	21 1.648 0.592 -0.956 -3.902 -13.46
	31 1.650 0.554 -1.079 -3.776 -11.20 -29.397
	41 1.650 0.543 -1.105 -3.673 -10.02 -24.01 -49.31
	(a, b) θ 1	θ 1	θ 2	θ 2	θ5	θ6	θ7	θ8
	22 1.649 0.656 -7.979 1.261 -0.559 -3.192
	32 1.652 0.589 -7.933 3.909 -0.835 -3.578 -27.67
	42 1.652 0.570 -7.635 4.083 -0.898 -3.626 -22.64 -47.78
	(a, b) θ 1	θ 1	θ 2	θ 2	θ 3	θ 3	θ7	θ8	θ9
	33 1.649 0.641 -6.703 2.097 -14.12 8.990 -0.512 -2.991
	43 1.651 0.603 -6.448 3.343 -13.94 10.91 -0.657 -3.287 -42.28
	(a, b) θ 1	θ 1	θ 2	θ 2	θ 3	θ 3	θ 3	θ 3	θ9	θ10
	44 1.650 0.630 -5.958 2.008 -12.88 7.966 -20.07 19.03 -0.513 -2.977
	(a, b)	λ * 0	λ * 2	λ * 4	λ * 6	λ * 8	λ * 10	λ * 12	λ * 14	λ * 16	ξ * 0
	10 0.01813 -0.1088								0.1060
	20 0.01880 -0.1343 -0.1561							0.1065
	30 0.01894 -0.1395 -0.1942 -0.2633						0.1066
	40 0.01898 -0.1407 -0.2032 -0.3284 -0.4998				0.1066
	50 0.01899 -0.1410 -0.2053 -0.3437 -0.6182 -0.9604			0.1066
	60 0.01899 -0.1411 -0.2058 -0.3472 -0.6452 -1.180 -1.826		0.1066
	70 0.01899 -0.1411 -0.2059 -0.3479 -0.6511 -1.228 -2.229 -3.380	0.1066
	80 0.01899 -0.1411 -0.2059 -0.3481 -0.6524 -1.238 -2.313 -4.091 -5.977 0.1066

TABLE IV :

 IV Position of Nontrivial FP in d = 3 for other truncations.

	(a, b) θ 1	θ 1	θ3	θ4	θ5	θ6	θ7	θ8	θ9	θ10
	10 1.659 0.753 -1.699
	20 1.675 0.745 -1.594 -10.59
	30 1.679 0.742 -1.485 -8.384 -22.41
	40 1.68 0.741 -1.434 -7.341 -18.19 -36.96
	50 1.68 0.741 -1.414 -6.84 -15.93 -31.12 -54.10
	60 1.68 0.741 -1.407 -6.609 -14.70 -27.47 -47.25 -73.60
	70 1.68 0.740 -1.405 -6.509 -14.02 -25.287 -42.062 -66.663 -95.172
	80 1.68 0.740 -1.405 -6.469 -13.67 -23.94 -38.77 -59.72 -89.58 -118.4

TABLE V :

 V Critical Exponents at Non trivial FP in d = 3 for other truncations.

Note that the notation used in[35] is opposite to the one used here: parameters with a tilde are dimensionful, those without tilde are dimensionless. The beta functions written in[35] contain a number of transcription errors, which however do not affect the subsequent results.

APPENDIX A: EXPLICIT BETA FUNCTIONS

In section II.B we have presented equations which in principle determine the beta functionals for V and F in d = 4 and in the gauge α = 0 and β = 1. At one loop the beta functionals can be immediately read off from there, but if one wants to get the "improved" beta functionals (meaning that no approximation is made beyond the truncation), then the system is too complicated to be solved. It can be solved if we assume that V and F are finite polynomials. We give here the algebraic equations that determine the beta functions in a five coupling truncation, including λ 0 , ξ 0 , λ 2 , ξ 2 , λ 4 . We give here linear equations for ∂ t λi and ∂ t ξi which can be easily solved using algebraic manipulation software. This exercise will also enable us to compare with familiar one loop results. These beta functions had been written previously in [35], but since we had left the cutoff generic, it was not possible to compute the integrals over momenta, which are contained in the expressions Q 2 and Q 1 . 1 Here the integrals have already been performed, using an optimized cutoff [36], so the beta functions are in closed form and completely explicit.

If one neglects the terms involving ∂ t ξ0 and ∂ t ξ2 in the r.h.s., these are the one loop beta functions for the couplings. One can recognize among them some familiar terms. The term containing -18 λ4 in the first line of eq. (A3) and the term containing 216 λ2 4 in the third line of eq. (A5) are the familiar beta functions of the mass and of the coupling in φ 4 theory in flat space. The term containing 72λ 4 (1 + 12ξ 2 ) in the third line of eq. (A4) is also known from earlier calculations [37,[START_REF] Buchbinder | Effective action in quantum gravity[END_REF].

Notice the ubiquitous appearance of the factors 1/(1 + 2 λ2 ), which represent threshold effects for the contributions of scalar loops: for k 2 λ 2 , λ2 1 and the denominator can be approximated by 1, whereas for k 2 λ 2 , λ2 1 and the term is suppressed. The denominators λ0 -ξ0 have a somewhat similar effect. When written in terms of the more familiar variables Λ and G they give rise to factors (1 -2 Λ). These can be approximated by 1 when Λ k 2 but they vanish when the dimensionless cosmological constant Λ tends to 1/2, corresponding to an infrared singularity in the RG trajectories. This is well documented in the literature.

The term proportional to 12 ξ0 λ4 in the third line of eq. (A5) is the leading gravitational correction (of order G) to the running of the scalar self coupling. Note that for small λ0 , λ2 and ξ2 , the denominator and the bracket to its right can be expanded as 1 + O( λ0 ) + O( λ2 ) + O( ξ2 ). The order of magnitude and sign of this term agree with the calculations done in [34]. One should not expect the results to agree exactly, because this term is scheme dependent and the calculations were done using different cutoffs, and also in different gauges. Notice that this term is proportional to λ4 , thus when we set λ4 to zero, the beta function for λ4 does not get any contribution from gravity, in agreement with the general statement that minimal coupling is self consistent. We observe that the same phenomenon happens in the case of the Yukawa coupling [49] and of the gauge coupling [48].


Mark Volfovich Losik

Anton S Galaev
email: galaev@math.muni.cz

Holonomy of Einstein Lorentzian manifoldsDedicated to

Keywords: numbers: 04, 20, Jb, 04, 40, Nr

The classification of all possible holonomy algebras of Einstein and vacuum Einstein Lorentzian manifolds is obtained. It is shown that each such algebra appears as the holonomy algebra of an Einstein (resp., vacuum Einstein) Lorentzian manifold, the direct constructions are given. Also the holonomy algebras of totally Ricci-isotropic Lorentzian manifolds are classified. The classification of the holonomy algebras of Lorentzian manifolds is reviewed and a complete description of the spaces of curvature tensors for these holonomies is given.

Introduction

In contrast to the case of Riemannian manifolds, where the classification of holonomy algebras is a classical result, which has a lot of consequences and applications both in geometry and physics (e.g. Riemannian manifolds with most of holonomy algebras are automatically Einstein or vacuum Einstein), see [START_REF] Alekseevsky | Riemannian manifolds with exceptional holonomy groups[END_REF][START_REF] Berger | Sur les groupers d'holonomie des variétés àconnexion affine et des variétés riemanniennes[END_REF][START_REF] Besse | Einstein manifolds[END_REF][START_REF] Joyce | Compact manifolds with special holonomy[END_REF][START_REF] Joyce | Riemannian holonomy groups and calibrated geometry[END_REF] and the references therein, the classification of the holonomy algebras of Lorentzian manifolds is achieved only recently [START_REF] Bergery | On the Holonomy of Lorentzian Manifolds[END_REF][START_REF] Leistner | On the classification of Lorentzian holonomy groups[END_REF][START_REF] Galaev | Metrics that realize all Lorentzian holonomy algebras[END_REF][START_REF] Galaev | Holonomy groups of Lorentzian manifolds: classification, examples, and applications[END_REF] (we recall it in Section 2). The most interesting case is when a Lorentzian manifold (M, g) admits a parallel distribution of isotropic lines and the manifold is locally indecomposable, i.e. locally it is not a product of a Lorentzian and a Riemannian manifold. In this case the holonomy algebra is contained in the maximal subalgebra sim(n) = (R ⊕ so(n)) ⋉ R n of the Lorentzian algebra so(1, n + 1) preserving an isotropic line (the dimension of M is n + 2). There is a number of recent physics literature dealing with these manifolds, see e.g. [START_REF] Brannlund | Holonomy, decomposability, and relativity[END_REF][START_REF] Brannlund | Supersymmetry, holonomy and Kundt spacetimes[END_REF][START_REF] Coley | Metrics with vanishing quantum corrections[END_REF][START_REF] Coley | Supergravity solutions with constant scalar invariants[END_REF][START_REF] Coley | Kundt spacetimes[END_REF][START_REF] Figueroa-O'farrill | Breaking the M-waves[END_REF][START_REF] Gibbons | Time-Dependent Multi-Centre Solutions from New Metrics with Holonomy Sim(n -2)[END_REF][START_REF] Gibbons | Holonomy Old and New[END_REF][START_REF] Hernandez | Supersymmetry and Lorentzian holonomy in various dimensions[END_REF]. In particular, in [START_REF] Brannlund | Holonomy, decomposability, and relativity[END_REF][START_REF] Brannlund | Supersymmetry, holonomy and Kundt spacetimes[END_REF][START_REF] Gibbons | Holonomy Old and New[END_REF] expressed the hope that Lorentzian manifolds with the holonomy algebras contained in sim(n) will find many applications in physics, e.g. they are of interest in M-theory and string theory.

The fundamental equation of General Relativity is the Einstein equation. In the absence of matter it has the form Ric = Λg, [START_REF] Alekseevsky | Riemannian manifolds with exceptional holonomy groups[END_REF] where g is a Lorentzian metric on a manifold M, Ric is the Ricci tensor of the metric g and Λ ∈ R is the cosmological constant. If the metric of a Lorentzian manifold (M, g) satisfies this equation, then (M, g) is called an Einstein manifold. If moreover Λ = 0, then it is called vacuum Einstein or Ricci-flat. In dimension 4 the solutions of this equation are obtained in [START_REF] Schell | Classification of four-dimensional Riemannian spaces[END_REF][START_REF] Ghanam | Two special metrics with R 14 -type holonomy[END_REF][START_REF] Hall | Holonomy groups and spacetimes[END_REF][START_REF] Kerr | Some applications of the infinitesimal-holonomy group to the Petrov classification of Einstein spaces[END_REF][START_REF] Kerr | Einstein spaces with four-parameter holonomy groups[END_REF]. In [START_REF] Bryant | Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor[END_REF][START_REF] Figueroa-O'farrill | Breaking the M-waves[END_REF] are found the holonomy algebras of vacuum Einstein Lorentzian spin manifolds up to dimension 11 admitting parallel spinors and some methods of construction of such manifolds are introduced. In dimension 11 these manifolds are purely gravitational supersymmetric solutions of 11dimensional supergravity. Other constructions are provided in [START_REF] Hernandez | Supersymmetry and Lorentzian holonomy in various dimensions[END_REF]. Recently in [START_REF] Gibbons | Time-Dependent Multi-Centre Solutions from New Metrics with Holonomy Sim(n -2)[END_REF] the Einstein equation on Lorentzian manifolds with holonomy algebras contained in sim(n) is studied and in some cases solutions are obtained. See [START_REF] Gibbons | Time-Dependent Multi-Centre Solutions from New Metrics with Holonomy Sim(n -2)[END_REF] for the discussion of the importance of this problem and the references there.

In the present paper we classify all possible holonomy algebras of Einstein and vacuum Einstein Lorentzian manifolds. First, in [START_REF] Galaev | Holonomy groups of Lorentzian manifolds: classification, examples, and applications[END_REF] it is proved that if (M, g) is Einstein, then its holonomy algebra coincides either with

(R ⊕ h) ⋉ R n ⊂ sim(n), or with h ⋉ R n ⊂ sim(n).
Here h ⊂ so(n) is the holonomy algebra of a Riemannian manifold. In general for such h ⊂ so(n) there is an orthogonal decomposition

R n = R n 1 ⊕ • • • ⊕ R ns ⊕ R n s+1 (2)
and the corresponding decomposition into the direct sum of ideals

h = h 1 ⊕ • • • ⊕ h s ⊕ {0} (3)
such that h annihilates R n s+1 , h i (R n j ) = 0 for i = j, and h i ⊂ so(n i ) is an irreducible subalgebra for 1 ≤ i ≤ s. Moreover, the Lie subalgebras h i ⊂ so(n i ) are the holonomy algebras of Riemannian manifolds. Consequently, each h i ⊂ so(n i ) is either one of the Lie algebras so(n i ), u( n i 2 ), su( n i 2 ), sp( n i 4 ), sp( n i 4 ) ⊕ sp(1), G 2 ⊂ so [START_REF] Bryant | Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor[END_REF], spin(7) ⊂ so [START_REF] Coley | Metrics with vanishing quantum corrections[END_REF], or it is a symmetric Berger algebra (the last are the Riemannian holonomy algebras such that any manifold with such holonomy is locally symmetric, these Lie algebras are listed e.g. in [START_REF] Besse | Einstein manifolds[END_REF]). In Section 4 we prove the following two theorems Theorem 4. If (M, g) is vacuum Einstein, then one of the following holds:

(1) The holonomy algebra of (M, g) coincides with (R⊕h)⋉R n , and in the decomposition (3) of h ⊂ so(n) at least one subalgebra h i ⊂ so(n i ) coincides with one of the Lie algebras so(n i ), u( n i 2 ), sp( n i 4 ) ⊕ sp(1) or with a symmetric Berger algebra. (2) The holonomy algebra of (M, g) coincides with h ⋉ R n , and in the decomposition

(3) of h ⊂ so(n) each subalgebra h i ⊂ so(n i ) coincides with one of the Lie algebras so(n i ), su( n i 2 ), sp( n i 4 ), G 2 ⊂ so [START_REF] Bryant | Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor[END_REF], spin(7) ⊂ so(8). Theorem 5. If (M, g) is Einstein and not vacuum Einstein, then the holonomy algebra of (M, g) coincides with (R ⊕ h) ⋉ R n , and in the decomposition (3) of h ⊂ so(n) each subalgebras h i ⊂ so(n i ) coincides with one of the Lie algebras so(n i ), u( n i 2 ), sp( n i 4 ) ⊕ sp(1) or with a symmetric Berger algebra. Moreover, in the decomposition (2) it holds n s+1 = 0.

In Section 5 we construct local Einstein metrics having holonomy algebras from Theorems 4 and 5. We consider metrics written in Walker coordinates of the form

g = 2dx 0 dx n+1 + n i,j=1 h ij (x 1 , ..., x n )dx i dx j + f (x 0 , ..., x n+1 )(dx n+1 ) 2 , (4)
and

g = 2dx 0 dx n+1 + n i=1 (dx i ) 2 +2 n i=1 u i (x 1 , ..., x n+1 )dx i dx n+1 +f (x 0 , ..., x n+1 )(dx n+1 ) 2 . ( 5
)
Here h = n i,j=1 h ij (x 1 , ..., x n )dx i dx j is a Riemannian metric. For the metric (4) we choose h to be an Einstein Riemannian metric with a cosmological constant Λ = 0 (resp., Λ = 0) and holonomy algebra h ⊂ so(n) as in Theorem 5 (resp., (2) of Theorem 4). Choosing an appropriate function f , we get that the metric g is Einstein with the cosmological constant Λ and the holonomy algebra (R ⊕ h) ⋉ R n (resp., h ⋉ R n ). Next, for a given holonomy algebra h ⊂ so(n) as in (1) (resp., (2)) of Theorem 4 we give an algorithm how to choose the functions u i and f in (5) to make the metric g vacuum Einstein with the holonomy algebra (R⊕h)⋉R n (resp., h⋉R n ). As examples, we provide vacuum Einstein Lorentzian metrics with the holonomy algebras G 2 ⋉ R 7 ⊂ so(1, 8) and spin(7) ⋉ R 8 ⊂ so [START_REF] Alekseevsky | Riemannian manifolds with exceptional holonomy groups[END_REF][START_REF] Coley | Supergravity solutions with constant scalar invariants[END_REF].

If (M, g) is a spin Lorentzian manifold and it admits a parallel spinor, then it is totally Ricci-isotropic (but not necessary vacuum Einstein, unlike in the Riemannian case), i.e. the image of its Ricci operator is isotropic [START_REF] Bryant | Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor[END_REF][START_REF] Figueroa-O'farrill | Breaking the M-waves[END_REF]. We prove the following two theorems Theorem 7. If (M, g) is totally Ricci-isotropic, then its holonomy algebra is the same as in Theorem 4.

Theorem 8. If the holonomy algebra of (M, g) is h ⋉ R n and in the decomposition (3) of h ⊂ so(n) each subalgebra h i ⊂ so(n i ) coincides with one of the Lie algebras su( n i 2 ), sp( n i 4 ), G 2 ⊂ so [START_REF] Bryant | Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor[END_REF], spin(7) ⊂ so [START_REF] Coley | Metrics with vanishing quantum corrections[END_REF], then (M, g) is totally Ricci-isotropic. Recall that an indecomposable Riemannian manifold with the holonomy algebra different from so(n) and u( n 2 ) is automatically Einstein or vacuum Einstein [START_REF] Besse | Einstein manifolds[END_REF]. This is not the case for Lorentzian manifolds. Indeed, using results of Section 5 it is easy to construct non-Einstein metrics having all holonomy algebras as in Theorems 4 and 5. On the other hand, Theorem 8 shows that Lorentzian manifolds with some holonomy algebras are automatically totally Ricci-isotropic.

To prove the above theorems we need the complete description of the space of curvature tensors for Lorentzian holonomy algebras, i.e. the space of values of the curvature tensor of a Lorentzian manifold, we provide it in Section 3. The study of these spaces is begun in [START_REF] Galaev | The spaces of curvature tensors for holonomy algebras of Lorentzian manifolds[END_REF] and finished recently in [START_REF] Galaev | One component of the curvature tensor of a Lorentzian manifold[END_REF].

Necessary facts from the holonomy theory can be found e.g. in [START_REF] Besse | Einstein manifolds[END_REF][START_REF] Brannlund | Supersymmetry, holonomy and Kundt spacetimes[END_REF][START_REF] Galaev | Holonomy groups of Lorentzian manifolds: classification, examples, and applications[END_REF][START_REF] Joyce | Compact manifolds with special holonomy[END_REF][START_REF] Joyce | Riemannian holonomy groups and calibrated geometry[END_REF]. Remark that by a Riemannian (resp., Lorentzian) manifold we understand a manifold with a field g of positive definite (resp., of signature (-, +, ..., +)) symmetric bilinear forms on the tangent spaces.

Classification of the Lorentzian holonomy algebras

Let (R 1,n+1 , η) be the Minkowski space of dimension n + 2, where η is a metric on R n+2 of signature (1, n + 1). We consider (R 1,n+1 , η) as the tangent space (T x M, g x ) to a Lorentzian manifold (M, g) at a point x. We fix a basis p, e 1 , ..., e n , q of R 1,n+1 such that the only non-zero values of η are η(p, q) = η(q, p) = 1 and η(e i , e i ) = 1. We will denote by R n ⊂ R 1,n+1 the Euclidean subspace spanned by the vectors e 1 , ..., e n .

Recall that a subalgebra g ⊂ so(1, n + 1) is called irreducible if it does not preserve any proper subspace of R 1,n+1 ; g is called weakly-irreducible if it does not preserve any non-degenerate proper subspace of R 1,n+1 . Obviously, if g ⊂ so(1, n + 1) is irreducible, then it is weakly-irreducible. From the Wu Theorem [START_REF] Wu | On the de Rham decomposition theorem[END_REF] it follows that any Lorentzian manifold (M, g) is either locally a product of the manifold (R, -(dt) 2 ) and of a Riemannian manifold, or of a Lorentzian manifold with weaklyirreducible holonomy algebra and of a Riemannian manifold. The Riemannian manifold can be further decomposed into the product of a flat Riemannian manifold and of Riemannian manifolds with irreducible holonomy algebras. If the manifold (M, g) is simply connected and geodesically complete, then these decompositions are global. This allow us to consider locally indecomposable Lorentzian manifolds, i.e. manifolds with weakly-irreducible holonomy algebras. For example, a locally decomposable Lorentzian manifold (M, g) is Einstein if and only if locally it is a product of Einstein manifolds with the same cosmological constants as (M, g). The only irreducible Lorentzian holonomy algebra is the whole Lie algebra so(1, n + 1) ( [START_REF] Berger | Sur les groupers d'holonomie des variétés àconnexion affine et des variétés riemanniennes[END_REF]), so we consider weakly-irreducible not irreducible holonomy algebras.

Denote by sim(n) the subalgebra of so(1, n + 1) that preserves the isotropic line Rp. The Lie algebra sim(n) can be identified with the following matrix algebra:

sim(n) =         a X t 0 0 A -X 0 0 -a    a ∈ R, X ∈ R n , A ∈ so(n)      . ( 6
)
The above matrix can be identified with the triple (a, A, X). We get the decomposition

sim(n) = (R ⊕ so(n)) ⋉ R n , which means that R ⊕ so(n) ⊂ sim(n) is a subalgebra and R n ⊂ sim(n) is an ideal,
and the Lie brackets of R ⊕ so(n) with R n are given by the standard representation of R ⊕ so(n) in R n . We see that sim(n) is isomorphic to the Lie algebra of the Lie group of similarity transformations of R n . The explicit isomorphism on the group level is constructed in [START_REF] Galaev | Isometry groups of Lobachevskian spaces, similarity transformation groups of Euclidean spaces and Lorentzian holonomy groups[END_REF].

If a weakly-irreducible subalgebra g ⊂ so(1, n + 1) preserves a degenerate proper subspace U ⊂ R 1,n+1 , then it preserves the isotropic line U ∩ U ⊥ , and g is conjugated to a weakly-irreducible subalgebra of sim(n). Let h ⊂ so(n) be a subalgebra. Recall that h is a compact Lie algebra and we have the decomposition h = h ′ ⊕ z(h), where h ′ is the commutant of h and z(h) is the center of h.

The next theorem gives the classification of weakly-irreducible not irreducible holonomy algebras of Lorentzian manifolds.

Theorem 1 A subalgebra g ⊂ sim(n) is the weakly-irreducible holonomy algebra of a Lorentzian manifold if and only if it is conjugated to one of the following subalgebras:

type 1. g 1,h = (R ⊕ h) ⋉ R n , where h ⊂ so(n) is the holonomy algebra of a Riemannian manifold; type 2. g 2,h = h ⋉ R n
, where h ⊂ so(n) is the holonomy algebra of a Riemannian manifold;

type 3. g 3,h,ϕ = {(ϕ(A), A, 0)|A ∈ h} ⋉ R n
, where h ⊂ so(n) is the holonomy algebra of a Riemannian manifold with z(h) = {0}, and ϕ : h → R is a non-zero linear map with ϕ| h ′ = 0;

type 4. g 4,h,m,ψ = {(0, A, X + ψ(A))|A ∈ h, X ∈ R m }, where 0 < m < n is an integer, h ⊂ so(m)
is the holonomy algebra of a Riemannian manifold with dim z(h) ≥ n-m, a decomposition R n = R m ⊕ R n-m is fixed, and ψ : h → R n-m is a surjective linear map with ψ| h ′ = 0.

The subalgebra h ⊂ so(n) associated to a weakly-irreducible Lorentzian holonomy algebra g ⊂ sim(n) is called the orthogonal part of g. Recall that for the subalgebra h ⊂ so(n) there are the decompositions (2) and (3). This theorem is the result of the papers [START_REF] Bergery | On the Holonomy of Lorentzian Manifolds[END_REF][START_REF] Leistner | On the classification of Lorentzian holonomy groups[END_REF][START_REF] Galaev | Metrics that realize all Lorentzian holonomy algebras[END_REF], see [START_REF] Galaev | Holonomy groups of Lorentzian manifolds: classification, examples, and applications[END_REF] for the whole history.

The spaces of curvature tensors

Let W be a vector space and f ⊂ gl(W ) a subalgebra. The vector space

R(f) = {R ∈ Λ 2 W * ⊗ f|R(u, v)w + R(v, w)u + R(w, u)v = 0 for all u, v, w ∈ W } is called the space of curvature tensors of type f. A subalgebra f ⊂ gl(W ) is called a Berger algebra if g = span{R(u, v)|R ∈ R(f), u, v ∈ W },
i.e. g is spanned by the images of the elements R ∈ R(f).

If there is a pseudo-Euclidean metric η on W such that f ⊂ so(W ), then any

R ∈ R(f) satisfies η(R(u, v)z, w) = η(R(z, w)u, v) (7)
for all u, v, z, w ∈ W . We identify so(W ) with Λ 2 W such that it holds (u ∧ v)(z) = η(u, z)v -η(v, z)u for all u, v, z ∈ W . E.g. under this identification the matrix from ( 6) corresponds to the bivector -ap ∧ q + A -p ∧ X, A ∈ so(n) ≃ Λ 2 R n . Equality [START_REF] Bryant | Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor[END_REF] shows that the map R : Λ 2 W → f ⊂ Λ 2 W is symmetric with respect to the metric on Λ 2 W . In particular, R is zero on the orthogonal complement f ⊥ ⊂ Λ 2 W . Thus, R ∈ ⊙ 2 f. In [START_REF] Galaev | The spaces of curvature tensors for holonomy algebras of Lorentzian manifolds[END_REF] the following theorem is proved.

Theorem 2 [START_REF] Galaev | The spaces of curvature tensors for holonomy algebras of Lorentzian manifolds[END_REF] It holds:

(1) each R ∈ R(g 1,h
) is uniquely given by

λ ∈ R, v ∈ R n , P ∈ P(h), R 0 ∈ R(h), and T ∈ End(R n ) with T * = T
in the following way

R(p, q) =(λ, 0, v), R(x, y) = (0, R 0 (x, y), P (y)x -P (x)y), R(x, q) =(η(v, x), P (x), T (x)), R(p, x) = 0 for all x, y ∈ R n ; (2) R ∈ R(g 2,h ) if and only if R ∈ R(g 1,h ), λ = 0 and v = 0; (3) R ∈ R(g 3,h,ϕ ) if and only if R ∈ R(g 1,h ), λ = 0, R 0 ∈ R(ker ϕ), and η(x, v) = φ(P (x)) for all x ∈ R n ; (4) R ∈ R(g 4,h,m,ψ ) if and only if R ∈ R(g 1,h ), λ = 0, v = 0, R 0 ∈ R(ker ψ), and pr R n-m •T = ψ • P .
Remark that the decomposition R(g

1,h ) = R ⊕ R n ⊕ ⊙ 2 R n ⊕ R(h) ⊕ P(h) is R ⊕ h- invariant, but not g 1,h -invariant.
Recall that for the subalgebra h ⊂ so(n) there are the decompositions (2) and (3). In addition we have the decompositions

P(h) = P(h 1 ) ⊕ • • • ⊕ P(h s ) and R(h) = R(h 1 ) ⊕ • • • ⊕ R(h s ).
The spaces R(h) for the holonomy algebras of Riemannian manifolds h ⊂ so(n) are computed by D. V. Alekseevsky in [START_REF] Alekseevsky | Riemannian manifolds with exceptional holonomy groups[END_REF]. Let h ⊂ so(n) be an irreducible subalgebra. The space R(h) admits the following decomposition into h-modules

R(h) = R 0 (h) ⊕ R 1 (h) ⊕ R ′ (h), (8)
where R 0 (h) consists of the curvature tensors with zero Ricci tensor, R 1 (h) consists of tensors annihilated by h (this space is zero or one-dimensional), R ′ (h) is the complement to these two spaces. Any element of R ′ (h) has zero scalar curvature and non-zero Ricci tensor. If R(h) = R 1 (h), then any Riemannian manifold with the holonomy algebra h is locally symmetric (such h ⊂ so(n) is called a symmetric Berger algebra).

Remark that any locally symmetric Riemannian manifold is Einstein and not vacuum Einstein. Note that R(h) = R 0 (h) if h is any of the algebras: su( n 2 ), sp( n 4 ), G 2 ⊂ so(7), spin(7) ⊂ so [START_REF] Coley | Metrics with vanishing quantum corrections[END_REF]. This implies that each Riemannian manifold with any of these holonomy algebras is vacuum Einstein. Next, R(u

( n 2 )) = R ⊕ R ′ (u( n 2 )) ⊕ R(su( n 2 )) and R(sp( n 4 ) ⊕ sp(1)) = R ⊕ R(sp( n 4
)). Hence any Riemannian manifold with the holonomy algebra sp( n 4 ) ⊕ sp(1) is Einstein and not vacuum Einstein, and a Riemannian manifold with the holonomy algebra u( n 2 ) can not be vacuum Einstein. Finally, if an indecomposable n-dimensional Riemannian manifold is vacuum Einstein, then its holonomy algebra is one of so(n), su( n 2 ), sp( n 4 ), G 2 ⊂ so [START_REF] Bryant | Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor[END_REF], spin(7) ⊂ so [START_REF] Coley | Metrics with vanishing quantum corrections[END_REF]. Now we turn to the space P(h), where h ⊂ so(n) is an irreducible subalgebra. Consider the h-equivariant map Ric :

P(h) → R n , Ric(P ) = n i=1 P (e i )e i .
This definition does not depend on the choice of the orthogonal basis e 1 , ..., e n of R n . Denote by P 0 (h) the kernel of Ric and let P 1 (h) be its orthogonal complement in P(h). Thus, P(h) = P 0 (h) ⊕ P 1 (h).

Since h ⊂ so(n) is irreducible and the map Ric is h-equivariant, P 1 (h) is either trivial or isomorphic to R n . The spaces P(h) for irreducible Riemannian holonomy algebra h ⊂ so(n) are computed recently in [START_REF] Galaev | One component of the curvature tensor of a Lorentzian manifold[END_REF]. In particular, P 0 (h) = 0 and P 1 (h) = 0 exactly for the holonomy algebras su( n 2 ), sp( n 4 ), spin( 7) and G 2 . Next, P 1 (h) ≃ R n and P 0 (h) = 0 exactly for the holonomy algebras so(n), u( n 2 ) and sp( n 4 ) ⊕ sp(1). For the rest of the Riemannian holonomy algebras (i.e. for the symmetric Berger algebras) it holds P 1 (h) ≃ R n and P 0 (h) = 0.

To make the exposition complete, we give a description of the space R(so(1, n + 1)) that follows immediately from [START_REF] Alekseevsky | Riemannian manifolds with exceptional holonomy groups[END_REF]. The space R(so(1, n + 1)) admits the decomposition [START_REF] Coley | Metrics with vanishing quantum corrections[END_REF]. The complexification R 0 (so

(1, n + 1)) ⊗ C of R 0 (so(1, n + 1)) is isomorphic to the so(n + 2, C)-module V 2π 2 (if n ≥ 3). Similarly, R 0 (so(1, 3)) ⊗ C ≃ V 4π 1 ⊕ V 4π ′ 1 , R 0 (so(1, 2)) = 0, and R 0 (so(1, 1)) = 0. It holds R ′ (so(1, n + 1)) ≃ (⊙ 2 R 1,n+1 ) 0 = ⊙ 2 R 1,n+1 /Rη, any R ∈ R ′ (so(1, n+1)) is of the form R = R S , where S : R 1,n+1 → R 1,n+1
is a symmetric linear map with zero trace and

R S (u, v) = Su ∧ v + u ∧ Sv. Similarly, R 1 (so(1, n + 1)) is spanned by the element R = Rid 2 , i.e. R(u, v) = u ∧ v.
This shows that an (n + 2)-dimensional Lorentzian manifold (n ≥ 2) with the holonomy algebra so(1, n+1) may be either vacuum Einstein, or Einstein and not vacuum Einstein, or not Einstein. Finally, there are no vacuum Einstein indecomposable Lorentzian manifolds of dimensions 2 or 3.

Applications to Einstein and vacuum Einstein Lorentzian manifolds

Now we are able to find all holonomy algebras of Einstein and vacuum Einstein Lorentzian manifolds.

Let R ∈ R(g 1,h ) be as in Theorem 2, then its Ricci tensor Ric = Ric(R) satisfies Ric(p, q) = -λ, Ric(x, y) = Ric(R 0 )(x, y),

Ric(x, q) =η(x, Ric(P ) -v), Ric(q, q) = tr T, [START_REF] Coley | Kundt spacetimes[END_REF] where x, y ∈ R n (recall that Ric(u, v) = tr(z → R(u, z)v)).

Obviously, these equations imply that there is no three-dimensional indecomposable Einstein Lorentzian manifolds with holonomy algebras contained in sim(1) = R ⋉ R. Thus we may assume that n ≥ 2.

Here is a result from [START_REF] Galaev | Holonomy groups of Lorentzian manifolds: classification, examples, and applications[END_REF].

Theorem 3 [START_REF] Galaev | Holonomy groups of Lorentzian manifolds: classification, examples, and applications[END_REF] Let (M, g) be a locally indecomposable Lorentzian Einstein manifold admitting a parallel distribution of isotropic lines. Then the holonomy of (M, g) is either of type 1 or 2. If the cosmological constant of (M, g) is non-zero, then the holonomy algebra of (M, g) is of type 1. If (M, g) admits locally a parallel isotropic vector field, then (M, g) is vacuum Einstein.

The classification completes the following two theorems.

Theorem 4 Let (M, g) be a locally indecomposable n + 2-dimensional Lorentzian manifold admitting a parallel distribution of isotropic lines. If (M, g) is vacuum Einstein, then one of the following holds:

(1) The holonomy algebra g of (M, g) is of type 1, and in the decomposition (3) of the orthogonal part h ⊂ so(n) at least one subalgebra h i ⊂ so(n i ) coincides with one of the Lie algebras so(n i ), u( n i 2 ), sp( n i 4 ) ⊕ sp(1) or with a symmetric Berger algebra.

(2) The holonomy algebra g of (M, g) is of type 2, and in the decomposition (3) of the orthogonal part h ⊂ so(n) each subalgebra h i ⊂ so(n i ) coincides with one of the Lie algebras so(n i ), su( n i 2 ), sp( n i 4 ), G 2 ⊂ so [START_REF] Bryant | Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor[END_REF], spin(7) ⊂ so(8). Theorem 5 Let (M, g) be a locally indecomposable n + 2-dimensional Lorentzian manifold admitting a parallel distribution of isotropic lines. If (M, g) is Einstein and not vacuum Einstein, then the holonomy algebra g of (M, g) is of type 1, and in the decomposition (3) of the orthogonal part h ⊂ so(n) each subalgebras h i ⊂ so(n i ) coincides with one of the Lie algebras so(n i ), u( n i 2 ), sp( n i 4 ) ⊕ sp(1) or with a symmetric Berger algebra. Moreover, h ⊂ so(n) does not annihilate any proper subspace of R n , i.e. in the decomposition (2) it holds n s+1 = 0.

Recall that the list of irreducible symmetric Berger algebras h ⊂ so(n) can be obtained from the list of the holonomy algebras of irreducible Riemannian symmetric spaces (this list is given e.g. in [START_REF] Besse | Einstein manifolds[END_REF]) amiting so(n), u( n 2 ) and sp( n 4 ) ⊕ sp(1). Proof of Theorems 4 and 5. Fix a point x ∈ M and let g be the holonomy algebra of (M, g) at this point. We identify (T x M, g x ) with (R 1,n+1 , η). By the Ambrose-Singer Theorem [START_REF] Besse | Einstein manifolds[END_REF], g is spanned by the images of the elements

R γ = τ -1 γ •R y (τ γ •, τ γ •)•τ γ ,
where γ is a piecewise smooth curve in M starting at the point x and with an end-point y ∈ M, and τ γ : T x M → T y M is the parallel transport along γ. All these elements belong to the space R(g) and they can be given as in Theorem 2. Suppose that (M, g) is vacuum Einstein. By Theorem 3, g is either of type 1 or 2. If g is of type 2, then from ( 9) and ( 10) it follows that each R γ satisfies λ = 0, v = 0, Ric(R 0 ) = 0, Ric(P ) = 0, and tr T = 0. Hence the orthogonal part h ⊂ so(n) of g is spanned by the images of elements of R 0 (h) and P 0 (h). From [START_REF] Galaev | One component of the curvature tensor of a Lorentzian manifold[END_REF] it follows that h ⊂ so(n) is spanned by the images of elements of R 0 (h). Thus h is the holonomy algebra of a vacuum Einstein Riemannian manifold. If g is of type 1, then each R γ satisfies λ = 0, v = Ric(P ), Ric(R 0 ) = 0, and tr T = 0. Hence for some element R γ it holds Ric(P ) = 0, i.e. at list for one h i ⊂ so(n i ) in the decomposition (3) it holds P 1 (h i ) = 0. If (M, g) is Einstein with the cosmological constant Λ = 0, then by Theorem 3, g is of type 1. We get that the curvature tensor R x at the point x given by ( 2) satisfies λ = -Λ, v = Ric(P ), and Ric(R 0 ) = Λη| R n ⊗R n . Hence for each h i ⊂ so(n i ) in the decomposition (3) it holds R 1 (h i ) = 0, and n s+1 = 0.

Remark. A simple version of Theorem 4 for Lorentzian manifolds with holonomy algebras of type 2 is proved in [START_REF] Galaev | Holonomy groups of Lorentzian manifolds: classification, examples, and applications[END_REF], where the possibility for h i ⊂ so(n i ) to coincide with the holonomy algebra of a symmetric Riemannian non-Kählerian space was not excluded.

Examples of Einstein and vacuum Einstein Lorentzian metrics

On an n + 2-dimensional Lorentzian manifold (M, g) admitting a parallel distribution of isotropic lines there exist local coordinates (the Walker coordinates) x 0 , ..., x n+1 such that the metric g has the form [START_REF] Figueroa-O'farrill | Breaking the M-waves[END_REF] where h(x n+1 ) = n i,j=1 h ij (x 1 , ..., x n+1 )dx i dx j is a family of Riemannian metrics depending on the coordinate x n+1 [START_REF] Walker | On parallel fields of partially null vector spaces[END_REF]. The parallel distribution of isotropic lines is defined by the vector field ∂ 0 (we denote ∂ ∂x a by ∂ a ). In [START_REF] Gibbons | Time-Dependent Multi-Centre Solutions from New Metrics with Holonomy Sim(n -2)[END_REF] the Einstein equations for the general metric [START_REF] Figueroa-O'farrill | Breaking the M-waves[END_REF] are written down and some solutions of this system under some assumptions on the coefficients are found. Of course, it is not possible to solve such system in general.

g = 2dx 0 dx n+1 + n i,j=1 h ij (x 1 , ..., x n+1 )dx i dx j + 2 n i=1 u i (x 1 , ..., x n+1 )dx i dx n+1 + f (x 0 , ..., x n+1 )(dx n+1 ) 2 ,
In this section we show the existence of a local metric for each holonomy algebra obtained in the previous section. It is easy to see that if the metric [START_REF] Figueroa-O'farrill | Breaking the M-waves[END_REF] is Einstein, then each Riemannian metric in the family h(x n+1 ) is Einstein with the same cosmological constant.

We will take some special h ij and u i in [START_REF] Figueroa-O'farrill | Breaking the M-waves[END_REF], then the condition on the metric g to be Einstein will be equivalent to a system of equations on the function f . In each case we will show the existence of a proper function f satisfying these equations. This will imply the existence of an Einstein metric with each holonomy algebra obtained above.

First consider the metric [START_REF] Figueroa-O'farrill | Breaking the M-waves[END_REF] with h ij independent of x n+1 and u i (x 1 , ..., x n+1 ) = 0 for all i = 1, ..., n. The holonomy algebras of such metrics are found in [START_REF] Bergery | On the Holonomy of Lorentzian Manifolds[END_REF]. If f is sufficiently general, e.g. its Hessian is non-zero, then the holonomy algebra g of this metric is weakly-irreducible. If ∂ 0 f = 0, then g = h ⋉ R n , where h is the holonomy algebra of the Riemannian metric h; if ∂ 0 f = 0, then g = (R ⊕ h) ⋉ R n . In addition we need to choose h and f to make g Einstein or vacuum Einstein. The Ricci tensor Ric(g) for such metric has the following non-zero components:

Ric 0 n+1 = 1 2 (∂ 0 ) 2 f, (12)
Ric i j = Ric i j (h), i, j = 1, ..., n, (13)
Ric i n+1 = 1 2 ∂ 0 ∂ i f, i = 1, ..., n, (14)
Ric n+1 n+1 = 1 2 f (∂ 0 ) 2 f -∆f , (15)
where ∆f

= n i,j=1 h ij ∂ i ∂ j f -n k=1 Γ k ij ∂ k f
is the Laplace-Beltrami operator of the metric h applied to f . Suppose that g is vacuum Einstein, then the metric h should be vacuum Einstein as well. Next, ∂ 0 f = 0 and ∆f = 0. Let f be any harmonic function with non-zero Hessian, locally such functions always exist. E.g. if h is flat, then we may take

f = (x 1 ) 2 + • • • + (x n-1 ) 2 -(n -1)(x n ) 2 . Thus, the metric g is vacuum Einstein and g = h ⋉ R n .
Suppose that g is Einstein with the cosmological constant Λ = 0, then h is Einstein with the same cosmological constant Λ. Next, 1 2

(∂ 0 ) 2 f = Λ, ∂ 0 ∂ i f = 0 and 1 2 f (∂ 0 ) 2 f -∆f ) = Λf .
We get that f = Λ (x 0 ) 2 + x 0 f 1 (x n+1 ) + f 0 (x 1 , ..., x n+1 ) such that ∆f 0 = 0. Taking f 0 to be harmonic with non-zero Hessian, and f 1 = 0, we get that the metric g is Einstein with the cosmological constant Λ and g = (R ⊕ h) ⋉ R n . It is impossible to construct in this way vacuum Einstein metrics with the holonomy algebras of type 1.

In [START_REF] Galaev | Metrics that realize all Lorentzian holonomy algebras[END_REF] for each weakly-irreducible not irreducible holonomy algebra is constructed a metric of the form [START_REF] Figueroa-O'farrill | Breaking the M-waves[END_REF] with h ij (x 1 , ..., x n+1 ) = δ ij , i.e. each Riemannian metric in the family h(x n+1 ) is flat. The Ricci tensor Ric(g) for such metric has the following components:

Ric 0 n+1 = 1 2 (∂ 0 ) 2 f, (16)
Ric i j =0, i, j = 1, ..., n, (17)
Ric i n+1 = 1 2 ∂ 0 ∂ i f - n j=1 ∂ j (∂ j u i -∂ i u j ) , i = 1, ..., n, (18)
Ric n+1 n+1 = 1 2 f - n i=1 (u i ) 2 (∂ 0 ) 2 f - n i=1 (∂ i ) 2 f + 2 n i=1 ∂ i ∂ n+1 u i (19)
+ n i,j=1 (∂ j u i -∂ i u j ) 2 + (∂ 0 f ) n i=1 ∂ i u i + 2 n i=1 u i ∂ 0 ∂ i f .
Now we recall the algorithm of the construction from [START_REF] Galaev | Metrics that realize all Lorentzian holonomy algebras[END_REF]. Let h ⊂ so(n) be the holonomy algebra of a Riemannian manifold. We get the decompositions (2) and (3). We will assume that the basis e 1 , ..., e n of R n is compatible with the decomposition of R n . Let n 0 = n 1 +• • •+n s = n-n s+1 . We see that h ⊂ so(n 0 ) and h does not annihilate any proper subspace of R n 0 . We will always assume that the indices i, j, k run from 1 to n, the indices î, ĵ, k run from 1 to n 0 , and the indices î, ĵ, k run from n 0 + 1 to n. We will use the Einstein rule for sums. Let (P α ) N α=1 be linearly independent elements of P(h) such that the subset {P α (u)|1 ≤ α ≤ N, u ∈ R n } ⊂ h generates the Lie algebra h. For example, it can be any basis of the vector space P(h). For each P α define the numbers P k α ĵî such that P α (e î)e ĵ = P k α ĵî e k. Since P α ∈ P(h), we have

P ĵ α kî = -P k α ĵî and P k α ĵî + P î α kĵ + P ĵ α îk = 0. ( 20
)
It holds Ric(P α ) = Ric(P α ) ke k, where Ric(P α ) k = î P k α îî . Define the following numbers

a k α ĵî = 1 3 • (α -1)! P k α ĵî + P k α îĵ . (21)
We have

a k α ĵî = a k α îĵ . (22)
From ( 20) it follows that

P k α ĵî = (α -1)! a k α ĵî -a ĵ α kî and a k α ĵî + a î α kĵ + a ĵ α îk = 0. ( 23
)
Define the functions

u î = a î α ĵk x ĵ x k(x n+1 ) α-1 (24)
and set u î = 0. We choose the function f to make the holonomy algebra g of the metric g to be weakly-irreducible. If ∂ 0 f = 0, then g is either of type 2 or type 4; if ∂ 0 f = 0, where G β is a polynomial of x î of degree at most 2. Thus we need to find solutions of a number of the Poisson equations

î (∂ î) 2 F = H, (27)
where H is a polynomial of x î of degree at most 2. Let us show a simple way to find a polynomial solution of such equation. Let

H 1 = H -1 2 (x î) 2 (∂ î) 2 H, then H = H 1 + 1 2 (x î) 2 (∂ î) 2 H and (∂ î) 2 H 1 = 0 for each î. Next, let H 2 = H 1 -x 1 ∂ 1 H 1 , then H 1 = H 2 + x 1 ∂ 1 H 1 , ∂ 1 H 2 = 0, and (∂ î) 2 H 1 = 0 for each î. Now it is obvious that the function F = 1 2 (x 1 ) 2 H 2 + 1 6 (x 1 ) 3 ∂ 1 H 1 + 1 24 (x î) 4 (∂ î) 2 H
is a solution of the equation [START_REF] Schell | Classification of four-dimensional Riemannian spaces[END_REF]. In order to make g weakly-irreducible we add to the obtained f 0 the harmonic function (

x 1 ) 2 + • • • + (x n-1 ) 2 -(n -1)(x n ) 2 .
Thus we get an example of the vacuum Einstein metric with the holonomy algebra (R⊕h)⋉R n , where h is the (not necessary irreducible) holonomy algebra of a Riemannian manifold such that P 1 (h) = 0, in other words, in the decomposition (3) at least one h i is the holonomy algebra of a not vacuum Einstein Riemannian manifold.

We have proved the following theorem.

Theorem 6 Let g be any algebra as in Theorem 4 or 5, then there exists an n + 2dimensional Einstein (resp., vacuum Einstein) Lorentzian manifold with the holonomy algebra g.

Example 1

In [START_REF] Galaev | Metrics that realize all Lorentzian holonomy algebras[END_REF][START_REF] Galaev | Holonomy groups of Lorentzian manifolds: classification, examples, and applications[END_REF] we constructed metrics with the holonomy algebras g 2,G 2 ⊂ so(1, 8) and g 2,spin (7) ⊂ so [START_REF] Alekseevsky | Riemannian manifolds with exceptional holonomy groups[END_REF][START_REF] Coley | Supergravity solutions with constant scalar invariants[END_REF]. In these constructions N = 1 and f = 0. Choosing in these constructions f = 1 3 n i,j=1 (P i jk (x k ) 2 ) 2 , we obtain vacuum Einstein metrics with the holonomy algebras g 2,G 2 ⊂ so(1, 8) and g 2,spin (7) ⊂ so(1, 9).

Lorentzian manifolds with totally isotropic Ricci operator

Let R ∈ R(g 1,h ) be as in Theorem 2. Consider its Ricci operator Ric = Ric(R) : R 1,n+1 → R 1,n+1 defined by η(Ric(x), y) = Ric(x, y), where x, y ∈ R 1,n+1 and on the right hand side Ric denotes the Ricci tensor of R. It is easy to check that Ric(p) = -λp, Ric(x) = η(x, Ric(P ) -v)p + Ric(R 0 )(x), [START_REF] Walker | On parallel fields of partially null vector spaces[END_REF] Ric(q) =(tr T )p + Ric(P ) -v -λq,

where x ∈ R n . A Lorentzian manifold (M, g) is called totally Ricci-isotropic if the image of its Ricci operator is isotropic, equivalently, η(Ric(x), Ric(y)) = 0 for all x, y ∈ R 1,n+1 .

Obviously, any vacuum Einstein Lorentzian manifold is totally Ricci-isotropic. If (M, g) is a spin manifold and it admits a parallel spinor, then it is totally Ricci-isotropic (but not necessary vacuum Einstein, unlike in the Riemannian case) [START_REF] Bryant | Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor[END_REF][START_REF] Figueroa-O'farrill | Breaking the M-waves[END_REF].

Theorem 7 Let (M, g) be a locally indecomposable n + 2-dimensional Lorentzian manifold admitting a parallel distribution of isotropic lines. If (M, g) is totally Ricciisotropic, then its holonomy algebra is the same as in Theorem 4.

The proof of this theorem is similar to the proofs of the above Theorem 3 from [START_REF] Galaev | Holonomy groups of Lorentzian manifolds: classification, examples, and applications[END_REF] and Theorems 4 and 5.

Using results of Section 3 it is easy to prove the following theorem.

Theorem 8 Let (M, g) be a locally indecomposable n + 2-dimensional Lorentzian manifold admitting a parallel distribution of isotropic lines. If the holonomy algebra of (M, g) is of type 2 and in the decomposition (3) of the orthogonal part h ⊂ so(n) each subalgebra h i ⊂ so(n i ) coincides with one of the Lie algebras su( n i 2 ), sp( n i 4 ), G 2 ⊂ so(7), spin(7) ⊂ so [START_REF] Coley | Metrics with vanishing quantum corrections[END_REF], then (M, g) is totally Ricci-isotropic.

Note that this theorem can be also proved by the following argument. Locally (M, g) admits a spin structure. From [START_REF] Galaev | Holonomy groups of Lorentzian manifolds: classification, examples, and applications[END_REF][START_REF] Leistner | On the classification of Lorentzian holonomy groups[END_REF] it follows that any manifold (M, g) with the holonomy algebra as in the theorem admits locally a parallel spinor, hence (M, g) is totally Ricci-isotropic.

Acknowledgments. I am thankful to D. V. Alekseevsky and Thomas Leistner for useful and stimulating discussions on the topic of this paper.

I wish to express my gratitude to Mark Volfovich Losik, my first teacher of Differential Geometry, for his guidance, mentoring and careful attention to my study and work since 1998 when I was at the very first course at Saratov State University.

The work was supported by the grant 201/09/P039 of the Grant Agency of Czech Republic and by the grant MSM 0021622409 of the Czech Ministry of Education.

then g is either of type 1 or type 3. We will make g to be vacuum Einstein, then g will be either of type 2 or type 1, i.e. it will equal either to h ⋉ R n or to (R ⊕ h) ⋉ R n .

Note that

(25) Suppose that g is vacuum Einstein. Then, first of all, the equality [START_REF] Galaev | One component of the curvature tensor of a Lorentzian manifold[END_REF] implies that f = x 0 f 1 + f 0 , where f 0 and f 1 are functions of x 1 , ..., x n+1 .

First suppose that g is of type 2, i.e. ∂ 0 f = 0 and f 1 = 0. Substituting this and (25) into the equation Ric = 0, we get the following equations

The first equation is equivalent to the condition P 1 (h) = 0. Clearly, the function

satisfies the second equation. In order to make g weakly-irreducible we add to the obtained f 0 the harmonic function

Thus we get a new example of the vacuum Einstein metric with the holonomy algebra h ⋉ R n , where h is the (not necessary irreducible) holonomy algebra of a vacuum Einstein Riemannian manifold.

Suppose now that g is of type 1. The equation Ric = 0 is equivalent to the following system of equations:

We may take f 1 = î 2 (α-1)! Ric(P α ) îx î(x n+1 ) α-1 . Substituting it into the last equation, we obtain the equation of the form i

where G is a polynomial of the variables x î and x n+1 , and it is of degree at most 2 in the variables x î and of degree at most 2(N -1) in x n+1 . Let f 0 = 2(N -1) β=0 f 0β • (x n+1 ) β , then each f 0β satisfies the equation