Nonlocal Quantum Gravity and M-Theory

arXiv:1404.2137v2 [hep-th] 23 Jun 2015 a cinfrtegaio n h olclextension nonlocal the and graviton nonlo- the effective for closed-SFT action the (in cal of nonlinear extension the metric) simultaneously which gravity the describe for to The theory proposed theory. field field is ten-dimensional string a of action such is that match linearized fix result to to the graviton is requiring the step by for infi- next uniquely an The factors with form derivatives. form operators of below) covariant number precise nite nonlocal made generic i.e., sense have a factors, models (in These in- well-behaved are from theory. but that start string models will of quantum-gravity we dependent nonlocal here, of employed the class summarize strategy a to the and of respect, this steps In effective dimen- of theory. ten properties field main string in the capturing gravity of capable of sions theory super-renormalizable a obtain, to expect path. might different one a result follow How- of we type unaccepted. the still see view knowledge, to our of ever, of point best technical the an a to chal- from to and, This demanding solution fields. rather its massless is reduce the lenge and for 2] action to Wilsonian [1, be exact theory would field possibility string obtained one in rigorously particular, equation Batalin–Vilkovisky In infinite the take by the terms. of fixed of dependence be that set cutoff then equation explicit should the renormalization-group action determines exact the massless an of the deriving form should for The one action question, Wilsonian fields. a effective such to an answer calculate properties? definite quantum (UV) a massless ultraviolet give its resulting the- To are the what string of and of theory form freedom field inte- the of has is degrees one what massive ory, that all assumed out Having grated sub- still exploration. territories to are sector ject gravity its of properties the HSCLRVE D REVIEW PHYSICAL ∗ † lcrncades [email protected] address: Electronic [email protected] address: Electronic nti ae,w xlr h osblt oachieve to possibility the explore we paper, this In and (SFT) theory field string of limit particle-field The 2 eateto hsc etrfrFedTer n Particle and Theory Field for Center & Physics of Department ASnmes 46.m 11.m 11.25.Yb l 11.10.Lm, particle-field 04.60.-m, unitary numbers: and PACS finite a for s candidate this, 11-dimensional a of of view is completion In result ultraviolet a out. th get selected t Comparing finally is of and factor amplitudes. poles form class new unique loops general a avoid the theory, a to of one from behavior allows start high-energy that We sector factor bosonic theory. form the field nonpolynomial of string completion by ultraviolet vated a construct We .INTRODUCTION I. 1 nttt eEtutr el aei,CI,Srao11 28 121, Serrano CSIC, Materia, la de Estructura de Instituto 91 209(2015) 124059 , olclqatmgaiyadM-theory and gravity quantum Nonlocal inuaCalcagni Gianluca Dtd pi ,2014) 7, April (Dated: 1, ∗ n enroModesto Leonardo and oe o1 iesos htispoete r,and the From are, M-theory. known of properties the spectrum with its and the compatible symmetries are what extend properties dimensions, further these whether can 11 we to whether model consists ourselves step asking third of The supergravity. ten-dimensional of oecek ftepoete fteresulting the of properties of the completion After of finite supergravity. checks to a 11-dimensional some get 10 of to sector from bosonic able process the are “oxidation” we dimensions, an 11 by model, nonlocal cin epooei sannoa-edter ii of limit nonlocal-field-theory a as it M-theory. propose we action, fiain ogaiy.Teter eepooe isto requirements: aims minimal proposed of here synthesis a theory mod- fulfill The infrared nonlocal gravity). for to [29–33] ifications Cornish Refs. gravity and also quantum (see in Moffat ideas [19–28] recent and more several 11], and Kras- [10, [12–18], [6–9], studies Tomboulis Efimov parallel by and theory and nikov field earlier quantum [3–5], nonlocal theory on string in 1980s 1 ii h hoyhst eprubtvl super- perturbatively be to has theory The (iii) i)Cascllclsprrvt hudb odap- good a be should supergravity local Classical (ii) egaiyi h anfcso h rsn paper. present the of of version focus nonlocal main describ A lim the be theory. is the can pergravity field M-theory as (quantum) of regarded freedom a of by among is degrees dualities light latter the of The where a web 11- theories supergravity. the these an between dimensional to and is refers theories leve quantum superstring also it the different at one model notions, “M-theory,” matrix a pos By M- as modern also formulation that what a to population admits about membrane bly a ambiguity According with of theory dimensional margin some means. is theory there general, In oti ups,w obn eut on ic the since found results combine we purpose, this To i pctm sacniumi hc oet invari- Lorentz which in continuum a is Spacetime (i) hsc,FdnUiest,203 hnhi China Shanghai, 200433 University, Fudan Physics, eunl,teeetv pctm ieso de- dimension energy). the spacetime with creases effective (con- level the quantum the sequently, at finite or renormalizable energy. low at proximation scales. all at preserved is ance prrvt ya xdto rcs.The process. oxidation an by upergravity mto M-theory. of imit emdf e-iesoa supergravity ten-dimensional modify we f1-iesoa uegaiymoti- supergravity 11-dimensional of s oeswt ffciesrn field string effective with models ese 1 ntepoaao n mrvsthe improves and propagator the in ere hrceie ya entire an by characterized heories 0 ard Spain Madrid, 006 2, † arXiv:1404.2137 D D 1su- 11 = D d11- nd 11 = 10 = si- ed it l. 2

(iv) The theory has to be unitary, without extra de- Although the model we present is a simplification of grees of freedom in addition to those present in the the most complete nonlocal 11D supergravity action one classical theory. can write compatibly with the M-theory content, it already displays a richness of implications that local su- (v) Typical classical solutions must be singularity free. pergravity does not possess, not only from the point of view of particle physics and the mathematical structure (vi) The proposal should be manifestly compatible with of the theory but also of cosmology. In particular, there the known features of string theory. is the possibility (classically) that the big-bang singu- Actually, there is an infinite class of theories compat- larity is removed and replaced by a finite bounce, near ible with properties (i)–(v), parametrized by different which a natural period of superinflation occurs. This nonpolynomial entire functions. We will see that, in or- has been shown in previous studies of the type of non- der to agree with string field theory and fulfill (vi), the locality in Eq. (1) [19–28, 34] (see also Refs. [35–39] for entire function will be singled out uniquely. The D = 11 earlier works on nonlocal gravity). Moreover, since the bosonic sector of the Lagrangian we are going to con- linearized limit of the nonlinear gravitational theory does struct step by step reads as not possess ghosts or other types of obvious instabilities, it is expected to provide a reliable model of cosmic evo- − 1 e Λ 1 1 lution all the way back to the highest densities. This = R + G − Rµν F e− Λ F L11 2κ2 µν − 2 4! 4 4 is in stark contrast with effective local gravitational ac- 11 × tions made of higher-order curvature invariants, for which + (A )+ (ψ) , (1) L 3 L their effective description breaks down at the scale of the highest-order term. Thus, the interest in a nonlocal field- where F4 is the field strength of the 3-form A3 and (A3) and (ψ) are nonpolynomial modifications of the Chern–L theory limit for M-theory is not merely academic as the L proposal would have sizable, testable consequences for Simons and fermionic actions. The metric tensor gµν has signature ( , +, +,..., +); the curvature tensor, Ricci the history of the early Universe. − tensor and curvature scalar are defined by, respectively, The gauge symmetry group of string field theory is Rα = ∂ Γα + ..., R = Rα , and R = gµν R . βγδ − δ βγ µν µαν µν extremely large and partly unknown. It includes space- Gµν = Rµν (1/2)gµνR is the Einstein tensor. The op- time diffeomorphisms and supersymmetry transforma- − 2 erator Λ = /Λ is the d’Alembertian divided by an tions [40], but the way in which such symmetries arise in 2 2 energy scale Λ. In momentum space, Λ k k , the low-energy field-theory limit is not straightforward. − → → E where in the second step we transformed to Euclidean For instance, to identify general coordinate transforma- momentum (hence the index “E”), k0 ik4. tions in closed string theory, one has to make field re- →− The form of the kinetic terms in the particle-field- definitions order by order in the string coupling [41]; the theory limit of SFT fixes the form factor of the above same holds for Abelian transformations in the open case gravity models uniquely among the infinitely many be- [42]. Also, fixing the gauge freedom of the full theory longing to this class. This, of course, is a much is necessary to compute amplitudes and S-matrix ele- lesser achievement than uniquely fixing the UV complete ments that, again, would require the Batalin–Vilkovisky particle-field limit of M-theory, but it indicates in a clear formalism. In this paper, we do not consider the problem way the form the gravitational sector should take if ten- of constructing a well-defined string field theory from a dimensional SFT is valid and, at the same time, if UV conformal field theory representation and, from that, de- finiteness is demanded. rive the effective action for gravity; therefore, we cannot Our Ansatz (1) has a number of properties meeting enter in the details of the above general and important with requirement (vi), all of which constitute important issues. Still, we offer a concrete particle-field proposal checks of its compatibility with string theory: that, regardless of its ultimate validity, can help at least in stimulating more interest in the subject. To obtain a (a) Contrary to local theories, the UV behavior does unique theory, it is sufficient to combine together three not get worse with dimensional reduction: both different ingredients: general covariance, the exponen- unitarity and renormalization survive after com- tial falloff of the propagator, and unitarity (ghost free- pactification (Sec. III). dom). We then ask what a completion of 11-dimensional (b) Type-II string amplitudes are recovered in the local supergravity, compatible with quantum mechanics and limit (Sec. IV B). which cannot be excluded in a multivacuum M-theory, should look like. Regarding the assumption of general (c) There exists an S-duality linking 11-dimensional covariance, we commented above about the difficulties in (11D) M-theory to ten-dimensional (10D) Type- understanding how diffeomorphism invariance arises in IIA nonlocal supergravity (Sec. IVC). the low-energy spacetime approximation of the theory. In a worst-case scenario, it might even happen that it (d) It is also argued (although not proved) that inclu- is not possible to write down a nonlinear and nonlocal sion of massive states does not spoil the UV prop- fully covariant action for gravity in closed SFT. However, erties of the theory (Sec. V). there is no negative proof in that direction yet and, if the 3 theory really reproduced gravity, it must satisfy general field content depend on the type of string. Open SFT covariance in some limit. Since we are able to formulate is the simplest example [51, 52]. The open SFT bosonic a quantum field theory of gravity when general covari- action is [3] ance is respected, we make such an assumption. This 1 1 1 field theory happens to display a class of nonlocal op- S = Ψ QΨ+ Ψ Ψ Ψ , (2) −g2 2α′ ∗ 3 ∗ ∗ erators that includes those of SFT, which leads us to o Z the second point. The exponential operator is a general where g is the open string coupling, α′ is the Regge feature of the particle-field limit of certain string field o slope, is a noncommutative product, and Q is the theories, but not of others (for instance, Berkovits’s non- Becchi–Rouet–Stora–Tyutin∗ (BRST) operator. The re- polynomial theory) in which the low-energy spacetime sulting equation of motion is QΨ+ α′Ψ Ψ = 0. The limit has never been derived while retaining nonlocal- action is invariant under the infinitesimal∗ gauge trans- ity. However, several dualities between different open formation δΨ = QΨ+Ψ Γ Γ Ψ, where Γ is a zero- SFTs are known, including a mapping between super- ghost-number state. The∗ gauge− group∗ of string field the- symmetric and bosonic classical solutions [43, 44], cubic ory is very large and not completely known. It includes and Berkovits’s supersymmetric SFTs [45, 46], and dif- at least worldsheet reparametrizations, spacetime diffeo- ferent polynomial supersymmetric SFTs [47, 48]. These morphisms, and (in the supersymmetric extensions) su- results have been found only for the open string, but persymmetry transformations [40, 53]. they are encouraging enough to put us in the position of Fixing the gauge and keeping only the tachyonic mode submitting a reasonable exploratory proposal. In fact, it φ, Eq. (2) gives rise to an effective spacetime action for is known that in closed SFTs such as the one reported φ that exhibits a nonlocal interaction [54, 55], in the next section the same type of nonlocality arises. ′ 2 This is perhaps not surprising if one recalls that the ex- 1 1 1 e3/(α Λ ) ponential derivative operator is a direct consequence of = dDx φ + φ φ˜3 , (3) S g2 2 α′ − 3 the existence of surface states in the Fock space of the o Z " # open-string case [49] and that states similar to these exist where also in the closed-string case [50]. Therefore, our under- 3/2 standing of the gauge group of SFT is already sufficient to 1 3 φ˜ := e Λ φ , = α′ ln 0.2616 α′ . (4) make an educated guess about the effective gravitational Λ2 4 ≈ action. We minimally review string field theory and renormal- The value of Λ is dictated by conformal invariance (which izable perturbative quantum-gravity models in Sec. II. is partly broken by level truncation). The other fields In Sec. III, we show in a five-dimensional example that of the level expansion, as well as the p-adic model [5], Kaluza–Klein compactification preserves the structure of display the same type of nonlocality. the form factors and, hence, the renormalizability and While there is only one proposal for the bosonic open unitarity properties of the theory. The main results in string, there are many for open superstring field theory. D = 11 and D = 10 dimensions are in Sec. IV, where we The first by Witten [56] was later modified in Refs. [57– provide various checks of the theory as detailed above. 59], depending on whether a certain picture-changing op- Inclusion of massive states is discussed in section V. In erator in the action was chiral and local [58, 59] or nonchi- Sec. VI, we collect some remarks on the issue of ultra- ral and bilocal [57, 60, 61]. A nonpolynomial version of violet finiteness in string theory and quantum gravity. open super-SFT was devised by Berkovits [62, 63]. The Section VII is devoted to a discussion. nonlocal effective action for the tachyon, representing an unstable brane, has been constructed only for the nonchi- ral version [57, 60, 61]. In that case, at lowest order the II. FROM STRING FIELD THEORY TO interaction is slightly more complicated than that of Eq. QUANTUM GRAVITY (4): U(φ)= e Λ φe Λ φ2. (5) The purpose of this section is to sketch the type of nonlocal operators arising in string field theory. Classical solutions of cubic open super-SFT can be mapped into those of Berkovits’s nonpolynomial SFT [45, 48], so we can regard the nonlocality (5) as a generic A. String field theory feature of open superstring field theory at the effective level. A string field Ψ is a mathematical object interacting The degrees of freedom corresponding to the graviton via certain composition rules and given by the superpo- are included in the spectrum of closed SFT. Consider the sition of string oscillators applied to the Fock vacuum bosonic case [1, 64–77] and the action

(or vacua, in the case of the closed string). The coeffi- +∞ ′ N−3 cients of the level expansion are particle-field modes of 1 − (g α ) N−1 S = Φ ⋆ Qb0 Φ+ g Φ ⋆ [Φ ] , (6) progressively higher mass. The interaction and precise α′ 2N−3N! Z NX=3 4 where Φ is the closed-string field, Q is the BRST opera- also allows one to find exact perturbative solutions (i.e., − ¯ tor, b0 = (b0 b0)/2 is a combination of antighost zero for free fields [88]) or very accurate approximate non- − N−1 modes, and the symbol [Φ ] = [Φ1 ⋆...⋆ ΦN−1] rep- perturbative solutions that can go, at the level of target resents the string field obtained combining N 1 terms spacetime, beyond marginal deformations [49, 91, 92]. − Φ1,..., ΦN−1 using the N-string vertex function. Again, These properties are a direct reflection of the gauge sym- for physical processes dominated by light states, one can metries of SFT [49]. All known exact solutions in (open) truncate the string field in terms of oscillators and par- SFT are superpositions of surface states which obey a ticle fields, Φ = Φphys +Φaux, where “diffusion” equation involving only Virasoro and ghost Φ = c− φ + A αµ α¯ν 0 ; (7) operators. This diffusion equation performs a change of phys 0 µν −1 −1 | i gauge that reparametrizes a trivial non-normalizable so- − c0 = c0 c¯0 is a combination of the left and right ghost lution of the equation of motion into a nontrivial normal- − µ µ zero modes; φ is the bosonic tachyon field; α−1 andα ¯−1 ized solution. At the level of spacetime fields, the same are the level-( 1) bosonic Virasoro oscillators; the sym- structure survives in Eq. (10) and, in fact, a spacetime − metric and the antisymmetric parts of the tensor Aµν are, diffusion equation holds for nonperturbative approximate respectively, the graviton hµν and a 2-form Bµν ; andΦaux solutions [49]. is made of various auxiliary fields. After gauge fixing and a rearrangement of the auxiliary fields, the resulting effective Lagrangian for the physical modes contains a kinetic B. Quantum field theories of gravity and an interaction part, = free + int. The kinetic and mass terms read L L L In this section, we review a class of gravitational the- 1 λ µν 1 λ 2 2 free = ∂λAµν ∂ A ∂λφ∂ φ φ , (8) ories constituting a “nonpolynomial” extension of the L −2 − 2 − α′ renormalizable quadratic theory by Stelle [93, 94]. In while there are 50 cubic interaction terms, D dimensions, the Lagrangian has the general structure ∼ g gα′ g g α′ [8, 10, 11, 22–27, 95] = φ˜3 + (∂φ˜)2A˜ + φÃ˜2 + φ˜(∂A˜)2 Lint 3!ǫ3 23ǫ2 2ǫ 24ǫ 2 µν g α′2 g α′ = R Rµν γ2(Λ)R Rγ0(Λ)R + φ˜ (∂2A˜)2 A˜2(∂2A˜)+ ..., (9) L κ2 − − 27ǫ − 23 4 3 µνρσ where ǫ =2 /3 and fields with a tilde are “dressed” by Rµνρσγ4(Λ)R , (11) the same nonlocal operator (4) except that the constant − 2 1/Λ2 has an extra 1/2 prefactor, which can be removed where κ = 32πGD is the Newton constant, the form 2 by a coordinate rescaling. factors γ0,2,4 are entire functions of the ratio Λ = /Λ , Supersymmetric versions of closed SFT, both heterotic and Λ is an invariant mass scale. and Type II [78–81], have been studied less, but one The path to reach Eq. (11) began with polynomial expects to find the same nonlocal structure in effective Stelle’s theory in four dimensions, later generalized to D- spacetime actions, Eqs. (4), (5), and (9). The same is dimensional spacetimes. The Lagrangian, with at most true for a more complete setting including both open and X derivatives of the metric, is closed strings, as proposals for the p-adic tachyonic action X X 2 2 2 indicate [82–86]. We conclude that, after a field redefi- poly = a2R + a4R + b4R + ... + aX R 2 + bX Rµν L µν nition [51, 54, 55, 67], the effective field actions on tar- X X 2 −2 get spacetime stemming from string field theory have the +cX Rµνρσ + dX R 2 R + ..., (12) following schematic structure for bosonic and fermionic where the first dots include a finite number of extra fields f˜ (e.g., Refs. [19, 87] and references therein): i terms with fewer derivatives of the metric tensor and 1 the second ones indicate a finite number of operators S = dDx f˜( m2)e− Λ f˜ U , (10) eff 2 i − i i − O(R2X/2−3R) with the same number of derivatives but i X Z higher powers of the curvature. Tensor indices are not ˜ where U is a potential (cubic in open SFT) of the fields fi contracted explicitly and we use the compact notation 2 µν in the bosonic case and, in supersymmetric cases such as R := Rµν R , and so on. An elementary power count- ˜ µν Eq. (5), of the fi’s further dressed by nonlocal operators. ing shows that the maximal superficial degree of diver- This is the point at which we will connect with quantum- gence of a Feynman graph is δ = D (D X)(V I)= gravity field theories. D + (D X)(L 1), where L, V ,− and−I are, respec-− From a field-theoretical point of view, the type of non- tively, the− number− of loops, vertices, and internal lines of locality in Eq. (10) is particularly benign. Not only is the a graph. For X = D, the theory is renormalizable since Cauchy problem well defined in general [88, 89],2 but it the maximal divergence is δ = D, and all the infinities can be absorbed in the operators already present in the Lagrangian (12). Unfortunately, however, the propaga- 2 The Cauchy problem can be rigorously defined also for the grav- tor contains at least one ghost (i.e., a state of negative itational theory presented here, as we will show elsewhere [90]. norm) that represents a violation of unitarity. 5

To overcome this problem, one considers a more gen- derivatives of the curvature tensor and containing also eral action combining curvature tensors and covariant nonpolynomial terms [96, 97]:

N+2 S = α ΛD−2n dDx g (∂ g )+ S 2n | | O2n ρ µν NP n=0 X Z p = dDx g α ΛDλ + α ΛD−2R + +ΛD−4(α1R2 + α2R Rµν + α3R Rµνρσ ) | | 0 2 4 4 µν 4 µνρσ Z +ΛD−6p(α1R3 + α2 R R + ...)+ΛD−8(α1R4 + α2R R R + α3 2R 2R + ...) 6 ... 6 ∇ ...∇ ... 8 ... 8 ...∇ ...∇ ... 8∇ ...∇ ... + ... +ΛD−2(N+2)(α1 RN+2 + α2 RN−1 R R + α3 R N R + ...) 2N+2 ... 2N+2 ... ∇ ...∇ ... 2N+2 ...... +R h ( )Rµν + Rh ( ) R . (13) µν 2 − Λ 0 − Λ

In Eq. (13), (∂ g ) denotes general-covariant oper- Lagrangian reads + =2−1hµν hρσ. Invert- O2n ρ µν Llin LGF Oµν,ρσ ators containing 2n derivatives of the metric gµν , while ing the operator [98], one finds the following two-point O µ SNP is a nonpolynomial action defined in terms of two function in the harmonic gauge (∂ hµν = 0) and in mo- entire functions h2,0 that we will fix later [11]. The max- mentum space: imal number of derivatives in the local part of the action is X = 2N + 4. From the discussion of the local the- P (2) P (0) −1(k)= + , (17) 2 2 2 2 ory (12), renormalizability sets 2N +4 > D. To avoid O −k h¯2(k ) (D 2)k h¯0(k ) fractional powers of the d’Alembertian operator, we take − 2N +4= D in even dimensions and 2N +4= D + 1 in where odd dimensions. k2κ2β(k2) n−2 µν ¯ 2 For N > 0 and n > 2, only the operators Rµν R , h2(k )=1+ , (18) n−2 n−2 µναβ 4 R R, and Rµναβ R contribute to the gravi- N ton propagator (with 2n derivatives in total), but using β(k2)=2 b (k2/Λ2)n +2 h (k2/Λ2) , (19) the Bianchi and Ricci identities the third can be elimi- n 2 n=0 nated. To leading order in the curvature, the Lagrangian X Dκ2β(k2)+4(D 1)κ2α(k2) density in (13) reads [22, 96] h¯ (k2)=1 k2 − , (20) 0 − 4(D 2) N − 2R N = λ¯ + [a R( )nR + b R ( )nRµν ] 2 2 2 n 2 2 L − κ2 − n − Λ n µν − Λ α(k )=2 an(k /Λ ) +2 h0(k /Λ ) . (21) n=0 X n=0 R h ( ) Rµν Rh ( )R + O(R3) , (14) X − µν 2 − Λ − 0 − Λ In Eq. (17), we omitted the tensorial indices of the oper- −1 (0) (2) where we have renamed the constants in Eq. (13) as λ¯ = ator and of the projectors P and P , defined as D −2 D−2 O (2) α Λ λ,2κ = α Λ , and so on. Finally, the entire [98, 99] Pµνρσ (k) = (θ θ +θ θ )/2 θ θ /(D 1), 0 2 µρ νσ µσ νρ − µν ρσ − −functions h (z) in Eq. (13) are (here z := /Λ2) (0) 2 0,2 − Pµνρσ (k)= θµν θρσ/(D 1), θµν = ηµν kµkν /k . We now assume that− the theory is− renormalized at 2[V (z)−1 1] κ2Λ2 z N ˜b zn h (z)= − − n=0 n , (15) some scale µ0. Setting 2 κ2Λ2 z PN a˜ = a (µ ) , ˜b = b (µ ) , (22) V (z)−1 1+ κ2Λ2 z a˜ zn n n 0 n n 0 h (z)= − n=0 n , (16) 0 − κ2Λ2 z the bare propagator only possesses the gauge-invariant, P physical massless spin-2 graviton pole, and h¯ = h¯ = ˜ 2 0 for general parametersa ñ and bn, where V (z) is a non- V −1. Choosing another renormalization scale, the bare polynomial entire function without zeros in the whole propagator would acquire poles that cancel with a shift in complex plane. The couplings and the nonlocal func- the self-energy in the dressed propagator. Thus, Eq. (17) tions of the theory have the following dimensions in mass reads D−4 2 2−D D units: [ãn] = [˜bn] = M ,[κ ] = M ,[λ¯] = M , [h ] = [h ]= M D−4. V (k2/Λ2) P (0) 2 0 −1(k)= P (2) . (23) The Lagrangian can be expanded at the second order O − k2 − D 2 in the graviton fluctuation gµν = ηµν + κhµν around the − Minkowski background. Adding a gauge-fixing term GF As shown by Efimov [8], a nonlocal field theory is unitary due to a local gauge symmetry under infinitesimalL co- and microcausal, provided the following properties are ordinate transformations [93], the linearized gauge-fixed satisfied by V (z): 6

(i) V (z) is an entire analytic function in the complex z N n n µν plane, and it has a finite order of growth 1/2 6 ρ< Zan an R ( Λ) R + Zbn bn Rµν ( Λ) R ρ − − − + , i.e., b> 0,c> 0 such that V (z) 6 ceb |z| . n=0 X h µν i ∞ ∃ | | Rh0( Λ) R Rµν h2( Λ) R 2 − − − − (ii) When Re(z) + (k + or N 2 → ∞ → ∞ k + ), V (z) decreases quite rapidly: (1) 2+n E Z (1) c R + ... . (25) → ∞ N cn n limRe(z)→+∞ z V (z) = 0, N > 0 [for example: − | | | | ∀ n=1 o V (z) = exp( zl) for l N ]. X − ∈ + (iii) [V (z)]∗ = V (z∗) and V (0) = 1. 2. Finite quantum gravity (iv) The function V −1(z) is real and positive on the real axis, and it has no zeros on the whole complex plane What we have said in the previous section applies to z < + . This requirement implies that there are spacetimes of any dimension D > 3, but in this paper we |no| gauge-invariant∞ poles other than the transverse are interested in defining a ultraviolet completion of 11- massless physical graviton pole. dimensional supergravity. In odd dimensions, there are no counterterms for pure gravity at the one-loop level in dimensional regularization, and the theory results in 1. Super-renormalizable quantum gravity being finite; it stays so also when matter is added to fill up the supergravity multiplet [101, 102]. We conclude that the amplitudes with an arbitrary number of loops We now specialize to a form factor V (z) = exp( zl) − are finite and all the beta functions vanish, (where l N+) satisfying the above properties. The high-energy∈ propagator takes the form −1(k) = 2 2 l 2 O βκ = βλ¯ = βan = βbn = βc(n) =0 , (26) exp[ (k /Λ ) ]/k . The m-graviton interaction has i −the same− scaling, since it can be written schemati- (m) m l 2l where n = 1,...,N and i runs from 1 to the number cally as h ηh exp[( η) /Λ ]h + ..., where µνL ∼ − of invariants of order N. In particular, we can fix all η = η ∂µ∂ν and “...” are other subleading interaction (n) terms. Placing an upper bound to the amplitude with L the coefficients ci to zero, while the couplings an(µ) loops, we find and bn(µ) do not run with the energy: an(µ)=ãn = const, bn(µ)= ˜bn = const. Using Eqs. (15) and (16), the 2 2 l I 2 2 l V “ (L) (dDk)L e−(k /Λ ) k−2 e(k /Λ ) k2 gravitational Lagrangian density (14) simplifies to A ∼ Z h i h i 2 µν 2 2 l L−1 = [R + G γ() R ] , (27) = (dk)DL e−(k /Λ ) k−2 ” . (24) L κ2 µν Z h i where we set the cosmological constant to zero and In the last step, we used the topological identity I = V + L 1. The L-loops amplitude is ultraviolet finite for e− Λ 1 − D γ() := − . (28) L> 1, and it diverges at most as “(k) ” for L = 1. Only one-loop divergences survive and, therefore, this theory 2 is super-renormalizable and unitary, as well as micro- The equations of motion up to order O(R ) are [95] causal [6–10]. A more rigorous power-counting analysis e− Λ G + O(R2)=8πG T . (29) will be done in a separate paper [100]. For a form factor µν D µν V = exp[ H(z)], where H(z) is an entire function with − logarithmic asymptotic behavior, power-counting renor- III. SELF-COMPLETE GEOMETRIC malizability has been shown already in Refs. [11, 22]. UNIFICATION Only a finite number of constants are renormalized in the action (14), i.e., κ, λ¯, a , b , and the finite number n n In traditional unification scenarios, gauge and gravity of couplings that multiply the operators in the next-to- fields descend from a higher-dimensional, local, purely last line of Eq. (13). On the other hand, the functions gravitational action via a Kaluza–Klein compactification. hi(z) are not renormalized. To see this, we write the en- +∞ r A negative feature of such theories is that divergences tire functions as a series, hi(z) = r=0 crz . Because worsen with the increase of the spacetime dimension- of the superficial degrees of divergence δ 6 D derived in P ality. A different situation arises in nonlocal models. Eq. (24), there are no counterterms that renormalize cr Namely, both renormalization and unitarity of the D- for r>N. Therefore, the nontrivial dependence of the dimensional theory are naturally inherited by, say, its entire functions hi(z) on their argument is preserved at (D 1)-dimensional compactification. As a simple ex- the quantum level. After multiplicative renormalization, ample− of this mechanism, in this section we study the the action becomes Kaluza–Klein compactification of the action (14) on a D −2 circle from D = 5 to D = 4. One can check that the S = d x g 2 Z κ R Z¯λ¯ | | κ − λ compactification from D = 11 to D = 10 follows similar Z p n 7 lines. In this section, we shall indicate with capital Ro- IV. M-THEORY FROM QUANTUM-GRAVITY man letters the indices in D dimensions and with Greek OXIDATION letters those in the reduced spacetime. Let us start with the following purely gravitational action in D = 5: We start by considering ten-dimensional Type-IIA su- 1 pergravity, and then we include the modifications sug- S = d5x g R + G γ() RAB . (30) gested by string field theory while maintaining general 5 2κ2 | | AB 5 Z covariance. Next, we show how to modify 11-dimensional p We can expand the five-dimensional line element as supergravity in order to get the previous action by dimensional reduction. To simplify the analysis, we only con- 2 A B ds = gABdx dx sider the bosonic sector of the theories. This is legitimate µ ν µ 5 µ ν = gµν dx dx +2aAµdx dx + aAµAν dx dx if we assume the existence of a supersymmetric extension of the nonpolynomial 11-dimensional or 10-dimensional +a(dx5)2, (31) pure gravity theories, since at the quantum level super- where a is a dimensionless constant, A, B = 0, 1, 2, 3, 5 symmetry does not worsen the counterterms of bosonic and µ,ν =0,..., 3. The curvature decomposes into gravity [103]. Actually, in 11 dimensions it is not so obvious that there are invariant counterterms with an odd R(5) = R + O(aA F )+ O(aF 2)+ O(a2A2F 2) , µν µν ∇ number of derivatives. One could imagine an invariant 1 counterterm containing the field strength Fµνρσ raised at R(5) = a2F F µν , 55 4 µν an odd power, but there is a discrete symmetry (time re- 1 1 versal together with Aµνρ Aµνρ) that excludes these R(5) = a F λ + a2A F F νρ , →− µ5 −2 ∇λ µ 4 µ νρ operators [101]. 1 R(5) = gABR(5) = R aF F µν , (32) AB 4 µν − A. Action where Fµν := ∂µAν ∂ν Aµ. Let us now expand the action − 4 (30) using (32) and omitting all the operators O(F ), The main feature of effective actions of string field the- O(R AF ), and O(A4F 4): ∇ ory is the exponential damping factor in Eq. (10). This form factor belongs to the class of operators indepen- 1 5 4 1 µν S = 2 dx d x√a g R aFµν F dently discussed below Eq. (23) in the context of nonlo- 2κ5 | | − 4 Z Z h cal quantum gravity. We propose to use the knowledge µν p(5)µ5 (5) +G γ() Rµν +2R γ() Rµ5 + vertices . (33) from the graviton action in string field theory to actually i fix γ( ) uniquely in the class of nonlinear gravitational Up to extra vertices, the last term reads actions reviewed in the previous section. The result is Eq. (28), and the action selected out of the infinite class 2 1 2R(5)µ5γ() R(5) = a F λµ (11) is Eq. (27). µ5 a −2 ∇λ Now we introduce the same modification for the other 1 fields that fill up the bosonic sector of supergravity. In γ() a F τ . (34) × −2 ∇τ µ 10 dimensions, we are led to write the action 10 −Λ Integrating several times by parts and using the Bianchi IIA d x −2Φ e 1 µν S = g e R + Gµν − R identity ρFµν + ν Fρµ + µFνρ = 0 for the curvature 10 2κ2 | | ∇ ∇ ∇ Z 10 ( Fµν , the operator (34) turns into p h −Λ µ 1 −Λ µνρ (5) a +4∂µΦ e ∂ Φ Hµνρ e H 2R(5)µ5γ() R = F e− Λ 1 F µν − 2 3! µ5 − 4 µν − × i 1 1 −Λ µν 1 −Λ µνρσ +vertices . (35) Fµν e F + F˜µνρσ e F˜ −2 2! 4! ) Replacing (35) in (33), the local F 2 terms cancel each 1 other, and we finally get 2 B2 F4 F4 + ..., (37) −4κ10 ∧ ∧ 1 Z 5 4 µν S = 2 dx d x√a g R + G γ( ) Rµν where Φ, H3 = dB2, F2, and F4 are, respectively, 2κ5 | | Z Z h the dilaton, the Neveu–Schwarz field strength, and the a p−Λ µν Fµν e F + vertices . (36) Ramond–Ramond 2-form and 4-form field strengths, − 4 ˜ i while F4 = dA3 A1 H3 and Λ is metric covariant (it The derivative order of the extra vertices is no greater does not contain− gauge∧ fields). To be in agreement with than that of the other operators in the action. Since string field theory, we have to fix Λ2 as in Eq. (4). We the operatorial structure of the form factors is preserved, know that 10-dimensional supergravity can be derived the theory keeps being super-renormalizable even after from 11-dimensional supergravity by dimensional reduc- compactification. tion. Similarly, the theory (37) can be derived from an 8

derivative gravity in N = 1, D = 4 filling up the super- 11D Supergravity α′ → 0 M-Theory gravity multiplets, paying attention to avoid extra poles 11D 11D in the propagators [104]. Of course, the extension of Type-IIA supergravity pre- Reduction String Field Theory Oxidation sented in Eq. (37) does not mimic all the features of string theory but only a general-covariant projection onto ′ → Massless states α 0 the massless sector. This is actually why the theory in 10D 10D ten dimensions is not finite but super-renormalizable. In Non-Polynomial Type IIA Supergravity α′ → 0 Type IIA Supergravity string theory, the infinite tower of massive particles is re- sponsible for the cancellation of the infinities in the loop amplitudes. However, the compactification of the 11- dimensional finite “M-theory” (38) should imply a finite FIG. 1. Diagrammatic representation of the completion of 11- ten-dimensional supergravity as well, when the Kaluza– dimensional supergravity. At the core of the graph is string Klein states are included [101]. field theory, whose limit α′ → 0 is Type-IIA ten-dimensional supergravity. The projection of SFT on the massless sector is nonpolynomial or semipolynomial Type-IIA supergravity. B. Consistency check of the conjecture We know that the dimensional reduction of 11-dimensional supergravity to ten dimensions is Type-IIA supergravity plus extra massive states; applying the reverse process, we can One way to verify the consistency of our conjecture is oxidate nonpolynomial Type-IIA supergravity to get a non- to look at the Type-II string theory amplitudes [4] (see polynomial completion of 11-dimensional supergravity (“M- also Ref. [105]). From the four-graviton amplitude in α′ → theory”). Taking the 0 limit in the M-theory action, we Type-II string theory, there follows the effective action find again the unique 11-dimensional supergravity. In parallel, the α′ → 0 limit of nonpolynomial Type-IIA supergravity S˜ = d10x g R + R4 , (39) is the usual Type-IIA supergravity. | | Z 4 µ1...µ8 pν1...ν8 R := t t Rµ1µ2ν1ν2 Rµ3µ4ν3ν4 extension of 11-dimensional supergravity by an oxidation Rµ5µ6ν5ν6 Rµ7µ8ν7ν8 , × process to be understood as the inverse of Kaluza–Klein where the symbol tµ1...µ8 is defined so that dimensional reduction. From what we learned in the previous section, the quantum finite 11-dimensional theory µν µ1...µ8 detΓ Fµν = t Fµ1µ2 ...Fµ7µ8 , (40) incorporating the SFT propagator as well as general covariance reads and Γ andpF are, respectively, the Lorentz generator and the Yang–Mills field strength. Although the Lagrangian −Λ 1 11 e 1 µν (9) for closed bosonic SFT includes other interaction ver- S11 = 2 d x g R + Gµν − R 2κ11 | | tices with four and six derivatives, these operators are Z p h 1 − µνρσ excluded in Type-IIA and Type-IIB string theory, and F e Λ F 4 −2 4! µνρσ the first correction to Einstein gravity is O(R ). How- ×1 i ever, the action (39) is not unique. Under a local field A3 F4 F4 +.... (38) redefinition g g + X , −12κ2 ∧ ∧ µν µν µν 11 Z → 10 µν 4 In Eqs. (37) and (38), we have left the Chern–Simons S˜ = d x g (R + Gµν X + R + ...) , (41) term untouched, but it should contain also nonlocal oper- | | Z p ators. Here, we concentrated on the modifications in the plus higher-order terms. In Ref. [4], Xµν was taken as a gravitational sector and do not discuss the possible modi- local functional of the metric and its derivatives. Here, fications to the Chern–Simons action. Figure 1 shows the we make the particular choice interrelations between local and nonlocal theories under dimensional reduction and oxidation. 2 X = γ()R = 1+ + ... R , (42) It is easy to show that the compactification on a circle µν µν − 2Λ2 − 6Λ4 µν IIA of the theory S11 gives S10 , if the Kaluza–Klein states are thrown away. Following Ref. [27], the reader can so that the action (41) agrees with the nonpolynomial easily convince themselves that the actions above in 10 action (37), eventually expanded in the form factor. Al- and 11 dimensions can be rendered supersymmetric at though we can conclude that our proposal is compatible least up to the quadratic order, because the form factor with the superstring-theory effective action at the first does not change the spectrum of the theory. A prelimi- order in the local truncation of the entire function, this nary analysis of supersymmetry has been done for N =1 result is somewhat weaker than a full physical equiva- supergravity in four dimensions and N = 1, N = 2 su- lence. In fact, it is well known that the S-matrix of any pergravity in three dimensions [27]. In that reference, Lagrangian field theory is invariant under field redefini- supersymmetry was realized for a nonpolynomial higher- tions that are nonlinear but local [106–108]. To the best 9

µ µ ν 2σ(xµ) 10 µ ν 2 of our knowledge, no analog of this theorem is known to = gµν (x )dx dx + e [dx + Aν (x )dx ] , hold for transformations with nonlocal operators such as (42), for which only a comparison with the local theory where M,N = 0,..., 10 and µ,ν = 0,..., 9. Replacing after truncation of the form factor is presently available. this decomposition into the 11-dimensional theory, we obtain 2 −ℓ11 1 10 σ e 1 µν SIIA = 2 d x g e R + Gµν − R C. S-duality 2κ | | " Z p 1 −ℓ2 1 −2σ −ℓ2 F e 11 F e H e 11 H All closed string theories defined in ten dimensions are −2 4! 4 4 − 2 3! 3 3 invariant under the rescaling × × 1 2 1 e2σF e−ℓ11 F B F F + ..., −4 2 2 − 4κ2 2 ∧ 4 ∧ 4 Φ Φ C, Z → − (46) g eCg , (43) s → s where we introduced the fundamental length ℓ11 = together with a possible rescaling of the other fields. 1/M11 = 1/Λ in 11 dimensions. In ten dimensions, the Here, Φ is the dilaton, gs is the string coupling, and C is gravitational coupling constant for all string theories is an arbitrary constant. Using this rescaling, one can reab- given by sorb gs in Φ; then, the coupling constant in string theory 2 7 8 2 is related to the dilaton expectation value Φ . There is 2κ = (2π) ℓs gs . (47) no potential for Φ at the perturbative level,h i and hence The gravitational couplings in 11 and 10 dimensions are Φ can take an arbitrary value. related by [109] h i The effective theory is also invariant under the rescal- 2κ2 = (2πR)2κ2 = (2π)8 ℓ9 g3, (48) ing 11 10 s s 2 where R is the radius of the 11th direction and ℓs = ′ ′ ′ 2 α λ α , α . Defining the 11-dimensional Planck mass as 2κ11 = → 8 −9 g λg , (44) (2π) M11 , we get µν → µν 1 3 again together with a rescaling of the other fields. This ℓ11 = gs ℓs , (49) ′ 3 invariance makes it possible to absorb α into the def- gs = (M11R) 2 , (50) inition of the metric gµν . As a further check that the 3 ℓ11 nonlocal theory under inspection has viable properties, ℓs = . (51) R we now show that it respects S-duality between Type- IIA theory and 11-dimensional supergravity. When com- As in the local case [110, 111], the 11D theory is the pactifying the latter to ten dimensions, one gets the low- nonperturbative decompactification limit g (R s → ∞ → energy limit of nonpolynomial Type-IIA string theory. ) of the 10D one. In fact, making use of the metric Upon dimensional reduction, the 11-dimensional metric rescaling∞ g decomposes into a scalar σ, a vector A , and the MN µ 2 ten-dimensional metric, g e−σg with σ = Φ , (52) µν → µν 3 2 M N ds = gMN dx dx (45) the action (46) becomes

2 2 ′ 3 Φ −α gs e 3 2 2 2 2 ′ 3 Φ ′ 3 Φ 1 10 −2Φ e 1 µν µ −α gs e 3 1 −α gs e 3 SIIA = 2 d x g e R + Gµν − R + 4( Φ) e µΦ F4e F4 2κ | | " ∇ ∇ − 2 4! Z p × 2 2 2 2 1 ′ 3 Φ 1 ′ 3 Φ 1 H e−α gs e 3 H F e−α gs e 3 F B F F + ..., (53) −2 3! 3 3 − 4 2 2 − 4κ2 2 ∧ 4 ∧ 4 × # Z where the dots also include other dilaton contributions coming from the d’Alembertian in the exponential operator between the ﬁeld strengths. This is the expression in the Jordan or string frame. Using (51) and the further rescaling g eΦ/2g , the action in the Einstein frame reads µν → µν

2 Φ ′ 3 −α gs e 6 2 Φ 2 Φ ′ 3 ′ 3 1 10 e 1 µν µ −α gs e 6 1 −α gs e 6 SIIA = 2 d x g R + Gµν − R + 4( Φ) e µΦ F4e F4 2κ | |" ∇ ∇ − 2 4! Z p × 10

2 Φ 2 Φ 1 ′ 3 1 ′ 3 1 H e−α gs e 6 H F e−α gs e 6 F B F F + .... (54) −2 3! 3 3 − 4 2 2 − 4κ2 2 ∧ 4 ∧ 4 × # Z

As an application of the scale invariance common to all the second-order variation of Smassive with respect to the string theories, we explicitly give the transformation rules massive fields. In compact notation, for fields and couplings of Type-IIA supergravity. The 2 action (54) is invariant under the transformations δ SIIA δχδ δτ δχτ HÎIA = + C , δhµν δhρσ δhµν δhρσ C/2 h=0 h=0 gµν e gµν , → 2 Φ Φ C, δ Smassive Hˆ (n) = . (58) → C− φ δφ(n)δφ(n) g e g , (n) s → s φ =0 C F2 e F2 , → In defining the effective action for the massless fields, all F eCF . (55) 4 → 4 the external massive states are fixed to zero, and the ′ one-loop effective action reads This invariance makes it possible to fix α = gs = 1. ∗ N (1) i i Γ = S (¯g)+ Tr ln(Hˆ )+ Tr ln Hˆ (n) IIA 2 IIA 2 φ V. EFFECTIVE ACTION INCLUDING n=1 X h i MASSIVE STATES i i Tr ln(Mˆ ) Tr ln(Cˆ) . (59) − − 2 In this section, we consider the infinite tower of massive states present in the string spectrum and we formally Evaluating the traces, we find the divergent contribution introduce the full effective action compatible with string (1) 1 10 field theory and general covariance. Apart from the mass- Γdiv d x g less sector, the ten-dimensional theory (37) should in- ∝− ǫ | |× Z p clude all the massive states. At quadratic order, the La- 2 µν (i) 5 β R + β 2 R + β 2 R R + ... + β (R ) grangian density reads R R Ric µν R5 i " i # X +∞ + β(j) (Φ,B , F , F , fermions) , (60) (n) 2 −Λ (n)µ1...µk (j) [2] [2] [4] massive = φ ( M ) e φ (56) O L µ1...µk − n j n=1 X X (1) (∞) where the index i indicates all possible combinations of + φ ,...,φ ,gµν ,B , Φ,H , A , V [2] [3] [1] curvature invariants of a given order. The other operators (Φ,B , F , F ) of dimension up to M D in- where the potential includes all the interactions with O(j) [2] [2] [4] V 2 3 4 the massless states as well as between the massive states volve the other massless fields, for example F[2], F[2], F[2], 5 2 2 compatible with string field theory. F , F ( F[2]) , and so on. [2] [2] ∇ We first truncate the infinite tower to a fixed integer The running coupling constants multiply all the opera- number N ∗ and calculate the one-loop effective action tors of dimension up to M D that we can add to the bare Γ(1). Here, we are interested in the beta functions coming action (37) from the beginning. However, the quantum from the divergent contribution to Γ(1). Integrating out divergences depend only on the vertices coming from the all the massive states, the effective action is form factors (the other vertices contribute to the finite part of the effective action). The beta functions β(i) turn (1) Γ = i ln Z (57) out to be independent of the running coupling constants − i but dependent on the nonrunning parameters defining = S (¯g)+ S (¯g)+ ln det(Hˆ ) IIA massive 2 IIA the form factor. Contributions to β(i) come from the ∗ N massless and massive states circulating in the one-loop i i + ln det(Hˆ (n) ) i ln det(Mˆ ) ln det(Cˆ) , amplitudes, namely, 2 φ − − 2 n=1 ∗ X N N∗ (i) (i) (i) (i) where we employed the background field method and β∗ := β0 + βn (Mn) =: βn (Mn) , decomposed the metric as g =g ¯ + h . The de- n=1 n=0 µν µν µν X X terminants for the operators Mˆ and Cˆ come from the M0 0 , (61) ≡ ghost contributions, HÎIA is the second-order variation of ˆ the ten-dimensional action with respect to all the mass- where the massless-states contribution from HIIA has ˆ (i) less fields (including the gauge-fixing term), and Hφ(i) is been denoted as β0 . The finite contributions to the 11

Lagrangian density in the ultraviolet regime read and four external states in string scattering amplitudes. Assuming that such finiteness held for all amplitudes, one µν fin βR2 R ln R + βRic2 Rµν ln R would like to understand the reasons beyond this prop- L ∝ −µ2 −µ2 erty. Could it be due to the infinite tower of higher-spin + ... + β(i) R3 ln R2 fields or to supersymmetry or both? If we focus on the R5 −µ2 i i ultraviolet divergences, we see that bosonic string theory X itself is finite, although the presence of a tachyon in the + β(j) Φ,B , F , F , fermions, ln , spectrum undermines the stability of the theory. The in- O(j) [2] [2] [4] −µ2 j troduction of supersymmetry allows one to remove the X (62) tachyon, but it is not able to tame the divergences, as evident in local supergravity. These data highlight the where µ is the renormalization scale. crucial role carried out by higher-spin fields in the string At this point, there are two possibilities. One is to spectrum in making the theory finite. However, in gen- assume that the beta functions vanish order by order in eral, it is not sufficient to introduce an infinite number the string spectrum, of particles: they also have to fit in a particular mass β(i)(M )=0 n, i . (63) spectrum. The finiteness of the theory is achieved if the n n ∀ One can thus achieve finiteness by introducing the min- mass spectrum implies a particular symmetry in the loop imal number of local operators needed to satisfy (63) at amplitudes that, in the string case, is modular invariance any mass-spectrum level. Such a conjecture could be [112]. This symmetry forces us to exclude the ultraviolet checked by considering a level-by-level truncation of the region from the integration domain in the loop ampli- action. This is not a trivial task because the exponen- tudes so that physical processes result in being finite. tially growing form factor is compatible with the presence Looking at the effective SFT action (10), there is an of many other interaction operators in the action, as long asymmetry between the kinetic and the interaction term, as they do not invalidate super-renormalizability [in other due to the presence of the exponential factor in the words, these operators in momentum space cannot grow quadratic operators. As mentioned in Sec. IIA, the ex- faster than any of the other operators present in (37)]. ponential operator is a consequence of the reparametriza- However, this order-by-order prescription is not a fea- tion symmetries of the full theory. Such asymmetry ture to be expected in string theory, where only the pres- would imply a suppression of interaction terms in the ence of an infinite number of fields makes the amplitudes ultraviolet and suggests that string theory does not finite. As we will comment in Sec. VI, string theory is manifest ultraviolet divergences, contrary to a general- possibly finite (and not only super-renormalizable) as a covariant effective action. However, due to the fact that consequence of modular invariance; therefore, we expect the Lagrangian is written in terms of curvature invari- the beta functions β(i) in (61) to be identically zero in ants, general covariance implies the same nonlocal op- the limit N ∗ + : erators in the kinetic term as well as in the interaction → ∞ +∞ vertices for gravity. Therefore, we are apparently able to β(i) := β(i)(M )=0 i . (64) get a finite theory only at second and higher orders in n n ∀ the loop expansion, but not at the one-loop level, since n=1 X we find the same scaling both in the graviton propagator This relation will need to be checked in the future. The and the vertices. On the other hand, general coordinate massive states in string theory play a similar role as of invariance in string theory can only be achieved through Pauli–Villars fields in quantum field theory, but with the cancellations among contributions from infinitely many crucial double difference of being infinitely many and 3 interaction terms [67]. Therefore, the role of the infinite physical (i.e., they do not include ghosts). number of massive states is twofold: to assure modular invariance (via the particular mass spectrum of the VI. FINITENESS IN QUANTUM GRAVITY string) and diffeomorphism invariance. AND STRING THEORY The nonuniqueness of string theory in ten dimensions reflects the compatibility of different mass spectra with both requirements. Let us consider again the gravity Evidence is known that, in string theory, loop ampli- sector of the nonpolynomial theory (37). Expanding the tudes are finite. This has been verified up to two loops action in powers of the graviton field and making the field redefinition 3 Λ Another way to make a ten-dimensional theory finite is to in- h := e 2 h˜ , (65) troduce a (non)local gravitational potential at least cubic in the Riemann tensor. Then, the theory is finite by a suitable choice of up to the three-graviton vertex we get the action the coefficients in the operators characterizing the potential [100]. At present, this mechanism seems incompatible with the previous S dDx h e− Λ h + h ∂2he− Λ h + ... (66) one involving an infinite number of massive states because the ∼ Z above-mentioned potential cannot be generated by integrating Λ Λ Λ D 2 −Λ out the massive states. = d x h˜ h˜ + e 2 h˜ ∂ e 2 h˜ e e 2 h , Z h i 12 which is very close to the structure in (10) and even more to 10 dimensions and has the correct local limit expected so to the structure in (5). However, in the present form, both in 11D and Type-II supergravity. the theory is no longer explicitly general covariant. These There are several open points that can be explored to arguments indicate that, possibly, also effective string verify the feasibility and consequences of the picture. field theory is not finite when general covariance is explic- Quantum theory and supergravity. Although our pro- itly recovered or implemented. If this was the case, then • the perturbative finiteness of string theory (in ten dimen- posal satisfies all the properties (a)–(d) simultaneously, sions) would mainly follow from modular invariance. an important consistency check would be to carry out the We mention a further complication in the picture that more ambitious program of the Wilsonian effective action illustrates the interplay of various perturbative and non- mentioned in the Introduction. Until then, our model can perturbative ingredients. The finiteness of nonpolyno- have some beneficial role in the advancement of the field mial supergravity in 11 dimensions, Eq. (38), is not a re- by constituting a practical alternative in which several flection of modular invariance in ten dimensions, since we cross-checks and avenues of exploration are readily avail- are not considering higher-spin fields in either theory. Ac- able. As outlined above, the role of higher-spin states in tually, following the considerations in Ref. [101], also the the UV behavior of the theory is still to be fully assessed. ten-dimensional theory (37) is expected to become finite Also, the theory in 11 dimensions has direct implications if we introduced the infinite tower of Kaluza–Klein modes for four-dimensional quantum gravity. In D = 4, there is coming from 11D nonpolynomial supergravity compact- optimistic evidence that local N = 8 supergravity be per- ified on a circle. These states are nonperturbative from turbatively finite (see Ref. [118] and references therein), the string-theory point of view [111]. but this view is not universally shared [119] and the the- However, ten-dimensional nonpolynomial supergravity ory may even have a divergence at seven loops [120–122]. could be made finite also in a different way, by introduc- Compactifying 11D nonpolynomial supergravity to four ing all the higher-spin fields of the perturbative string dimensions would give rise to a nonlocal version of N =8 spectrum. (Other possible states to be introduced could supergravity. Possibly, also the four-dimensional theory come from the quantum membrane in 11 dimensions would be finite, provided we retain the infinite tower of [101].) Given that in both cases the ten-dimensional the- massive states coming from the Kaluza–Klein compacti- ory might become finite, it is permissible to ask whether fication. Moreover, we showed that our proposal respects the two different completions are actually related by a S-duality between Type-IIA supergravity and M-theory. nonperturbative duality. The role of the exponential Various other checks of S-duality as in Ref. [123] and of form factor in this and other dualities should be checked T-duality [124–126] could be devised at this stage. in the future. Another possibility is to make the 10D Singularities. Point (v) in Sec. I deserves further at- theory finite by applying the Pauli–Villars regularization tention,• too. Explicit solutions, found via the diffusion scheme [113–117]. It is indeed well known that the pres- equation method [49] or perturbative techniques [88], can ence of the infinite tower of resonances in string theory indicate whether infinities in the theory are removed also acts, in some sense, like a Pauli–Villars regulator. In- in the sense of spacetime singularities. The cosmology frared divergences survive, but they can be controlled by of our model in D = 4 has been considered in Ref. [127] using supersymmetry. A last option we wish to mention in the local limit and in Ref. [34] exploiting the asymp- is to introduce an O(R3) local potential to make all the totic freedom of the theory. In particular, the results beta functions vanish [100]. of Ref. [34] indicate that the cosmic big-bang singularity might be indeed removed, and that other nontrivial phenomenology (such as primordial acceleration) is VII. DISCUSSION induced by the same nonlocal mechanism, as generally found in nonlocal cosmologies with exponential opera- In this paper, we began by reviewing an extended the- tors [19–21, 28, 128]. It is likely that also black holes ory of gravity capable of overcoming the renormalizabil- in M-theory nonlocal gravity would show, perhaps not ity problem of Einstein gravity in the ultraviolet regime. surprisingly, interesting physics. At the quantum level, the theory is super-renormalizable p-adic string and complex poles. For completeness, in even dimensions. In odd dimensions, at one loop, there we• briefly comment on the arising of p-adic features in are no local invariant counterterms with an odd num- the quantum theory. p-adic models are based on propaga- ber of derivatives, and the theory turns out to be finite. tors of the form appearing in p-adic string theory, which All the other loop amplitudes are finite because of the shares some properties with string field theory [5, 82–86]. nonpolynomial nature of the action. Then, we fixed the For instance (and setting all couplings to 1), a free p-adic nonlocal kinetic terms of the theory according to what scalar field is characterized by a Lagrangian of the form was found in effective string-field-theory actions and pro- = φ(e− 1)φ, i.e., without an overall (polynomial posed the resulting supersymmetric extension as an 11- Lof the) multiplying− the entire function. Contrary to dimensional quantum supergravity, i.e., as a candidate string field theory, however, such models have a tachy- field-theory limit of the massless sector of M-theory. The onic mode (not to be confused with the tachyon of the proposal is consistent with the compactification from 11 bosonic string). In fact, the dispersion relation associated 13 with the p-adic scalar is where γ(∂τ ) is a form factor in one dimension. In Ref. 2 [130], it was shown that the open string emerges natu- ek 1=0 , (67) − rally from Eq. (69) for a particular choice of the form whose solutions are k2 = (k0)2 + k 2 = 2πin, where factor not dissimilar from ours, while in Ref. [132] we n Z. If the spatial momentum− is real,| | then k0 is com- gave a precise determination of the dependence of the plex∈ valued, and there is an instability in Lorentzian sig- dimensionality of the object described by (69) from the nature. The latter represents false-vacuum excitations resolution scale at which it is observed. In the context of of the p-adic string. On the other hand, the form fac- quantum gravitational theories, an entire function with tor typical of string field theory gives rise, for a massless asymptotically polynomial behavior in a region around field, to the dispersion relation the real axis is [11] 2 +∞ k2ek =0 , (68) ( 1)n+1 p(z)2n V −1(z)= eH(z) = exp − 2 2 2nn! which is not invariant under k k +2πin. Therefore, "n=1 # → X in the SFT case, there are no extra complex or real poles 1 Γ[0,p(z)2]+γ +ln[p(z)2] = e 2 E , in the propagator. For this and other reasons (p-adic { } actions are not derived as effective actions from proper where γE is Euler’s constant and p is a polynomial of a string theory), p-adic systems are regarded as simpli- certain degree. When expanded for small z (small mo- fied nonlocal toy models of some more fundamental SFT menta), the function H does not include the linear term structure. Nevertheless, they can be employed to obtain of string field theory. To obtain it, here we modify the some useful insight in SFT. above expression as H H′, where The theory (27) with form factor (28) is of SFT type → rather than p-adic. As we have seen, at quadratic order H′(z)= H(z)+1 e−z (70) − in the graviton fluctuations the in the denominator 1 =1 e−z + Γ[0,p(z)2]+ γ + ln[p(z)2] compensates two of the four derivatives in the numer- − 2 E ator, and the outcome is the bare propagator G(k2) l 1 2 exp( k2)/k2 around flat spacetime. The advancement∼ = 0 + z + ... + z + ... + ln[p(z) ] . − − 2 in the techniques for understanding the dynamics of non- particle string ? local theories does not actually require to use p-adic toy ? |{z} |{z} |{z} models, since such techniques can be applied directly.4 With this choice of form factor, the theory| is{z well} de- However, our theory might possess some p-adic features fined and perturbatively calculable [100]. Equation (70) at the quantum level. To be more precise, p-adic-like dis- shows a local point-particle phase, a string phase, and persion relations may arise as a result of quantum cor- other phases that, in 11 dimensions, one might conjec- rections to the bare propagator. In fact, the theory (27) ture to represent extended objects such as membranes is tree-level unitary and, at the quantum level, we do not as well as high-energy corrections to the particle phase. expect extra real poles apart from those of the local spec- The spectral dimension dS can be a useful tool to un- trum. However, other operators will be generated in the derstand the nature of the underlying objects. For H′, quantum action and the dressed dispersion relation for d = 11 when z 0 (Λ + , IR), d = 0 in all the S → → ∞ S the graviton will contain extra k-dependent terms with higher-derivative intermediate regimes, and dS > 0 in respect to Eq. (68). In general, the solutions of such the high-energy regime z + . A possible interpreta- a dispersion relation will be complex, similarly to what tion of this result is that the→ theory∞ interpolates between happens for Eq. (67). This proliferation of complex con- two point-particle regimes feeling different dimensions, jugate poles has been recently considered in Ref. [129]. passing through multiple phases dominated by extended In our case, however, the theory is finite at the quantum objects with vanishing extension along one or more direc- level, and one can show that such complex poles can be tions in the ambient spacetime. However, the relation be- removed by a field redefinition with a trivial Jacobian. tween nonlocality and extended objects is more delicate The actual presence and physical interpretation of these than this direct identification. In fact, nonlocal operators poles deserve further attention. do not change the poles of standard local propagators, Extended objects. Finally, it is worth noticing that which means that the string and membrane spectrum of • nonlocality has been often advocated as a description of local and nonlocal versions of the theory is the same. In extended objects. This interpretation is suggested, inter particular, the insertion of nonlocal form factors would alia, by studies on the nonlocal particle action [130, 131] not change the spectrum of matrix or noncommutative models in an essential way. The presence of membranes µ Sparticle = dτ x˙ µ γ(∂τ )x ˙ , (69) in the 11-dimensional theory is unrelated to nonlocality: Z it is rather signaled by the presence of p-forms in effective target actions (local or nonlocal). However, nonlocal operators could thicken the branes of the local theory, as 4 We will do so for the theory presented in this paper in a forth- the results of Ref. [132] suggest. This effect could be coming publication [90]. verified, for instance, with the techniques of Ref. [132] 14 when applied to a brane effective action. Moreover, to checked in the future, that the class of theories presented check whether M2- and M5-branes actually are included in this paper can provide a general framework for a de- in our theory, one should also verify that metrics repre- scription of extended objects such as strings and mem- senting M2- and M5-branes are solutions of the nonlocal branes, thus strengthening our tenet that the action (1) dynamical equations in 11 dimensions (see, e.g., Sec. 4.9 correctly captures the main degrees of freedom of M- of Ref. [133] for the local case). We expect that the brane theory in the field-theory limit. metric will be singularity free as in the black hole case [95]: the exponential form factor would smear the Dirac delta source in the transversal directions into a Gaussian, and the solution would be regular everywhere. To check ACKNOWLEDGMENTS the existence and validity of such solutions, and whether they are exact or approximate, would require the full We thank R. Brandenberger, H. Verlinde, and E. Wit- equations of motion, which will be presented and studied ten for useful discussions and acknowledge the i-Link co- in Ref. [90]. Preliminary calculations indeed show that operation programme of CSIC (Project No. i-Link0484) M2- and M5-branes are exact dynamical solutions. for partial sponsorship. The work of G.C. is under a These hints are suggestive of the possibility, to be Ramón y Cajal contract.

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