Letters in High Energy Physics LHEP-186, 2021

String Theory, the Dark Sector, and the

Per Berglund,1 Tristan H¨ubsch,2 and Djordje Minic3 1Department of Physics and Astronomy, University of New Hampshire, Durham, NH 03824, USA 2Department of Physics and Astronomy, Howard University, Washington, DC 20059, USA 3Department of Physics, Virginia Tech, Blacksburg, VA24061, USA

Abstract

We discuss , , and the hierarchy problem in the context of a general noncommuta- tive formulation of theory. In this framework, dark energy is generated by the dynamical geometry of the dual while dark matter, on the other hand, comes from the degrees of freedom dual to the visible matter. This formulation of is sensitive to both the IR and UV scales, and the Higgs scale is radiatively stable by being a geometric mean of radiatively stable UV and IR scales. We also com- ment on various phenomenological signatures of this novel approach to dark energy, dark matter, and the hierarchy problem. We find that this new view on the hierarchy problem is realized in a toy model based on a nonholomorphic deformation of the stringy . Finally, we discuss a proposal for a new nonperturbative formulation of string theory, which sheds light on M-theory and F-theory, as well as on and .

Keywords: string theory, dark energy, dark matter, hierarchy ical signatures stemming from this correlation. We also com- problem ment about other phenomenological signatures of this new ap- DOI: 10.31526/LHEP.2021.186 proach to dark energy, dark matter, and the hierarchy problem in the context of string theory. In particular, we discuss this new view on the hierarchy problem within a toy model based on a 1. INTRODUCTION nonholomorphic deformation of the stringy cosmic string. Fi- nally, we present a proposal for a new nonperturbative formu- The deep foundations and real world implications of string the- lation of string theory, which sheds light on M-theory as well ory [1] are still shrouded in mystery. In particular, the problem as F-theory and illuminates the emergence of supersymmetry of dark energy has excited a lot of recent discussion within var- and holography. ious aspects of string theory [2,3,4]. Motivated in part by some prescient work on various aspects of string theory [5,6,7,8], as well as some recent developments in the noncommutative field 2. GENERAL STRING THEORY AND DARK theory [9], double field theory [10], [11], and ENERGY quantum foundations [12, 13], a generic, noncommutatively generalized geometric phase-space formulation of string the- We begin our discussion with a review of the generic noncom- ory [14] has been recently developed. Within this framework, mutative and doubled formulation of string theory and its rela- which clarifies many foundational issues found in the classic tion to dark energy. Following the recent discussion in [14], the textbook discussion of string theory [1], it was recently shown starting point is the chiral string worldsheet description that dark energy is naturally induced from the overall curva- Z ture of the dual of the observed spacetime [15]. This effect is ch 1 h A A i B Sstr = ∂τX (ηAB + ωAB) − ∂σX HAB ∂σX , (1) the leading (zeroth) order term in an expansion of the non- 4π τ,σ commutative scale, λ, and is realized in certain stringy-cosmic- A string-like toy models that rely only on the ubiquitous - with τ, σ worldsheet coordinates. Furthermore, X (τ, σ),(A = a (“axilaton”) system and gravity [16]. In particular, it 1, . . . , 26, for the critical bosonic string) combine the sum (x ) was pointed out that the Higgs scale is radiatively stable by and the difference (x˜a) of the left- and right-moving chiral 1 being the geometric mean of the radiatively stable UV (Planck) on the string. The mutually compatible dynamical fields are and IR (dark energy) scales, respectively. the antisymmetric symplectic structure ωAB, the symmetric po- larization metric ηAB, and the doubled symmetric metric HAB, In this paper, we consider the inclusion of the dual matter respectively, defining the so-called Born geometry [14]. Quan- which, to first order in λ, can be interpreted as the dark matter tization renders the doubled “phase-space” operators Xˆ A = as well as providing an intrinsic stringy resolution to the hier- archy problem. Furthermore, in this new approach, the known fields and their dark matter duals are dynam- 1This formulation also leads to a natural proposal for a nonperturbative string ically correlated, and we comment on various phenomenolog- theory which will be discussed in Section7 of this paper. 1 Letters in High Energy Physics LHEP-186, 2021

a (xˆ /λ, x˜ˆa/λ) inherently noncommutative, inducing [14] produces a positive cosmological constant Λ > 0. Hence, the weakness of gravity is determined by the size of the canonically ˆ A ˆ B AB [X , X ] = iω , (2) conjugate dual space, while the smallness of the cosmological constant is given by its curvature. In particular, we have or, in components, for constant nonzero ωAB, p R −g(x) R(x) + ...  a ˆ 2 a a b ˆ ˆ S¯ = X + . . . (6) [xˆ , x˜b] = 2πiλ δb, [xˆ , xˆ ] = 0 = [x˜a, x˜b], (3) R p X −g(x) where λ denotes the fundamental length scale, such as the fun- which leads to a seesaw-like formula for the cosmological con- damental Planck scale, e = 1/λ is the corresponding funda- stant, discussed below. mental energy scale, and the string tension is α0 = λ/e = λ2; see also Section6. Note that the Hamiltonian and diffeomor- So far, we have omitted the matter sector explicitly. In what A B A B follows, we argue that the dual part of the matter sector ap- phism constraints, ∂σX HAB∂σX = 0 and ∂σX ηAB∂σX = 0, respectively, are treated on the equal footing. In particular, pears as dark matter, which is, in turn, both sensitive to dark the usual spacetime interpretation of the zero-mode sector of energy [18] and also dynamically correlated with the visible string theory [1] is tied to the solution of the diffeomorphism matter. We next focus on this dark matter and emphasize the constraint by level matching. In this more general and gener- unity of the description of the entire dark energy and mat- ically noncommutative formulation, the spacetime interpreta- ter sector, induced and determined by the properties of the tion is replaced by a modular spacetime (or quantum space- dual spacetime, as predicted by this general, noncommuta- time) realization (also found in the context of quantum founda- tively phase-space doubled formulation of string theory. tions) [14]. Thus, all effective fields must be regarded as bilocal φ(x, x˜) [14], subject to (3), and therefore inherently non-local in the conventional xa-spacetime. Such noncommutative field 3. DARK SECTOR AND THE HIERARCHY theories [9] generically display a mixing between the ultravi- PROBLEM olet (UV) and infrared (IR) physics with continuum limits de- fined via a double-scale renormalization group (RG) and the We have already emphasized that in this generic noncommu- self-dual fixed points [9], [14]. This has profound implications tative formulation of string theory, all effective fields must be for the generic physics of string theory and, in particular, the regarded a priori as bilocal φ(x, x˜) [14], subject to (3). More- problems of dark energy, dark matter, and the separation of over, the fields are doubled as well, and thus for every φ(x, x˜), scales that goes beyond the realm of effective field theory. there exists a dual φ˜(x, x˜). This can be easily seen in the back- ground field approach, which we will consider in the next sec- In [15], we have argued that the generalized geometric for- tion. Therefore, in general, to the lowest (zeroth) order of the ex- mulation of string theory discussed above provides for an ef- pansion in the noncommutative parameter, λ Snc takes the fol- fective description of dark energy that is consistent with a de eff lowing form (that also includes the matter sector and its dual), Sitter spacetime. This is due to the theory’s chirally and non- which generalizes equation (5) (and where, once again, the cor- commutatively (3) doubled realization of the target space and responding Planck lengths are set to one): the stringy effective on the doubled noncommutative (3) a spacetime (x , x˜a) ZZ q h i g(x) g˜(x˜) R(x)+R˜ (x˜)+Lm(A(x, x˜))+L˜ dm(A˜(x, x˜)) . (7) ZZ q nc = ( )  ( ) + ( ) +  Seff Tr g x, x˜ R x, x˜ Lm x, x˜ ... , (4) Here, the A fields denote the usual Standard Model fields, and the A˜ fields are their duals, as predicted by the general formu- L including the matter Lagrangian m and with the correspond- lation of quantum theory that is sensitive to the minimal length ing Planck lengths set to one. (The ellipses denote higher-order (the noncommutative parameter λ [14]). In the following sec- curvature terms induced by string theory.) This result can be tion, we will elaborate more explicitly on the dual matter de- understood as a generalization of the famous calculation by grees of freedom. Right now, we are concerned with the gener- Friedan [17]. Using (3), Snc clearly expands into numerous eff alization of the discussion summarized in the previous section. terms with different powers of λ, which upon x˜-integration and from the x-space vantage point produce various effective After integrating over the dual spacetime, and after taking 2 terms. Dropping Lm, for now, to the lowest (zeroth) order of the into account T-duality, equation (6) now reads: expansion in the noncommutative parameter λ of Snc takes the p eff R −g(x) R(x) + L (x) + L˜ (x) ¯0 X m dm form S = p + . . . (8) ZZ q q R −g(x) S = − −g(x) −g˜(x˜) R(x) + R˜ (x˜), (5) X The proposal here is that the dual sector (as already indicated in a remarkable result which was first obtained almost three the previous section) should be interpreted as the dark matter decades ago by Tseytlin [7], effectively neglecting ωAB in (2) by assuming that [xˆ, x˜ˆ] = 0 [7]. 2Tseytlin proposal has been further explored by Davidson and Rubin who in In this leading limit, the x˜-integration in the first term of (5) particular showed that the cosmological constant is necessarily nonnegative def- defines the gravitational constant GN and in the second term inite [19]. 2 Letters in High Energy Physics LHEP-186, 2021 sector, which is correlated with the visible sector via the dark the dual labels x˜ and also λ (in a coarsest approximation) by the energy sector, as discussed in [18]. We emphasize the unity of global dynamical scale s ∼ ∆x˜ ∼ λ2∆x−1. Also, normal order- the description of the entire dark sector based on the properties ing produces σ. This is an effective realization of the sequester of the dual spacetime, as predicted by the generic formulation mechanism in a noncommutative phase of string theory. One of string theory (as a quantum theory with a dynamical Born important lesson here is that the low-energy effective descrip- geometry) [14]. tion of the generic string theory is, to the lowest order, a se- questered effective field theory and, more generally, a noncom- Let us turn off the dynamical part of gravity and consider mutative effective field theory [9] of a new kind, which is de- the hierarchy problem. First, we have fined within a doubled RG, which is covariant with respect to ZZ q the UV and IR cutoffs and which is endowed with a self-dual S = − g(x)g˜(x˜) L (A(x, x˜)) + L˜ (A˜(x, x˜)), (9) 0 m dm fixed point [14]. which leads to the following nonextensive action to lowest or- der in Λ R p   4. DARK MATTER AND STRING THEORY −g(x) Lm(x) + L˜ dm(x) S¯ = X + . . . (10) 0 R p In this section, we elaborate on the dual matter degrees of free- X −g(x) dom and the explicit appearance of dark matter in the gen- after integrating over the dual spacetime. eral formulation of string theory [14]. The previous discussion This implies a seesaw formula which involves the matter looked at the stringy effective action to the lowest order in Λ, scale from the matter (and dark matter) part of the action and neglecting the noncommutative aspect of the generalized geo- metric description of the string worldsheet (1). In order to see the scales related to the UV (Planck scale, MP) and the IR (dark the effect of the leading order correction in Λ of (7), we con- energy scale, MΛ): 2 sider the zero modes of Sch . The associated action is MΛ MP ∼ MH, (11) str fixed by the symmetries of the chiral worldsheet in terms of the where MH denotes the characteristic matter scale. Completely symplectic form ω, the O(D, D) metric η, and the double met- generally, this relation follows from the diffeomorphism con- ric H—the so-called Born geometry—and takes the following straint of the chiral string worldsheet, which is controlled by form (following the general results for the chiral string world- the O(d, d) biorthogonal metric η and which implies, in the sheet description [14]): limit of zero modes (and zero momenta), that EE˜ = M2. Here, Z ˜ 2  E and E are the energy scales in the observed and dual space- SMP = px˙ + p˜x˜˙ − λ pp˜˙ − Nh − Nd˜ . (13) time, respectively, and M is a new (mass) parameter. Since the τ IR (dark energy) scale can be interpreted as the vacuum en- Here, the Hamiltonian constraint, fixed by the double metric, is ergy, which in the matter sector is controlled by the Higgs po- given by h = p2 + p˜2 + m2, and the diffeomorphism constraint, 2 tential, M naturally sets the Higgs scale, MH, which is indeed fixed by the O(D, D) metric, is d = pp˜ − M . These constraints the case numerically. If we remember that the geometry of the are inherited from the quantization of chiral worldsheet theory. dual spacetime is responsible for the origin of dark energy, then Finally, the symplectic structure fixes the λ2 pp˜˙ term. Note that the dual energy E˜ can be set to the dark energy scale. Then, the m2 should not be confused with the (mass)2 of a particle exci- fundamental energy scale E in the observed spacetime is the tation. In parallel with the usual discussion found in introduc- Planck scale. This mixing of the UV and IR scales in a fully tory chapters of textbooks on quantum field theory, one has to covariant formulation is a unique feature of the chiral string understand the representation theory associated with the sym- worldsheet theory. (For other approaches to the hierarchy prob- metries of the underlying Born geometry and interpret m2 and lem which mix the UV and IR scales but which violate covari- M2 in terms of the relevant Casimirs in the full representation ance, consult [20].) theory of this description [23]. We are not going to pursue this question in what follows, but we alert the reader that M is a Note that both the UV and IR scales are radiatively stable. new parameter not found in the context of the effective field First, we note that the MP is the UV scale and the issue of ra- theory description, while m2 can be interpreted as a particle diative stability does not apply to it. Second, let us recall the mass only in a very degenerate limit in which p˜ = 0 and M = 0. radiative stability of the dark energy scale MΛ, the IR scale: in particular, as we have noticed, in our previous work, the effec- For concreteness, let us start with a vector background, by tive action of the sequester type [21] (see also [22]) shifting the momenta and the dual momenta by the standard, minimally coupled gauge field and its dual [14]: Z h R Λ i  Λ  p−g + s4L(s−2gab) + + σ 4 4 , (12) x 2G G s µ pa → pa + Aa(x, x˜), p˜a → p˜a + A˜ a(x, x˜). (14) where L denotes the combined Lagrangians for the matter and These gauge fields have the usual Abelian gauge symmetries. ( Λ ) dark matter sectors, µ is a mass scale, and σ s4µ4 is a global in- Thus, in the target space action for the gauge fields, we end teraction that is not integrated over [21]. This can be provided up with canonical Maxwellian terms (with the obvious in- by our set-up: start with bilocal fields φ(x, x˜) [14], and replace dex structure) plus a characteristic coupling inherited from the

3 Letters in High Energy Physics LHEP-186, 2021 symplectic structure case, the Hubble scale) has been observed in astronomical data Z and has been studied in the context of modified dark matter  2 2 ˜ ˜2 ˜  λ8 F − aλ [[ A,A ]] + F + FF + ... , (15) in [18]. However, the ratio L10 should not be naively consid- x,x˜ ered to be trans-Planckian, because that would require that λ is where the λ2[[ A,A˜ ]] term stems directly from the λ2 pp˜˙-term the Planck length, and L is the Hubble length. Instead, the two in (13) by the “minimal coupling” shift (14), its dimensionful scales should be considered as effective UV and IR scales, the coefficient a to be determined below. This “mixing” term may joint appearance of which is a direct evidence of a departure be expressed as from effective field theory, as further discussed in Section6. Having illustrated the general idea with the vector fields Z τ ˜ 0 def 0 0 dA(x˜(τ )) a [[ A,A˜ ]] = dτ A(x(τ )) , (16) Aa and A˜ , the form of the corresponding action for a scalar φ dτ0 τ∗ and its dual φ˜ is immediate (and similarly for pseudo-scalars): h iτ Z τ h i = 1 A(x)A˜(x˜) + 1 dτ0 A(x)A˜˙ − A˙ (x)A˜(x˜) . (17) 2 2 8 τ∗ τ∗ Z h λ i (∂φ)2 − [[ φ,φ˜ ]] + (∂φ˜)2 + ... , (20) L10 The first of these includes a (worldline-local) mixing term, to- x gether with its value at some “reference” proper time, τ∗, the as this would be forced in every dimensional reduction frame- integral of which (15) evaluates to an irrelevant additive con- work. Similarly, the corresponding action for the (by ˙ stant. The second part, [AA˜ − A˙ A˜], is far more interesting, as taking the “square root” of the propagating part of the scalar it provides a telltale “Zeemann”-like coupling of (A, A˜)-pairs action to accommodate the -statistics theorem) is to corresponding “external/background” fluxes, scaled by the coefficient “aλ2.” This relies on identifying A, A˜ as canonical Z h λ9 i (ψ∂ψ¯ ) − [[ ψ¯,ψ˜ ]] + (ψ∂˜¯ ψ˜) + ... , (21) coordinates in the target-spacetime (classical) field theory. Al- x L10 ternatively, in the underlying worldsheet quantum field theory, where [[ ψ¯,ψ˜ ]] = ψ¯ψ˜ are nonderivative bilinear terms, accompa- the A, A˜ are coefficients of certain quantum states, for which the nied by “external” fluxes as in (19). This result may be justified [[ A,A˜ ]]-term likewise accompanies a Berry’s phase-like quan- by target-spacetime supersymmetry, even if supersymmetry is tity. ultimately broken: the indicated terms are restricted to free The second quantization of the action (15) in the Coulomb fields in flat spacetime. In particular, the omitted interaction gauge (and its dual) produces the following structure fixed by terms here also include metric and curvature deviations from the Born geometry: flat spacetime. From the underlying worldline, or even world- Z sheet [14] point of view, such terms are induced from gen- (∂A)2+(∂˜ A)2−aλ2[[ A,A˜ ]]+(∂˜ A˜)2+(∂A˜)2+∂A ∂˜ A˜ + ... . eralizing (13) along the standard construction of GLSMs [24]; x,x˜ (18) the “free-field-limit” terms shown herein however remain un- Integrating over the dual spacetime (x˜), and setting p˜ → 0 in changed. The same prefactor is also implied from the world- ˙ the observable spacetime for consistency, poduces (with indices line point of view: the of each pp˜-term in the La- on the respective gauge fields fully restored) grangian (13) is a fermionic bilinear, χ¯χe, which couples to the same “external/background” flux (20), giving rise to a world- Z h 8 i line super-Zeemann effect [25]. 2 λ a ˜ b ˜ 2 (∂[a Ab]) − fb [[ Aa,A ]] + (∂[a Ab]) + ... , (19) x L10 λ9 The L10 scaling coefficient in (21), forced on dimensional def b a grounds in this mass-mixing term, sets the scale in this novel where A˜ a = ηab A˜ and f encode “background fluxes,” natu- b “seesaw mechanism,” in principle tunable to induce naturally rally at the fundamental scale λ from (3). Properly normalized a small masses. In fact, the inclusion of several mass on the worldline (17), the “mixing” term [[ Aa,A˜ ]] in spacetime scales enables concrete models to incorporate an entire hierar- must be renormalized by the volume of the primordial observ- chy of seesaw mechanisms, with a more reasonable chance to able spacetime, L10, prior to any compactification and inflation. approach the intricacies of a realistic mass spectrum. This term is thereby sensitive to both the fundamental UV cutoff λ and the primordial IR cutoff L. To summarize, the leading “kinetic” parts in the actions (20) and (21), together with the “mixing terms,” [[ φ,φ˜ ]] and [[ ψ¯,ψ˜ ]], In other words, in the observable spacetime, the visible sec- are seen to be natural: (a) by dimensional reduction from (19) tor and its dual/dark counterpart are “mixed/correlated” (19), to (20) and (b) by supersymmetry from (20) to (21). Space- in a way that is sensitive to both the UV and IR cutoffs. This cor- time supersymmetry is broken by interaction terms, explicitly relation becomes invisible in effective field theory, and it van- omitted from (19), (20), and (21), such as in the “axilaton” sys- ishes as either λ → 0 or L → ∞. Note that with a particular tem [16], and not at all unlike the Polonyi mechanism [26]; for double scaling, this term can be finite! This is consistent with a recent discussion, see also [27]. the general set-up of the chiral string worldsheet theory which has two cutoffs (UV and IR) and which generically should be In addition, pseudo-scalars such as can be viewed defined with self-dual RG fixed points. Such a correlation be- as boost generators (at least for constant profiles) between the tween visible and dark matter involving an IR scale (in this observed and dual [14]. First, note that the constant

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Kalb-Ramond field can be absorbed into a nontrivial symplec- doubled, and noncommutative formulation of string theory tic form (on its diagonal) after an O(d, d) rotation [14]. Thus, with a fundamentally bosonic and noncommutative formula- the Kalb-Ramond two-form enters into an explicit noncommu- tion, wherein spacetime and matter (and supersymmetry at the tativity of the modular spacetime, and it can be used to ro- Planck scale) can be viewed as emergent phenomena. tate between observed and dual spacetime coordinates (as an Next, we comment on the seesaw formula, which mixes UV explicit illustration of relative, or observer-dependent, local- and IR scales, and the neutrino sector. Such a seesaw would ity) [14]. Since the Kalb-Ramond two-form field dualizes into involve the neutrino and its dual partner. Quite generically, a pseudo-scalar in 4d, the 4d axion has the same features. More the dual sector acts as a source for the visible sector, and the generally, nonconstant Kalb-Ramond profiles imply nonasso- overall effect is to make the visible sector essentially mas- ciative structure [14]. Thus, axions are indicators of noncom- sive. This immediately provides a curious—and ubiquitous— mutative (when constant) and nonassociative (when propagat- mixing “mass” effect, where the length scale ratio λ9/L10 may ing) structures in modular spacetime, respectively. well be phenomenologically relevant; see Section 6. For every Finally, we note that the peculiar correlation between the -dual fermion pair, there is a mass matrix of the general h m λ9/L10 i visible and dark sectors, discussed for scalar, pseudo-scalar, form 9 10 , which induces well-known seesaw rela- λ /L me fermionic, and vector degrees of freedom, can be also found in tions. This includes all Dirac mass terms, as well as a mix of the gravitational and dual gravitational sectors. Thus, the ob- Dirac and Majorana terms, and may well provide a new way of served gravity and dark energy are correlated via the scale of generating neutrino masses. noncommutativity. This might have interesting observable ef- Next, we comment on other phenomenological issues asso- fects for the so-called H0 tension [28]; see [29]. ciated with the dual Standard Model. In the dual QED sector, we should find a dual that is correlated with the visible photon and that is distinct from the of effective 5. COMMENTS ON PHENOMENOLOGI- field theory. The correlation is proportional to the fundamental CAL IMPLICATIONS length and is finite even in the limit of zero momenta (deep in- frared). Also, the usual visible photon/dark photon coupling is The most important general predictions of the chiral string subdominant to this term that is inherent in our story. Similarly, worldsheet theory [14] are as follows: (1) the geometry of in the dual of the weak sector of the Standard Model, we have the dual spacetime determines the dark energy sector [15], a dual of the visible Z. This dual of Z should be distinguished and (2) the dual matter degrees of freedom naturally ap- from the usual Z0 by its sensitivity to the fundamental length pear as dark matter candidates, as discussed in the preced- and by its correlation to the visible Z. These types of correla- ing section. We note that, quite explicitly, the dark matter sec- tions might be found in correlated events in the accelerators, tor provides “sources” for the visible matter sector. This fol- which are not products of any standard particle decays. 8 lows from the coupling λ [[ φ,φ˜ ]], as predicted by the dou- L10 Finally, in the dual QCD, we should find interesting phe- bled/noncommutative set-up, and provides an explicit correla- nomenology in the deep infrared, even though that is a very tion between the dark matter sector and a visible sector. Given difficult region to study in QCD. In particular, given the new the seesaw formula for the dark energy which relates the dark view of the axion field in the above discussion, we have a pos- energy scale to the fundamental length, which could be taken sible new viewpoint of the strong CP problem in QCD. The first to be the Planck energy scale, then the dark matter is also sensi- observation here is that, according to [14], the constant Kalb- tive to the dark energy. So, the visible matter, dark matter, and Ramond field mixes x and x˜ spacetimes (it acts as a boost that dark energy are all related. This is consistent with the observa- linearly combines the spacetime and its dual in the context of tional evidence presented in [18], as we have already alluded a larger doubled and noncommutative quantum spacetime). to in this article. Also, the commutator of dual spacetime coordinates is given We emphasize that in our discussion of the hierarchy prob- by the constant B Kalb-Ramond 2-form field. So, for B = 0, we lem, the UV and IR scales are radiatively stable, and so is their get just the observed (4d) spacetime. Also, its H = dB field product, the Higgs scale. This new view on the hierarchy prob- strength is trivially zero. But H is dual (in 4d) to the axion lem goes beyond the usual tools of effective field theory due to (a), which is also constant. But B is zero (there is no preferred explicit presence of the widely separated UV and IR scales. The background direction), and so, this constant axion may be in- usual suggested approaches to the hierarchy problem: techni- terpreted as a uniform distribution for the axion (whose con- color, SUSY, and are all within the canonical stant values can be positive and negative). Now, focus on the effective field theory. In the context of string theory, effective QCD axion, relevant to the strong CP problem, and appearing field theory (and the approach to the hierarchy problem via a in the CP violating term aF ∧ F. Averaging this term, linear in SUSY effective field theory) can be naturally found via string the axion, over a uniform distribution for this axion produces compactifications, but in that case, one is faced with the is- R k zero: ( −k da a = 0, with k → ∞). For a complete argument, sue of supersymmetry breaking (and the fundamental question we would have to study small fluctuations of the axion field in of “measures” on the string landscape/swampland [2,3,4]). order to understand the robustness of this new viewpoint on We claim that these issues are transcended in the general, the strong CP problem.

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6. DARK ENERGY SEESAW AND THE HI- to [7]. Therefore, certain generic features of this doubled de- ERARCHY PROBLEM scription, such as the intensive effective action, directly trans- late into certain geometric features of our models, discussed The preceding discussion about dark energy, dark matter, and below. the hierarchy problem is based on the generic noncommuta- In these string models, the cosmological constant within the tive formulation of string theory. We now present a more con- codimension-2 world is determined by the anisotropy ventional realization of the above analysis within a class of a ∆ω of the axion-dilaton system whose effective energy momen- specific discretuum of toy models [30, 31, 32, 33, 16] that aim to tum tensor is given via realize de Sitter space in string theory. In particular, several fea- 1 ρσ 1 2 −2 tures of the above noncommutatively generalized phase-space Tµν− 2 gµν g Tρσ = Gττ¯ ∂µτ∂ντ¯ = diag[0, ···, 0, 4 ω ` ], (25) reformulation of string theory are naturally captured due to the 2 essential stringy nature of the models. with Gττ¯ = −1/(τ−τ¯) , and where ` is the characteristic length 2 scale in the transversal 2-plane Y⊥ [32, 33, 16]: This family of models is constructed by starting with an F-theoretic [34] type-IIB string theory spacetime, W3,1 × Y4 × ∆ω2 ∼ ∼ 2 2 2 Λ 2 implies MΛ M /MP, (26) Y⊥(×T ), where the complex structure of the zero-size “hid- ` den” T2 fiber of F-theory is identified with the axion-dilaton relating the mass scales of the vacuum energy/cosmological τ =def α + ie−Φ modulus. Specifically, we compactify on Y4 = K3 constant (MΛ), , i.e., Standard Model (M), and or T4 and let the observable spacetime W3,1 (via warped metric) the Planck scale (MP). This seesaw formula can be seen to vary over Y2 , and Y2 → S1 × Z, with the polar parametriza- ⊥ ⊥ arise in two ways: first, the formula (29) may be understood z+iθ iθ def 4 tion `e = re , where z = log(r/`) ∈ Z, while Y pre- as a consequence of dimensional transmutation, whereby the serves supersymmetry. Finally, we deform τ to vary nonholo- (modified) logarithmic nature of the transversal Green’s func- morphically, only over S1 ⊂ Y2. By cross-patching two distinct tion [30] (characteristic only√ of codimension-2 solutions) re- solutions and by deforming the apparently singular metric into lates the length scales ` and Λ [32]. Alternatively, the see- de Sitter space, we get the final nonsupersymmetric solution. saw formula (29) follows from adapting Tseytlin’s result for 3,1 1 The codimension-2 solution W o (S × Z) has a positive S¯ to the models of [32, 33, 16]: in the denominator of the 3,1 cosmological constant, Λ, along W , and the warped metric above formula, the volume of the transversal 2-plane produces is [35] R p 2 2 def the length scale Y2 −g(x) ∝ ` ; the numerator (with ∆ω = ⊥ 2 2 a b 2 2 2 2 ω2 − ω2 A2(z = 0) ) ds = A (z) g¯ab dx dx − ` B (z)(dz + dθ ), (22) c √ Z q  a b 2 2 Λ x0 2 2 2 − ( ) ( ) + 2 where g¯ab dx dx = dx0 − e (dx1 + dx2 + dx3) is the met- g x R x Lm ∝ ∆ω (27) Y2 ric on W3,1. The two explicit solutions for τ are [30] ⊥ reproduces the anisotropy variance of these axion-dilaton pro- τ (θ)=b + i g−1 eω(θ−θ0) I 0 s , (23) files, whereas the remaining volume-integration renormalizes −1 −1  gs the Newton constant as required in [30, 32]. The anisotropy τII (θ)= b0 ± gs tanh[ω(θ−θ0)] ± i . (24) cosh[ω(θ−θ0)] ω determines the above axion-dilaton stress tensor for the de Sitter solution and asymptotes to the Minkowski cosmic ( Z) Given the stringy SL 2; monodromy of the axion-dilaton → 2 brane limit ωc at z 0. Note that in the F-theory limit, system over a transversal 2-plane Y⊥ in the spacetime, these ω → 0 and ωc → 0. This singular supersymmetric configu- toy models exhibit S-duality. In generalizations where various ration is deformed into a de Sitter background by turning on moduli fields replace the axion-dilaton system, this directly an anisotropic axion-dilaton profile (23)–(24). Thus, Λ that fig- implies T-duality, which is covariantly realized in the generic ures in the seesaw formula can be understood as being related phase-space approach (1)–(3). to the cosmological breaking of supersymmetry. We stress that We emphasize that these models are a deformation of our discussion gives an argument for the existence of de Sitter the stringy cosmic string (D7-brane in IIB string theory) [36] background in string theory, albeit in its generic generalized ge- and as such represent effective stringy solutions (in the sense ometric and intrinsically noncommutative formulation, which of [37]) and not just IIB solutions. That is, our from the effective spacetime description is described by our solutions are indeed found as deformations of certain classic stringy models. One of the features of this doubled and general- F-theory backgrounds, but as codimension-2 solutions, they ized geometric description is that the effective spacetime action can be viewed as effective stringy solutions with an effective is intensive (as opposed to extensive), which directly translates “worldsheet” description that is, to lowest order, doubled and into the seesaw formula for the cosmological constant (29). generically noncommutative (as described by equation (1)). Note that more explicitly Thus our deformed stringy cosmic string solutions are natu- rally equipped with a generalized geometric (and noncommu- ∆ω2  π 2  M 2D−4 ∼ = D−2 (` )2 D ΛD−2 2 MD−2 MD−2 . tatively doubled) spacetime structure, which to the lowest or- ` D−3 MD−2 der of the doubled target space description directly connects (28)

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In our primary case of interest, of a minimal (simple) Λ ∝ ∆ω2 → 0 is necessary, but not sufficient. Letting Λ → 0 by supersymmetry-preserving compactification to D = 6 dimen- sending M4 → ∞ can be forced by letting z0 → ∞ in (31), the 2 1 sions, this becomes geometrical meaning of which is that Y⊥ → S “at infinity,” indicating some singular dimensional collapse. 2 2 8 ∆ω π 2 M6 1/` ∼ M4 7→ MP 2 Λ ∼ = ` , −−−−−−−−−→ MΛ ∼ M /MP This seesaw formula can be rewritten in a form that is even `2 2 M 7→ M 9 M4 6 (29) more appealing by relating the scale M to the Planck scale via = (− ) and relates the mass scales of the vacuum energy/cosmological an exponential factor M Mp exp const. Mp/Mi , and thus 2 the vacuum energy density MpΛ is given as constant (MΛ), particle physics/Standard Model (M6 7→ M), and the Planck scale (MP). 4 4 MΛ = Mp exp{−8 const. Mp/Mi}, (34) More precisely, the D- and the (D−2)-dimensional char- where Mi correspond to what is roughly an effective scale that acteristic (Planck) mass scales are related by the exact expres- numerically corresponds to the inflation scale (see also [3]). 3 sion [30, 31, 16] Note that (33) comes close to this, except that it is hard to set − D−1   z0 → M4/Mi except by hand. Nevertheless, our toy model, D−4 D−2 2 2(D−2) z0 D−3 1 M = M 2π` |z0| e Γ± ; , (30) D−2 D 2(D−2) |z0| even though realized in the conventional spacetime interpre- tation of string theory, does illustrate the main features of the where z0 is the radial distance (in units of `) from the (D−2)- 2 generic noncommutative and doubled formulation of string dimensional brane world to the boundary of Y⊥, and Γ± de- 1 theory, at least when it comes to the dark energy seesaw and notes the “[0, ]-incomplete gamma function” for z0 < 0 and |z0| its relation to the separation of scales associated with the hier- > = its complement for z 0. For D 6, this yields archy problem. Finally, we comment on the UV and IR scales 2 in the section on dark matter and string theory. The UV (or the 5 p 5 M 2 − 8 z0 4 2 − 16 z0/2 6 M4 = ζ0 |z0| e M6 ` , M4 = ζ0 |z0| e , (31) noncommutativity scale) should be considered as an effective M` scale to be empirically determined. In turn, L is the primordial def 3 1  3  where 0 ζ0 = 2πΓ ; 2πΓ ≈ 14.89, (32) 6 8 |z0| 6 8 IR cutoff. focusing on the z0 > 0 case of (30), since that enables the ex- 4 ponential hierarchy M4  M6. In turn, solving (31) for M6, the cosmological constant (29) becomes 7. A NONPERTURBATIVE FORMULATION OF STRING THEORY π2 M 2 π2 ∼ | |5/4 −2z0 4 = | |5/4 −2z0 2 2 Λ z0 e 2 z0 e M4 M` , (33) 9ζ0 ` 9ζ0 In view of the preceding discussion regarding the generic for- mulation of string theory and its relation to the problem of dark which reveals the effect of the z0-driven exponential hierarchy energy and dark matter, we now propose a nonperturbative in the “axilaton” models [30, 31, 32, 33, 16]. Therefore, the pri- formulation of string theory (and its M- and F-theory avatars). mordial “size-of-the-universe-scale” L in (19)–(21) is free to be Indeed, the chiral string worldsheet theory offers such a new naturally within one or two orders of magnitude of the stringy view on the fundamental question of a nonperturbative formu- fundamental length scale λ, resulting in a phenomenologically lation of quantum gravity [14] by noting the following: in the realistic scale introduced by the mixing terms (19)–(21). chiral string worldsheet description, the target space is found The seesaw expression appears to be technically natural. to be a modular space (quantum spacetime), but the same can be also said of the worldsheet. If the string worldsheet is made That is, when MP → ∞, the cosmological constant scale goes to zero, and in that case, the dual space curvature is zero, and modular in its chiral formulation, by doubling of τ and σ, so we get a flat dual space and thus enhanced symmetry. This is that X(τ, σ) → Xb (τ, σ) can be in general viewed as an infinite- precisely what ’t Hooft naturalness requires: when the phys- dimensional matrix (acting on the basis of Fourier components ical cutoff in some theory goes to infinity, the small parame- of τ˜ and σ˜ , the doubles of τ and σ, respectively), then the cor- ters in the theory vanish and should be protected by some hid- responding chiral string worldsheet action becomes Z den symmetry. It is tempting to relate that hidden symmetry  A B A B Tr ∂τXb ∂σXb (ωAB + ηAB) − ∂σXb HAB∂σXb , (35) to supersymmetry. However, this appears to be a bit too naive. τ,σ Such conjectural supersymmetry restoration requires the van- where the trace is over the (suppressed) matrix indices. The ma- ishing of the Ricci tensor, which requires ω → 0: the vanishing trix elements then emerge as the natural partonic degrees of freedom. We arrive at a nonperturbative quantum gravity by re- placing the σ-derivative with a commutator involving one ex- 3Note: the exponential factor [31, Eq. (15)] was inadvertently omitted in [16, def  X26 A = 5 Eq. (3.3)]. Also, note that 0 6 Γ(x; y) = Γ(x) − γ(x; y) 6 Γ(x), and both incom- tra b (with 0, 1, 2, . . . , 25) : plete gamma functions range from 0 to Γ(x). ∂ XA → [X26 XA] 4In the “naked singularity to brane-world” coalescing limit σ b b , b . (36) 3 1 −5/8 z 3 1 limz →0 Γ( ; )|z0| e 0 = 0 since Γ( ; ) vanishes faster than any (neg- 0 8 |z0| 8 |z0| ative) power can diverge. In turn, moving the naked singularity away from the 5That the canonical worldsheet of string theory might become noncommuta- z brane-world by keeping z0 6= 0 makes the hierarchy grow exponentially, ∼ e 0 . tive in a deeper, nonperturbative formulation, was suggested in [6].

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This dictionary suggests the following fully interactive and underlying bosonic formulation. This formulation realizes the nonperturbative formulation of the chiral string worldsheet SL(2, ZZ) symmetry of IIB string theory. In this noncommutative theory in terms of an (M-theory-like) matrix model form of the matrix model formulation of F-theory, in general, fabcs(Xb ) is a above chiral string action (with a, b, c = 0, 1, 2, . . . , 25, 26) dynamic background, determined by the matrix analog of the Z vanishing of the relevant beta function. By T-duality, the new  a b c a b c d  Tr ∂τXb [Xb , Xb ]ηabc − Hac[Xb , Xb ][Xb , Xb ]Hbd , (37) τ covariant M-theory matrix model reads as where the first term is of a Chern-Simons form and the sec- Z 1 i j k i j k l  SncM = ∂τXb [Xb , Xb ]gijk − [Xb , Xb ][Xb , Xb ]hijkl , (39) ond of the Yang-Mills form, and ηabc contains both ωAB and 4π τ ηAB. This is then the nonperturbative “gravitization of the quantum” [14]. We remark that in this nonperturbative ma- with 27 bosonic Xb matrices, with supersymmetry emerging trix theory-like formulation of the chiral string (and quantum in 11 dimensions. Once again, the backgrounds gijk(Xb ), and gravity), the matrices emerge from the modular worldsheet, hijkl (Xb ) are fully dynamical and should be determined by a and the fundamental commutator from the Poisson bracket matrix-analog of the Renormalization Group (RG) equation with respect to the dual worldsheet coordinates (of the mod- and the vanishing of the corresponding beta function. ular/quantum worldsheet)—that is, quantum gravity “quan- Note that in this approach holography [43] (such as tizes” itself, and thus quantum mechanics originates in quan- AdS/CFT [44], which can be viewed as a “quantum Jarzynski tum gravity. (However, this formulation should be distin- equality on the space of geometrized RG flows” [45]) is emer- guished from Penrose’s “gravitization of the quantum” and gent in a particular semiclassical “extensification” of quantum gravity-induced “collapse of the wave function” [38]. Also, spacetime, in which the dual spacetime degrees of freedom note some similarity of the chiral string worldsheet formula- are also completely decoupled. The relevant information about tion, in its intrinsic noncommutative form, to the most recent ω , η , and H is now contained in the new dynamical proposal by Penrose regarding “palatial” twistor theory [39].) AB AB AB backgrounds fabcd in F-theory and gijk and hijkl in M-theory. At this point, we also recall that the authors of [6] explicitly This proposal offers a new formulation of covariant Matrix state in the conclusion of their paper the following. “(1) The theory in the M-theory limit [46], which is essentially a par- density of gauge invariant degrees of freedom, per unit energy, tonic formulation: strings emerge from partonic constituents in per unit spacetime volume, is much less in the proper formula- a certain limit. This new matrix formulation is fundamentally tion of string field theory than in any ordinary relativistic field bosonic, and thus, it is reminiscent of bosonic M-theory [47]. theory.. . (2) The translation degree of freedom of the string The relevant backgrounds gijk and hijkl should be determined center of mass is in some sense doubled. . . (3) The familiar con- by the matrix RG equations. Also, there are lessons here for the tinuous world sheets should be replaced, in the proper formu- new concept of “gravitization of quantum theory” as well as lation of the classical theory, by some less continuous structure, the idea that dynamical Hilbert spaces or 2-Hilbert spaces (here perhaps related to continuum worldsheets the way quantum represented by matrices) are fundamentally needed in quan- phase space is related to classical phase space.” This prescient tum gravity [48]. This matrix-like formulation should be under- prediction is remarkably close to our nonperturbative formula- stood as a general nonperturbative formulation of string the- tion. ory. In this partonic (quantum spacetime) formulation, closed In thinking about nonperturbative matrix model formu- strings (as well as ) are collective excitations, in turn con- lations of string theory, it is natural to invoke the IIB ma- structed from the product of open string fields. Similarly, our trix model [40], based on D- as well as the matrix toy model can be understood as a collective excitation in this model of M-theory [41], based on D0-branes. However, these more fundamental “partonic” formulation. The observed clas- matrix models lack very important covariant properties associ- sical spacetime emerges as an “extensification” [14], in a partic- ated with F-theory and M-theory. In our proposal, we can do ular limit, out of the basic building blocks of quantum space- better. Given our new viewpoint, we can suggest a new covari- time. Their remnants can be found in the low-energy bilocal ant noncommutative matrix model formulation of F-theory, by quantum fields, with bilocal quanta, which were a motivation C C also writing in the large N limit ∂τXb = [Xb , Xb ], in terms of for our discussion of dark matter in string theory. XC XC commutators of two (one for ∂σ b and one for ∂τ b ) extra Finally, it is an old realization that the 10d superstring can × X N N matrix-valued chiral b ’s. Notice that, in general, we do be found as a solution of the [5]. The not need an overall trace, and so the action can be viewed as authors [5] explicitly state in their abstract that “consistent a matrix, rendering the entire nonperturbative formulation of closed ten-dimensional superstrings, i.e., the two N = 1 het- F-theory as purely quantum in the sense of the original matrix erotic strings and the two N = 2 superstrings, are contained formulation of quantum mechanics by Born-Jordan and Born- in the 26-dimensional bosonic closed string theory. The latter Heisenberg-Jordan [42] thus appears as the fundamental string theory.” This is pre- 1 cisely what we have in our proposed nonperturbative formula- S = [Xb a, Xb b][Xb c, Xb d] f , (38) ncF 4π abcd tion. (Such a bosonic formulation is also endowed with higher where instead of 26 bosonic Xb matrices one would have 28, mathematical symmetries, as already observed in [8].) Super- with supersymmetry emerging in 10(+2) dimensions from this symmetry (in the guise of M- and F-theory) is emergent from

8 Letters in High Energy Physics LHEP-186, 2021 our nonperturbative and seemingly entirely bosonic formula- ACKNOWLEDGMENTS tion in a similar fashion. This should allow going around the obvious problems raised by the apparent falsification of super- We would like to thank J. A. Argyriadis, D. Edmonds, L. Frei- symmetry at the observable LHC energies. del, V. Jejjala, M. Kavic, J. Kowalski-Glikman, Y.-H. He, R. Leigh, and T. Takeuchi for discussions. PB would like to thank the Simons Center for Geometry and Physics and the 8. CONCLUDING REMARKS CERN Theory Group, for their hospitality, and TH is grateful to the Department of Physics, University of Maryland, and the In this paper, we have related the problems of dark energy Physics Department of the University of Novi Sad, Serbia, for and dark matter to the hierarchy problem, in the context of a hospitality and resources. 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[53] J. A. Argyriadis, Y. H. He, V. Jejjala and D. Minic, “Dynamics of genetic code evolution: The emergence of universality,” arXiv:1909.10405 [q-bio.OT], to appear in PRE.

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