Planckian Axions in String Theory
PPlanckian Axions in String Theory
Thomas C. Bachlechner, Cody Long, and Liam McAllister
Department of Physics, Cornell University, Ithaca, NY 14853 USA
We argue that super-Planckian diameters of axion fundamental domains can naturallyarise in Calabi-Yau compactifications of string theory. In a theory with N axions θ i , thefundamental domain is a polytope defined by the periodicities of the axions, via constraintsof the form − π < Q ij θ j < π . We compute the diameter of the fundamental domain interms of the eigenvalues f ≤ . . . ≤ f N of the metric on field space, and also, crucially,the largest eigenvalue of ( QQ (cid:62) ) − . At large N , QQ (cid:62) approaches a Wishart matrix, due touniversality, and we show that the diameter is at least N f N , exceeding the naive Pythagoreanrange by a factor > √ N . This result is robust in the presence of P > N constraints, whilefor P = N the diameter is further enhanced by eigenvector delocalization to N / f N . Wedirectly verify our results in explicit Calabi-Yau compactifications of type IIB string theory.In the classic example with h , = 51 where parametrically controlled moduli stabilization wasdemonstrated by Denef et al. in [1], the largest metric eigenvalue obeys f N ≈ . M pl . Therandom matrix analysis then predicts, and we exhibit, axion diameters > M pl for the precisevacuum parameters found in [1]. Our results provide a framework for achieving large-fieldaxion inflation in well-understood flux vacua.December 4, 2014 a r X i v : . [ h e p - t h ] D ec ontents A.1 Classical ensembles 33A.2 Approach to universality 36
B Q in Calabi-Yau Hypersurfaces in Toric Varieties 37 Introduction
An important class of inflationary models are those involving super-Planckian dis-placements of the inflaton field. These ‘large-field’ scenarios yield a detectably-largespectrum of primordial gravitational wave fluctuations, and can therefore be tested inthe coming generation of CMB polarization experiments. The predictions of large-fieldmodels depend sensitively on the couplings of the inflaton φ to the degrees of freedomcomprising the ultraviolet completion of gravity (see [2] for a review). As a result, toformulate a large-field model one must make explicit or implicit assumptions aboutquantum gravity.A leading proposal for controlling the ultraviolet sensitivity of large-field inflation isto incorporate a weakly broken shift symmetry, φ → φ + const., in order to protect theinflaton potential over a super-Planckian range. From the viewpoint of the low-energyeffective field theory for φ , the shift symmetry is an internally-consistent assumptionthat renders small renormalizable couplings of φ radiatively stable. However, generalreasoning about the absence of exact continuous global symmetries in quantum gravity,and specific results from string theory, strongly suggest that not every shift-symmetriceffective field theory coupled to gravity admits an ultraviolet completion. To providea microphysical foundation for large-field inflation, one must therefore establish theexistence of a suitable symmetry in a computable regime of quantum gravity.A well-motivated strategy is to take the inflaton field(s) to correspond to one ormore axions in a compactification of string theory. Axions are numerous in Calabi-Yau compactifications, and in the absence of specific fluxes and branes that introducemonodromy, the potential for each axion vanishes to all orders in perturbation theory.The leading potential then arises from nonperturbative effects, and is sinusoidal. Fora single dimensionless axion θ , the Lagrangian takes the form L = 12 f ( ∂θ ) − Λ (1 − cos( θ )) , (1.1)where Λ is a dynamically-generated scale, and the parameter f is known as the axiondecay constant. The canonically-normalized field with mass dimension one is then φ ≡ f θ . In vacua of string theory involving small numbers of axions, the axion decayconstants are typically small, f (cid:28) M pl , in the regime of weak coupling and large volumewhere perturbative computation of the effective action is valid [3] (see also [4]). Thefundamental domain for φ has diameter 2 πf , and as a result (1.1) does not give rise toa realistic inflationary model in the absence of monodromy.The purpose of this paper is to compute the diameter of the fundamental domainin an extension of (1.1) to a totally general system with N (cid:29) N . This may sound straightforward, but influential earlyworks [5, 6] as well as more recent analyses [7–15] — including our own works on thesubject — captured only fragments of the full field range that is present in generic large N systems, including explicit string compactifications. In this work we unifythe field range enhancements arising in N-flation [6], including kinetic alignment fromeigenvector delocalization [14], with the full field range arising from the decay constantalignment mechanism of Kim, Nilles, and Peloso [5]. We then argue that enhancementsof the field range by a factor ∼ N compared to the naive expectation are automaticallypresent in a broad class of theories. Finally, we illustrate our results in a completelyexplicit compactification of type IIB string theory.The organization of this paper is as follows. In § N axions and give an intuitive estimate for itsdiameter, along with an overview of our results. In § §
4. In § § § §
8. In appendix A we briefly review afew facts about random matrix theory that are needed in this work. In appendix B wegive examples of nontrivial fundamental domains arising in string compactifications onCalabi-Yau hypersurfaces in toric varieties.
Consider a theory of N axions θ i that at the perturbative level enjoy the continuousshift symmetries θ i → θ i + const., so that a general two-derivative action for the θ i canbe written L = 12 K ij ∂θ i ∂θ j , (2.1)where K ij is a metric on the field space M , which is diffeomorphic to R N .Nonperturbative contributions from instantons give rise to a potential that is asum of sinusoidal terms, L = 12 K ij ∂θ i ∂θ j − N (cid:88) i =1 Λ i (cid:2) − cos (cid:0) Q ij θ j (cid:1)(cid:3) , (2.2) In a supersymmetric theory, K ij arises from the K¨ahler metric on field space, but our argumentsapply with or without supersymmetry. Q is an N × N matrix with integer entries. This potential breaks the continuousshift symmetries to discrete shifts. The associated periodic identifications,Γ i : Q ij θ j ∼ = Q ij θ j + 2 π , (2.3)define N pairs of identified hyperplanes in R N . By the fundamental domain, we meanthe intersection of all the identifications, M Γ ≡ M / Γ ∩ · · · ∩ M / Γ N ⊂ R N , i.e. theregion inside all pairs of hyperplanes. For the problem of large-field inflation, an in-teresting invariant quantity is the diameter of M Γ , measured in units where M pl = 1(which we shall use for the remainder). This diameter, which we will denote by D ,corresponds to the magnitude of the maximal rectilinear displacement that the canon-ical field Φ can undergo (in the absence of monodromy, which would allow traversingmultiple copies of M Γ , as in [16, 17].) As such, D is a proxy for the field range rel-evant for large-field axion inflation. Clearly, D depends on the identifications Γ i : thefundamental domain is bounded by adjacent maxima of each of the sinusoidal terms.To compute D , it is convenient to first perform the GL ( N, R ) transformation φ = Q θ . (2.4)In the φ i basis, the hyperplanes defining the identifications are orthogonal, and formthe faces of an N -cube of side 2 π . The kinetic matrix is then given by Ξ = ( Q − ) (cid:62) K Q − , (2.5)and the Lagrangian takes the form L = 12 ∂ φ (cid:62) Ξ ∂ φ − N (cid:88) i =1 Λ i [1 − cos ( φ i )] , (2.6)At the perturbative level, the metric on field space is independent of the axions, so Ξ isa constant matrix, up to nonperturbatively small corrections. However, Ξ is in generalnot diagonal in the φ i basis. Thus, the φ are related to the canonically-normalized fields Φ by a further GL ( N, R ) transformation (i.e., a diagonalization of Ξ by an orthogonaltransformation, combined with a rescaling by the eigenvalues ξ i of Ξ ).We should stress the elementary but crucial point that writing Q = in the θ i basis is not equivalent to beginning with a theory for which Q (cid:54) = in the θ i basis, and Throughout this work we will assume that the number P of nonperturbative terms is at least N ;that is, all axions are stabilized. In the present discussion we take P = N for simplicity, describingthe case P ≥ N in § M Γ hypercubic. In the formercase, the metric on M Γ is K , while in the latter case it is Ξ = ( Q − ) (cid:62) K Q − . Because Q is generally not orthogonal, the eigenvalues of Ξ differ from those of K . To summarize, the task is to determine the invariant diameter D of the fundamentaldomain M Γ . To do so, one must specify the identifications Γ, but these are not invariantunder changes of coordinates: there is a preferred ‘lattice’ basis φ i in which the periodicidentifications are defined by the faces of a hypercube of side length equal to (say) 2 π .This matters, because GL ( N, R ) transformations in systems with N (cid:29) N . One must therefore be careful to specify themetric K and the identifications Γ in the same basis, and then proceed to compute theinvariant distance D .Thus far all of our statements have been deterministic, and amount to saying thatin a theory that specifies Q (cid:54) = and K , it is obviously incorrect to take Q = whencomputing D . We now turn to making statistical arguments, based on the phenomenaof universality, eigenvalue repulsion, and eigenvector delocalization in random matrixtheory. We will argue that D (cid:29) D| Q = , with an enhancement that is parametric in N . The precise degree of enhancement depends on the forms of Q and K , as we willexplain below.Because the argument involves a number of independent computations of the be-havior of N × N matrices at large N , here we will give an accessible overview of mainsteps of the calculation. The complete calculation follows in §
3, while background onrelevant results from random matrix theory appears in appendix A.To compute D , it is convenient to work in the φ i basis, where M Γ is an N -cubeof side 2 π . Hypersurfaces in M of constant invariant distance r from the origin areellipsoids E r defined by φ (cid:62) Ξ φ = r . (2.7)The diameter D is then given by D = 2 r max , where r max is the largest value of r forwhich E r intersects M Γ .The largest possible D arises if the shortest principal axis of E r , corresponding tothe eigenvector Ψ Ξ N of Ξ with the largest eigenvalue ξ N , is parallel to a diagonal of the N -cube. In that case we have D max = 2 πξ N √ N . (2.8) The fact that the axion field range is large when the smallest eigenvalue of Q (cid:62) Q is small is the coreof the Kim-Nilles-Peloso (KNP) mechanism for decay constant alignment [5], which was generalizedto the case N > §
5n general, Ψ Ξ N will not point precisely along a diagonal, but due to vast number ofdiagonals in a hypercube, Ψ Ξ N is with high probability very nearly parallel to a diagonal,so that (2.8) is an accurate estimate.In order to estimate the typical diameter, we first assume that the metric on fieldspace is trivial, K = f , and that the matrix Q is sparse and contains random integerswith r.m.s. size σ Q . Even though Q is sparse, when a fraction (cid:38) /N of its entries arenon-vanishing the matrix Q (cid:62) Q approaches its universal limit of a random matrix inthe Wishart ensemble. In this random matrix ensemble, strong eigenvalue repulsionforces the smallest eigenvalue λ to obey λ (cid:46) σ Q /N . If the non-vanishing entries of Q have scale O (1), the minimum scale of the matrix Q is given by σ Q ≈ / √ N . In thiscase, from (2.8) we find that D (cid:38) N / f . (2.9)In § P , exceeds the number of axions, so that Q is rectangular. Furthermore, we will showin § In the previous section we outlined our strategy for determining the diameter of M Γ .We now turn to a more detailed analysis and derive the main results of this work.Let us consider an action for N axions whose potential is generated nonperturba-tively, and is periodic in the axions. This action will be further motivated in §
5, whenwe discuss embeddings of our results in supergravity theories that arise as effectivetheories in string compactification. We assume that there are P ≥ N nonperturbativeterms in the potential, so that the most general Lagrangian for the axions θ is givenby L = 12 K ij ∂θ i ∂θ j − P (cid:88) i =1 Λ i (cid:2) − cos (cid:0) Q i j θ j (cid:1)(cid:3) , (3.1)where we chose units such that each of the axions has the shift symmetry Q i j θ j →Q i j θ j + 2 π , and the entries of the P × N matrix Q are integers. Without loss ofgenerality we can decompose Q as Q = (cid:18) QQ R (cid:19) , (3.2)6here Q is a square, full rank matrix and Q R is a rectangular ( P − N ) × N matrix.Now, define fields φ as φ = Q θ , (3.3)such that Q θ = (cid:18) Q R Q − (cid:19) φ . (3.4)Here we are making a field redefinition to simplify N terms in the potential, while P − N terms will depend on linear combinations of the φ i . Therefore, the fundamentaldomain is given by an N -cube of side length 2 π , cut by 2( P − N ) hyperplanes thatconstitute the remaining constraints: − π ≤ (cid:0) Q R Q − φ (cid:1) i ≤ π ∀ i . (3.5)Some comments are in order. If the matrix Q were square, then this field redefinitionwould be unique, and would uniquely define what we mean by an axion: a field thatappears in the potential as the argument of a cosine. In the rectangular case there aremore cosines than fields, so the definition of an axion is not physical, but depends ona choice of basis. However, the diameter of the fundamental domain is physical andbasis-independent. In terms of the axions φ i the Lagrangian becomes L = 12 ∂ φ (cid:62) Ξ ∂ φ − N (cid:88) i =1 Λ i (cid:2) − cos (cid:0) φ i (cid:1)(cid:3) − P − N (cid:88) i =1 Λ i (cid:104) − cos (cid:16)(cid:0) Q R Q − φ (cid:1) i (cid:17)(cid:105) , (3.6)where, as before, Ξ = ( Q − ) (cid:62) K Q − (3.7)is the kinetic matrix of our choice of axions φ i , with eigenvalues ξ i . So far, we haveperformed a field redefinition so that the fields φ appear as the arguments of N of thecosines. Finally, the canonically normalized fields are given by Φ = diag( ξ i ) S (cid:62) Ξ φ , (3.8)where S (cid:62) Ξ diagonalizes Ξ , S (cid:62) Ξ Ξ S Ξ = diag( ξ i ) . (3.9)We will use (3.8) in order to determine canonically normalized distances on modulispace.In general, no closed form expression is available for the maximal diameter of thepolytope defining the fundamental domain M Γ . Instead, to obtain a lower bound on7he maximal diameter, we will compute the diameter D of M Γ along the directionof a particular unit vector ˆ v in the φ basis. A useful choice is to take ˆ v to be thedirection defined by a linear superposition of kinetic matrix eigenvectors Ψ Ξ i , weightedin proportion to the square roots ξ i of the corresponding eigenvalues ξ i : v = (cid:88) i ξ i Ψ Ξ i . (3.10)We now define an operator (cid:36) Q ( w ) that rescales a vector w to saturate the constraintequations (3.5) of the fundamental domain: (cid:36) Q ( w ) ≡ π Max i (cid:0) {| ( Q Q − R w ) i |} (cid:1) × w . (3.11)In the geometric picture of § w ends on an ellipsoid E w at invariant distance r w fromthe origin, and (3.11) rescales w → (cid:36) Q ( w ) so that E (cid:36) Q ( w ) just intersects M Γ .Using the rescaling operator and (3.8), we find that the canonically normalizeddiameter of the fundamental domain along the direction ˆ v is D = (cid:13)(cid:13) diag ξ i S (cid:62) Ξ (cid:36) Q (ˆ v ) (cid:13)(cid:13) = (cid:107) (cid:36) Q (ˆ v ) (cid:107) (cid:118)(cid:117)(cid:117)(cid:116) N (cid:88) i =1 ξ i , (3.12)where we used that the eigenvectors are orthonormal: (cid:80) i S (cid:62) Ξ ξ i Ψ Ξ i = S (cid:62) Ξ S Ξ ξ = ξ . Asa check, in the special case of Q = and K = f , we can evaluate (3.12) analyticallyand obtain the familiar N-flation result: D = 2 π √ N f .While (3.12) gives an analytic expression for the diameter of the fundamental do-main along an arbitrary direction, it is only useful once the periodicities and the kineticmatrix are defined. We now turn to evaluating the diameter of a generic fundamentaldomain. To that end, we assume that the integer entries of the matrix Q are indepen-dent and identically distributed (i.i.d.). For a sufficiently large number of non-vanishingentries, the matrix Q (cid:62) Q then approaches its universal limit of a Wishart distribution [21–25]. In particular, assuming the entries of Q are of similar scale, the universallimit is reached when a fraction (cid:38) /N of the entries in Q are non-vanishing. In thefollowing, we will assume that the universal limit has been reached and Q consists ofrandom integers of similar scale. We will consider three different models for the metricon field space: the identity matrix, a Wishart matrix, and a heavy-tailed matrix.The above assumptions are motivated by compactifications of type IIB string the- See also appendix A for a brief review of basic facts from random matrix theory. §
6. Furthermore, metrics of Wishart and heavy-tailed type arecompelling models for metrics on K¨ahler moduli spaces [26].
In order to evaluate the diameter of the fundamental domain (3.12), we need an es-timate for the quantity (cid:107) (cid:36) Q (ˆ v ) (cid:107) that corresponds to the dimensionless diameter inthe direction ˆ v . In general, we can compute the diameter directly from the entries ofthe matrix Q and the metric K . In order to obtain the typical diameter for a genericmatrix Q , we assume that its integer entries are i.i.d. random variables. The scale ofthe matrix Q is set by σ Q = (cid:104)Q(cid:105) r.m.s. . In the resulting ensemble of kinetic matrices Ξ , which is approximately rotationally invariant, the eigenvectors Ψ Ξ i are uniformlydistributed on the unit sphere, so that the unit vector (3.10) has normally distributedentries with standard deviation 1 / √ N ,ˆ v i ∈ N (0 , / √ N ) . (3.13)This phenomenon is known as eigenvector delocalization. The median size of the largestentry evaluates to Max( {| ˆ v i |} ) = √ − (2 − /N ) √ N ≡ (cid:96) N √ N . (3.14)For the case of a square matrix Q = Q , the constraints for the fundamental domainsimply become Max | v i | ≤ π and therefore (3.11) immediately becomes (cid:107) (cid:36) Q (ˆ v ) (cid:107) = (cid:13)(cid:13)(cid:13)(cid:13) π Max i ( {| ˆ v i |} ) × ˆ v (cid:13)(cid:13)(cid:13)(cid:13) = 2 π(cid:96) N √ N . (3.15)The result in (3.15) can be understood intuitively from the fact that a high-dimensionalhypercube has vastly more diagonal directions than faces, and therefore a randomly-selected direction is nearly aligned with a diagonal direction, giving a diameter en-hanced by √ N .For the case where the number of constraints P is larger than the number of axions, Q is rectangular. The first N constraints are again Max | v i | ≤ π , while the remainingconstraints are given by Max( {| Q R Q − ˆ v | i } ) ≤ π . By extensive numerical simulationwe observe that the entries of Q R Q − ˆ v for fixed Q are Gaussian distributed and thetypical standard deviation is given by √
2, independent of σ Q , N , and P . Therefore,the typical size of the largest-magnitude entry of the vector Q R Q − ˆ v is given byMax( {| ( Q R Q − v ) i |} ) ≈ − (2 − P − N ) ≈ (cid:112) P − N ) ≡ l P − N . (3.16)9 igure 1 . The fundamental domain in the presence of P (cid:29) N constraints, for N = 2.The square shown is the domain | v , | ≤ π , and the lines are 100 hyperplanes defined by | ( Q Q − v ) , | = π , where the elements of Q Q − are Gaussian distributed with standarddeviation √
2. The black circle illustrates the typical location of hyperplanes, while thedashed, red circle illustrates the analytic estimate (3.17) for the size of the fundamentaldomain.
These entries are typically much larger than the entries of v , so whenever the number ofconstraints is larger than the number of axions, the diameter is limited by the additionalconstraints. The typical diameter then is given by (cid:107) (cid:36) Q ( v ) φ (cid:107) = (cid:13)(cid:13)(cid:13)(cid:13) π Max i ( {| v i |} ) × ˆ v (cid:13)(cid:13)(cid:13)(cid:13) = 2 πl P − N , (3.17)and the enhancement of the diameter originating from the presence of diagonals is lost.The loss of enhancement from the presence of diagonals can be understood geo-metrically, as illustrated in Figure 1. The addition of a large number of constraints isdefined in terms of P − N hyperplanes, typically located a distance 1 / ( √ N σ Q ) fromthe origin and with normal vectors uniformly distributed on the sphere. The result-ing fundamental domain is described by an approximately spherical region around theorigin, of diameter 2 π/ √ We now proceed to evaluate the diameter of the fundamental domain in physical units,first assuming the metric to be the identity matrix K = f . Then, we have for the10inetic matrix Ξ : Ξ = f ( QQ (cid:62) ) − = f S Q (cid:62) Q diag( Q − i ) S (cid:62) Q (cid:62) Q , (3.18)where S (cid:62) Q (cid:62) Q Q (cid:62) Q S Q (cid:62) Q = diag( Q i ) . (3.19)In the second equality in (3.18) we have used the fact that eigenvectors do not changeupon inversion. Therefore, we can use eigenvector delocalization of the Wishart en-semble. Considering the diameter in the direction Ψ Ξ N , from (3.12) we obtain theconservative bound D ≥ ξ N (cid:107) (cid:36) Q ( Ψ ) Q (cid:62) QN (cid:107) . (3.20)The largest eigenvalue of Ξ , ξ N , obeys (see appendix A) ξ N = f Q i ) . (3.21)In the large N limit, the median size of the smallest eigenvalue of the Wishart matrix Q (cid:62) Q is given by Q = C σ Q /N , where C ≈ . D ≈ f √ N √ C σ Q (cid:107) (cid:36) Q ( Ψ Q (cid:62) QN ) (cid:107) (cid:46) (cid:40) f N / for P = Nf N for
P > N , (3.22)where we used (A.13) in the last inequality to set σ − Q (cid:46) N . To consider a more general metric, let K be a Wishart matrix that is diagonalized by S K and has maximum eigenvalue f N . The kinetic matrix Ξ is then given by Ξ = Q (cid:62) diag f i Q , (3.23)where Q = S K ( Q − ). While (3.23) is not an inverse Wishart matrix, a reasonable guessfor the kinetic matrix is to approximate it as a rescaled inverse Wishart matrix, Ξ = Q (cid:62) diag( f i ) Q ∼ σ f i ( Q (cid:62) Q ) − , (3.24)where the scale of the eigenvalues is given by σ f i = (cid:104) f i (cid:105) r.m.s. ≈ f N /
4. Therefore, Ξ − is approximately an inverse Wishart matrix of scale σ Ξ − = σ Q / √ σ f i , and the typical11argest eigenvalue of Ξ is given by ξ N = 1 (cid:114) σ Q σ f i C N = (cid:112) N/ C σ Q f N , (3.25)where we used again that σ f i ≈ f N /
4. The physical field range is then given by
D ≈ f N √ N √ C σ Q (cid:107) (cid:36) Q (Ψ Q (cid:62) QN ) (cid:107) (cid:46) (cid:40) f N N / for P = Nf N N for P > N , (3.26)We have verified this result through extensive simulations.
For a heavy-tailed metric K , the eigenvalues f i are distributed with a polynomialfluctuation probability, so the scale σ f i is not defined. While there are many distinctensembles of matrices exhibiting heavy tails, a simple model that we will adopt isone where one of the metric entries dominates over all others, so the metric takes theschematic form K = f N , K ij (cid:28) f N ∀ i (cid:54) = 1 or j (cid:54) = 1 . (3.27)See [26] for examples of heavy-tailed K¨ahler metrics in explicit string compactifications.Thus, for the matrix Ξ = ( Q − ) (cid:62) KQ − , Q − j / (cid:107) Q − j (cid:107) is a unit eigenvector corre-sponding to the eigenvalue f N (cid:107) Q − j (cid:107) , while all other eigenvalues are much smaller.The matrix Q has entries of scale σ Q and is otherwise random, so that the elements ofthe inverse matrix obey (cid:88) i ( Q − ) i Q i = 1 , (3.28)where we can approximate the entries of Q as Gaussian random variables with vanishingmean and standard deviation σ Q . The entries of the matrix Q − are then approximatelydistributed according to the inverse Gaussian distribution with standard deviation √ N in order to satisfy (3.28). It is then plausible that the sum σ Q (cid:107) Q − j (cid:107) = σ Q (cid:80) i ( Q − ) i is inverse chi-squared distributed with unit standard deviation: σ Q | ˜ Q − j | = σ Q N (cid:88) i =1 ( Q − ) i ∈ χ − (1) . (3.29)While we will not prove this relation, we have verified (3.29) numerically, finding an12
10 10050 2001510501005001000 1510501005001000 D / π D / π P /N Figure 2 . Left: Diameter versus the number of fields for a fixed number P = 4 N of non-vanishing entries in Q . Right: Kinematic range vs. P /N for fixed N = 100. Dashed linesillustrate numeric results, and the solid lines are the analytic results. From top to bottom,red: unit metric (3.22); green: Wishart metric (3.26); gray: non-square Q matrix (3.26) with P − N = 3; orange: heavy-tailed metric (3.32); blue: √ N for comparison. excellent match. The median of (cid:107) Q − j (cid:107) is then given by˜ λ = (cid:18) √ σ Q erfc − (1 / (cid:19) . (3.30)Therefore, we have for the square root of the largest eigenvalue of Ξ ξ N ≈ √ − (1 / f N σ Q . (3.31)Using Eq. (3.12) we find the diameter D ≈ f N √ − (1 / σ Q (cid:107) (cid:36) Q (Ψ) Q (cid:62) QN (cid:107) (cid:46) (cid:40) f N N for P = Nf N √ N for P > N . (3.32)Finally, Figure 2 illustrates numerically the approach to universality and the scalingof the kinematic range with N . So far we have evaluated the typical diameter of the fundamental region, which we foundto be parametrically larger than the typical scale of the metric eigenvalues. However,13n order to realize large field chaotic inflation within one fundamental domain, thediameter in the light directions of the potential is required to be large. In this sectionwe consider the diameter for a displacement of the lightest canonical field. We willfind that universality generically leads to an alignment of the largest direction with thelightest canonical field.Let us again consider the Lagrangian (3.6) for the fields φ . Well inside the funda-mental domain, with − π (cid:28) (cid:0) QQ − R φ (cid:1) i (cid:28) π , we can expand the potential to quadraticorder, L = 12 ∂ φ (cid:62) Ξ ∂ φ − φ (cid:62) M φ φ , (4.1)where M φ = diag(Λ ,...N ) + ( Q R ) (cid:62) diag(Λ N +1 ,...P ) Q R , (4.2)is the mass matrix in the φ basis. The canonically normalized fields Φ are given by Φ = diag( ξ i ) S (cid:62) Ξ φ , (4.3)and the Lagrangian becomes L = 12 ∂ Φ (cid:62) ∂ Φ − Φ (cid:62) M Φ , (4.4)where M = diag(1 /ξ i ) S (cid:62) Ξ M φ S Ξ diag(1 /ξ i ) . (4.5)To obtain a lower bound on the typical arc length traversed during the approach tothe vacuum, we consider a scan over random initial conditions, uniformly distributedover the boundary of validity of the quadratic approximation, i.e. we examine an initialpoint (cid:36) Q (ˆ v ) , (4.6)where ˆ v is a unit vector with uniform probability density on the sphere S N − . Thesemidiameter of the fundamental domain in the direction ˆ v is a lower bound for the Inflation could proceed beyond one fundamental domain, as we will discuss, and could span manyfundamental domains in the presence of monodromy. In [14] it was shown that for a trivial Q matrix the direction of largest field space diameter isgenerically misaligned with respect to the lightest canonical field. Note that most of the volume of an N -polytope is concentrated at the boundary, so a scan overinitial positions that is uniform throughout the polytope would yield displacements similar to thosefrom a scan over the boundary. D ˆ v = 12 (cid:107) diag ξ i S (cid:62) Ξ (cid:36) Q (ˆ v ) (cid:107) . (4.7)Because the initial points (cid:36) Q (ˆ v ) are uniformly distributed on S N − , the displacements (cid:36) Q (ˆ v ) will typically have overlaps of 1 / √ N with the direction corresponding to themaximum diameter of the fundamental domain. Thus, the typical displacement fromthe vacuum in a scan over random initial conditions is given by D ˆ v ≈ √ N D . (4.8)In the above estimate we considered the typical field range when scanning overinitial conditions uniformly distributed in the fundamental domain. However, one mightalso be interested in the maximum field range over which the quadratic approximationis valid, along the direction of the lightest field. To analyze this, we assume that thehierarchy in the eigenvalues of the kinetic matrix Ξ is much larger than the hierarchy ofthe entries in the rotated mass matrix S (cid:62) Ξ M φ S Ξ . The mass matrix for the canonicallynormalized fields Φ is then dominated by the ξ contribution: M = diag(1 /ξ i ) S (cid:62) Ξ M φ S Ξ diag(1 /ξ i ) ≈ Λ M φ ξ i ξ j , (4.9)where Λ M φ is the typical scale of the entries of S (cid:62) Ξ M φ S Ξ , so that the lightest directionis given approximately by ˆ v Φ = v Φ | v Φ | ∼ (0 , . . . , , , (4.10)which approximately coincides with the direction giving the maximum diameter. Thisalignment occurs because in the φ basis the light direction corresponds to Ψ Ξ N , theeigenvector corresponding to the largest axion decay constant. Using (4.7), we findthat the diameter in the direction of the lightest field is D light ≈ D . (4.11)We have observed a generic enhancement to the diameter of a single fundamental For the case of a Wishart metric K we have verified numerically that the hierarchy of the entriesof S (cid:62) Ξ M φ S Ξ is parametrically smaller than the hierarchy in the matrix ξ i ξ j , by a factor of order N independent of the Λ i , leading to dynamic alignment. igure 3 . Contour plot of a two-dimensional axion potential, along with the region of validityof the quadratic expansion and a set of randomly chosen inflationary trajectories. The axesare canonically normalized fields. domain of the potential, due to eigenvector delocalization and the nontriviality of the Q matrix. This is a promising setting for realizing chaotic inflation. Starting thesystem with a displacement along the lightest direction can lead to single-field slow rollinflation in a quadratic potential: V (Φ) = 12 m Φ , (4.12)which yields a large number of e -folds, N e = 14 | ∆Φ | (cid:38) N D , (4.13)where we used the estimate from (4.11). For example, taking the metric on modulispace to be a Wishart matrix, we find the scaling N e ∝ N f N M . (4.14)Although single-field inflation is a possibility in this system, it is not a genericoutcome. Instead, the more massive fields will decay first, with the lighter fields settlinginto their minima later. This process is illustrated in Figure 3. A number of features16re worth noting. While kinetic alignment allows the diameter of one lattice domainto be super-Planckian at large N , this does not imply that the inflationary trajectoryremains within a region where a quadratic approximation to the potential is valid.In particular, although the large hierarchy in the axion decay constants leads to anapproximate alignment of the least massive direction with the kinematically largestdirection, a slight misalignment can lead to an evolution into a neighboring minimum.This does not spoil the possibility of inflation: there is still a large field displacement,and inflation can proceed driven during the approach to the neighboring minimum.These effects, in particular the multifield dynamics during the onset of inflation, cangive rise to interesting physical phenomena, such as non-adiabatic perturbations oreven domain walls. A full analysis of these effects is beyond the scope of this work. Our discussion so far has been at the level of a low-energy effective field theory contain-ing N axions. However, because high-scale inflation is extremely sensitive to physicsat the Planck scale, it is important to inform the effective description with the data ofan ultraviolet completion. We will therefore explain how our considerations extend toaxions in string theory. As a bridge between our general analysis and specific stringtheory constructions, we now discuss axions in four-dimensional N = 1 supergravitytheories, incorporating the structures of the effective supergravity theories that arisein the flux compactifications of type IIB string theory described in §
6. The effectivesupergravities presented here generically exhibit kinetic and dynamic alignment.
We will now examine the scalar potential in an N = 1 supergravity theory, with an eyetowards the K¨ahler moduli sector of Calabi-Yau compactifications of type IIB stringtheory. The Lagrangian of the chiral superfields φ A is given by L = K A ¯ B ( φ C , ¯ φ ¯ D ) ∂ µ φ A ∂ µ ¯ φ ¯ B − V ( φ C , ¯ φ ¯ D ) , (5.1)with the F-term potential V ( φ C , ¯ φ ¯ D ) = e K (cid:16) K A ¯ B D A W ¯ D ¯ B W − | W | (cid:17) . (5.2) We omit the D-term potential, because in the constructions that we will discuss, the D-terms donot involve the axions to leading order, and can be safely ignored in analyses of inflationary dynamics.
17n the above equations K A ¯ B is the K¨ahler metric on moduli space, which is independentof the axions at the perturbative level, and W is the holomorphic superpotential. In thecase of type IIB string theory, the indices A and ¯ B run over the dilaton, the complexstructure moduli, and the K¨ahler moduli, such that A = 1 , . . . , h , + h , + 1. As statedbefore, we will concern ourselves with the case in which the complex structure moduliand dilaton are integrated out supersymmetrically at a high scale, so we will henceforthrestrict ourselves to an effective theory for the K¨ahler moduli T j = τ j + iθ j , labeled bythe indices i, j . A consistency requirement for our analysis is that the motion of theinflaton does not destabilize any fields that we have assumed to be set at their minima.We will therefore examine the cross-coupling terms in the Hessian, and ensure thatthese are not large enough to push a previously-stable saxion away from its minimumso as to destabilize the configuration. At a supersymmetric critical point we can writethe potential in terms of small fluctuations as V ( T, ¯ T ) = V ( T ) + (cid:88) ij ∂ i ∂ j V T i T j = V ( T ) + (cid:0) ¯ T T (cid:1) H (cid:18) T ¯ T (cid:19) , (5.3)where i, j run over unbarred and barred indices and T denotes the fluctuations aboutthe minimum. The Hessian matrix is given by H = (cid:18) ∂ i ¯ V ∂ ij V∂ ı ¯ V ∂ ıj V (cid:19) = H Z − | W | (cid:18) K i ¯ K ¯ ıj (cid:19) , (5.4)where H Z = (cid:18) Z ¯ Ai ¯ Z ¯ ı ¯ A − Z ij W − ¯ Z ¯ ı ¯ W ¯ Z A ¯ ı Z jA (cid:19) , (5.5)and Z AB = Z BA ≡ D A D B W . Here D A V B = ∂ A V B + K A V B − Γ CAB V C , and we have usedK¨ahler transformations to set K = 0 at the critical point.We can transform the Hessian matrix into a ( τ θ ) basis via (cid:18) T ¯ T (cid:19) = (cid:18) i − i (cid:19) (cid:18) τθ (cid:19) = U (cid:18) τθ (cid:19) , (5.6)such that V ( τ, θ ) = V ( τ ) + (cid:0) τ θ (cid:1) U † H U (cid:18) τθ (cid:19) . (5.7) A discussion of this problem in the context of N-flation appears in [27]; see also [28].
18e then have H τθ = U † H U , (5.8)which evaluates to H τθ = 2 (cid:18) Z ¯ Z − | W | K − (cid:0) W Z + W ¯ Z (cid:1) i (cid:0) W Z − W ¯ Z (cid:1) i (cid:0) W Z − W ¯ Z (cid:1) ¯ ZZ − | W | f K + (cid:0) W Z + W ¯ Z (cid:1) (cid:19) . (5.9)Here Z ¯ Z is contracted using the K¨ahler metric. Let us now consider the couplingsbetween the saxions τ i and the axions θ i . In [29] it was shown that tachyons allowedby the Breitenlohner-Freedman bound are ubiquitous in AdS vacua, and will renderan uplifted solution unstable, unless | W | (cid:28) m susy /N . Here, m susy is the scale of thesupersymmetric fermion mass matrix Z ij . Therefore, the scale of the masses of τ i is given by Z ¯ Z ∼ M τ ∼ m susy , while the couplings between τ and θ are given by M τθ ∼ W ¯ Z ∼ W m susy . Then the constraint | W | (cid:28) m susy /N leads to M τ (cid:29) N M τθ , . (5.10)To leading order in τ and θ , the displacement of the minimum for τ can be estimatedby solving ∂ τ V ( τ, θ ) | τ = τ min = 0, which gives (cid:107) ∆ τ min (cid:107) = (cid:107) (cid:0) M ττ (cid:1) − M τθ ∆ θ (cid:107) ∼ N (cid:107) ∆ θ (cid:107) . (5.11)Here we have considered only the leading order contributions to the τ - θ mixing termsin the Hessian. In general there will be higher-order contributions, but when ourexpansion is valid these are not large enough to destabilize the vacuum.We now turn to a more specific effective supergravity theory, in which the super-potential takes the form W = W ( S, χ ) + (cid:88) j A j ( χ a ) e − q ji T i = W ( S, χ ) + (cid:88) j A j ( χ a ) e − q ji ( τ i + iθ i ) . (5.12)In the last equality we have expressed the complex chiral scalar in terms of its realsaxion and axion components. If the K¨ahler potential is independent of the axions, atleast to the order at which we are working, then the axion potential can be written V = C + (cid:88) j B j cos( q ji θ i − θ W ) + (cid:88) j 6, we19ill consider the KKLT moduli stabilization scheme in type IIB string theory, whichrequires solving the F-flatness constraints F i = 0, ∀ i . In general the A i prefactorsin each nonperturbative term will be complex, and will contribute a phase to eachexponential. When we have N axions we can simply perform a shift to absorb each A i phase, and can therefore take the A i to be real. In addition, we can perform a K¨ahlertransformation to make W real and negative. For the remainder of this work we willassume that these transformations have been performed.To extract the axion-saxion coupling at the supersymmetric minimum we needto compute the matrix Z AB = D A D B W , where D A is the geometrically covariantand K¨ahler covariant derivative, and D B is the K¨ahler covariant derivative. At asupersymmetric minimum D A W ≡ F A = 0, so we can write Z AB = D A D B W = ∂ A F B + K ,A F B + Γ CAB F C = ∂ A F B , . (5.14)Writing F B = ( ∂ B + K ,B ) W , we have Z AB = ∂ AB W + K ,B ∂ A W + K ,AB W . (5.15)This is not manifestly symmetric in the induced A and B , but we can fix that bymultiplying the critical point equation by K ,A : K ,A ∂ B W = − K ,A K ,B W . (5.16)Therefore, we find Z AB = ∂ AB W + K ,AB W − K ,A K ,B W . (5.17)Applying this to (5.12) we find Z ij = (cid:88) k A k (cid:0) q ki q kj (cid:1) e − q ki T i + ( K ,ij − K ,i K ,j ) W . (5.18)The scale of the inflaton mass is approximately set by the scale m susy /N . If the axionsare stabilized at θ i = 0, then the superpotential will be real at the minimum, as willthe matrix Z . Therefore, from the form of equation (5.9), the axions and the saxionswill be decoupled to leading order, and we do not need to worry about destabilizingthe saxions during inflation, as long as each axion does not move too much. For thisreason we will focus on the θ i = 0 vacuum.20 Figure 4 . Normalized probability distribution of the eigenvalue spectrum of Q (cid:62) Q alongwith the analytic Wishart eigenvalue spectrum. The full effective potential in (5.13) has P cosine terms appearing due to the nonper-turbative superpotential and an additional N terms of the form cos( φ i − φ j ), appearingas cross terms with Q matrix Q cross . The full Q matrix is then given as Q = (cid:18) QQ cross (cid:19) . (5.19)Note that the additional constraints on the fundamental domain originating from thecross terms decrease the diameter found by considering only superpotential periodicitiesby at most a factor of 2, because only differences φ i − φ j appear. Therefore, the crossterms contain no new physical enhancement of, or limitation on, the diameter of thefundamental domain. However, the effective potential contains the full matrix Q andpicking an arbitrary full rank N × N matrix can be used to define the axions. The metricon field space and its decay constants, however, do depend on the choice of axions. Inparticular, because there is a large number of possible full rank matrices, with essentiallyrandom entries, the metric on field space approaches that of an inverse Wishart matrix,independent of the periodicities in the nonperturbative superpotential. This approachto universality is illustrated in Figure 4. Here we chose Q = , K = f , N = 51and defined the axions φ = Q θ in terms of a full rank matrix Q that consists of N randomly chosen rows of the full matrix Q . Due to universality, the metric on modulispace approaches an inverse Wishart distribution with potentially large eigenvalues.This observation is purely due to the fact that the definition of the axions and theassociated metric is arbitrary. Despite the presence of very large metric eigenvalues,in this example the field range is not enhanced compared to the trivial case Q = .21his is a consequence of the fact that the axion lattice domains are defined by theperiodicities of the superpotential. It will be instructive to verify that the kinetic alignment mechanism we have describedcan occur in a UV-complete theory, at large N . Weakly-coupled string theory is, at themoment, our best tool for testing whether a particular mechanism is consistent with atheory of quantum gravity. In this section we will discuss an explicit compactificationof type IIB string theory with moderately large N and a nontrivial Q matrix. Ourfindings suggest that the kinetic alignment discussed above can occur very naturally incompactifications of type IIB string theory on certain Calabi-Yau orientifolds.We will examine a state-of-the-art string compactification, with h , = 51, that wasintroduced by Denef, Douglas, Florea, Grassi, and Kachru (DDFGK) [1]. Their con-struction is almost completely explicit: quantized flux values are specified to stabilizethe complex structure moduli and dilaton at weak coupling, and the K¨ahler moduli arestabilized by nonperturbative effects, which are known to be present and to providenon-vanishing contributions to the superpotential. The only piece that is not com-pletely explicit are the Pfaffian prefactors of the nonperturbative superpotential terms,which are set to unity. Type IIB string theory compactified on a Calabi-Yau threefold X yields an N = 2 d = 4 effective theory. In the absence of branes, the massless fields are the h , vectormultiplets, which include the complex structure moduli, and h , hypermultiplets, whichinclude the K¨ahler moduli. We are interested in a N = 1 theory, which can be obtainedby orientifolding, resulting in an N = 1 supergravity theory with an internal space ˆ X ,the orientifold of the threefold X . For simplicity we will assume that all of the divisorsare even under the orientifold action (general at present, but specified in the exampleof § K = − V ) , V = 16 (cid:90) J ∧ J ∧ J . (6.1)Here the K¨ahler form is expanded as J = t i ω i , where ω i ∈ H , ( X, Z ). The K¨ahlermoduli have a natural interpretation as the volumes of four-cycles. These volumescombine with periods of the Ramond-Ramond four-form to form chiral superfields.The complex scalar components take the form: T j = 12 (cid:90) D j J ∧ J + i (cid:90) D j C (4) ≡ τ j + iθ j . (6.2)We will write the general nonperturbative superpotential as W = W + (cid:88) i A i e − q ij T j . (6.3)Here W is the value of the flux superpotential with the complex structure fields set attheir minima, and the A i are one-loop determinants. The geometry (before orientifolding) is a resolution of the orbifold T / Z × Z , whichhas 51 K¨ahler moduli and 3 complex structure moduli. T = ( T ) has three K¨ahlermoduli, which descend to the so-called “sliding divisors” { R i } , i = 1 . . . 3. The orbifoldaction is z z z α + − − β − + − α ◦ β − − +There are 48 fixed lines under the orbifold action, whose resolution introduces 48exceptional divisors, denoted by { E iα,jβ } , where i = 1 . . . , α = 1 . . . , i < j . Wewill consider what DDFGK refer to as the “symmetric resolution.” There are 12 fixeddivisors under the orientifold action, resulting in 12 O7-planes. An SO(8) stack of D7-branes is placed on each O7-plane. The D7-brane divisors will be denoted by D iα . Inthe compact model the D iα are disjoint, so there is no massless bifundamental matterarising from intersections of D7-branes. In addition, the D iα are rigid, so there is no23djoint matter, and the gauginos will condense. The D iα can be expressed in terms ofthe sliding divisors and exceptional divisors. For example, D α = R − (cid:88) β E α, β − (cid:88) γ E γ, α . (6.4)Each exceptional divisor is rigid, and supports a Euclidean D3-brane, which generatesa superpotential of the form ∆ W ∼ e − πτ iα,jβ . (6.5)The gaugino condensates generate superpotentials of the form∆ W ∼ e − πτ iα / , (6.6)where we have used the fact that the dual Coxeter number of SO(8) is 6. Expandingthe K¨ahler form as J = r i R i − t α, β E α, β − t β, γ E β, γ − t γ, α E γ, α , (6.7)the volume of the orientifold can be written as V = r r r − (cid:32) r i (cid:88) βγ t β, γ + . . . (cid:33) − (cid:32)(cid:88) αβ t α, β + . . . (cid:33) + 14 (cid:32)(cid:88) αβγ t α, β t β, γ + t α, β t γ, α + . . . (cid:33) − (cid:88) αβγ t α, β t β, γ t γ, α . (6.8)The areas of the generators of the Mori cone are A i,jβ = r i − (cid:88) α t iα,jβ ,A ++ − = 12 ( t α, β + t β, γ − t γ, α ) , (6.9)plus cyclic permutations of the latter. DDFGK found a particularly symmetric criticalpoint by setting t iα,jβ = t, r i = r , (6.10)24hrough which the curve areas and divisor volumes simplify greatly: V = r − rt + 48 t ,V iα,jβ = V E = rt − t ,V iα = V D = r − rt + 16 t ,A i,jβ = A r = r − t ,A αβγ = A t = t . (6.11)Under the assumption that the one-loop determinants can be set to unity, a minimumwas sought where the phases vanish. The superpotential can then be written as W = W + 48 e − π ( tr − t ) + 12 e − π ( r − rt +16 t ) / . (6.12)DDFGK explicitly stabilized the complex structure moduli using flux, finding that W ∼ − . 3, which gives a supersymmetric local minimum with the K¨ahler parameters r ≈ , t ≈ . , (6.13)yielding volumes of V ≈ , V E ≈ , V D ≈ , A r ≈ . , A t ≈ . . (6.14)These values are not parametrically large, and one should ask whether additionalperturbative and nonperturbative effects are important in this regime of parameters.DDFGK directly demonstrated that the leading known corrections are controllablysmall, as we now explain. There are nonperturbative corrections to both the K¨ahlerpotential and the superpotential. There could be a contribution to the superpotentialfrom multi-wrapped or fluxed instantons, but these contributions will be suppressedby higher-order powers of the exponential that is already present. Since the values ofthese exponentials are e − πV E ∼ × − and e − πV D / ∼ × − , these contributionsare expected to shift the minimum by a very small amount. The corrections to theK¨ahler potential are a bit more complicated, especially given the small volumes of theexceptional curves. First, there are perturbative α (cid:48) effects, which correct the K¨ahlerpotential to K = − (cid:32) V + ξg / s (cid:33) , ξ ≡ − χ ( Y ) ζ (3)8(2 π ) ≈ − . . (6.15)25n this formula, χ ( Y ) is the Euler characteristic of the “upstairs” Calabi-Yau. This cor-rection gives a percent-level correction to the volume, and can therefore be consistentlyneglected. Nonperturbative corrections can be estimated through the correspondingcorrection to the underlying N = 2 prepotential:∆ F ≈ π ) ∞ (cid:88) n =1 n e − π √ g s (2 A t ) n ≈ − . (6.16)Here we have restricted to a sum over worldsheets wrapping exceptional curves, sincethese will give the leading order contribution. We have also included a factor of tworelevant in moving from the upstairs space to the downstairs space. There are 192 min-imal exceptional curves, which in turn provide a percent-level correction to the K¨ahlerpotential. In addition, since g s ≈ . 27 is moderately small, string loop correctionsshould not significantly shift the minimum. More details on these results can be foundin [1].We will consider this point in moduli space as a toy model. It is, of course, nota realistic model for inflation, as the minimum is a supersymmetric AdS vacuum.However, it is still instructive to demonstrate kinetic alignment in a completely explicitand well-controlled string compactification. To compare this example to the rest of thepaper, we write the nonperturbative contributions to the superpotential in the form (cid:80) i A i e − q ij T j , so that the eigenvalues of the kinetic matrix correspond to axion decayconstants. The matrix q is then given by 26 = π − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − 10 1 0 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − 10 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 . (6.17) We are now in a position to determine the diameter of the fundamental domain in theDDFGK compactification. The diameter is given by (4.7), D light = 2 π | diag ξ i S (cid:62) Ξ (cid:36) Q (ˆ v ) | , (6.18)27here ˆ v = S Ξ diag( ξ − )Ψ M Φ . Using the Hessian matrix for the axions in (5.9), the q matrix (6.17), and the K¨ahler metric on moduli space, we numerically find that thediameter along the lightest direction is D light = 1 . M pl . (6.19)This can be compared to the results of § 3, where the field space diameter wasobtained analytically. As we argued in § 3, it is reasonable to approximate the kineticmatrix Ξ as an inverse Wishart matrix. We can test this assumption by comparing thelargest eigenvalue of the kinetic matrix obtained from the K¨ahler potential (6.15) to thetypical largest eigenvalue of a Wishart matrix given in (3.25). Using the scale of the q matrix (6.17), σ Q ≈ . 18, and the largest metric eigenvalue f N ≈ . M pl , (3.25) gives ξ Wishart N ≈ . M pl , while numerically we typically find ξ DDFGK N ≈ . M pl . Accord-ing to (3.26), the field space diameter is obtained by rescaling the largest eigenvalue ofthe kinetic matrix by (cid:107) (cid:36) Q (ˆ v ) (cid:107) , which takes into account the additional P − N = 9 con-straints. From (3.17) we expect that for random choices of constraints, (cid:107) (cid:36) Q (ˆ v ) (cid:107) ≈ . v corresponding to the lightestcanonically normalized field, (cid:107) (cid:36) Q (ˆ v ) (cid:107) ≈ . 3. It is encouraging that our large N esti-mates based on universality and eigenvector delocalization are accurate, in this example,to within factors of order a few.Finally, (3.26) gives an analytic estimate for the field space diameter from randommatrix theory of D = (cid:115) 512 + log(4) − (cid:112) π − (2 − / ) f N σ Q ≈ . M pl . (6.20)This matches the actual diameter (6.19) rather well. Our results unify a number of effects identified in prior works, as we will now explain. The very special case K = diag( f i ), Q = corresponds to the simplest construction Note that by using different choices of q , corresponding to different coordinates, the eigenvaluesof the kinetic matrix change. We have observed examples in which ξ N ≈ M pl , which might naivelybe interpreted as a super-Planckian axion decay constant. However, as the definition of the axions isambiguous in this example, this does not correspond to a physically large diameter. Note again that the kinetic matrix is basis dependent. We obtained a typical value by evaluating ξ N for a large number of random basis choices. For simplicity of presentation we take P = N in this discussion. 28f N-flation [6] (a version of assisted inflation [31]), for which the field range is givenby the Pythagorean sum ∆Φ = 2 π (cid:112)(cid:80) i f i . In the much more general circumstancewhere K is not diagonal in the basis where Q = , eigenvector delocalization causes theeigenvector Ψ KN with the largest eigenvalue f N to point in an approximately diagonaldirection, leading to the range ∆Φ = 2 π √ N f N [14]. The result of the present workis closely parallel to that of [14]: we have seen that when Ξ = ( Q − ) (cid:62) K Q − is notdiagonal in the basis where Q = , eigenvector delocalization causes the eigenvectorΨ Ξ N with the largest eigenvalue ξ N to point in an approximately diagonal direction,leading to the range ∆Φ = 2 π √ N ξ N .To understand the crucial distinction between ξ N and f N , it is useful to workin the concrete case of Calabi-Yau compactifications of type IIB string theory. Inthis setting we notice that K can be computed in terms of classical data, namelythe intersection numbers. At this level, the axion field space is R N ; the axions havevanishing potential, have infinite range, and do not decay. Meaningful statementsabout axion decay constants require specifying the nonperturbative effects that breakthe continuous shift symmetries to discrete shifts, which are encoded in Q . For thisreason, for any N > 1, a computation of the eigenvalues f i of the K¨ahler metric K defined by the classical K¨ahler potential does not determine the physical field range. In particular, an upper bound on f N does not provide an upper bound on the possibleaxion displacement during inflation, for two reasons. First, ∆Φ /f N is parametricallylarge at large N — as large as O ( N / ) — for generic K and Q . Second, even for N = 2,there is the possibility that the smallest eigenvalue λ Q (cid:62) Q of Q (cid:62) Q is accidentally smallin comparison to its expected size (cid:104) λ Q (cid:62) Q (cid:105) in an ensemble of Q matrices with the samesymmetries and with entries of the same r.m.s. size.The possibility that λ Q (cid:62) Q (cid:28) (cid:104) λ Q (cid:62) Q (cid:105) is the foundation of the Kim-Nilles-Peloso(KNP) mechanism of decay constant alignment [5]. The proposal of KNP, describedfor N = 2 in [5] and generalized to N > K = diag( f i ) in abasis where Q is nontrivial, and to take Q (cid:62) Q to have an accidentally small smallesteigenvalue. Such an accidental enhancement is plausibly realizable in the landscapeof string vacua, but for N = 2 — and indeed for any N that is not large — thisoccurs infrequently [19]. The increased likelihood at large N of large enhancementsfrom small λ Q (cid:62) Q / (cid:104) λ Q (cid:62) Q (cid:105) was observed by Higaki and Takahashi in [19] (see also [20]),and a slightly different perspective on enhancements at large N, also building on [5],was given by Choi, Kim, and Yun in [18].Here we have not relied on λ Q (cid:62) Q (cid:28) (cid:104) λ Q (cid:62) Q (cid:105) , but have instead shown that for Q The bound on the diameter of axion moduli space obtained for simplicial K¨ahler cones in [32] usesonly the data of K , taking Q = , and does not apply in the general case where Q (cid:54) = . (cid:104) λ Q (cid:62) Q (cid:105) itself is small,because of eigenvalue repulsion. Thus, the field range computed in this work is the thegeneric circumstance , not a fine-tuned possibility. A potential obstruction to achieving a super-Planckian displacement in a theorywith an extremely large number of axions is that renormalization of the Planck mass(cf. [6]) reduces the effective range ∆Φ, measured in renormalized Planck mass units.General reasoning suggests( M ren . pl ) − ( M barepl ) ≡ δM ∼ N π Λ , (7.1)where Λ UV is the ultraviolet cutoff. However, (7.1) is manifestly ultraviolet sensitive,and a more meaningful approach is to examine the leading correction that arises instring theory. Compactifying type IIB string theory on a six-manifold X with Eu-ler characteristic χ ( X ) and volume V , and including the four-loop σ -model couplingquartic in ten-dimensional curvature [33, 34], one finds δM = M × ζ (3) χ ( X )8(2 π ) g / s ( l s ) V , (7.2)where V is the Einstein frame volume of the orientifold, and we are using the conventionsof [1]. If the axions in question arise in the K¨ahler moduli sector, so that N = h , ,the correction (7.2) has the same parametric scaling as (7.1), if h , is taken largewith h , fixed. However, in typical Calabi-Yau compactifications, (7.2) is a modestcorrection, δM (cid:46) M , and does not parametrically alter the field range. In theexample of DDFGK, δM /M = 0 . N needed in N-flation models [6]with Q = . The range we have exhibited is an ‘enhancement’ compared to prior expectations, but it wouldbe more accurate to say that those prior works that considered only the f i , rather than the ξ i ,underestimated the typical diameter of field space. (cid:62) Q ( P = N ) Q (cid:62) Q ( P > N ) K Unit Wishart WishartUnit √ N f N / f N f Wishart √ N f N N / f N N f N Heavy Tailed f N N f N √ N f N Table 1 . Parametric scaling of the maximum diameter of the axion fundamental domain fordifferent choices of metrics K and axion constraints Q . P is the number of constraints, N isthe number of axion fields, and f N is the largest eigenvalue of K . We have computed the diameter D of the axion fundamental domain in a general fieldtheory with N axions, with the Lagrangian L = 12 K ij ∂θ i ∂θ j − N (cid:88) i =1 Λ i (cid:2) − cos (cid:0) Q ij θ j (cid:1)(cid:3) , (8.1)where Q is a P × N matrix of integers defining the periodic identifications of theaxions. One key result is the diameter (3.12) along a particular direction, which givesa deterministic lower bound on the maximal diameter. We evaluated (3.12) in variousregimes using results from random matrix theory, leading to approximate lower boundsthat hold with high confidence at large N . The resulting scalings with N are shown inTable 1.We substantiated our general findings by computing the diameter of the axionfundamental domain in explicit Calabi-Yau compactifications of string theory. We fo-cused on the vacuum of F-theory constructed in [1], where all moduli are fixed in aregime where known higher-order corrections are controllably small. The nonperturba-tive superpotential generated by Euclidean D3-branes and by gaugino condensation onD7-branes defines a specific 51 × Q matrix (6.17) for the h , = 51 Ramond-Ramondaxions that complexify the K¨ahler moduli. For the precise vacuum parameters takenin [1], where higher order corrections are parametrically controlled, the largest metriceigenvalue obeys f N ≈ . M pl . Our random matrix results predict D (cid:38) M pl , and bydirect computation we have confirmed that D (cid:38) . M pl .Let us close by discussing the potential implications of our results. There are anumber of arguments against the possibility of arbitrarily large displacements ∆Φ of31calar fields in effective theories that admit completions in quantum gravity. How-ever, it has proved difficult to sharpen general quantum gravity arguments to placeaccurate limits ∆Φ < M pl , as contrasted with ∆Φ < ∞ : the maximal ∆Φ in a giventheory depends on the details of the ultraviolet completion, and existing general ar-guments are not precise enough to capture factors of order π . Moreover, there aremechanisms implying the plausible existence of counterexamples — constructions oflarge-field inflation in string theory — based on effects such as decay constant align-ment [5], N-flation [6], or monodromy [16, 17]. These proposals have not yet led touniversally acknowledged existence proofs of large-field inflation in string theory, be-cause of the difficulty of embedding these mechanisms into explicit and parametricallycontrolled compactifications with stabilized moduli.Our findings present a way forward: they provide a framework for exhibiting super-Planckian axion displacements in well-understood vacua of string theory, without fine-tuning of parameters, and without working at extremely large N (cid:38) . By unifying thedecay constant alignment effect of KNP [5] with the eigenvector delocalization describedin [14], and arguing that both effects are generically present, we have shown that thediameter of axion field space is parametrically larger in N (cid:29) Q is a somewhat sparse matrix, and we argued that many flux compactificationson Calabi-Yau orientifolds fall into this category. While our field theoretic argumentsapply for any N (cid:29) 1, in this work the largest number of axions we have examinedin an explicit vacuum of string theory is N = h , = 51, in the case of the DDFGKcompactification of F-theory [1]. Because D ≈ M pl in this example, we anticipatethat displacements suitable for large-field inflation, ∆Φ (cid:38) M pl , could be achievedin a compactification with similar structures but with h , of order a few hundred,comfortably inside the range of known Calabi-Yau threefolds. Exhibiting an exampleof this sort is an important problem for the future.We have argued that in a theory consisting solely of N axions, inflationary evo-lution can rather naturally proceed along the super-Planckian diameters that we haveidentified. However, in compactifications with spontaneously broken supersymmetry,including the example of [1], the couplings of saxions to axions may lead to instabilitiesthat preclude inflation. This is a general difficulty: even in vacua of string theory thatadmit super-Planckian axion displacements, the uncontrolled evolution of moduli fieldspresents a challenge for any candidate construction of large-field inflation. The theo-ries we have described here are a promising arena for grappling with this fundamentalproblem. See e.g. [3, 35], as well as the recent review [2]. cknowledgements We thank Raphael Bousso, Thorsten Rahn, John Stout, and Timm Wrase for usefuldiscussions, and we thank Diederik Roest and Cliff Burgess for sharing their relatedresults with us. This work was supported by NSF grant PHY-0757868 and by a SimonsFellowship. A Results from RMT In the study of theories with N (cid:29) A.1 Classical ensembles Random matrix ensembles can be classified by their symmetry properties. Two classesof physical relevance are the Hermite (Wigner) and Laguerre (Wishart) β -ensembles.Consider a random N × N matrix A with entries that are independent, identicallydistributed (i.i.d.) random numbers of variance σ . The ensemble of Wigner matriceswith β = 1 , M H = A + A † , (A.1)while the Wishart ensemble is defined in terms of an M × N matrix A M L = A · A † , (A.2)where β = 1 corresponds to real entries in A , while β = 2 corresponds to complexentries. These are rotationally invariant ensembles of random matrices. In the large N limit, the precise probability distribution for the entries of A loses relevance (aslong as its variance is sufficiently bounded), and a universal limit is approached. Inthis limit, the symmetry properties of the ensemble define statistical observables suchas the eigenvalue and eigenvector distributions. Table 2 lists some properties of theWigner and Wishart ensembles in large N limit [38].Note in particular that the joint eigenvalue distribution of both the Wigner and theWishart ensemble can be interpreted as the probability distributions of a classical, one-dimensional gas at finite temperature 1 /β with Coulomb interactions. The probability33 Invariance Joint eigenvalue distribution Hermite GOE Wigner 1 M → Q (cid:31) MQ GUE Wigner 2 M → U † MULaguerre Real Wishart 1 M → Q (cid:31) MQ Complex Wishart 2 M → U † MU Table 2 . Caption Consider a random N × N matrix A with entries that are independent, identicallydistributed (i.i.d.) random numbers of variance σ . The ensemble of Wigner matriceswith β = 1 , M H = A + A † , (A.1)while the Wishart ensemble is defined in terms of an M × N matrix AM L = A · A † , (A.2)where β = 1 corresponds to real entries, while β = 2 corresponds to complex entries in A . These are rotationally invariant ensembles of random matrices that we will makeextensive use of. In particular, in the large N limit, the details of the probabilitydistribution by which the entries of A are distributed lose relevance (as long as itsvariance is sufficiently bounded) and a universal limit is approached. In this limit,the symmetry properties of the ensemble define the statistical observables such as theeigenvalue and eigenvector distribution. Table B lists some properties of the Wignerand Wishart ensembles in large N limit [ ? ].Note in particular that the joint eigenvalue distribution of both the Wigner and theWishart ensemble can be interpreted as the probability distribution of a classical, one-dimensional gas at finite temperature 1 /β with Coulomb interactions. The probabilityis given by ρ ( λ i ) = e βH . In the large N limit, the eigenvalue spectrum of the Wignerensemble approaches the famous Wigner semicircle law ρ ( λ ) = 12 πN σ √ N σ − λ , (A.3)30 β Invariance Joint eigenvalue distribution Hermite GOE Wigner 1 M → Q (cid:31) M Q C e − β ( σ (cid:31) Ni =1 λ i − (cid:31) j Laguerre Real Wishart 1 M → Q (cid:31) M Q C e − β ( σ (cid:31) Ni =1 λ i − (cid:31) j Hermite GOE Wigner 1 M → Q (cid:31) M Q C e − β ( σ (cid:31) Ni =1 λ i − (cid:31) j Laguerre Real Wishart 1 M → Q (cid:31) M Q C e − β ( σ (cid:31) Ni =1 λ i − (cid:31) j It is important to address the conditions under which random matrices approach theuniversal regime. In particular, the classical random matrix ensembles are definedin terms of non-heavy tailed entries, i.e. the cumulative distribution function of theentries decays at least exponentially. However, there are more general matrices thatstill approach universality.Let us consider the example of a N × N unit matrix that is perturbed by a ma-trix δ Q , where δ Q is a random matrix with real i.i.d. elements with the Gaussiandistribution N (0 , σ δQ ): Q = + δ Q . (A.10)The matrix Q (cid:62) Q is given by Q (cid:62) Q = + ( δ Q σ δQ + δ Q (cid:62) σ δQ ) + δ Q (cid:62) σ δQ δ Q σ δQ , (A.11)in which the first term has eigenvalues of order 1, the second term is a Wigner matrixwith eigenvalues of order (cid:113) N σ δQ , and the third term, which is a Wishart matrix,has eigenvalues of order 4 σ δQ N . The matrix Q (cid:62) Q is well approximated by a Wishartmatrix for σ δQ (cid:38) / (2 √ N ). While this parametric scaling is confirmed by numericalstudies, we observe that the smallest eigenvalue actually approaches the Wishart resultfor σ δQ (cid:38) √ / √ N .Let us consider the example of Q = + δ Q , where δ Q consists of a matrix with N δQ random entries equal to one, with all other entries vanishing. The quadratic normof the entries of δ Q evaluates to (for N δQ (cid:28) N ) σ δQ = N δQ /N . (A.12)As we noted above, the matrix Q (cid:62) Q approaches universality for σ δQ (cid:38) N , whichis satisfied for N δQ (cid:38) N . (A.13)Therefore, we have a lower limit for σ δQ (assuming only unit entries of Q ), σ δQ (cid:38) (cid:114) N , (A.14)which corresponds to N δQ = 2 N . Thus, the universal regime is approached by perturb-ing a unit N × N matrix by 2 N random elements.36 x x x x x x x x x x p − − − − − − − − − − Table 3 . Charges for V A . B Q in Calabi-Yau Hypersurfaces in Toric Varieties In this appendix we briefly explore the form of q in two examples of Calabi-Yau hy-persurfaces in toric varieties. We will simply demonstrate the nontriviality of certain q , and not concern ourselves with explicit orientifold involutions, etc. All reflexivepolytopes in four dimensions are available in the Kreuzer-Skarke database [44]. Tri-angulation of the corresponding polytope yields a simplicial toric variety with at mostpointlike singularities [45], which are missed by a generic Calabi-Yau hypersurface. Weuse the algorithm presented in the appendix of [26] to triangulate the polytopes anddefine the toric variety. For each ray v i in the fan that defines the toric variety there is acorresponding homogeneous coordinate x , the vanishing of which defines a divisor D i .The D i are called the toric divisors, and define irreducible hypersurfaces in the toricvariety. In the following we will refer to both the divisor and its cohomological dual as D i . A subset of the D i form a basis for H , ( X , Z ). To see which linear combinations of D i contribute to the nonperturbative superpotential we need to compute certain Hodgenumbers of the toric divisors. This can be done using the program cohomCalg [46],an implementation of the algorithm suggested and proved in [47–49], which uses theKoszul sequence to calculate line bundle topology in toric varieties. We then calculatethe leading order contributions to the nonperturbative superpotential, which definesthe q matrix. As a first example, we consider the Calabi-Yau hypersurface in the toricvariety given in Table B, which we denote by V A .37he Stanley-Reisner ideal is given bySR = { x x , x x x x , x x x x , x x , x x , x x , x x , x x , x x , x x ,x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x } . (B.1)This toric variety defines a Calabi-Yau hypersurface with h , = 7. We take D i = { D , D , D , D , D , D , D } as a basis for divisors. The divisors { D , D , D , D , D } are rigid toric divisors. Moreover, the combinations D + D , D + D are rigid. The q matrix is then given by q = D D D D D D D W W W W W W W . Here each W i , i = 1 . . . 7, denotes the i th contribution to the nonperturbative su-perpotential. We have kept only the leading contributions to the nonperturbativesuperpotential, neglecting higher-order terms, e.g. from the rigid cycle D + D .The K¨ahler cone conditions present a difficulty in this example. If we demand thateach of the holomorphic curves, given by generators of the Mori cone, has area of atleast 1, then the four-cycles that appear in the superpotential are forced to becomevery large. As a result, the nonperturbative superpotential — and correspondingly,the scalar potential — become extremely small in Planck units, precluding modulistabilization near the GUT scale. For the purpose of constructing models of large-field inflation, one would like to find Calabi-Yau manifolds with “mild” topology, bywhich we mean that the divisor volumes do not grow rapidly with the curve areas. TheDDFGK compactification described in § V B . The Stanley-Reisner ideal is given bySR = { x x , x x , x x , x x , x x , x x , x x , x x , x x , x x , x x ,x x , x x , x x , x x x , x x x , x x x , x x x , x x x , x x x } . (B.2)38 x x x x x x x x x x p − − − − − − − − − − − − − Table 4 . Charges for V B . This toric variety defines a Calabi-Yau hypersurface with h , = 7. We take D i = { D , D , D , D , D , D , D } as a basis for divisors. The divisors { D , D , D , D , D } are rigid toric divisors, while { D , D } are exact Wilson divisors with h , = 1. More-over, the combinations D + D , D + D , and D + D are rigid. The q matrix isthen given by q = D D D D D D D W W W W W W W W . 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