Rigid Limit in N=2 Supergravity and Weak-Gravity Conjecture
aa r X i v : . [ h e p - t h ] J u l arXiv:0706.2114UT-07-21YITP-07-35 Rigid Limit in N = 2 Supergravityand Weak-Gravity Conjecture
Tohru Eguchi † and Yuji Tachikawa ‡ † Department of Physics, University of Tokyo,Tokyo 113-0033, Japan,and † Yukawa Institute for Theoretical Physics, Kyoto University,Kyoto 606-8502, Japanand ‡ School of Natural Sciences, Institute for Advanced Study,Princeton, New Jersey 08540, USA abstract
We analyze the coupled N = 2 supergravity and Yang-Mills system using holomorphy,near the rigid limit where the former decouples from the latter. We find that there appearsgenerically a new mass scale around gM pl where g is the gauge coupling constant and M pl is the Planck scale. This is in accord with the weak-gravity conjecture proposed recently.We also study the scale dependence of the gauge theory prepotential from its embeddinginto supergravity. 1 Introduction
Quantization of general relativity has been one of the most serious challenges for theoreticalphysics for a long time. Its coupling constant is dimensionful, which makes the theoryapparently non-renormalizable. Thus, we need to complete the theory in the ultraviolet(UV) to make it into a consistent quantum theory. The prime candidate for quantizedgravity is the superstring theory, and the progress we made during the last decade makesus confident that there exist many consistent four-dimensional theories with a high degreeof supersymmetry containing quantized graviton in their spectrum. These low energyfield theories coupled to gravity have a consistent UV completion and are obtained viacompactification of superstring theory on suitable internal manifolds.When we come to theories with a smaller number of supersymmetries the situationbecomes somewhat delicate. Recent developments suggest that there exists an enormousnumber of N = 1 supersymmetric four-dimensional models with negative cosmologicalconstant (for a review, see e.g. [1]). This landscape of superstring vacua, if taken atface value, predicts a disturbingly huge number, 10 or larger, of solutions with varyinggauge groups and matter contents. Then it is natural to ask which theory is realized as alow-energy effective description of a consistent theory with quantized gravity [2]. Severalcriteria have been already proposed in [3, 4] which characterize models in the swampland which cannot be UV completed to a consistent theory of quantum gravity.The criterion we will focus in this article is the weak-gravity conjecture proposed in [3];one way to state the conjecture is that if a consistent theory coupled to gravity with thePlanck scale M pl contains a gauge field with the coupling constant g , then there shouldnecessarily be a new physics around the mass scale gM pl . We refer the reader to theoriginal article for the arguments which led to this proposal [3]. Our objective in thisarticle is to show how this conjecture will generically hold within the framework of N = 2supersymmetric Yang-Mills coupled to N = 2 supergravity.The system of N = 2 supersymmetry is well suited to the analysis of the effects ofquantum gravity on the gauge theory. One advantage is that the dynamics of N = 2 su-persymmetric Yang-Mills theories has been studied in great detail since the pioneering workof [5]. Another advantage is that the limit where the N = 2 Yang-Mills theory decouplesfrom the N = 2 supergravity is fairly well understood in the context of the string com-pactification on Calabi-Yau (CY) manifold with a fiber of ADE singularities. This limit isknown as the rigid limit or decoupling limit since supersymmetry becomes rigid and gravitydecouples from the gauge theory in the limit. It is also called the geometric-engineeringlimit [6, 7, 8], since non-Abelian gauge symmetry is generated by ADE singularities.In this paper we consider a type II string theory on CY manifolds which possess K CP and thus has a dual heterotic string description. At the geometricengineering limit ǫ → K ǫ . We shall show that the ratioof these periods leads to the hierarchy of gauge and gravity mass scales which has exactlythe form of the weak gravity conjecture. Since the geometric engineering limit is the onlyway to generate non-Abelian gauge symmetry in type II theory, the weak gravity conjecture2eems to hold generically in N = 2 gauge theory coupled to N = 2 supergravity. Actuallyas is well-known, M het = gM pl is the mass scale of heterotic string theory and thus the weakgravity conjecture seems to fit very nicely with the type II-heterotic duality.In our analysis the holomorphy and the special geometry of N = 2 theories play thebasic role. Holomorphic functions are determined by their behavior at the singularities, inparticular by the monodromy properties around the singular locus.The organization of the paper is as follows: In Section 2 we discuss an example of atype II string theory compactified on a CY manifold with a K ǫ → ǫ predicts a kinetic term for a field S ∂ µ S∂ µ S (Im S ) . (1.1) S corresponds to the gauge coupling constant S = θ/ (2 π )+4 πi/g and maps to the heteroticdilaton under the type II/heterotic duality. In Section 3 we will discuss generalization ofthe weak gravity hypothesis. We discuss in Section 4 the mechanism of how the logarithmicperiods necessarily appear in a CY manifold with a K renormalization group formulafor the dependence of the prepotential F gauge on the dynamical scale Λ [9, 10, 11]. Wederive for any gauge theory of ADE group, a relation ∂F gauge ∂ log Λ = hπi u . (1.2)Here u = h tr φ i and φ denotes the adjoint scalar in the vector multiplet. h is the Coxeternumber of the group. We conclude this note with some discussions in Section 6. Let us start with an example from the string theory. As is well-known, in the type IIAsuperstring theory, an N = 2 supergravity system in four dimensions can be obtained bycompactification on a CY manifold M . It is also known that the SU ( n ) N = 2 gaugesymmetry arises if M has a sphere of A n − type singularities. In the simplest case of A singularity such a CY manifold has at least two K¨ahler parameters: one for the size of thesphere of the singularities, and the other for the size of resolution of singularities. Oneexplicit example is given by a CY manifold X which is a degree 8 hypersurface in theweighted projective space WCP , , , , with Hodge numbers h = 2, h = 86.Our analysis is facilitated by going to the mirror type IIB theory where world-sheetinstanton corrections in IIA theory are summed up by mirror transformation. Mirror pair3f X and X ∗ has been extensively studied in the literature (e.g. [12, 13, 14]). We firstbriefly review their properties. Defining equation of the mirror X ∗ is given by X ∗ : W = B x + B x + 14 x + 14 x + 14 x − ψ x x x x x − ψ ( x x ) = 0 (2.1)in an orbifold of WCP , , , , . [ B : ψ : ψ ] parametrizes the complex structure moduli of X ∗ . We first note that this hypersurface has a structure of a K CP : by achange of variables x = x x , ζ = x /x , W is rewritten as W = B ′ x + 14 x + 14 x + 14 x − ψ x x x x = 0 , (2.2) B ′ = B ζ + 1 ζ ) − ψ . (2.3) ζ parametrizes the base of the K K ζ ) has singularitiesat B ′ = 0; large complex structure limit , (2.4) B ′ = ψ ; conifold singularity . (2.5)These are located by imposing equations W = 0 , ∂W/∂x i = 0 , i = 0 , , , ζ , we find B ′ = 0 = ⇒ ζ = e ± , where e ± = ψ B ± s(cid:18) ψ B (cid:19) − , (2.6) B ′ = ψ = ⇒ ζ = e ± , where e ± = ( ψ + ψ ) B ± s(cid:18) ψ + ψ B (cid:19) − . (2.7)Singularities of the total space X ∗ are located by further imposing ∂B ′ /∂ζ = 0 ∂B ′ ∂ζ = ⇒ B = 0 or ζ = ± . (2.8)Substituting ζ = ± X ∗ B = ± ψ , B = ± ( ψ + ψ ) . (2.9)These coincide with the locations where e ± , e ± become degenerate.Thus the discriminant of the mirror CY manifold is given by∆ = B ( B − ψ )( B − ( ψ + ψ ) ) . (2.10)Three components of the discriminant loci are depicted in Figure 1. The first and thesecond factor intersect tangentially at the large complex structure point, and the third4igure 1: Discriminant loci of the moduli of the CY X ∗ , before the blowup.Figure 2: Discriminant loci of the moduli of the CY X ∗ after the blowup. LCS stands forthe Large Complex Structure point.factor is the conifold locus. The conifold locus and the locus B = 0 also meet tangentiallyat the rigid limit , so that the moduli space needs to be blown up at these points.We now concentrate on the region near the rigid limit. The blowing up introduces anexceptional curve which is a CP parametrized by [Λ : u ] via the relation ǫ Λ = B, ǫu = ψ + ψ . (2.11)The exceptional curve is at ǫ = 0. The discriminant loci after the blowup are shown inFigure 2.The defining polynomial W in the limit ǫ → W = ǫ (cid:20)
12 ( w + Λ w ) + x + y + z − u (cid:21) + O ( ǫ ) . (2.12)after a suitable redefinition of the coordinates. This is a fibration of A singularity over CP parametrized by w . It is in fact the Seiberg-Witten geometry of the N = 2 supersymmetricpure SU (2) Yang-Mills theory with the modulus u = h tr φ i and the dynamical mass scale The parameter sets (
B, ψ , ψ ) and ( − B, ψ , ψ ) describe the same complex structure, and so thenatural coordinate of the moduli is B rather than B .
5. Thus, the exceptional curve we have introduced is identified as the u -plane of SU (2)gauge theory: the u -plane is naturally compactified at u = ∞ into a sphere. We call thissphere the rigid limit locus.Note that before taking the rigid limit ǫ →
0, the theory contains h + 1 = 3 gaugefields: they are the graviphoton, the gauge partner of the scalar field S and the U (1)(Cartan-subalgebra) part of SU (2) gauge field. Here S denotes the scalar field whichcorresponds to the gauge coupling constant in field theory, S = θ π + 4 πig . (2.13)We recall that when CY manifold M possesses a K CP , there exist a dualitybetween type IIA on M and heterotic theory on K × T [15]. The field S correspondsto the size of the base CP of K ǫ →
0, two of the gauge fields, thegraviphoton and the partner of S , disappear and we are only left with the (Cartan partof) SU (2) gauge field. Let us next quickly recall the structure of vector multiplet scalars in the N = 2 theories.First, in the case of field theories of rigid N = 2 supersymmetry with the gauge group U (1) n , there exist n complex scalar fields φ i , ( i = 1 , . . . , n ). Their K¨ahler potential is givenby K = Im X i ( a Di ) ∗ a i (2.14)where a i and a Di are holomorphic functions of the VEV’s of φ i . a i and a Di are called thespecial coordinates or the periods of the theory. Dual periods are related to each other as a Di = ∂F gauge ∂a i , i = 1 , · · · , n (2.15)where F gauge denotes the prepotential.Secondly, in the case of N = 2 supergravity with N vector multiplets, there exist2( N + 1) periods X a , F a , a = 1 , · · · , N + 1. The K¨ahler potential is given by e − K = Im X a F ∗ a X a . (2.16)The periods X a , F a are holomorphic functions of scalars Φ i , ( i = 1 , . . . , N ). Under theK¨ahler transformation K → K − f − f ∗ periods are transformed as X a → e f X a , F a → e f F a .The mass squared of a BPS-saturated soliton with charges ( q a , m a ) is then given by m = e K | X a ( q a X a + m a F a ) | , (2.17)6hich is invariant under the K¨ahler transformation. An important property of the super-gravity periods is the transversality condition: X a X a ∂F a ∂ Φ i − X a ∂X a ∂ Φ i F a = 0 , (2.18)which guarantees the existence of the prepotential. Prepotential of N = 2 supergravity isa homogeneous function of degree 2 in X a .In the case of CY compactification of type IIB string theory, the periods are given by X a = Z A a Ω , F a = Z B a Ω (2.19)where Ω is the (3 , A a , B a are the canonical basis of H ( M ∗ , Z )of CY manifold. In this case the condition (2.18) comes from the Griffiths transversality R Ω ∧ ∂ Φ i Ω = 0.Now let us go back to the example of the previous section, type IIB string theorycompactified on X . In the field theory limit we have only one gauge field ( n = 1) andtwo periods a and a D of SU (2) Seiberg-Witten theory. At the level of supergravity thereexist three gauge fields (two vector multiplets, N = 2) and six periods X a , F a , a = 1 , , X and F , are converted to the gauge theoryperiods in the rigid limit. They behave as X = ǫ / a + O ( ǫ ) , F = ǫ / a D + O ( ǫ ) . (2.20)Remaining four periods behave as X , X = 1 + O ( ǫ / ) , F , F = 12 πi log ǫ + O (1) . (2.21)The origin of logarithmic behaviors in F , F will be discussed in Section 4: they comefrom the geometry of K e K behaves as log 1 / | ǫ | . Therefore the supergravityK¨ahler potential is expanded as K = log(log 1 / | ǫ | ) + | ǫ | log 1 / | ǫ | Im( a D ) ∗ a + · · · . (2.22)as ǫ →
0. Note that Im (cid:0) a D (cid:1) ∗ a is the K¨ahler potential of the field theory (2.14). Thus wecan clearly see that SU (2) super Yang-Mills theory decouples from gravity.The factor | ǫ | in front of the K¨ahler potential of the field theory determines the hierarchybetween the Planck scale and the scale of the gauge theory: it is basically in accord withthe expectation [8] with | ǫ | / being identified with the dynamical mass scale Λ gauge of thegauge theory. The existence of an extra factor of log 1 / | ǫ | in the denominator was first7ecognized by the authors of [14]. We will see in the following that this factor implies theweak-gravity conjecture in the present context.Let us now consider the weak coupling region of gauge theory for the sake of simplicity.There the periods a and a D behave as a ≈ √ u, a D ≈ iπ √ u log u. (2.23)Using the relation of periods to the low-energy gauge coupling constant τ : τ = θ π + 4 πig ( m W ) = ∂a D ∂a , (2.24)we find e − π /g ( m W ) = u − / . (2.25)The coupling constant g in the above equation is to be evaluated at the scale of the mass m W of the massive gauge boson where the coupling stops running. m W is, in turn, givenby the formula (2.17) m W = e K | X | = | ǫ | log 1 / | ǫ | u. (2.26)From (2.25) and (2.26), we find the dynamical scale of the gauge theoryΛ gauge = m W e − π /g ( m W ) = | ǫ | / (log 1 / | ǫ | ) / M pl (2.27)where we reinstated the Planck scale to recover the correct mass dimension.Let us next introduce a chiral superfield S = θ/ π + 4 πi/g via the relation S = 1 πi log ǫ. (2.28)Then, the monodromy around ǫ = 0 is generated by the shift S → S + 2. Im S , which is thepartner of the dynamical theta angle, is the natural bare gauge coupling constant in thesupergravity. Furthermore, S coincides with the heterotic dilaton which we have discussedat the end of Section 2.1. There will be subleading corrections to (2.28) if one goes outsidethe region of weak coupling or small ǫ . Another notable fact is that, because of the K¨ahlerpotential (2.22), the field S in fact has the standard kinetic term for the dilaton, g SS ∗ ∂ µ S∂ µ S ∗ = ∂ µ S∂ µ S ∗ (Im S ) . (2.29)Using the field S = θ/ π + 4 πi/g , the relation (2.27) now becomesΛ gauge = e − π /g · gM pl . (2.30)There exists an extra factor of g in front of M pl in the above equation, which means thatthe ultraviolet gauge coupling g is defined not at the Planck scale M pl but at a lower energyscale gM pl . The running of the gauge coupling from the value at low energy Im τ to theone at high energy Im S is schematically depicted in Figure 3. The existence of the newscale gM pl is what the weak gravity conjecture has predicted. Thus the analysis of the N = 2 SU (2) gauge theory coupled to supergravity supports the weak gravity conjecture.8igure 3: Running of the coupling in the gauge theory coupled to supergravity. Let us consider what happens in the generic N = 2 gauge theory coupled to N = 2supergravity. Suppose the gauge theory has n vector multiplets. In the coupled gauge-gravity system the gauge coupling constant is promoted to a scalar field S . Thus there isat least one extra vector multiplet in the locally supersymmetric theory. Let us considerthe minimal situation; i.e. the total number of the U (1) vector multiplets being equal to n + 1. Then, altogether there are n + 2 gauge fields including the graviphoton and therewill be 2 n + 4 supergravity periods. Therefore, by coupling the gauge theory to the N = 2gravity we should have at least four extra periods.We assume that there is a locus E in the suitably blown-up moduli space given bythe local parameter ǫ = 0 around which some Ω i , ( i = 1 , . . . , n ) of the periods Ω I ,( I = 1 , . . . , n + 2)) become parametrically small, Ω i ∝ O ( ǫ /h ) for some power h . Thisstatement itself is not invariant under K¨ahler transformation, so we also demand that therewill be periods which stay constant near E .The monodromy around E may also be logarithmic: thus there might be periods be-having as ∝ (log ǫ ) k . Let p be the largest power k of such periods. There is a mathematicaltheorem which then states the K¨ahler potential behaves as e − K = Im X a F ∗ a X a ∝ (log | ǫ | ) p . (3.1)Let us define the chiral field S by S = 1 πi log ǫ = θ π + 4 πig (3.2)as before. Repeating the argument presented in the last section, we readily obtain a relationΛ gauge ∼ e − π / ( hg ) · g p M pl . (3.3)We will see in the next section that h equals the quadratic Casimir of the gauge group inthe case of pure N = 2 gauge theory. see e.g. Appendix A of [16] and references therein. S is given by ∂ µ S∂ µ S ∗ (Im S ) or ∂ µ S∂ µ S ∗ (3.4)depending on p = 0 or p = 0, respectively. Thus, the weak gravity conjecture in N = 2supergravity coupled to super Yang-Mills follows from the existence of a logarithmic period ∼ (log ǫ ) p , p ≥
1. Furthermore, the appearance of such logarithmic periods is related tothe field S = log ǫ corresponding to the dilaton in heterotic theory. For a CY which is a K CP , 3-cycles can be constructed explicitly. Wefollow the approach of [14] and Appendix in [17] . Consider a CY with a defining equation w + µ w + W K ( x, y, z ; t ℓ ) = 0 (4.1)where t ℓ denote the moduli of the K
3. The holomorphic 3-form is given byΩ = dww ∧ Ω K , Ω K = dx ∧ dy∂ z W K . (4.2)3-cycles of CY are made of the product of a 1-cycle of the CP base and a 2-cycle of K K K
3, since holomorphic cycles have the representative which are of the (1 , K must vanish. Holomorphic cycles of K Pic Λ Pic = H , ( K ∩ H ( K , Z ) (4.3)and its dimension is called the Picard number ρ ( K K Pic ⊕ Λ tr . (4.4)It is well-known that the lattice Λ has a signature of (3 , K , ρ ( K − tr becomes (2 , − ρ ( K K X , the Picardnumber is ρ ( K
3) = 19 and thus there are three transcendental cycles with signature(2 , A singularity. Two 2-cycles of the positive signature generate periodswhich have logarithmic behavior in ǫ as we see below. Ref. [14] discusses another exampleof CY manifold X which also possesses a K SU (3) gauge10igure 4: Cuts in the base CP .defining eq. h d x d y d z deg. of Casimirs A n − x + y + z n n n/ n/ , , . . . , nD n +1 x + y z + z n n n n − , , . . . , nE x + y + z
12 6 4 3 2 , , , , , E x + y + yz
18 9 6 4 2 , , , , , , E x + y + z
30 15 10 6 2 , , , , , , , , A singularity. In the case of general A r singularity there will be 2 + r transcendentalcycles with the signature (2 , r ). As we shall see below, two transcendental cycles of K K
3, whose moduli dependon w . Suppose a transcendental two-cycle S i degenerates at w + µ /w = k i . For a small µ , this happens at w + i ∼ k i and w − i ∼ µ /k i , see Figure 4. Let C be the circle around theorigin | w | = | µ | , and D i denote the path connecting w ± i . Then C × S i and D i × S i areclosed 3-cycles of CY manifold.In general, Yang-Mills gauge theories are geometrically engineered by fine-tuning theparameters { t ℓ } of K K SU ( n ), [18]for SO ( n ) and [19, 20] for E n groups. Suppose we have a singularity of type G withrank G = r around x = y = z = 0. The moduli { t ℓ } of K { u , · · · , u h } , { v , v , · · · } , (4.5)where u i corresponds to the degree i Casimir invariant of the group G . u i are tuned tovanish as ǫ i/h in the geometric engineering limit and we rescale them as ǫ i/h · u i . Here h isthe dual Coxeter number of G . v j are the moduli which remain finite in the engineering11imit. We also introduce the rescaled coordinates as w = ǫ ˜ w, x = ǫ d x /h ˜ x, y = ǫ d y /h ˜ y, z = ǫ d z /h ˜ z. (4.6) d x,y,z are the degrees of x, y, z (see Table 1). We also set µ = ǫ Λ h . Then the definingequation (4.1) of the CY becomes ǫ (cid:18) ˜ w + Λ h ˜ w + W ADE (˜ x, ˜ y, ˜ z ; u i ) + O ( ǫ /h ) (cid:19) = 0 . (4.7)The holomorphic 3-form is given byΩ = dww ∧ dx ∧ dy∂ z W K = ǫ ( d x + d y + d z ) /h − d ˜ w ˜ w ∧ d ˜ x ∧ d ˜ y∂ ˜ z W ADE = ǫ /h d ˜ w ˜ w ∧ Ω ADE (4.8)where we used the fact d x + d y + d z = h + 1.There are r independent two-cycles S i of K r A i = C × S i and ¯ B i = D i × S i , i = 1 , · · · , r of the CY as explained above. We can take their linear combinations, A i and B i , so thatthey have the canonical intersection form, ( A i , A j ) = ( B i , B j ) = 0, ( A i , B j ) = δ ij . Then a i = Z A i d ˜ w ˜ w ∧ Ω ADE , a Di = Z B i d ˜ w ˜ w ∧ Ω ADE , (4.9)are identified with the special coordinates of Seiberg-Witten theory. Corresponding super-gravity periods behave as X i = Z A i Ω = ǫ /h a i + O ( ǫ /h ) , F i = Z B i Ω = ǫ /h a Di + O ( ǫ /h ) . (4.10) K T a , a = 1 , S i andstay at finite values of x , y and z . Now the defining equation of CY near the 3-cycles T a is given by w + ǫ Λ h w + W K ( x, y, z ; 0 , v j ) = 0 (4.11)Thus from the cycle U a = C × T a we obtain the periodΩ U a = Z U a Ω = I C dww Z T a Ω K = 2 πi c a ≈ O (1) where c a = Z T a Ω K ( u i = 0; v j ) . (4.12)In the case of the cycles V a = D a × T a , the end points of the w integration become w − a ∼ ǫ Λ h k a , w + a ∼ k a (4.13)12here k a = w + ǫ Λ h /w is the value at which the 2-cycle T a degenerates. Then we findthe logarithmic behaviorΩ V a = Z V a Ω = Z w + a w − a dww Z T a Ω K ≈ − c a log ǫ. (4.14)The analysis of the monodromy under the phase rotation of ǫ suggestsΩ V a ≈ − πi log ǫ · Ω U a + O ( ǫ /h ) , (4.15)although the precise form of this expression will depend on the intersection form of T a . Thuswe have established the existence of periods behaving logarithmically near the engineeringlimit. As an application of the above analysis, we shall derive the relation ∂F gauge ∂ log Λ = hπi u (5.1)for pure N = 2 Yang-Mills theory with gauge groups G = A, D, E from its embeddinginto supergravity. Here u = h tr φ i is the second order Casimir and is a monodromy-invariant coordinate of the moduli space. The relation describes the scaling violation ofthe prepotential of gauge theory and is called the renormalization group equation.Before we start describing our derivation, let us recall how the equation (5.1) wasobtained from the point of view of the gauge theory. Originally it was derived for SU (2) in[9] using the Picard-Fuchs equation for Seiberg-Witten curve, and later it was generalized tothe classical gauge groups in [10, 11] using the property of the hyperelliptic curve describingthe dynamics of the theory. For the E -type gauge groups the relation has not been givenfrom the SW curve because of its complexity; thus our method gives the first verificationof the relation for the E -type gauge groups.The relation has been used in the analysis of the geometrical engineering limit in oneof the earliest papers on the subject [6]; here instead, we derive it from the study of theperiods near the engineering limit. It was conjectured already in [11] that the relationshould have a natural interpretation in supergravity since log Λ is no longer an externalparameter but becomes a VEV of a field in supergravity.The relation should also follow from the microscopic calculations: Recall the funda-mental relation in the path integral which states that h ∂ λ L i = ∂ λ L eff (5.2)where L is the bare Lagrangian and L eff is the low-energy effective Lagrangian includingthe quantum correction. λ denotes some coupling constant of the theory. In the case ofsupersymmetric theories one can likewise show h ∂ λ W i = ∂ λ W eff , h ∂ λ F i = ∂ λ F eff (5.3)13here W and F are the super and prepotential, respectively. Now in the N = 2 theory F = τ tr φ . Then the relation (5.1) follows because log Λ ∝ τ is the bare couplingconstant and u = h tr φ i . It can be seen more explicitly in the framework of multi-instanton calculation [21]. So the prepotential constructed from the SW curve shouldsatisfy the relation.Let us now turn to our derivation. Instead of the relation (5.1) itself, we shall showthat its derivative with respect to the moduli satisfies ∂ F gauge ∂u j ∂ log Λ = hπi δ j . (5.4)Then (5.1) follows by integration. The integration constant is zero by virtue of the homo-geneity of F gauge . The homogeneity can be used to rewrite LHS of (5.4) and rewrite it asfollows: ∂ F gauge ∂u j ∂ log Λ = ∂∂u j F gauge − X i a i ∂F gauge ∂a i ! = X i ∂a i ∂u j a Di − X i a i ∂a Di ∂u j . (5.5)Now we use the relation (4.10) between the periods of rigid and local theory and obtain X i F i ∂X i ∂u j − X i X i ∂F i ∂u j = ǫ /h ∂ F gauge ∂u j ∂ log Λ + O ( ǫ /h ) (5.6)One of the fundamental properties of the special geometry in supergravity is the transver-sality condition (2.18). We decompose the periods into two sets as ( X i , F i ; X a , F a ) where X i , F i are the periods which become those of the gauge theory of the rigid limit, and X a , F a are the extra periods in supergravity. We have X i F i ∂X i ∂u j − X i X i ∂F i ∂u j = − X a F a ∂X a ∂u j + X a X a ∂F a ∂u j . (5.7)As shown in the previous section, X a and F a have at most log ǫ singularity and therest are analytic in ǫ i/h u i . Furthermore, the logarithmic terms cancel in the RHS of (5.7)because the LHS is analytic in ǫ /h . Therefore we have X i F i ∂X i ∂u j − X i X i ∂F i ∂u j = − X a F a ∂X a ∂u j + X a X a ∂F a ∂u j = const · ǫ /h δ j + O ( ǫ /h ) . (5.8)Comparing with (5.6), we obtain (5.4) up to a constant factor.Two comments are in order: first, the constant factor is non-trivial to determine in gen-eral but should be straightforward to fix in specific cases. It then fixes the proportionalityfactor between u entering in the geometry and h tr φ i . Second, the derivation above wasso simple that it makes us suspicious why a similar analysis cannot be done in the fieldtheory limit. Indeed, RHS of (5.5) is a monodromy-invariant quantity of mass dimension14 − j . Thus, it is a rational function of u j ’s of dimension 2 − j and it is forced to be δ j once one can argue it does not have poles. This is precisely the hard part because thespecial coordinates a i and a Di are complicated functions of u j ’s with a lot of cuts. In ourderivation, we utilize the fact that the extra supergravity periods are analytic in ǫ j/h u j ,which does the job. In this article, we have seen how the holomorphy inherent in N = 2 supersymmetry canbe effectively used to study the effect of gravity upon the running of gauge theory. Morespecifically, we showed how the monodromy of the periods around the locus of the rigidlimit translates to the hierarchical separation of the dynamical scale of gauge theory andthe Planck scale. We have argued that, as compared to the naive relationΛ gauge ≈ e − π /hg M pl (6.1)there is generically an extra factor of the gauge coupling constant g in the right hand side,Λ gauge ≈ e − π /hg · gM pl (6.2)supporting the weak gravity conjecture. We have also seen how the scaling violation of theprepotential of the gauge theory, (5.1), can be naturally understood from the embeddinginto supergravity.The result presented here is only a small step in utilizing the holomorphy to understandthe dynamics of the coupled N = 2 supergravity-gauge systems. We believe many moreproperties can be learned in a similar manner. It would also be interesting to make acomparison with the result in [22] where the authors calculated the one-loop effect of gravityto the beta function of the gauge theory. It was argued in [23] that the beta function in[22] alone leads to the weak gravity conjecture. We will have to supersymmetrize the resultof [22] to carry out the comparison to our case.It will be very important to see if it is possible to extend our results to the realm of N = 1 supersymmetric theories. In the case when N = 1 theories are obtained fromthose of N = 2 by introducing fluxes, branes etc. many of the structures of the lattersurvive. Hopefully we will have enough control over mass scales of these theories to derivethe characterization of consistent N = 1 field theories coupled to gravity. Acknowledgements
The authors would like to thank Nima Arkani-Hamed and Seiji Terashima for discussions.TE would like to thank the Institute for Advanced Study where a part of the work wasdone. Research of TE is supported in part by a Grant-in-Aid from the Japan Ministry ofEducation and Science. Research of YT is supported by DOE grant DE-FG02-90ER40542.15 eferences [1] M. R. Douglas and S. Kachru, “Flux compactification,” arXiv:hep-th/0610102.[2] C. Vafa, “The string landscape and the swampland,” arXiv:hep-th/0509212.[3] N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, “The string landscape, black holesand gravity as the weakest force,” arXiv:hep-th/0601001.[4] H. Ooguri and C. Vafa, “On the geometry of the string landscape and the swampland,”
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