On the 3-form formulation of axion potentials from D-brane instantons

Journal of High Energy Physics, Feb 2017

The study of axion models and quantum corrections to their potential has experienced great progress by phrasing the axion potential in terms of a 3-form field eating up the 2-form field dual to the axion. Such reformulation of the axion potential has been described for axion monodromy models and for axion potentials from non-perturbative gauge dynamics. In this paper we propose a 3-form description of the axion potentials from non-gauge D-brane instantons. Interestingly, the required 3-form field does not arise in the underlying geometry, but rather shows up in the KK compactification in the generalized geometry obtained when the backreaction of the D-brane instanton is taken into account.

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On the 3-form formulation of axion potentials from D-brane instantons

Received: December formulation of axion potentials from Eduardo Garc a-Valdecasas 0 1 2 3 4 Angel Uranga 0 1 3 4 up the 0 1 4 -form 0 1 4 Open Access 0 1 4 c The Authors. 0 1 4 0 Campus de Cantoblanco , 28049 Madrid , Spain 1 C/Nicolas Cabrera 13-15, Campus de Cantoblanco , 28049 Madrid , Spain 2 Departamento de F sica Teorica, Universidad Autonoma de Madrid 3 Instituto de F sica Teorica UAM-CSIC 4 eld does not arise in the The study of axion models and quantum corrections to their potential has experienced great progress by phrasing the axion potential in terms of a 3-form eld dual to the axion. Such reformulation of the axion potential has been described for axion monodromy models and for axion potentials from non-perturbative gauge dynamics. In this paper we propose a 3-form description of the axion potentials from non-gauge D-brane instantons. Interestingly, the required 3-form underlying geometry, but rather shows up in the KK compacti cation in the generalized geometry obtained when the backreaction of the D-brane instanton is taken into account. D-branes; Flux compacti cations; Gauge-gravity correspondence 1 Introduction and main results 2 Review of 3-forms and monodromy. 3 3-forms from D-brane instanton backreaction D-brane instanton backreaction The 3-form and its coupling Some toroidal examples Gauge non-perturbative e ects Introduction and main results Axions have become an essential template to describe physics of scalar elds whose potential enjoys special protection properties due to an underlying symmetry principle. Naively, the symmetry corresponds to the perturbative global symmetry shifting the value of the scalar eld, which is violated by non-perturbative e ects, as originally proposed for the QCD axion [1]. However, it has recently become clear that the most fundamental symmetry structure is that of the dual 2-form. Contributions to the axion potential which spoil the shift symmetry must arise from the existence of a 3-form which eats up the dual 2-form to make it (and so the dual axion) massive. The gauge symmetry of the 3-form constrains the form of these contributions in an advantageous way for many phenomenological applications. The description of the axions in terms of forms and their duals has also been key to the use of the weak gravity conjecture [2] to constrain transplanckian axion model This formulation has been well understood: For the QCD axion, in [21{23], where the 3-form is actually the Chern-Simons composite 3-form built out of the QCD gauge elds. In string compacti cations producing axion monodromy [24, 25], as described in [26] connecting it to the earlier description in [27, 28].1 In these cases, the 3-form is a fundamental eld, and its couplings arise from di erent sources, ranging from ChernSimons couplings to uxes in the 10d action [26, 47] (see also [48]), torsion homology [26] (see also [49]) or topological brane-bulk couplings [43]. 1For other works related to axion monodromy, see [15, 29{46]. The two above phenomena, in particular the presence of uxes and non-perturbative e ects on D-brane gauge sectors, play an important role in several scenarios of moduli stabilization (and thus of their axion components), along the lines in [50]. Actually, the gauge non-perturbative e ects can be described in string theory as particular cases of D-brane instanton e ects wrapping the same cycle as the gauge D-branes in the compact space. In general, in string theory there are other non-perturbative e ects from D-brane instantons not wrapping such cycles (sometimes dubbed stringy or exotic D-brane instantons [51{53], see [54, 55] for reviews), and contributing to the stabilization of axions as well. It is therefore natural to wonder about the 3-form description of these latter e ects. Interestingly, there is no known description of this kind: since there is no gauge group associated to the cycles, we cannot use any composite Chern-Simons 3-form; on the other hand, for e.g. a D3-brane instanton on a 4-cycle, there is not any obvious corresponding harmonic form able to produce a 3-form in the 4d theory upon compacti cation. In this paper we solve this question and provide the 3-form description for the stabilization of an axion by non-gauge D-brane instanton e ects. The key idea is to notice that the stabilization occurs when the non-perturbative e ect is included in the theory, so it is only then that we can hope to nd a suitable 3-form. Therefore, the internal form supporting the 4d 3-form must arise only in the geometry backreacted by the presence of the D-brane instanton, in the sense discussed in [56, 57]. In general, these correspond to generalized geometries, so the corresponding form need not be harmonic with respect to the underlying CY metric, rather it corresponds to (a piece of) a generalized calibration. We study this in the particular example of D3-brane instantons on 4-cycles, but the lesson is general (as expected from T-duality / mirror symmetry). Also, we show that the picture is compatible with D-brane instantons corresponding to gauge nonperturbative e ects. The paper is organized as follows. In section 2 we review the 3-form description of axion stabilization and its interplay with axion monodromy and non-perturbative gauge dynamics. In section 3 we provide the 3-form description of axion potentials induced by non-gauge D-brane instantons: after posing the question in section 3.1, we review the D3brane instanton backreacted geometry in section 3.2, and obtain the 4d 3-form and its couplings in section 3.3. A simple example is displayed in section 3.4. Section 3.5 describes the generalization, in particular the mirror picture of D2-brane instantons in type IIA compacti cations. In section 4 we discuss the case of gauge D-brane instantons. Finally, section 5 contains our nal remarks. Review of 3-forms and monodromy. Consider an axion , regarded just as a scalar taking values in a circle (i.e. with discrete periodicity2 2 ) and with an (approximate) shift symmetry. In many applications one is interested in generating a non-trivial potential for this axion, violating precisely this shift symmetry. For simplicity we consider the potential expanded at quadratic order around 2For simplicity we set the axion decay constant to f = 1. a minimum, as for instance arises in moduli stabilization; the general picture is however more general. Hence we have the lagrangian A potential of this kind is naively not compatible with the axion periodicity, but may be made so by including multivalued branches for the potential, thus inducing axion monodromy. The description of the periodicity properties are automatically implemented by using a dual formulation, in terms of a 3-form c3 eating up the 4d dual 2-form db2 = d , in 4d). This is described by the following lagrangian3 where F4 = dc3. This theory has the gauge invariance As emphasized in [26], it is this gauge symmetry that protects the atness of the axion potential against uncontrolled corrections even in transplanckian eld ranges. Dualizing the 2-form back into the axion , we obtain the Kaloper-Sorbo description of axion monodromy [27, 28, 35, 59] Integrating out the non-dynamical 4-form eld strength F4 one recovers a potential with the structure (2.1). The idea can be generalized to other potentials by considering corrections, which due to the gauge invariance, (2.3) must be in terms of higher powers in F4. This leads for instance to the attening in [60]. At this point we would like to emphasize an important clari cation: the description of the massive axion in terms of a 3-form eating up the dual 2-form, or in terms of a coupling between (a function of) the axion and the 4-form eld strength, is not necessarily a signature of axion monodromy, but rather of the existence of a non-trivial axion potential. Axion monodromy arises in cases when the potential does not naively satisfy the axion periodicity. On the other hand, the 3-form description should also hold for non-monodromic potentials (i.e. single-valued and consistent with the axion periodicity). Our main interest in this note is indeed the 3-form description of single instanton non-perturbative potentials, which indeed are periodic in the axion. Before entering this discussion, we conclude this review with a brief recap of the 3-form description of axion potentials induced by non-perturbative gauge dynamics. This has been SYM, which will be useful later on. The axion belongs to a chiral multiplet = Im T , and we have a coupling SSYM = 3We allow for a Zn discrete gauge symmetry, see [26, 48, 58] for discussions of such system and the corresponding Zn charged domain walls. The non-perturbative dynamics produces a gaugino condensate superpotential W = ! is the SYM dynamical scale, regarded as a where ! = e2 i=n and = exp The theory has n gapped vacua, which di er by the value of h i (the phase of the gaugino condensate), in the sense that a shift (k + 1)th. Therefore the potential has a periodicity of + 2 changes the kth vacuum to the + 2 n. On the other hand, the axion should actually have a periodicity of 2 ; in fact, this is ensured because there is a 3-form description of the above axion potential. This follows from realizing that the coupling (2.5) describes a coupling to a 4-form F4 structure, with F4 = tr F 2. Namely, In other words, although the potential is periodic with a period 2 n, the periodicity of 2 is achieved via a monodromic structure, but with a nite number n of branches. This is structure of domain walls among vacua is naturally described in terms of the (composite) 3-form description [21{23]. 3-forms from D-brane instanton backreaction In string theory, there are non-perturbative contributions to the superpotential beyond the above non-perturbative gauge dynamics contributions. These come from, for instance, euclidean D-brane instantons which do not correspond to gauge theory instantons. A prototypical example is provided by D3-branes instantons wrapped on 4-cycles in CY compacti cations,4 such as those used in moduli stabilization in [50]. In our discussions we will assume the instantons to indeed produce superpotential terms; it would be interesting, but beyond our scope, to develop an understanding of a dual description for instanton e ects generating higher F-terms due to extra fermion zero modes [61, 62]. Denoting by T the complex modulus associated to the wrapped 4-cycle, we have a superpotential Wnp = A e T where A is a prefactor depending on complex structure moduli (and possibly also on open string moduli), whose detailed structure is not essential for our present purposes. We expect the axion potential induced by these instantons to admit a description in terms of a 3-form eating up the dual 2-form. On the other hand, there is no obvious candidate for such a 3-form in the 4d CY compacti cation: the only available RR are even-degree gauge potentials, so the 3-form should arise from the RR 6-form integrated over a 3-cycle. However, there is no natural pairing between a 3-cycle and a 4-cycle in a CY so as to support the topological coupling F4 ultimately responsible for the axion 4In general, these may carry world-volume uxes, but for simplicity we will restrict to the case of trivial potential. Therefore there is no natural candidate for the 3-form coupling to the axion to reproduce its potential. The solution to this problem is to follow the intuition gained in the discussion of the SYM superpotential. In that case, the 3-form arises only when the existence of nonperturbative sectors in the gauge theory are taken into account; namely, the fact that implementation of a similar concept for non-gauge D-brane instantons requires proposing that the 3-form describing the axion stabilization should be looked for not in the original CY geometry, but rather in the geometry perturbed by the gravitational backreaction of the D-brane instantons. This perturbation of the geometry has been studied in the literature in [56, 57] by exploiting the technology of generalized geometry. D-brane instanton backreaction In this section we review ideas in [56, 57] to describe the backreaction of D-brane instantons in terms of generalized geometry. The uninterested reader may wish to jump to section 3.3. The e ect of D-brane instantons can be encoded in the underlying CY geometry by means of a deformation turning the SU(3) holonomy into (in general) an SU(3) structure, associated to the existence of two spinors (not covariantly constant due to the already in the type IIB case, the two spinors are written 1 = + here + and + are complex conjugate of N = 1 supersymmetry, and satisfying r , and + is the 4d spinor specifying the +, where W0 is the superpotential The spinors (1;2) can be used to de ne two polyforms, 2 = + Noticing the chirality of the spinors in the sandwich, the polyform + contains even degree contains odd degree forms. A common alternative notation is (for type IIB) The familiar case of SU(3) structure corresponds to (2) . For SU(3) holonomy the spinors are covariantly constant and the polyforms 1 = 2 = The compacti cation ansatz is ds2 = e2A(y)g (x) dx dx The 10d elds can be organized in complex quantities, in agreement with the 4d susy structure. One holomorphic quantity is is the 10d dilaton (not to be confused with the 4d axion). This is motivated because it provides the calibration for BPS domain wall D-branes (notice that in IIB, calibrates odd-dimensional cycles). In other words, the tension of a D-brane BPS domain wall is obtained by integrating the above form over the wrapped cycle. For instance, for standard CY compacti cations, a D5-brane on a supersymmetric 3-cycle BPS domain wall, whose tension is given by R the 3-form part of the polyform The second quantity is , where the subindex (3) denotes C describes the RR backgrounds not encoded in the background the generalized calibration of BPS D-brane instantons (notice that in IIB, even-dimensional cycles). For instance, for standard CY compacti cations, a D3-brane wrapped on a holomorphic 4-cycle provides a 4d BPS instanton whose action is given The 3-form and its coupling From the supersymmetry conditions recast in terms of the 10d version of the 4d elds Z, T , one can show that, in a weak coupling expansion, we have dH Z = backreaction of the instanton e ect on the 10d geometry in terms of the appearance of a 1-form component Z(1) of Z, which was absent in the CY geometry. This equation de nes an special 1-form Z(1). It is associated to the globally de ned supersymmetric spinor, in the presence of the non-perturbative correction to the geometry. Its structure can be obtained by integrating the above equation. For the particf = 0, one obtains [56] where the tilded superpotential has the dependence on open string degrees of freedom One intuitive way to understand the above expression is to notice that, in a theory containing gauge D3-branes (i.e. D3-branes sitting at a point in the internal space), the 4d superpotential as a function of the D3-brane position is obtained by considering a 1chain L joining two di erent points in the CY and integrating Z(1) over it. This follows because Z(1) is the calibrating form for a D3-brane wrapped on the 1-chain L, which de nes a domain wall interpolating between the two con gurations of the D3-branes at the two (end)points. We thus have Z(1) = f W~ np where we regard f as the D3-brane position in the direction normal to the instanton 4cycle. The result therefore reproduces the familiar dependence on open string moduli, microscopically associated to Ganor strings [63] (see also [64]). In the following we recast (3.7) as d 1 = 2, and we use both 1 and 2 as internal pro les for the KK reduction of higher-dimensional form elds in the backreacted geometry. This is similar to the non-harmonic forms used in KK reduction of massive U(1)'s, and studied in [49] in compacti cation spaces with torsion (co)homology. We may use these forms to perform the KK reduction of the 10d RR 4-form C4 as C4 = This produces a 3-form in 4d spacetime, naturally associated to the non-perturbative e ect, and a 2-form, dual of . Moreover, is it clear that the 3-form is eating up the 2-form, by noticing that the eld strength F5 has a term F5 = (1 + 10d) ( 2 ^ (c3 + db2) where 10d is added to take the self-duality of F5 into account. This clearly has the gauge This implies that the 3-form is eating up the 2-form to become massive, and correspondingly provides a dual description of the axion becoming massive, as in section 2. It is also straightforward to show that this 3-form has a Kaloper-Sorbo coupling to the axion. We focus just on the leading term = R C4, and F4 = dc3. We simply massage the kinetic term of the (self-dual) 4-form and focus on the components in (3.11) F5 ^ F5 = C4 ^ 2 ^ F4 = where we used 2 Some toroidal examples To esh out this somewhat abstract description, let us now consider a toroidal compact T6, where for simplicity we take a factorizable, T6 = T2 local complex coordinates be z1; z2; z3. Let us study the backreaction caused by a instantonic D3-brane wrapping the 4-cycle in [56, 65], the complex structure Z = structure with a 1-form piece 4 de ned by z3 = 0. Using the general formulas gets corrected, becoming a generalized complex This is the 1-form to be used to produce the 4d 3-form upon compacti cation of the 10d Notice that it actually corresponds to dz3, a harmonic 1-form already present in the underlying toroidal geometry. Therefore there seems to be essentially no new geometric structure associated to the backreacted geometry, namely, no axion potential due to the instanton e ect. This feature is clearly related to the existence of extra harmonic forms in the T6 geometry, which are not present in generic CYs. However, it nicely dovetails the expectation that D3-brane instantons in toroidal geometries have additional fermion zero modes, and do not produce non-perturbative superpotentials for the corresponding moduli. In order to actually get non-trivial structure, we can consider orbifolds which remove the extra harmonic forms, and produce genuine CY geometries. Consider for instance T6=(Z2 Z2), where the generators of the orbifold group act as ( z1; z2; z3), and ! : (z1; z2; z3) ! (z1; z2; z3). To describe the quotient, we inu3 = 0, so f = u3, and we have It is now clear that the 1-form supporting the 3-form in the compacti cation of the 10d 4-form is non-harmonic with respect to the original CY geometry. Although we have focused on the case of axions with potential arising from D3-brane instantons on 4-cycles, the ideas hold for general RR axions associated to other cycles, and with potentials arising from the corresponding wrapped D-brane instantons. In order to illustrate this, we consider the mirror con guration of type IIA with axions arising from the RR 3-form on a 3-cycle, stabilized by D2-brane instantons. Let us thus study the mirror dual to the con guration of D3-branes on a 4-cycle, in the setup of a general CY (alternatively the main ideas can already be illustrated in the toroidal examples in section 3.4). Consider the CY in the large complex structure limit, where it can be regarded as a T3 (parametrized by coordinates yi, 1; 2; 3;) bered over a 3d The mirror dual can be obtained by applying three T-dualities [66], along the coordinates yi. The D3-brane instanton thus turns into a D2-brane wrapped on the 3-cycles locally spanned by x1, x2, y30 (with the prime denoting the T-dual coordinate). One further sees that the complex structure deformation Z(1) df = dx3 + idy3 gives rise to a polyform T = T(2) + T(4); This follows from the fact that Z(1) is eventually use to expand the RR forms and obtain the 4d 3-form. Hence, these are the 2- and 4-form components of (3.6) produced by the backreaction of the D2-instanton, as we argue later on. Before that, let us conclude that 5A global construction is easily produced by using Weierstrass equations for the 2-tori, but we will not need this extra complication. C = T ^ c3 ! C7 = T(4) ^ c3 ; C5 = T(2) ^ c3 where the c3 in the last two expansions is understood to be the same 4d 3-form. Let us nish by arguing further that the above T is indeed the backreaction corresponding to the D2-brane instanton. In the original picture in section 3.3 we considered the superpotential in the theory in the presence of a gauge D3-brane, given by the integral of the calibrating form over a 1-chain. Under the mirror transformation we must consider the theory in the presence of a gauge D6-brane wrapped on the 3-cycle 3 (i.e. the mirthe calibrating form T over a (generalized) 4-chain generalized) 3-cycles (see e.g. [67]) interpolating between two (possibly WD6 = We are interested in the superpotential as a function of one D6-brane complex modulus, given by one deformation of the special lagrangian 3-cycle, and the corresponding Wilson line along one 1-cycle in 3. The actual components turned on in this T are T(4), which will be integrated along the 4-chain produced by the deformation of the 3-cycle component T2 integrated over the 2-chain spanned by the 1-cycle in such deformation. The latter accounts for the contribution to the superpotential of the induced D4-brane charge arising from possible D6-brane worldvolume uxes on the 2-forms Poincare dual to the The resulting variation of the superpotential is WD6 = T(2) ^ F = where F is the magnetic eld induced in the D6-brane. Therefore, the only way to reproduce the D6-brane open string moduli dependence described by the Ganor zeroes is that the instanton backreaction indeed produces the deformation T with components T(2) and T(4). Gauge non-perturbative e ects There are D3-brane instantons which admit the interpretation of gauge theory instantons. This happens when the D3-brane instanton wraps precisely the same 4-cycle as a stack of 4d spacetime- lling D7-branes. The description of stabilization of axions coupling to non-abelian gauge interactions has been recast in terms of coupling to the composite Chern-Simons 3-form in [21{23] cf. section 2. It is natural to ask about any possible interplay between this and the 3-form discussed in earlier sections. Actually, the 3-form in earlier sections arises only when the D-brane instantons backreact, in other words when the gauge dynamics is geometrized. This implies that we must instantons) is de ned by condensate hSi = h To this order there is no deformation of Z, and moreover no 1-form that can support the 3-form. The latter is however generated when the non-perturbative gaugino condensate (which is described by (fractional) euclidean D3-branes) is included. The gaugino i = e2 ik=n 3 is a non-zero vev for the gaugino bilinear. In [65] it Z = T = e exp(ie =2J ) dZ = i`s4hSi 2( ) consider the geometry that results when the D7-branes, together with the euclidean D3brane instantons, are backreacted on the geometry. The resulting con guration no longer contains open string degrees of freedom, as everything is encoded in the backreacted geometry. Therefore the relation between the 3-forms is essentially holographic: on one side there are open string degrees of freedom, and the axion stabilization mechanism is described as in [21{23] in terms of a 3-form constructed out of the open string sector gauge elds; on the other side, there is a backreacted geometry, and no open string degrees of freedom, and the axion stabilization arises from a 3-form supported by the distorted geometry. esh out the latter claim, let us consider the description of the backreaction of D7-branes and their euclidean D3-brane instanton e ects. We concentrate in the case To describe the backreacted geometry, we borrow results from [65]. The backreaction of the D7-branes, at the perturbative level (i.e not including the euclidean D3-brane This is exactly as in (3.7), thereby con rming the anticipation that the backreacted D7/D3 system can support a 3-form in agreement with the mechanism in earlier sections. In this note we have provided the description of the axion potential from non-gauge D-brane instanton e ects in terms of a 3-form eating up the 2-form dual to the axion. The 3-form arises from the KK reduction of higher-dimensional RR elds in the generalized geometry arising when the D-brane instanton backreaction is taken into account. The mechanism also holds for D-brane instantons corresponding to gauge instantons, in which case the generalized geometry description of the axion couplings can be regarded as holographically related to earlier 3-form descriptions of stabilization of QCD-like axions. Our works puts axion potentials from non-perturbative e ects in a similar footing to other stabilization mechanisms, like ux compacti cations. We hope this can improve the study of the interplay between di erent stabilization mechanisms in string theory. In another line, given recent results in applying the Weak Gravity Conjecture to axion models in terms of their dual 3-forms [15], we expect our analysis to allow similar analysis for nonperturbative axion potentials from instantons. We hope to come back to these questions in the near future. Acknowledgments We would like to thank L. Iban~ez, F. Marchesano, A. Retolaza and G. Zoccarato for useful discussions. E. G. and A. 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Eduardo García-Valdecasas, Angel Uranga. On the 3-form formulation of axion potentials from D-brane instantons, Journal of High Energy Physics, 2017, DOI: 10.1007/JHEP02(2017)087