Heterotic T-fects, 6D SCFTs, and F-Theory
Anamaría Font, Iñaki García-Etxebarria, Dieter Lust, Stefano Massai, Christoph Mayrhofer
MMPP-2016-61LMU-ASC 15/16
Heterotic T-fects, 6D SCFTs, and F-Theory
Anamar´ıa Font a , I˜naki Garc´ıa-Etxebarria b , Dieter L¨ust b,c ,Stefano Massai c , and Christoph Mayrhofer ca Departamento de F´ısica, Centro de F´ısica Te´orica y ComputacionalFacultad de Ciencias, Universidad Central de VenezuelaA.P. 20513, Caracas 1020-A, Venezuela b Max-Planck-Institut f¨ur PhysikF¨ohringer Ring 6, 80805 M¨unchen, Germany c Arnold Sommerfeld Center for Theoretical Physics,Theresienstraße 37, 80333 M¨unchen, Germany [email protected], [email protected], [email protected],[email protected], [email protected]
Abstract
We study the (1 ,
0) six-dimensional SCFTs living on defects of non-geometric heteroticbackgrounds (T-fects) preserving a E × E subgroup of E × E . These configurationscan be dualized explicitly to F-theory on elliptic K3-fibered non-compact Calabi-Yauthreefolds. We find that the majority of the resulting dual threefolds contain non-resolvable singularities. In those cases in which we can resolve the singularities weexplicitly determine the SCFTs living on the defect. We find a form of duality in whichdistinct defects are described by the same IR fixed point. For instance, we find that asubclass of non-geometric defects are described by the SCFT arising from small heteroticinstantons on ADE singularities. a r X i v : . [ h e p - t h ] M a r ontents − II ∗ ] model and E singularity . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.1 Adding five-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Five-branes on C / Z k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.1 [I n − p − ] model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2.2 [I n − p − q ] model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 − III] model . . . . . . . . . . . . . . . . . . . . . . 224.2 A global model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Ogg-Namikawa-Ueno classification 52E Matter representation analysis 60
String theory admits a rich set of supersymmetric compactifications, giving rise to a vastspace of lower dimensional field theories. Most of the study of these compactifications focuseson regimes where the background can be understood geometrically by considering a classicalsupergravity reduction on the geometry, supplemented with knowledge of the dynamics onbrane stacks. This is far from being the only possibility, but it is very convenient and veryamenable to concrete analysis. Nevertheless, it would be interesting to go away from this geo-metric class of backgrounds, both to learn more about the non-classical and non-geometricalproperties of string theory, and to gain some insight about the broader set of possible stringvacua.In this paper we focus on a class of compactifications of the E × E heterotic stringwhich are very non-classical, involving compactifications on “spaces” that cannot be globallydescribed as geometries, while remaining accessible thanks to duality with F-theory. We can,in this way, probe many of the properties of the heterotic string away from the classical regimewhere it is conventionally studied.More concretely, we will focus on cases where the compactification space for the heteroticstring is at a generic point locally geometric, and described by a T fibration. The non-classical nature of the background arises from the patching between local descriptions, whichwe choose to involve non-trivial elements of the T-duality group acting on the T [1]. Theresulting total space is usually referred to as a non-geometric T-fold [2]. In the context of theheterotic string one should note that there is additional gauge bundle data (denoted by E T in the following) which mixes with the geometric data of the T under generic elements ofthe T-duality group of the T . The patching will send ( τ, ρ, E T ) → ( τ (cid:48) , ρ (cid:48) , E (cid:48) T ), with τ thecomplex structure of the torus, ρ = (cid:82) T B + iJ its complexified K¨ahler modulus, and E T theWilson line data along the two cycles of the torus. The primed values arise from the actionof the O (2 , , Z ) T-duality on the T .Such fibrations will in general have defects, i.e. subloci of the compactification space wherea local description in terms of the heterotic string on a smooth T × R n with a smooth bundleis no longer possible. For concreteness, we consider the compactification of the heterotic stringto six dimensions. In this case, we have locally a T fibration over a complex one-dimensionalbase. At certain points of the base we have defects, which will induce a monodromy action on( τ, ρ, E T ) as we go around them. Our goal in this paper is to describe, for a particular classof bundles E T , the low energy dynamics living on the defect itself.We will do this by dualizing the configuration to F-theory, where the dynamics on thedefect can be characterized by purely geometric means. In order to do so in the most explicitway possible, we restrict the bundle E T to have SU (2) structure, so it will break E × E E × E . The bundle data on the T is then described by a single complex number,whose real and imaginary parts are given by the Wilson line of the SU (2) Cartan around theone-cycles of the T . We denote this complexified Wilson line by β in the rest of the paper.With a single Wilson line turned on, the T-duality group is O (2 , , Z ) and an order foursubset of this group can be identified with Sp (4 , Z ), which is the action of the mapping classgroup of a genus-two curve on the homology. In this paper we restrict to monodromies in this Sp (4 , Z ) subgroup, so we have a formulation in terms of monodromies of genus-two curves.This correspondence is in fact very deep: as shown recently in [3, 4], there is a very closeconnection between the moduli space of genus-two Riemann surfaces and the moduli spaceof elliptically fibered K3 surfaces having an E and an E point. By duality with F-theory,this is precisely the moduli space of the heterotic string on T with a single Wilson line.Furthermore, the map has been explicitly worked out in [4, 6–8] (generalizing previous workin the case with unbroken E × E symmetry [3, 9]): given a genus-two Riemann surface,parameterizing the moduli of a heterotic compactification with unbroken E × E , there areexplicit expressions — to be reviewed below — for the moduli of the dual K3.In fact, the existence of the genus-two description for the heterotic vacua on T with a singleWilson line gives us a formal, but geometric, description of the very non-geometric heteroticcompactifications of interest in this paper. This viewpoint is particularly fruitful since thereexists a classification of the possible degenerations of genus-two fibers over a complex one-dimensional base, obtained by Ogg-Namikawa-Ueno [10, 11]. This is analogous to, but moreinvolved than, the Kodaira classification of degenerations of genus one fibrations, which areextensively used in F-theory.We can now summarize the main results of this paper. For each of the possibilities allowedby the classification of genus-two degenerations — or equivalently, for every defect preserving E × E and with monodromy in Sp (4 , Z ) — we will apply the heterotic/F-theory dualitymap to express the heterotic backgrounds in terms of F-theory compactifications. Generically,the F-theory background dual to a given 5-brane defect on the heterotic side will be highlysingular. In some cases (the exact criterion is stated in section 6) we can resolve the singularityby performing a finite number of blow-ups in the base of the fibration. For all the caseswhere this resolution is possible we construct the resulting smooth geometry. The blow-upscorrespond to giving vevs to tensor multiplets of the 6d (1,0) theory on the defect, such thatit flows to a Lagrangian description in the IR. In this way, from the knowledge of the smoothgeometry one can understand some aspects (such as anomaly polynomials [12, 13]) of thestrongly coupled CFT living at the origin of the tensor branch in terms of more ordinaryquantum field theories. Let us note that as one might have expected, for the cases that wecan resolve we obtain theories that fall into the recent classification of [14–16].In order to test our approach we will first consider local genus-two models that correspondto geometric ADE singularities of a K3 surface, together with a monodromy ρ → ρ + n for thecomplexified K¨ahler modulus. As expected, from the resolution of the dual F-theory models The connection between the heterotic moduli space with one Wilson line and the associated Siegel modularforms of genus-two Riemann surfaces was first noted in [5].
4e find a non-perturbative enhancement of the gauge algebra which agrees with the theoryof pointlike instantons hitting the orbifold singularity determined in [17], with n related tothe number of instantons at the singular point ( n = 0 corresponds to local cancellation of themodified Bianchi identity, and thus to having as many small instantons as the degree of theADE singularity). We also determine the matter content from the dual F-theory geometry,and verify explicitly that it agrees with the expectation from anomaly cancellation [18].We then move on to non-geometric models that involve monodromies in the K¨ahler modu-lus ρ with a non-trivial action on the torus volume. We find a form of duality, in that distinctdefects can give rise to the same SCFTs. For instance, we often encounter the same SCFTsas those describing pointlike instantons on ADE singularities, even for defects arising fromnon-geometric configurations. Understanding the origin of these dualities is an importantopen problem. We stress that we also find non-geometric degenerations which are not dualto pointlike instantons on ADE singularities, and give SCFTs which are genuinely new in theheterotic context.This paper is organized as follows. In section 2 we review the formulation of heterotic/F-theory duality in terms of a map between genus-two curves and K3 surfaces, and we discusshow it can be used to study non-geometric heterotic backgrounds in terms of K3 fiberedCalabi-Yau three-folds. In section 3 we apply our formalism to study local heterotic degenera-tions which admit a geometric description in some duality frame. In section 4 we discuss trulynon-geometric models and we show how to construct a global model with such degenerations.We also explicitly describe various dualities between different non-geometric and geometricdefects. In section 5 we list the resolutions of the remaining non-geometric models, consid-ering in particular a class of models that do not admit a limit with vanishing Wilson line.Finally in section 6 we provide the details of the classification of all possible local heteroticmodels, both geometric and non-geometric, admitting F-theory duals that can be resolvedinto smooth Calabi-Yau three-folds. We conclude with a discussion in section 7. We relegateto appendix A the resolutions of geometric models that correspond to pointlike instantons onADE singularities. In appendix B we discuss the heterotic/F-theory duality for the case ofvanishing Wilson line. In appendix C we show the expressions of the Igusa-Clebsch invariantsin terms of coefficients of a sextic that describe a given genus-two curve. In appendix D wereproduce the Namikawa-Ueno classification of singular genus-two fibers, and for each entrywe compute the order of vanishing of the Igusa-Clebsch invariants. Finally, in appendix Ewe explain how to extract the matter content from the F-theory resolutions for an explicitexample. In this section we review the formulation of F-theory/heterotic duality recently discussedin [4, 8]. We first discuss the duality in eight dimensions and then we show how to fiber itover a common base to study non-geometric heterotic compactifications to six-dimensions in5erms of F-theory on Calabi-Yau three-folds.
It is well known that the E × E heterotic string compactified on T is dual to F-theorycompactified on an elliptically fibered K3 surface [19]. For the heterotic compactification witha Wilson line that breaks the gauge group to E × E the corresponding K3 is described by aWeierstraß model of the form y = x + ( a u v + c u v ) x w + ( b u v + d u v + u v ) w = 0 , (2.1)where [ u : v ] ∈ P and [ y : x : w ] ∈ P , , are the homogeneous coordinates of the base andthe Weierstraß equation, respectively. For generic values of the coefficients the fiber has aKodaira singularity of type III ∗ ( E ) at u = 0 and a singularity of type II ∗ ( E ) at v = 0. Byvirtue of the F-theory/heterotic duality, there must be a map relating the heterotic moduli ρ , τ and β to the K3 coefficients a , b , c and d .To obtain an understanding for this map, we study certain limits thereof. Consider first thespecial case c = 0. One can immediately see from (2.1) that this implies that both singularitiesare of type II ∗ ( E ). Thus, c = 0 corresponds to vanishing Wilson line, i.e. to β = 0. In thislimit, the coefficients a , b and d are related to the heterotic moduli τ and ρ in the followingway j ( τ ) j ( ρ ) = − a d , (2.2)( j ( τ ) − j ( ρ ) − b d , where j is the SL (2 , Z ) modular invariant function. The map for this specific configurationwas originally obtained in [9]. Note that we can interpret the moduli τ and ρ as complexstructures of two elliptic curves (one of which is the physical heterotic torus) which are gluedtogether at one point, i.e. a degenerated genus-two curve. The map thus can be read as arelation between SL (2 , Z ) modular forms and the K3 coefficients, cf. appendix B. As we willnow discuss, we can extend this relation to encompass a non-vanishing Wilson line.In the general setup, with c (cid:54) = 0, the map has been recently established in [4], usingprevious findings about K3 surfaces related to curves of genus two [7, 20, 21]. The threeheterotic complex parameters ρ , τ , β live on the Narain moduli space M het = D , /O (2 , , Z ) with D , := O (2 , , R ) O (2 , R ) × O (3 , R ) , (2.3) Recall that the compactification of the E × E heterotic string on T comprises eighteen complex moduli:the sixteen Wilson line moduli β i with i = 1 , . . . ,
16, the complex structure τ of the torus and the complexifiedK¨ahler modulus ρ of the torus. Since throughout this article we are only interested in compactifications withan unbroken E × E non-abelian subgroup, we will drop the superscript of β . O + ( L , ) of the Narain U-duality group O (2 , , Z ) which preserves orientations, because we will be ultimately interestedin fibering the duality group holomorphically over a base.A crucial observation is that there is an isomorphism D , ∼ = H between the symmetricspace and the genus-two Siegel upper half-plane [22]: H = (cid:26) Ω = (cid:18) τ ββ ρ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) τ, ρ, β ∈ C , det Im (Ω) > , Im ( ρ ) > (cid:27) . (2.4)Since on the same grounds a (two-to-one) relation between O + ( L , ) and Sp (4 , Z ) can be estab-lished, there is a correspondence between the moduli space of the heterotic compactificationand the quotient of H by the genus-two modular group Sp (4 , Z ). The action of M ∈ Sp (4 , Z )on Ω is given by M (Ω) = ( A Ω + B )( C Ω + D ) − , M = (cid:18) A BC D (cid:19) (2.5)where A , B , C and D are 2 × M ∈ Sp (4 , Z ). More details about thisquotient and the relation to M het can be found in [4] and [22].A genus-two curve has four linearly independent cycles that can be chosen to span acanonical basis such that the intersection form has a symplectic structure (see e.g. [23]). Weindicate the symplectic basis as ( a , a , b , b ) in figure 1. The matrix Ω introduced in eq. (2.4)can be determined from integrals of the two holomorphic one-forms over the a i , b i cycles [11].The transformations in Sp (4 , Z ) are induced by changes of homology basis that preserve theintersection form.Coming back to the dual F-theory description (2.1), it has been found that the dualitymap can be expressed in terms of genus-two Siegel modular forms as [4, 7, 20, 21] a = − ψ (Ω) , b = − ψ (Ω) , c = − χ (Ω) , d = χ (Ω) . (2.6)The definition and properties of the relevant Siegel modular forms can be found in [4], seealso [8].We also note that in eight dimensions the heterotic/F-theory map we use naturally geo-metrizes the extra massless string states appearing at self-dual points on the moduli space interms of degenerations of the dual K3 surface [9, 24]. A recent discussion on this from thedouble field theory point of view appeared in [25]. Genus-two curves
As we have discussed above, the heterotic moduli can be put in correspondence with themoduli of a hyperelliptic genus-two curve. In turn such curve, denoted Σ, can be represented7y a sextic: y = f ( x ) = (cid:88) i =0 c i x i = c (cid:89) i =1 ( x − θ i ) . (2.7)In order to connect the c i coefficients with the a , b , c , d in the dual K3 fibration (2.1), weneed to determine the Siegel modular forms appearing in the map (2.6) in terms of the c i ’s.This can be done in a convenient way by first computing the Igusa-Clebsch invariants of thesextic (2.7) and then relating them to the Siegel modular forms of the corresponding genus-twocurve. The Igusa-Clebsch invariants are defined in terms of the six roots θ i of (2.7) as: I = c (cid:88) (12) (23) (45) ,I = c (cid:88) (12) (23) (31) (45) (56) (64) ,I = c (cid:88) (12) (23) (31) (45) (56) (64) (14) (25) (36) ,I = c (cid:89) i