Contents
Abstract

We study a class of 6d non-geometric vacua of the heterotic string which can be understood as fibrations of genus-two curves over a complex one-dimensional base. The 6d theories living on the defects that arise when the genus-two fiber degenerates at a point of the base are analyzed by dualizing to F-theory on elliptic K3-fibered non-compact Calabi-Yau threefolds. We consider all possible degenerations of genus-two curves and systematically attempt to resolve the singularities of the dual threefolds. As in the analogous non-geometric vacua of the heterotic string, we find that many of the resulting dual threefolds contain singularities which do not admit a crepant resolution. When the singularities can be resolved crepantly, we determine the emerging effective theories which turn out to be little string theories at a generic point on their tensor branch. We also observe a form of duality in which theories living on distinct defects are the same.

Non-Geometric Vacua of the

[3mm] Heterotic String and Little String Theories

Anamaría Font and Christoph Mayrhofer

Departamento de Física, Centro de Física Teórica y Computacional

Facultad de Ciencias, Universidad Central de Venezuela

A.P. 20513, Caracas 1020-A, Venezuela Arnold Sommerfeld Center for Theoretical Physics,

Theresienstraße 37, 80333 München, Germany

Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut,

Am Mühlenberg 1, D-14476 Golm, Germany

afont@fisica.ciens.ucv.ve, christoph.mayrhofer@lmu.de

1 Introduction

The construction and understanding of string compactifications beyond the supergravity approximation are important open problems that deserve investigation. For one reason, non-geometric string vacua exist and are in principle on the same footing as the more widely explored geometric vacua that can be interpreted in terms of supergravity reductions on some internal spaces. Moreover, it is conceivable that they could have appealing phenomenological features. In fact, in the article that originally contemplated the class of non-geometric vacua that we will consider, one motivation was to search for compactifications with a reduced number of massless moduli in the low-energy theory [1].

In [1] the key idea was to build vacua as fibrations by letting the moduli of type II strings compactified on vary over a base. A further essential ingredient was to allow for monodromies in the duality group when going around points on the base where the moduli become singular. Since among these monodromies there are transformations that invert the torus volume, the compactifications are intrinsically non-geometric. The scheme of fibrations of moduli was later extended to heterotic strings where duality with F-theory can be used to extract properties of the resulting non-geometric vacua [2]. Taking the base to be complex one-dimensional leads to six-dimensional non-geometric heterotic vacua that have received attention more recently [3, 4, 5, 6, 7].

In this paper, we will further examine six-dimensional non-geometric heterotic vacua described locally as fibrations over a base parametrized by . As in recent works, we focus on configurations in which the heterotic gauge background is chosen to have structure so that the gauge group is broken to or , depending on whether one starts with the or the heterotic string. In this situation the heterotic moduli comprise one complex Wilson line together with the complex structure and the complexified Kähler modulus of . The T-duality group acting on the space spanned by these three moduli is [8]. Restricting this group to , the subgroup of order four which can be identified with , gives an isomorphism between the heterotic moduli space and the moduli space of genus-two curves [3, 4]. Hence, the non-geometric heterotic vacua can be defined equivalently as fibrations of a genus-two Riemann surface over the base . In general, a non-trivial holomorphic fibration will degenerate at certain points on and encircling them will induce an T-duality transformation on the moduli, thereby signalling the presence of defects, dubbed T-fects in [9], such as NS5s or more exotic 5-branes. Now, the possible degenerations of genus-two fibers over the -plane have been classified by Ogg, and Namikawa and Ueno [10, 11]. Our objective is to continue the study, initiated in [5], of the six-dimensional theories living on the T-fects corresponding to degenerations in the Namikawa-Ueno list. This program is carried out by dualizing the configuration to F-theory, where the T-fects can be characterized geometrically.

The fundamental heterotic/F-theory duality relates the heterotic string compactified on and F-theory compactified on an elliptically fibered K3 surface [12, 13, 14]. At the moduli level, the explicit map when there are no Wilson lines, i.e. the gauge group on the heterotic side is unbroken, was found in [15, 2]. In the case at hand, when there is one Wilson line breaking the gauge group to or , the map from the heterotic moduli to the dual K3 moduli was established lately in [16, 17, 3, 4]. This map can be expressed in terms of Siegel modular forms of the genus-two curve encoding the heterotic moduli. Thus, in F-theory the non-geometric heterotic vacua described as genus-two fibrations over a base correspond to specific K3 fibrations over the same base. Moreover, to preserve supersymmetry the total space of the F-theory fibration must be a Calabi-Yau—a threefold in the case that the base of the K3-fibration is complex one-dimensional. Since in F-theory there is a well-defined geometric formalism to analyze degenerations of the fiber along the base, the heterotic/F-theory duality enables us to infer properties of the T-fects of non-geometric heterotic vacua.

The T-fects connected to the genus-two degenerations in the Namikawa-Ueno list [11] were surveyed in [5] in the context of the heterotic string. The purpose of the present work is to extend the analysis to the heterotic string. One motivation is to check for the existence of dualities among defects observed in [5]. The ultimate goal is to discover the main features of the theories living on the T-fects. The study of such theories in the heterotic string actually started with the seminal treatise of Witten [18] who showed that a heterotic 5-brane, or equivalently a small instanton, supports a six-dimensional (1,0) supersymmetric gauge theory with group and 16 hypermultiplets in the fundamental representation. Already exploiting tools of F-theory, the theories arising from small instantons sitting at ADE singularities in K3 were later analyzed in great detail in [19, 20, 21]. These generic theories were also derived from the dual perspectives of type I D5-branes [22, 23], and type IIA configurations of D6, D8 and NS5-branes [24, 25]. In the heterotic string, F-theory methods were also used early on in [26]. Various aspects of non-geometric vacua of the heterotic string have been considered more recently in [2, 3, 6, 7].

In the following, we will present the results of a systematic study of heterotic T-fects associated to the genus-two degenerations in the Namikawa-Ueno classification [11]. As in the case addressed in [5], we will apply the duality map to every genus-two degeneration in the Namikawa-Ueno list in order to obtain the dual F-theory background. Since this background turns out to have a elliptic fibration with a non-minimal singularity, we will attempt to turn the singularity into a minimal one by performing a series of blow-ups in the base of the fibration. When the resolution can be accomplished we will determine the emerging smooth geometry. Introducing blow-ups is equivalent to giving generic vevs to scalars in tensor multiplets of the 6d theory on the defect, i.e. we move onto the tensor branch of the theory. Thus, knowing the smooth geometry allows to deduce the gauge groups and matter content characterizing the IR limit, valid in the tensor branch, of the theory living on the defect. Analogous techniques have actually been employed in the recent classification of SCFTs [27, 28, 29, 30, 31] and little string theories (LSTs) [32]. Actually, the theories that we obtain fall into known configurations whose UV completions are conjectured to be LSTs [33, 32]. The theories indeed have a mass scale and enjoy T-duality upon circle compactification, both typical properties of LSTs [34].

Let us finally give an overview of the rest of the article. The six-dimensional non-geometric heterotic vacua of interest are described in more detail in Section 2. There we recall the basics of heterotic compactifications on and review the formulation of heterotic/F-theory duality in terms of a map between genus-two (sextic) curves and elliptically fibered K3 surfaces. In addition, we explain how the map connects degenerations of sextics over a complex one-dimensional base, classified by Namikawa and Ueno, with degenerations of K3 fibered Calabi-Yau threefolds. In Section 3 we first sketch the procedure to resolve singularities and then apply the method to local heterotic degenerations which have a geometric description in some duality frame. We also discuss truly non-geometric singularities that exhibit a kind of duality with geometric defects. In Section 4 we catalog all possible local heterotic degenerations admitting F-theory duals that can be resolved into smooth Calabi-Yau threefolds. We conclude with further observations about the results. Appendix A contains the resolutions of several models corresponding to small instantons on ADE singularities.

2 Non-Geometric Heterotic Vacua and F-Theory

This section is devoted to outlining the construction of the six-dimensional non-geometric heterotic vacua studied in this paper. We will first explain the structure of the vacua and then discuss how to exploit F-theory/heterotic duality to analyze their properties.

2.1 Heterotic Vacua in 8 and 6 Dimensions

The starting point is the compactification of the heterotic string on a torus . The emerging eight-dimensional theory contains moduli fields encoding the geometric and gauge bundle data. The geometric moduli consist of the complexified Kähler modulus, , with the Kalb-Ramond two-form and the holomorphic one-form of the torus which follows from the metric on , and the complex structure modulus given by , where and are the two generators of the non-trivial one-cycles of the torus. Furthermore, from the gauge bundle data we have 16 complex Wilson line moduli from the Cartan generators of the non-Abelian gauge group of the heterotic string, i.e. , . In the following, we restrict ourselves to background gauge bundles which only have structure, so it will break down to , or to . With this choice there is only a single complex modulus, called in the following, whose real and imaginary parts are given by the Wilson line of the Cartan around the one-cycles of the as defined above. It is well known that the three complex parameters , and live on the heterotic moduli space [8]

(2.1)

where is the local moduli space of the compactification and the duality group which identifies physically equivalent theories.

Having the eight-dimensional moduli from the torus compactification we construct, in the next step, six-dimensional vacua by letting the moduli fields vary along two real, or one complex, dimension. Since we allow in this construction for identifications of the moduli under the duality around paths of non-trivial homotopy, it is very cumbersome to work directly with the moduli fields. To circumvent this difficulty we use, like in F-theory, a geometric object which has (almost) the same moduli space as the fields we want to describe. The variation of the fields becomes then a fibration of the object along the complex one-dimensional base. In our case, the geometrification is done via a genus-two curve such that we end up with a genus-two fibration.

Since the description of the heterotic moduli space in terms of genus-two curves will be crucial in the following, we briefly review it. To every point of the moduli space characterized by , and , there is an associated genus-two curve whose period matrix belongs to defined by

(2.2)

The four independent one-cycles of can be chosen to span a canonical homology basis, and with , such that intersection form is symplectic. The elements of the period matrix are , where the are holomorphic one-forms of with normalization . The moduli space is obtained by taking the quotient of by the action

(2.3)

The action of is induced by changes of the homology basis that preserve the intersection form. In [35] it was shown that is isomorphic to which is an index four subgroup of the full Narain duality group .111Note that this also induces a four-to-one map between the moduli space of the genus-two curve and the true heterotic moduli space . Hence the moduli spaces are only almost identical as mentioned above.

The geometrification of the heterotic moduli in terms of a genus-two curve suggests, as mentioned already, to build lower-dimensional vacua by considering genus-two fibrations. The idea is simply to let the moduli vary adiabatically along a base . Since we consider six-dimensional vacua, has to be complex one-dimensional, locally parametrized by a coordinate . For the moduli to fulfill the (BPS) equations of motion the fibration must be holomorphic in . Hence, a non-trivial fibration has to degenerate at co-dimension one loci on the base. Encircling such a degeneration point, the genus-two fiber returns to itself but undergoes an monodromy transformation. Thus, upon transport around a non-contractible loop, the heterotic moduli return to their values only up to a duality transformation. Since includes transformations such as , which exchanges large and small volume, non-geometric vacua are part of these compactifications. A generic genus-two degeneration will induce a monodromy involving all three moduli , and . The localized physical objects that lie at the center of the genus-two degenerations are dubbed T-fects, which is short for T-duality defects [9]. In the case of the heterotic string, the six-dimensional theories that live on T-fects were studied in [5]. In this paper we extend the analysis to the heterotic string.

Although genus-two fibrations have monodromies only in , they have the great advantage that their degenerations are classified. This was done more than forty years ago by Ogg [10] and Namikawa and Ueno [11], who gave a classification of all possible holomorphic degenerations of genus-two fibers over a complex one-dimensional base. In particular, Namikawa and Ueno (NU) provide explicit local equations of the possible degenerations together with the corresponding monodromies. The NU list supplies a large number of T-fects. However, we do not know how to study them directly in terms of the heterotic string. Therefore, we use the duality with F-theory and analyze them in that setting, as we discuss next.

2.2 F-Theory and Vacua with Varying Moduli

Since the inception of F-theory [12], it has been known that both heterotic strings compactified on and F-theory compactified on an elliptically fibered K3 surface are dual to each other [13, 14]. This duality is best understood in the large volume/stable degeneration limit [36, 13]. In this limit on the heterotic side, whereas on the F-theory side the K3 degenerates into two surfaces which intersect each other along a . The heterotic modulus is the complex structure of the F-theory at the intersection, while the Wilson lines are encoded in the intersection points (spectral cover data [37]) of the respective nine exceptional curves of the two ’s with the . Such a precise identification with all Wilson line moduli turned on exists so far only in this special limit. On the other hand, the duality map is known along the whole moduli space when there is none [15, 2] or only one non-vanishing Wilson line [16, 17, 3, 4]. In this article we focus on the latter case where the Wilson line breaks down to , or to .

For the duality with the heterotic string the hypersurface describing the elliptically fibered F-theory K3 takes the form

(2.4)

where , , and , are the homogeneous coordinates of the fiber ambient variety and the base , respectively. This K3 has a singularity at and a at which correspond to and gauge groups, respectively.

For the heterotic compactification the dual K3 is described by [19, 20, 2]

(2.5)

where and depend on the base coordinates according to

(2.6)

The fibrations (2.5) and (2.4) are birationally equivalent to each other as shown for instance in [2, 3], and for an earlier account of this in terms of toric geometry see [26]. In the latter reference it is described how the two different fibrations are realized as two different two-dimensional reflexive sections of the same polytope. In Figure 1 we indicated, for our case, the two different sections of the K3 which give (2.4) and (2.5), respectively.

(a) Reflexive section dividing polytope into - and -top

(b) Reflexive section dividing polytope into - and trivial-top
Figure 1: The two different reflexive sections through the (dual) polytope defined via equation (2.4) and (2.5), respectively. The red dots indicate toric divisors which correspond to the resolution divisors of the , , and gauge groups. Divisors which intersect each other lie next to each other along the red edges. Note the appearance of the extended Dynkin diagram structure. Furthermore, the of on the right hand side is given by the divisor corresponding to the green interior point on the edge of the reflexive section.

Although a given fan of the polytope allows at most for one fibration, they are connected via birational flops. The double fibration structure is expected from the known T-duality of the two heterotic strings upon circle compactification [21, 32].

To analyze the K3 in (2.5) we bring it to Weierstraß form. In the patch we obtain

(2.7)

The singularities of this fibration are located at the vanishing locus of the discriminant

(2.8)

We observe that the fiber has Kodaira singularities of type () at , and of type () at , for generic coefficients. The gauge group is actually . When the group enhances to .

Heterotic/F-theory duality requires the existence of a map relating the heterotic moduli (2.2) to the coefficients , , and of the dual K3 surfaces (2.4) and (2.5). The duality map has been established recently [16, 17, 3]. It can be written as

(2.9)

where , , , and are genus-two Siegel modular forms [38] of weight indicated by the subscript. Modularity is meant with respect to the transformation (2.3).

From the previous discussion we conclude that there is a well-defined relation between the moduli space of the heterotic string compactified on with one complex Wilson line, the moduli spaces of genus-two curves and elliptically fibered K3 surfaces with or singularities. Thus, non-geometric heterotic vacua encoded in terms of genus-two fibrations over a base can be realized, in F-theory, as K3 fibrations over the same base. The advantage of this correspondence is that in F-theory there is a proper procedure to analyze degenerations of the fiber along the base. In this way, heterotic/F-theory duality can be applied to explore the physics of T-fects associated to genus-two degenerations.

We are mostly interested in six-dimensional non-geometric heterotic vacua given by genus-two fibrations over a base with local coordinate . In particular, we want to consider the genus-two degenerations compiled by Namikawa and Ueno (NU) [11]. In the NU classification the genus-two singularities are described in terms of fibrations of hyperelliptic curves represented by sextics of the form

(2.10)

where the are functions (or sections) of . Furthermore, the hyperelliptic curve fibrations of [11] are in a canonical form with the singularity located at . Determining the K3 coefficients , , , , as functions of is facilitated by having the genus-two fibrations in the form of (2.10). To begin we compute the modular forms of the genus-two curve from the Igusa-Clebsch invariants222See appendix C of [5] for the explicit form of the Igusa-Clebsch invariants in terms of the coefficients of the sextic. , , , [39]

(2.11)

Combining these relations with (2.9) then gives

(2.12)

for the complex structure coefficients of the K3. Since the Igusa-Clebsch invariants are polynomials of the ’s so will be the coefficients of the dual K3. Hence, in the end we obtain for every genus-two singularity in the NU list a K3 fibration over the same -plane with the K3 fiber degenerating at as well. Let us remark here that understanding the map from the F-theory to the heterotic setup is much more involved. Some progress in this direction has been achieved recently in [7].

In the next section we will look at the F-theory singularities that arise from the map and attempt to resolve them. It turns out that whether a resolution is possible or not depends on the vanishing order of the coefficients , , and , denoted , , and , at . Notice that in all cases we have .

To conclude this section let us briefly consider the fibration of the (by itself elliptically fibered) F-dual K3 over a compact base, concretely over a . In the case of the heterotic string, cf. (2.7), imposing that the total space is a Calabi-Yau threefold implies that the latter can be understood as an elliptic fibration over the Hirzebruch surface .333Starting with (2.4) in the heterotic string leads to an elliptic fibration over [3]. Moreover, , , , and must be polynomials of degree 8, 12, 20 and 24, respectively, in the homogeneous coordinates of the base. Now, it is known that this F-theory compactification is precisely dual to the heterotic string compactified on K3 with group broken to and 20 half-hypermultiplets in the representation [13, 14].

3 Resolution of Singularities: Formalism and Examples

After establishing the duality map between heterotic and F-theory vacua the next task is to tackle the resolution of singularities. To this end we first review a general formalism based on toric techniques [5] in this section. Afterwards we apply the method to a class of NU models whose degenerations correspond to small instantons on ADE singularities. Comparing with the known resolutions in these cases [21, 23] allows to verify the validity of our approach. We then consider examples in which there is no initial interpretation of the singularities.

3.1 Formalism

We work systematically with a Weierstraß model all along. This means that an elliptic fibration is always represented by a hypersurface equation of the form

(3.1)

where , , are again the homogeneous coordinates of and and are sections of some line bundles over the base , . The crucial requirement in F-theory is that the elliptic fibration has to be Calabi-Yau. This condition constrains the line bundles of and to be and , respectively, with the canonical bundle of the base. The elliptic fiber becomes singular when the discriminant vanishes. We will refer to a model as resolved if the elliptic curve has only minimal singularities (or Kodaira type singularities) [40] along the base, i.e. there are no non-minimal points along the discriminant locus where vanishes to order four or higher and simultaneously to order six or higher.

In the F-theory framework the six-dimensional vacua of interest are obtained by taking a K3 fibration over a base parametrized by . Thus, at the start we have a hypersurface equation such as (2.4) or (2.7), with and depending on . Recall that the dependence on is dictated by the particular NU model under study.

To sketch the resolution procedure let us first examine the F-theory dual of the heterotic string, which is simpler yet captures the essentials. Since the coefficient in front of the term in in (2.4) is constant, there is no non-minimal singularity along . Therefore, we only have to look in the patch for such points and in the beginning it turns out that there is just one non-minimal singularity at . To get rid of this non-minimal point we follow [21] and blow-up the base at this point. However, as we will explain momentarily, we proceed in a rather toric manner by introducing the maximal amount of 444Crepant in the sense that the proper transform of the hypersurface equation (2.4), or (2.7), after the base blow-up is still Calabi-Yau. We do not claim that the canonical class of the base does not change which is obviously wrong. blow-ups at the non-minimal point at once, and not blow-up after blow-up. Subsequently, we search for non-minimal points along the newly introduced exceptional curves and, if necessary, apply the toric blow-up method also to these points. The procedure stops when we do not find any new non-minimal points anymore.

We next turn to the dual F-theory K3 of the heterotic string. In the defining equation (2.7) the coefficients of in and in are both constant so that non-minimal points along are absent. Thus, again it suffices to work in the patch. A hallmark of this string is that in all NU models there is a singularity along . Moreover, the vanishing degrees of and the monodromy cover along this curve indicate a singular fiber of type supporting an algebra , where is identical to the vanishing order of at . Additionally, in most models there is an enhancement to a non-minimal singularity at the point which in some exceptional cases is shifted to , . To resolve these non-minimal points we proceed as above, i.e. in cycles of introducing base blow-ups and checking for additional non-minimal points.

To be more concrete, let us now briefly review the torically inspired blow-up procedure of [5]. In the first step, we choose local affine coordinates on such that the non-minimal singularity lies at . Then, we expand the sections and in these coordinates,

(3.2)

and collect the minimal exponents and , i.e. the vertices of the Newton polytopes of and which can be connected to the origin without passing through the respective polytopes. Next we search for all toric blow-up [41] directions which are crepant. For the elliptic fibration to remain Calabi-Yau, the blow-up must involve the fiber coordinates and too, i.e.

(3.3)

Hence, the canonical class of the ambient variety after the blow-up is given by times the last column in the weight table

(3.4)

where is the divisor class of the exceptional divisor and we have dropped primes to simplify notation. Imposing that the resolution of the Weierstraß equation has to be crepant implies that must factor off the hypersurface equation (3.1) when its proper transform is taken after applying (3.3). This amounts then to the constraints

(3.5)

which must be fulfilled for all and . If the constraints (3.5) are fulfilled for the minimal exponents then all the remaining ’s and ’s fulfill them trivially.

The set of toric blow-ups that need to be introduced consists of the solutions to the inequalities (3.5) which have coprime entries. Knowing the it is straightforward to compute the vanishing orders of , denoted , and , along the corresponding exceptional divisors. This data determines the fiber type at the degeneration [40]. For the reader’s convenience, we reproduced the Kodaira classification in Table 1. The gauge algebra supported on each divisor is uniquely identified analyzing the monodromy covers [42].

type singularity gauge algebra monodromy
0
0 0 1
0 0 or
1 2 cusp
1 3
2 4 or
6 or or
2 3 or
4 8 or
3 9
5 10
Table 1: Kodaira classification of degenerate elliptic fibers.

After this toric resolution step we still have to check whether there are no non-minimal points along the exceptional curves just introduced. If there are any of them, we must repeat the resolution procedure, which we just described, at these points. The process of resolving and checking stops when all non-Kodaira type singularities have been removed.

As realized in [5], there might be cases when the resolution cannot be accomplished. This occurs when there is an infinite number of solutions to (3.5). Moreover, it can be shown that the set of blow-ups is finite if and only if or or or .

To obtain the self-intersection numbers of the (blow-up) divisors in the base, it is too naïve to take the respective toric resolution and calculate from

(3.6)

where the ’s are the lattice vectors corresponding to blow-up divisors. The reason for this is that when we do the cycles of toric resolutions and checking for non-minimal points we change the self-intersection number of the (toric) divisors on which we still find non-minimal points after the toric resolution, because we have to blow-up these points in the next cycle. The self-intersection of the divisor changes by for each non-minimal point which lies on that divisor and we have to blow-up. Note that this is not only true for the blow-up divisors but also for the rational curve at . Although it has self-intersection number in the beginning, its self-intersection number becomes after resolving the non-minimal point at ,555Since the divisor is non-compact, we cannot define a self-intersection number and, therefore, no change in it. cf. Figure 2.

Figure 2: Schematic drawing of the toric resolutions of the singularity, cf. Section 4.4. The resolution starts on the left hand side with the base blow-up of . After this blow-up there are two non-minimal points in the --patch which are indicated by the two crosses. Both singularities lie along but non-vanishing. Therefore, we have to do a coordinate redefinition such that the singularities lie at and , respectively, to apply again our toric machinery. These cycles of toric resolution and searching for non-minimal points continue in the obvious way until all non-minimal singularities are removed.

3.2 Geometric Models: Small Instantons on ADE Singularities

In this section we discuss resolutions of models which on the genus-two side have a NU degeneration , and , with [11]. Here we use the notation . These models are expected to correspond to heterotic compactifications with small instantons sitting at ADE singularities based on the monodromy action on the moduli and the Bianchi identity . For example, in the model the monodromy is

(3.7)

When the Wilson line value is turned off, , whereas the monodromy in is precisely that of a Kodaira type fiber of the fibration. Indeed, shortly we will see that the model describes instantons on an singularity.

As explained in Section 2.2, the starting point is the genus-two model given in the NU classification. The next step is to compute the Igusa-Clebsch invariants that determine the and coefficients entering in the dual K3 on the F-theory side. In Table 2 we collect the defining equations of the ADE NU models together with the vanishing degrees at of the coefficients and . From the latter we can infer the behavior of the functions and , cf. (2.6), which control the loci of singularities and small instantons on the heterotic side [19, 20, 21]. In particular, it follows that there are small instantons on top of the -type singularity, where

(3.8)

is precisely the vanishing degree of at .

sing. NU type local model
Table 2: Genus-two models for ADE singularities.

On the F-theory side there are non-minimal points which we seek to resolve following the procedure described in Section 3.1. In general the resolution consists of a series of base blow-ups. Each divisor can be characterized by an integer equal to minus its self-intersection number, and by the gauge algebra factor it supports. This algebra is derived from the vanishing orders of along the blow-up divisors, cf. Table 1, and the study of the monodromy covers following the formalism of [42]. In order to determine the matter content it is also important to give the intersection pattern of the blow-ups. All this information can be efficiently found using the toric geometry techniques reviewed in the preceding section.

In the end, to each model admitting a resolution we can associate a full local gauge algebra, denoted , and a number of blow-ups, denoted . In turn counts the massless tensor multiplets [21]. In all models the full algebra and the total number of blow-ups agree with the results obtained originally in [21]. We complete these results by providing the complete pattern of the curves supporting the algebras. In fact, each pattern fits the predictions based on the analysis of the theory of small instantons on singularities, where is the discrete subgroup of associated to the ADE group [23]. More precisely, for a singularity of type the structure of the resolution is dictated by the extended Dynkin diagram of . At the nodes of the diagram, labelled by , there are algebras of type , , or , according to whether the representation of associated to the respective node is real, pseudoreal, or complex. A node with the conjugate of a complex representation does not give a new algebra factor. For example, for an singularity with odd, the resolution has one and factors because there is only one real, together with complex nodes plus their conjugates. The values of the depend on data of the extended diagram. We refer to [23] for details. In particular, it follows that the extended node of the diagram, which is always real, gives an algebra , where is the number of instantons on the singularity. This is just the factor due to the singularity at in the ADE NU degenerations.

The matter content can also be determined from the resolution output and it agrees with the predictions in [23] as well. Concretely, each , , , sitting at a curve of self-intersection , , , respectively, has altogether , , fundamentals. A at a curve of self-intersection has fundamentals plus an extra hypermultiplet in the antisymmetric representation. Also in other cases, it can be shown that the matter necessary for anomaly cancellation is present. For instance, if we take the at a curve of self-intersection , the analysis of the monodromy cover indicates that there is one hypermultiplet in the fundamental and four in the spinor representation.

There is also matter due to intersections of the global with the local base blow-ups and the rational curve at as we now explain. The discriminant (2.8) shows that the fiber () is at whereas the () occurs at the vanishing of . Recall further that from the term there factors off the () at , with . At lowest order in we can write and , where and are some non-zero constants. In the case that , as in the examples in Table 2, the locus of the fiber intersects the singularity at , in the patch . This is in contrast to the case , then the intersection locus will be at where we have to introduce the base blow-ups. As it turns out, the intersection pattern of the -locus with the blow-up divisor is model-dependent. The upshot is that in either case the only intersects the and there is matter at this point. Concerning the locus, for it intersects and there is an additional , whereas for it will intersect one of the divisors introduced by the resolution of the non-minimal singularity at . The two situations are depicted in Figure 3. In all examples, including matter from all intersections, the has altogether hypermultiplets in the fundamental representation.

a)

b)

Figure 3: Schematic representation of blow-ups and intersections of characteristic divisors

We will display the results using a notation such that each blow-up divisor introduced in the resolution is identified by the algebra it supports written above an integer which is equal to minus its self-intersection number. In the heterotic, below the universal , supported along the curve with self-intersection , we will write , adding the asterisk to indicate that is not a blow-up divisor. Besides, adjacent divisors intersect and when necessary this is made clear by drawing an explicit link. Thus, a generic point on the tensor branch of the 6d theory corresponding to a resolvable degeneration will be captured by a tree-like diagram with nodes.

Anomaly cancellation gives significant information about the resulting 6d theories. In all models it happens that the matter content is such that the irreducible gauge quartic anomaly cancels for each gauge factor. Moreover, the remaining pure gauge contribution to the anomaly polynomial takes the form666We use the conventions of [43] for the anomaly polynomial and those of [42] for the traces involved.

(3.9)

Here is the field strength of the gauge factor at the node, with , where refers to , and the so-called adjacency matrix is equal to minus the self-intersection matrix. If there is no algebra at the node we set . The adjacency matrix can be read off from the diagrams representing the theories, for an example see e.g. (3.15). Concretely, the diagonal elements of are the integers under the nodes in the diagram while the off-diagonal elements are or 0 depending on whether the nodes are linked or not. In all models one can check that is positive semi-definite, with only one zero eigenvalue. In consequence, can be cancelled by the Green-Schwarz-Sagnotti mechanism [44, 45] involving just tensor multiplets [23]. The null eigenvalue further implies that a linear combination of gauge couplings is independent of the scalars in the tensor multiplets and therefore it defines a mass parameter.

The existence of a mass scale suggests that the UV completion of the theories arising from the resolutions are little string theories (LSTs) [34]. In fact, the theories that we obtain have appeared in the recent classifications of LSTs [33, 32]. Moreover, dropping the node corresponding to in the diagrams, i.e. deleting the corresponding column and row in , gives the tensor branch of 6d SCFTs embedded in the LSTs [32]. In this case the remains as a flavor symmetry of the 6d SCFTs as observed originally in [23]. In all cases we find that the residual adjacency matrix, denoted , , is positive definite, has determinant one, and further satisfies

(3.10)

This property enters in the computation of the anomaly polynomial of the SCFTs applying the methods developed in [43, 46].

An interesting feature of the theories emerging from the resolution of NU degenerations is that they can be characterized by some quantities that match in the and the heterotic strings. For instance, for a concrete degeneration with resolution , the quantity

(3.11)

can be shown to be the same for both heterotic strings by virtue of duality upon further compactification on a circle [21]. We have found that this indeed occurs, which actually provides an useful check of the results. Moreover, for the particular case of the models in Table 2, corresponding to small instantons on ADE singularities, it turns out that for above a minimum value the resolution satisfies

(3.12)

where is the Coxeter number of the ADE group , given by , for , respectively. This fact was observed in [47].

To each resolution we can assign a second intrinsic quantity that takes the same value for both heterotic strings. Knowing the local algebra and the matter content it is easy to compute the number of vector multiplets given by and the total number of hypermultiplets . The number of tensor multiplets and the instanton number are also inherent properties of the theory derived from the concrete resolution. With this data we define

(3.13)

An indication that depends only on the underlying NU degeneration, so it matches in both heterotic strings, is the fact that in all models corresponding to small instantons on ADE singularities , as pointed out in [48]. One way to derive the relation (3.13) is to consider a global heterotic model constructed as a compactification on K3 with large instantons breaking the gauge group to or [49], plus small instantons on the ADE singularity giving the local theory. Imposing cancellation of the pure gravitational anomaly leads to (3.13).

It is worthwhile to compare the resolutions of the same NU model in both heterotic strings, for instance to check the matching of the quantities and defined above. To this end, we will give in the current section the resolutions in the string too. In the diagrams representing the resulting theories we will also include the divisor with label , but in the string it does not support any gauge algebra. The pure gauge anomaly in the resulting theories again takes the form (3.9). In all cases the self-intersection matrix is positive semi-definite with a single null eigenvalue. Hence, also these theories potentially complete to LSTs in the UV. Similar claims have been made in [32] for the theories associated to small instantons on ADE singularities that we consider in this section. Notice that T-duality upon circle compactification, reflected in the double fibration structure of the F-theory duals, requires that the resulting theories in both heterotic strings be LSTs [32]. Again, dropping the node gives the tensor branch description of 6d SCFTs embedded in the LSTs. This is the situation which was implicitly assumed in [5].

Below we will present the resolutions of three examples of Table 2 which are relevant for the ensuing discussion. The remaining models can be found in Appendix A.

3.2.1 Model and Singularity

The number of small instantons on the singularity is . For the resolution in the heterotic string gives

(3.14)

A systematic analysis reveals that at both the leftmost and rightmost divisors with singular type fiber, supporting algebra , there is one additional non-minimal point which requires an extra blow-up with fiber and hence no algebra. In [5] these divisors were reported with self-intersection . However, it is understood that a single curve with algebra and self-intersection comes with one small instanton [31]. The resolution shown in (3.2.1) makes this explicit. Similarly, one can readily verify that , since the only matter are hypermultiplets transforming as in each cluster.

In the heterotic string, for , , we obtain the resolutions

(3.15)

Notice that the structure of the intersections mimics the extended Dynkin diagram of , in agreement with the analysis of [23]. The algebra factor arises from the singularity at . The total number of base blow-ups is and the rank of the full algebra is such that , which is also the value obtained for the resolution in (3.2.1). It is also straightforward to check that because the matter hypermultiplets comprise