On the Frozen F-theory Landscape

We study $6$d $\mathcal{N} = (1,0)$ supergravity theories arising in the frozen phase of F-theory. For each of the known global models, we construct an F-theory compactification in the unfrozen phase with an identical non-abelian gauge algebra and massless matter content. Two such low energy effective theories are then distinguished through gauge enhancements in moduli space. We study potentially new global models obtained via compact embeddings of a plethora of $6$d $\mathcal{N}= (1,0)$ superconformal field theories and little string theories constructed using frozen $7$-branes. In some cases, these provably do not exist, and in other cases, we explicitly construct a compact embedding, yielding $6$d supergravity theories with new massless spectra. Finally, by using gravitational anomaly cancellation, we conjecture the existence of localized neutral hypermultiplets along frozen $7$-brane loci.


Introduction
The landscape of F-theory compactifications [1][2][3] on elliptic Calabi-Yau manifolds has given rise to the largest class of supersymmetric string vacua to date.Compactifications yielding 8d N = 1 vacua correspond to elliptic K3-surfaces, which are completely classified.On the other hand, 4d N = 1 vacua correspond to the choice of an elliptic Calabi-Yau fourfold, together with the specification of fluxes and D3-brane data.In stark contrast, our understanding of such vacua is largely bottom-up; there exists only algorithmic and computational approaches towards classifying the bases of such fourfolds, together with recent mathematical results on boundedness [4].Moreover, in such ensembles, we have control only on the coarsest aspects of the physics, such as the gauge symmetry.Finally, 6d N = 1 vacua occupy a rather rich middle ground; there is a sharp link between the physics and the geometry of an elliptic Calabi-Yau threefold.The classification of surface bases is significantly more tractable, the massless content of the corresponding supergravity theories can be completely determined, and there is a rich, growing interplay between swampland constraints and the known boundaries of the landscape.
Nevertheless, there are known string compactifications in other duality frames which involve intricate non-geometric data, even in higher dimensions.For example, in an M-theory background on a C 2 /Γ singularity, an additional C 3 -field period can be specified on the boundary, leading to frozen singularities [5] with a restricted deformation space.The classification of compact backgrounds with such singularities and their low energy effective physics have largely been ignored, due in part to a lack of understanding of frozen singularities, as well as the fact that such vacua are rather special and are somewhat insignificant when searching for generic phenomena in the landscape.
A resurgence in their study was initiated in [6], which demonstrated that the only new supersymmetric defect in F-theory from frozen singularities is a strongly coupled version of the orientifold plane, O7 + , with positive D7-brane charge.The study of 8d F-theory vacua with O7 +planes first appeared in [7], and a systematic classification of such compact models was investigated more recently in [8].The effects of incorporating such defects in 8 dimensions are markedly simple; they lead to either a rank 4 or a rank 12 vector multiplet moduli space, reduced from the usual 20 dimensional space of vacua.In addition, they give rise to sympletic (sp(n)) gauge algebras, which does not occur in conventional 8d F-theory compactifications.
The study of F-theory compactifications to 6-dimensions with O7 + -planes was initiated in [9], and a number of consistent compact and non-compact constructions [10] were given based on dualities to other frames.On the other hand, the complete list of conditions that are necessary and sufficient to specify a frozen 6d F-theory vacua and the resulting supergravity spectrum is still unknown.In this paper, we take a first step towards understanding the massless spectra of frozen F-theory compactifications to six dimensions.Our aim is to identify a proxy to distinguish the frozen phase from the unfrozen phase of 6d F-theory purely from the low energy spectrum, to construct compact embeddings of all known local frozen models, and to compare the resulting supergravity spectrum with the conventional 6d F-theory landscape.In the process, we will demonstrate the existence of 6d supergravities with new massless spectra arising from the frozen phase.
We emphasize that this is merely a first step towards understanding 6d F-theory compact-ifications with frozen 7-branes.The ultimate goal in this direction is the compilation of precise necessary and sufficient conditions for specifying such compactifications, which would lead to a systematic construction of frozen vacua.It would also be desirable to investigate phase transitions among such models, and to make precise the relation with other interpretations of frozen vacua as appearing in [11] through the specification of B-field fluxes.

Summary of Results
In this paper, we study the massless supergravity spectrum in the frozen phase of F-theory.We emphasize that there are no known necessary and sufficient conditions guaranteeing the existence of a given F-theory model with frozen 7-branes.Nevertheless, we will study a plethora of both local and global constructions, whose existence is strongly evidenced either through dualities or through a weakly coupled type IIB limit.
As explained above, a critical new feature in six dimensions is that the frozen and unfrozen phases of F-theory are, a priori, virtually indistinguishable at the level of the low energy effective physics.This leads us to the central question of study in this paper: To what extent can we distinguish between the frozen and unfrozen phases of F-theory from the non-abelian gauge sector and massless matter in the effective 6d N = (1, 0) supergravity theories?1. Section 3: For each known frozen 6d F-theory model, we explicitly construct an unfrozen 6d F-theory model with an identical massless spectrum.We give a criterion for distinguishing a frozen and an unfrozen supergravity theory with identical massless spectra by gauge enhancements in moduli space.

Section 4:
We prove that most of the known new constructions of superconformal theories using frozen 7-branes do not admit a compact embedding.In the remaining special cases, we explicitly construct a compact embedding.As a consequence, we construct compact 6d F-theory vacua with new massless spectra in the frozen phase.

Section 5:
We conjecture the existence of localized neutral hypermultiplets along intersections of residual I 1 -loci with frozen 7-branes.
Our bottom-up analysis of the known constructions in the frozen phase of F-theory lead us to the following statements, which comprise the main results of this paper.We first study the known compact frozen F-theory models, whether they can be realized in the unfrozen phase, and what criteria are sufficient to distinguish between the two supergravity theories.
Claim 3.1.Let T 1 , T 2 be two 6d N = (1, 0) supergravity theories with the same massless spectrum, engineered via a frozen/unfrozen F-theory compactification respectively.Assume that there exists an sp(n) factor in both theories, which is localized on a frozen 7-brane in the case of T 1 .
Then T 2 admits a limit in moduli space enhancing sp(n) to so(2n), while the analogous limit for T 1 realizes a (4, 6)-divisor.
The existence of a (4, 6)-divisor corresponds to an infinite distance limit, often realizing a decompactification of the effective 6d theory.We note that such limits have been analyzed and sharpened in the frozen phase of F-theory in 8 dimensions in [8].
A natural question is whether the massless spectrum of a 6d supergravity theory realized in the frozen phase of F-theory, can be realized via an F-theory compactification in the unfrozen phase as well.Given the wealth of constructions in [10], we answer this question.Claim 4.1.Most, but not all of the new superconformal field theories and little string theories exhibited in [10] do not admit a compact embedding.
Nevertheless there exists compact frozen F-theory compactifications whose massless spectrum cannot be realized in the unfrozen phase of F-theory.
Finally, based on our study of compact examples together with gravitational anomaly cancellation, we make the following conjecture.
Conjecture 5.1.Consider a 6d F-theory compactification with a frozen 7-brane localized along a divisor D with residual discriminant ∆.Then there exists at least 1 2 D • ∆ localized neutral hypermultiplets.
We conclude that the presence of O7 + -planes in 6d F-theory compactifications lead to a number of implications for the low energy effective physics.Most notably, as demonstrated in Claim 3.1, these lead to a certain obstruction for gauge enhancement in comparison with the unfrozen phase of F-theory, and this will be explored further in future work.In addition, we conjecture that the presence of such orientifold planes could potentially be a new source of localized neutral hypermultiplets, and it would be interesting to investigate this by establishing a weakly coupled type IIB limit in the frozen phase.

Consistency conditions for frozen F-theory models
In this section, we review the necessary conditions in specifying a compact, frozen 6d F-theory model, first appearing in [9].In Section 2.1, we review the basics of 6d F-theory compactifications, emphasizing a perspective which generalizes well in defining compactifications with frozen 7-branes.In Sections 2.2 and 2.2.1, we review the basics of frozen singularities, their realization in F-theory models, and present an example.

6d F-theory compactifications
In this section, we review the basics of F-theory compactifications to six dimensions.
An F-theory compactification to six dimensions is specified by the choice of an elliptically fibered Calabi-Yau threefold π : X → B over a smooth complex algebraic surface B. Up to a sequence of birational transformations [12], this data can be specified by a Weierstrass equation where f and g are global sections of −4K B and −6K B respectively, with −K B denoting the anti-canonical divisor on B. The torus fiber degenerates along the algebraic curve ∆ ⊂ B, often called the discriminant locus, and in generic situations, the discriminant decomposes into a union of smooth irreducible algebraic curves ∆ = ∆ 1 ∪ . . .∪ ∆ n .In particular, loops around the codimension-1 irreducible components ∆ i correspond to elements in SL(2, Z), which naturally induce a monodromy action on the 1-cycles of the torus fiber.The low energy effective physics is specified by a 6d N = (1, 0) supergravity theory, and such theories occupy a particularly rich corner of the string landscape.On one hand, they are strongly constrained by gauge, gravitational, and mixed anomalies, and on the other hand, even the possible combinations of non-abelian gauge algebras and massless matter content has evaded a complete classification.Our knowledge of the possible supergravity theories resulting from string compactifications has progressed in tandem with our understanding of elliptic Calabi-Yau threefolds, whose singular limits have not been completely classified.We will not attempt to summarize the various constraints on 6d N = (1, 0) supergravity theories and we point to the vast literature [13][14][15][16][17][18][19][20][21][22] for a comprehensive treatment.
The details of the effective 6d N = (1, 0) supergravity theory rely crucially on the singularities of the total space X.Indeed, codimension-1 irreducible components ∆ i , specify the location of stacks of generalized (p, q) 7-branes, which can support non-abelian gauge algebras.At codimension-2, pairwise intersections of the discriminant components ∆ i ∩ ∆ j support localized modes corresponding to jointly charged matter.The precise non-abelian gauge group can be deduced from the standard classification of Kodaira-singular fibers together with an analysis of their global monodromies, and we refer to [23, Tables 1, 2] for details.The precise counting of jointly charged hypermultiplets is somewhat more subtle and depends on details of the intersections, and we refer to [24] for details.
In general, the gauge group and matter content of a 6d F-theory compactification is often studied via M-theory compactified on a crepant resolution X → X, and then inferred by taking the singular limit.Motivated by the non-existence of crepant resolutions in certain examples as well as an intrinsic study of the gauge theory at the singular point, a theme [25,26] has emerged which aims to develop techniques to study F-theory directly on X.In particular, the 6-dimensional physics is sensitive only to the associated Weierstrass model (2.1), and not to the details of the total space X.This is exemplified through the fact that the SL(2, Z)-monodromy associated to an irreducible component of the discriminant ∆ i admits a (unique up to conjugation) factorization [25, Section 3.2] and [27]: Put differently, every discriminant component ∆ i can be decomposed uniquely into a combination of (p, q) 7-branes and we will use this perspective in our analysis of the frozen phase of F-theory.

Frozen 7-branes
A somewhat less understood class of singularities in M-theory arise from singularities of the form C 2 /Γ, together with the specification of a fractional C 3 -field period on a boundary 3-sphere.These so-called frozen singularities [5] have a restricted deformation space and exhibit a smaller gauge algebra than expected from the singularity itself.
Though understood for quite some time [28], O7 + -planes (orientifold planes with positive RRcharge), were demonstrated in [6] to be the only class of frozen singularities arising in F-theory.This suggests that O7 + -planes are the only remaining class of supersymmetric defects in F-theory compactifications, and motivates a more systematic study of their effects on compactifications to lower dimensions.We briefly review their critical properties, summarizing the similarities and differences with the usual combinations of (p, q) 7-branes, and we refer to [9] for details.
A single O7 + -plane shares a number of similarities with a configuration consisting of a single O7 − -plane parallel to 8 D7-branes.As explored at length in [7], both exhibit +4 units of D7-brane charge, and a D3-brane encircling such defects exhibit a monodromy SL(2, Z)-conjugate to that of an I * 4 -fiber: Nevertheless, there are a number of critical differences; for one, an O7 + -plane on top of n D7brane yields an sp(n) gauge symmetry, while an O7 − -plane on top of n + 4 D7-branes yields an so(2n + 8) gauge symmetry.As a consequence, a single D3-brane on top of a single O7 + -plane may fractionate into two, which does not happen in the latter case, due to a factor of two in the relative Chan-Paton indices.Moreover, a D3-brane colliding with an O7 + -plane yields an O(2)-gauge theory with two charge 2 massless hypermultiplets, while a collision with the latter configuration yields SU (2) N f = 8 Seiberg-Witten theory.These observations are in line with the results of [8], which concluded that (p, q)-strings must end on an O7 + -plane with both even p and q charges in contrast to the latter case, where p and q can take arbitrary integer values.

F-theory constructions
As emphasized in Section 2.1, we will specify an F-theory construction through the choice of a smooth algebraic surface B, together with the location of (p, q) 7-branes localized along curves in B. Similarly, an F-theory construction in the frozen phase will be specified through the same information, together with the choice of replacing a stack of (p, q) 7-branes with the monodromy of an I * n+4 -fiber by an O7 + -plane with n D7-branes.Following [9], we will denote the latter singularity with I * n+4 .On the other hand, as argued via duality in [9], such an F-theory construction does not admit a canonical assignment of divisors supporting individual gauge algebras.In this section, we will review the necessary conditions in specifying a frozen 6d F-theory compactification, but we emphasize that these are not sufficient.Throughout this section, we will fix a smooth compact algebraic surface B with canonical divisor K, and Weierstrass polynomials f and g.The associated discriminant ∆ contains irreducible components ∆ a , with corresponding monodromies M a .
A 6d F-theory compactification in the frozen phase is specified by the following data: (a) For each irreducible component ∆ a with monodromy matrix M a conjugate to M I * n , the choice of an unfrozen/frozen 7-brane, designated by I * n / I * n respectively.We denote by F , the sum of irreducible divisors D a supporting a frozen 7-brane.
(b) A collection of gauge algebras g i , together with disjoint embeddings ρ i,a : We define the associated gauge divisor by where µ i,a = 0 if ρ i,a = 0, and is 1 if ρ i,a is non-trivial.The coefficient o i,a is the Dynkin index of the embedding ρ i,a and we refer to [29] for a precise definition.By invoking the Green-Schwarz mechanism in the presence of O7 + -planes, one finds the following modified anomaly polynomial where p 1 (T ) is the pontryagin class of the tangent bundle of B and ν(F i ) is the instanton number density of the field strength F i valued in g i .This agrees with the usual anomaly polynomial when F = 0 and strongly suggests replacing the usual assignment of matter content with the simple substitution K → K + F which leads to the following additional consistency conditions.We will demand that the following must hold to define a consistent frozen F-theory configuration: 3. For each pair (g i , Σ i ), the total number of all localized charged hypermultiplets must add up to the total number of hypermultiplets of corresponding representation, dictated by [9, Table 3.1].
Roughly, the first three conditions correspond to the cancellation of gauge anomalies, while the last imposes gravitational anomaly cancellation.We will assume that all intersections of gauge divisors are transversal, each of which hosts a single bi-fundamental, aside from so − sp intersections which host a half bi-fundamental.
We briefly summarize the Aspinwall-Gross model, which first appeared in [30], and admits a frozen variant described in [9].Consider the base B = F 4 , with Weierstrass polynomials f = e 2 f , g = e 3 g, ∆ = e 18 q 48 ∆ where f , g, ∆ are generic members of the prescribed class.Moreover, e denotes the exceptional divisor, and q denotes the fiber class of B. We note that a concrete tuning realizing such polynomials on B can be found in [30].
In the conventional F-theory phase, e supports an I * 12 -singularity with an so(32)-algebra, q supports an I ns 48 -singularity with an sp(24)-algebra, and there is a single bi-fundamental localized at the intersection.Following the above prescription, we now assume that e supports a frozen I * 12 -singularity.In addition, we define the following frozen and gauge divisors where Σ 1 supports an sp(8) gauge algebra, and Σ 2 supports a su(24) gauge algebra.We note that the canonical embedding su(24) − → su( 48) is an embedding of index 2, contributing a factor of 2 in definition of Σ 2 .Finally we note that anomaly cancellation implies the following which gives 1 bifundamental of sp(8) × su( 24) and 2 antisymmetrics of su (24).We note that the gauge group appearing on the fiber class q in this compact case, is different from the expected gauge group of transverse D7-branes intersecting an O7 + -plane.

Frozen models via ordinary F-theory embeddings
In this section, we study in detail the massless spectrum of the compact models constructed in the frozen phase of F-theory in [9].These models admit alternative constructions via duality to either the heterotic or type I string, and thus are particularly credible examples with which to explore consistency conditions for constructing frozen F-theory compactifications.In order to begin identifying universal features appearing in the frozen phase, for every such example, we will explicitly produce an unfrozen F-theory model which engineers a 6d N = (1, 0) supergravity theory with precisely the same massless content.For simplicity, we will ignore potential contributions from u(1) factors, as well as the global forms of the gauge group.
In Section 3.2.1,we will construct an F-theory compactification on F 1 reproducing the Aspinwall-Gross model with a frozen 7-brane on F 4 .In Section 3.2.2,we will perform a similar construction for the frozen compact P 1 × P 1 examples.Such constructions are necessarily more involved, and will proceed in two steps: 1. We first tune the required gauge algebras on curves with the correct mutual intersections on P 2 using a Tate model.

2.
We then demonstrate that there exists blowups to a compact base with the correct number of tensor multiplets and matter content.
In Section 3.3, we identify a criterion to distinguish between unfrozen and frozen F-theory compactifications realizing the same massless spectrum.

Tate forms and (4, 6)-points
In this section, we briefly review a more generalized form of the Weierstrass equation (2.1) in engineering elliptic Calabi-Yau threefolds.The main utility of such constructions is that it allows us to engineer rather degenerate singular configurations using the results of [31].
In general, one can construct an elliptic Calabi-Yau threefold using a Tate model: where the above coefficients are global sections of appropriate powers of the anti-canonical class on B, in particular, a n ∈ H 0 (X, O(−nK B )).Using [31, Tables 2, 3], we can easily engineer Kodaira I n and I * m -fiber types which are critical in constructing frozen singularities.In order to interpolate between the Tate and Weierstrass form, we will rely on the results described in [32].The Weierstrass polynomials corresponding to the Tate form (3.1) are given by the following The associated discriminant ∆ can then be readily computed using (2.1).
In practice, we will often engineer the desired singularities using a Tate form, and then achieve the desired model through a sequence of phase transitions by blowing up the base of the elliptic fibration.In order to preserve the Calabi-Yau condition, it is well understood that one needs to first engineer a point p ∈ B such that the orders of vanishing of the Weierstrass polynomials satisfy (ord p (f ), ord p (g), ord p (∆)) ≥ (4, 6, 12).We may then blow up B at p and together with a change of variables, this yields a Calabi-Yau threefold over the blowup.Physically, such orders of vanishing indicate a conformal matter point, and blowing up corresponds to moving out on the tensor branch of the theory.For details, we refer to [33, Appendix A].

Aspinwall-Gross model
We will take the base surface of the F-theory compactification to be the Hirzebruch surface F 4 with a unique curve of self-intersection −4.In [9, Section 5.1], a frozen variant of the F-theory compactification [34] on F 4 with a tuned I * 12 fiber along the zero section was given.The massless spectrum consists of the following gauge algebra sp(8) uncharged hypermultiplets 32 tensor multiplets 1 Table 1: Massless matter content of the generic Aspinwall-Gross model in the frozen phase.Throughout, we will omit the supergravity multiplet.
We will reproduce the same massless content with a conventional F-theory compactification with base surface B = F 1 , i.e. the Hirzebruch surface with a unique curve of self-intersection −1.
We first consider the following tuning with a Tate model where the p i 's denote generic representatives of the corresponding global section.This particular tuning implies that we must have p 4 ∈ |12f |.This leads to the following Weierstrass model From the form of the discriminant ∆, one verifies straightforwardly that this indeed reproduces the desired massless spectrum.

Compact P 1 × P 1 with one frozen
In this section, we construct an unfrozen F-theory model reproducing the same massless content as that of the P 1 × P 1 model with a single frozen 7-brane exhibited in [9, Section 5.3].At a perturbative level, these coincide with the models of [35,36], obtained by compactifying type IIB on T 2 /Z 2 × T 2 /Z 2 , and flipping a single O7 − to an O7 + -plane.The model we consider is obtained from the Higgs branch of an F-theory realization of such a compactification.The massless spectrum for this model is given in In a conventional F-theory compactification, anomaly cancellation then implies that su(8) must be supported on a (0)-curve and sp(4) i must be supported on (−1) curves, with all three intersecting pairwise.We will reproduce this massless content with a base surface B = P 2 .
We first consider the following tuning with a Tate model on P 2 where h 1 , h 2 , h 3 ∈ |h| are general lines.From the degrees of the Tate coefficients, p 1 must be a degree 3 polynomial, and q is a degree 2 polynomial.We may choose polynomials p 1 and q such that they vanish simultaneously at a single point u on the line h 1 , two points v 1 , v 2 on h 2 , and two points w 1 , w 2 on h 3 .Moreover, we may choose such points away from the pairwise intersections Using the ansatz given in equation (3.2), this leads to the following Weierstrass model In particular, one verifies straightforwardly that this realizes an su(8) algebra supported on h 1 and sp(4) algebras supported on h 2 , h 3 , with all three intersecting pairwise.
Next, we perform the required blowups to arrive at a model with precisely 5 tensor multiplets.From the choices of p 1 and q, as well as the above Weierstrass model, we note that the five points u, v 1 , v 2 , w 1 , w 2 are all (4, 6)-points.Blowing up all five points, we conclude that the exceptional divisors do not support any additional gauge algebras, and that this gives the desired self-intersections required by anomaly cancellation.

Compact P 1 × P 1 with two frozen
Finally, in this section, we construct an unfrozen F-theory model reproducing the same massless content as that of the P 1 × P 1 model with two frozen 7-branes.
The massless spectrum now consists of the following gauge algebra so(8) ⊕ tensor multiplets 8 Table 3: Massless content of a frozen F-theory model on P 1 × P 1   We note that anomaly cancellation requires that the so(8) factor is supported on a (−4)-curve, while the sp(2) factors are all supported on (−1)-curves.
We consider the following tuning with a Tate model, again on P 2 where again, h 1 , h 2 , h 3 , h 4 ∈ |h| are four general lines on P 2 .Moreover, again from the Tate coefficients, q and p 2 are general quadrics.Converting to a Weierstrass model, we obtain the following it follows straightforwardly that this realizes an sp(2) factor on each line h i all intersecting pairwise, as well as an so(8) factor supported on the quadric q.
As in Section 3.2.2,we perform the required blowups to arrive at a model with 8 tensor multiplets.We note from the form of the Weierstrass model, that the 8 points of intersections q ∩ h i for i = 1, 2, 3, 4 are all (4, 6)-points.Blowing up all these points, we again conclude that the exceptional divisors do not support any additional gauge algebras, and that this gives the desired self-intersections required by anomaly cancellation.

Distinguishing frozen from unfrozen supergravity theories
Our goal in Section 3.2 was to construct, for every compact model of [9], an F-theory compactification without O7 + -planes realizing the same massless spectrum.As a consequence, we conclude that for such models, it is a priori impossible to distinguish between the frozen and unfrozen phases purely from the non-abelian gauge algebras and massless matter content in the low energy effective physics.
In this section, we will use our concrete F-theory models to propose an answer to Question 1.1 for 6d F-theory compactifications.We will exploit the distinction in the moduli spaces of two Ftheory compactifications with the same massless spectrum, with one in the unfrozen phase, and with the other in the frozen phase.Our main claim is the following: Claim 3.1.Let T 1 , T 2 be two 6d N = (1, 0) supergravity theories with the same massless spectrum, engineered via a frozen/unfrozen F-theory compactification respectively.Assume that there exists an sp(n) factor in both theories, which is localized on a frozen 7-brane in the case of T 1 .
Then T 2 admits a limit in moduli space enhancing sp(n) to so(4n), while the analogous limit for T 1 realizes a (4, 6)-divisor.
For simplicity, we will illustrate the claim in the example of Section 3.2.1, but we note that this can also be done for the examples computed in Sections 3.2.2 and 3.2.3.
In Section 3.2.1,we considered the following Tate model on the base B = F 1 .In particular, we note that the I ns 16 fiber of Section 3.2.1 admits an enhancement to an I * 12 fiber localized along the exceptional curve e of F 1 , leading to an so(32) algebra supported on e.
We now illustrate that a completely analogous enhancement in the frozen realization of the Aspinwall-Gross model leads to a (4, 6) divisor, and hence an infinite distance limit [37] in moduli space.We recall that the frozen Aspinwall-Gross model is given by the following Tate model on the base B = F 4 , as considered at the end of Section 2.2.1: a 1 = 0, a 2 ∼ ep 2 , a 3 = 0, a 4 ∼ e 8 q 48 , a 6 = 0 which leads to the Weierstrass model: e 3 p 2 (−2p 2 2 + 9e 6 q 24 ) ∆ = e 18 q 48 (−p 2 + 2e 3 q 12 )(p 2 + 2e 3 q 12 ) We demand that e supports an I * 12 -fiber, and hence supports a frozen 7-brane.Subject to the prescription of Section 2.2.1, we define the gauge divisors which leads to the massless spectrum of Section 3.2.1 in the limit that all the sp(1) singular fibers collide.
We now consider an analogous enhancement as performed in the unfrozen phase with the same massless spectrum; namely, we attempt to tune an additional orientifold onto the exceptional curve.This can be understood as attempting to tune the mass of open string states between the 7-brane on the exceptional curve and the 7-brane on the residual I 1 to zero.In the low energy effective physics, there is an identification of unlocalized neutral hypermultiplets between the two supergravity theories; the two tunings can then be identified with giving non-zero vacuum expectation values to the corresponding massless multiplets.
Geometrically, this corresponds to the following replacement: which leads to the Weierstrass model: e 6 p 2 (−2p 2 2 + 9e 4 q 24 ) ∆ = e 20 q 48 (−p 2 + 2e 2 q 12 )(p 2 + 2e 2 q 12 ) Clearly, this induces a (4, 6) divisor on the exceptional curve e, signaling an infinite distance limit.Such limits were first analyzed in [8], which demonstrated how infinite distance limits can be realized in 8d N = 1 vacua through affinizations of the gauge algebra using string junctions.The appearance of (4, 6)-points in 8d F-theory lead to infinite dimensional Kac-Moody algebras, which result in decompactifications to either 9 or 10 dimensions.The analogous statements in 8d F-theory with frozen singularities can then be obtained by embedding the O7 + -plane into the 9d affine algebras.By studying our model in 6-dimensions adiabatically, we expect that our infinite distance limit corresponds to the 9d CHL vacua compactified on a P 1 with an appropriate twist.

Frozen LSTs, SCFTs and the Swampland
In this section, we investigate whether 6d N = (1, 0) supergravity theories which have not been constructed in ordinary F-theory, can be realized in the frozen phase of F-theory.Our main source of examples stems from a compilation [10] of local embeddings of new little string theories and superconformal field theories using frozen 7-branes.Our goal is to study whether such local models can be completed into a compact F-theory model with frozen 7-branes.
For unfrozen 6d superconformal field theories, only a number of top-down constructions and constraints based on global symmetries [38,39] are known for the existence of compact embeddings.Nevertheless, we emphasize that for the new SCFT's realized in the frozen phase, most of the models do not admit a compact embedding based purely on their intersection constraints, independent of the ranks of the individual gauge algebras.
The gauge theory sectors summarized in [10, Section 3.1] which do not admit a conventional F-theory embedding are given by the following: 1. su(n) with 1 symmetric hypermultiplet and n − 8 fundamental hypermultiplets 2. su(n) with 2n fundamental hypermultiplets, with a subset transforming under so(m) 3. sp(n) with 2n + 8 fundamental hypermultiplets, with 3 subsets transforming under three factors so(m 1 ), so(m 2 ), so(m 3 ) Our central claim can be summarized as follows: Claim 4.1.Case (1) never admits a compact embedding, but cases (2), (3) do admit compact embeddings in frozen F-theory models In Section 4.1, we review our conventions for specifying gauge theory chains, and we summarize a critical Lemma.In Section 4.2, we use the Kodaira condition to demonstrate that most of the above local embeddings can not be completed into a global model.In Section 4.3, we demonstrate the existence of new 6d N = (1, 0) supergravity theories via explicit constructions with frozen 7-branes in F-theory.In the process, we explicitly realize an argument [40] exploiting unitarity of string worldsheet theories in a compact F-theory model, which may be of independent mathematical interest.

6d N = (1, 0) theories
We will primarily be interested in whether certain 6d N = 1 gauge theory sectors can be completed into supergravity models.In this section, we review our conventions and our central argument.
Let B be a smooth complex algebraic surface.We will specify a local gauge theory sector through its non-abelian gauge algebra and matter content.This will be portrayed through intersections between singular Kodaira fibers yielding non-abelian gauge algebras with diagrams of the following type where K i denote the associated Kodaira type, n i denotes the negative of the self-intersection of the curve on which the fiber is supported, a line between two Kodaira fibers indicates that corresponding curves intersect transversely in B, and C and D denote the curves of support.If such a gauge theory sector indeed admits an F-theory embedding, then it must satisfy the following necessary condition.
Lemma 4.1.Let ∆ ⊂ B be the discriminant locus of an elliptic fibration, admitting a factorization into irreducible components

F-theory swampland
In this section, we prove that most of the new superconformal field theories (SCFTs) and little string theories (LSTs) engineered via frozen 7-branes in [10] do not admit a compact embedding.Our main technique will be to analyze the intersecting Kodaira fibers from [10], focusing on particular sub-blocks of such constructions.We begin by assuming that there exists an embedding into a compact, frozen F-theory model subject to the rules of [9].In particular, when each Î * 4+nfiber is flipped to an I * 4+n -fiber, the resulting configuration should still be a consistent 6d F-theory model.Our main technique will be to analyze the contradictions in the resulting model by applying Lemma 4.1.
In the following three sections, we will analyze three particular sub-blocks which feature in every new frozen 7-brane construction in [10].We will find that most of these sub-blocks do not admit a compact embedding.

Tangential Intersections
We begin by demonstrating that a gauge theory sector which engineers an su(n) gauge theory with 1 symmetric tensor and n − 8 fundamentals does not admit a compact embedding.We remark that this was demonstrated earlier in [40], though such arguments are not a priori, obviously independent of the choice of basis for the anomaly lattice.On the other hand, our argument is geometric and assumes an F-theory realization, and so we view our results as complementary.
We engineer this gauge theory using the following configuration of singular fibers t where the double line denotes a tangential intersection.Subject to the rules of Section 2.2.1, there is a single gauge divisor Σ = C supporting an su(n) gauge algebra.In particular, this configuration leads to an su(n) gauge theory with 1 symmetric and n − 8 fundamentals.We claim that the above configuration cannot be realized on any compact algebraic surface B, but for a rather subtle reason.Indeed, let C, D ⊂ B be (−4) and (−1) curves respectively, intersecting tangentially at a single point.Consider a contraction π : B → B 0 , blowing down the (−1)-curve D. Then π(C) ⊂ B 0 is a cuspidal curve with self-intersection 0. In particular, its corresponding linear system induces a morphism |π(C)| : B 0 → P 1 , with generic fiber a smooth elliptic curve.On the other hand, any elliptic ruled surface giving a 6d (1, 0) F-theory model must be a rational elliptic surface, and we claim that this cannot be the case.
To see the claim, we note that the anti-canonical bundle of B 0 satisfies −K B 0 = q, with q the fiber class.Let f 0 denote the cuspidal curve π(C).Then the induced Weierstrass model on B 0 must satisfy the following f = f 2 0 f , g = f 3 0 g, ∆ = f 10 0 ∆ where the vanishing loci of f , g, ∆ are localized on distinct fibers of B 0 not intersecting f 0 .Blowing up the singular point on f 0 with exceptional divisor e, the proper transform is given by the class f 0 = π * f 0 − 2e.We claim that the singular fiber localized on the exceptional divisor e must be non-split.
Indeed, the splitness condition is given by checking the quantity where f 0 denotes the proper transform of the class on the blowup.On the other hand, as the vanishing loci of g, f correspond to distinct fibers on B 0 , their proper transform on B evaluated along e = 0 cannot contain any powers of the proper transform f 0 .Thus we conclude that this quantity cannot factorize, and the singular fiber localized on the (−1)-curve must be non-split.We further note that a compact engineering of such a model is possible in the case n = 8 realizing the following intersection. t We next consider case (2) of Claim 4.1.A simple example of a configuration of singular fibers giving rise to such a gauge theory is given by the following Using the rules of Section 2.2.1, we consider the gauge divisors Σ 1 = 2C, Σ 2 = D, supporting so(m) and su(n) respectively.Altogether, this leads to the following massless multiplets: Table 4: Massless content of a 6d N = (1, 0) gauge theory which cannot be engineered in the unfrozen phase of F-theory We briefly review the argument for why such a gauge theory cannot be realized in the unfrozen phase of F-theory.In a conventional F-theory compactification, we must have the so(m) supported on a (−4) curve C 1 and the su(n) supported on a (−2)-curve C 2 , such that C 1 and C 2 intersect transversely.In particular, the Weierstrass polynomial f must take the following form where c 1 = 0 defines the curve C 1 .On the other hand, this implies that and thus f must vanish to order at least 1 along C 2 .Together with a similar argument for the Weierstrass polynomial g, we conclude that this contradicts the assumption of C 2 supporting an su(n) algebra.
Finally, we argue that many SCFTs and LSTs constructed in the frozen phase do not admit a compact embedding.We consider a more generalized sub-block given by the following The discriminant must then factorize into the following form: The intersection number of the residual discriminant with the divisor D 1 is given by: Applying Lemma 4.1, we demand that ∆ • D i ≥ 0 for all i, and this implies that the only valid solution is m = 16, n = 8, k = 4, in which case ∆ • D i = 0 for all i.In Section 4.3.1,we will explicitly construct a compact embedding of such a model.

SU (n) and triple intersections
In this section, we consider case (3) of Claim 4.1.We first illustrate a simple configuration of singular fibers giving rise to such a gauge theory: Again, using the rules of Section 2.2.1, we consider the assignment of gauge divisors supporting the gauge algebras so(a), sp(k − 4), so(b), so(c) respectively.Altogether, this leads to the massless content in Table 5.The non-existence of a conventional F-theory realization follows gauge algebra so(a) ⊕ so(b) ⊕ so(c) ⊕ sp(k − 4) charged hypermultiplets Table 5: Massless content of a 6d N = (1, 0) gauge theory which cannot be engineered in the unfrozen phase of F-theory by a similar argument as in the previous section, which we briefly illustrate.To realize such massless content in the unfrozen phase of F-theory, we must have so(a), so(b), so(c) supported on individual (−4)-curves C 1 , C 2 , C 3 transversely intersecting the curve C 4 which supports sp(k − 4).
In particular, the Weierstrass polynomial f must take the following form In particular, we have the intersection and hence f must vanish to at least order 2 along C 4 .Arguing similarly with the Weierstrass polynomial g, this yields a contradiction with the assumption that C 4 supports an I ns 2k−8 -fiber.
Finally, we consider the general sub-block given by the following intersections: We analyze the discriminant locus, which takes the following form Computing the intersection of ∆ with D 1 , we obtain the following: Similarly, we obtain the following intersections Applying Lemma 4.1 again and imposing the non-negativity of all the above inequalities implies the following unique solution a = b = c = 16, l = m = 4, k = 12.In Section 4.3.2,we will explicitly construct a compact embedding of such a model.

New massless spectra in 6d F-theory
In the previous section, we focused on ruling out infinite families of potential compact embeddings of frozen 7-brane configurations by applying Lemma 4.1 to universal sub-blocks of such constructions.Nevertheless, we discovered special cases which were consistent with such a condition.In this section, we will demonstrate that such cases do indeed admit compact embeddings with frozen 7-branes, leading to 6d N = (1, 0) vacua with new massless gauge and matter content.Along the way, we find a potential infinite family in Section 4.3.3,which we rule out by relying on string unitarity arguments introduced in [40,41].As a consequence, we find a concrete geometric realization of their argument, which may be of independent mathematical interest.We will construct a compact realization of such a model using the Tate form described in Section 3.1.Let B be the Hirzebruch surface F 2 with a unique (−2)-curve E defined by e = 0. Recall that the anti-canonical class satisfies −K B = 2e + 4f and consider the following tuning a 1 ∼ qe, a 2 ∼ qep 2 , a 3 = 0, a 4 ∼ e 4 q 4 , a 6 = 0 where q = 0 defines a general curve Q in the linear system |e + 4f |.In particular, we note that the polynomial p 2 must take values in the system |2e + 4f |.This leads to the following Weierstrass model: e 3 q 3 (4p 2 + eq)(−16p 2 2 − 8ep 2 q + 71e 2 q 2 ) ∆ = 1 16 e 10 q 10 (−4p 2 + 7eq)(4p 2 + 9eq) We may assume that Q defines a curve in F 2 tangentially intersecting the zero section E. In addition, we note that generically, the curve defined by p 2 = 0 intersects with Q at 8 points, as we have (2e + 4f ) • (e + 4f ) = 8.From the Weierstrass model, we conclude that there exists 8 (4, 6) points localized along the intersections Q ∩ p 2 , as well as a single (4, 6)-point localized at the tangential point of intersection Q ∩ E. We perform the following phase transitions, moving along the tensor branch of an F-theory compactification specified by the above Weierstrass model.We first blowup all 8 (4, 6) points corresponding to the intersection of p 2 = 0 with Q.As Q has self-intersection 6 on F 2 , its proper transform has self-intersection 0. Next, we perform a blowup at the intersection Q ∩ E, leading to an exceptional curve E 1 supporting an I ns 8 -fiber, as can be seen from the form of the discriminant ∆.As this is a tangential intersection, the proper transforms of Q and E intersect E 1 at a common point.Blowing up this common point once more leads to an exceptional curve supporting an I ns 16fiber.A direct computation of the self-intersections leads precisely to diagram (4.11), and we note that the proper transform of E 1 must support a split fiber since it has self-intersection (−2).The

An infinite family
Finally, we note that there exists an additional, a priori infinite family of such models with the following intersecting Kodaira fibers I * m+4 (4) We briefly provide an alternative argument to show that such models can only exist for m = 0, demonstrating that our construction in Section 4.3.2 is sharp.Based on the analysis of [41], [40] used consistency of non-instantonic string probes in 6d supergravity backgrounds to prove the finiteness of many classes of models.We refer to [40] for details, and we briefly summarize the necessary aspects in applying the argument to our setting.Our geometric construction in Section 4.3.2gives a natural embedding of the anomaly vectors of the gauge algebras associated with the unflipped version of (4.3).A D3-brane wrapped on an effective curve C in a compact algebraic surface B yields a 2d N = (0, 4) supersymmetric worldsheet theory in the noncompact 6-dimensional spacetime.The condition that this corresponds to a non-instantonic string with non-negative tension translates into the following geometric constraints: where D i denotes the divisors supporting the non-abelian gauge algebras in (4.3).Unitarity of the string worldsheet theory then imposes the following constraint where c G i denotes the central charge associated with the current algebra of the gauge algebra G i , and h ∨ i denotes the dual Coxeter number of G i .
Taking the three outer (−4)-curves in (4.3), we recall that these were obtained as the proper transform of hyperplane classes on P 2 by blowing up three points each on three intersecting lines.Choosing two out of these nine points such that these do not lie on the same of the three lines, we take a separate line passing through these two points on P 2 .We then take C to denote the proper transform of this curve.A straightforward computation implies that C only intersects with one of the outermost vertices corresponding to the Kodaira fiber I * m+4 .Evaluating equation (4.4) with C, we obtain the bound c G = 8 + m ≤ 8 Thus, it must be the case that m = 0 and this implies that our geometric construction of the model in Section 4.3.2 is sharp.It would be interesting to interpret more generally, the condition on the unitarity of the string worldsheet as a geometric constraint on engineering singular limits of compact Weierstrass elliptic Calabi-Yau threefolds.

Remaining models
In this section, we list the remaining diagrams in [10] which admit a compact embedding in the frozen phase of F-theory.We note in passing that this can be done using the same techniques carried out in the previous sections, but we do not carry out the analysis here.
In each entry, we list the configuration of Kodaira singular fibers and the corresponding equation in [10].We refer the reader to the relevant section in [10] for the corresponding gauge theory configuration.

Conjectures on neutral hypermultiplets
In Sections 3 and 4, we analyzed a number of compact F-theory models with frozen 7-branes.Such concrete analyses are potentially helpful in deducing general properties of frozen 7-branes in 6-dimensional compactifications.In this section, we analyze the simplest compact embeddings of frozen 7-branes and make a simple observation based on gravitational anomaly cancellation.
Conjecture 5.1.Consider a 6d F-theory compactification with a frozen 7-brane localized along a divisor D with residual discriminant ∆.Then there exists at least 1 2 D • ∆ localized neutral hypermultiplets.
We note that none of the compact examples considered thus far exhibit non-trivial intersections between the residual I 1 -locus and a frozen O7 + -plane.
For a 6d N = (1, 0) supergravity, we recall that gravitational anomalies impose the following constraint where H is the total number of hypermultiplets, charged and uncharged, V is the number of vector multiplets, and T is the number of tensor multiplets.
We consider an otherwise generic Weierstrass model on a compact algebraic surface B with an I * s n+4 fiber localized along a divisor D of self-intersection (−4).In the unfrozen phase of Ftheory, this realizes an so(2n + 16) gauge algebra with 2n + 8 fundamental hypermultiplets.In the frozen phase, this realizes a sp(n) gauge algebra with 2n + 8 fundamental hypermultiplets.For 6d supergravity theories with such matter content, we compute the difference H ch − V for every value of n in the following table.We note that for n ≥ 9, there does not exist any number of tensor multiplets such that the resulting model satisfies gravitational anomalies.In the last row, we subtract the second row from the first row.Using gravitational anomalies, and assuming that the number of tensor multiplets is the same, this computes the difference in neutral hypermultiplets between the two models.Table 6: Table of the difference between the number of charged hypermultiplets and vector multiplets of a generic F-theory model with an so(2n+16) and an sp(n) gauge algebra respectively.The difference in the counting of neutral hypermultiplets is encoded in the last row.
In general, the total number of neutral hypermultiplets can be split into localized and unlocalized hypermultiplets.We expect that unlocalized neutral hypermultiplets should coincide with complex structure deformations of the total space of the Weierstrass model that do not deform the singularities, and hence that this should yield the same answer between the unfrozen and the frozen phase of F-theory on identical Calabi-Yau threefolds.Moreover, the intersections with the residual discriminant are given by the following In the unfrozen phase, there are 2n+8 fundamental hypermultiplets, and so there is 1 fundamental hypermultiplet for every double intersection1 with the residual discriminant.We conclude that in the frozen phase, there must be 1 localized neutral hypermultiplet for every double intersection of the frozen 7-brane with the residual discriminant.
It would be interesting to confirm such a result directly via a weakly coupled type IIB limit in the frozen phase of F-theory.The existence of such localized neutral hypermultiplets could perhaps be explained via a similar mechanism as in the I 1 -conifold model explored in [42] and [43,Section 4.1].