The frozen phase of F-theory

We study the interpretation of O7+ planes in F-theory, mainly in the context of the six-dimensional models. In particular, we study how to assign gauge algebras and matter content to seven-branes and their intersections, and the implication of anomaly cancellation in our construction, generalizing earlier analyses without any O7+ planes. By including O7+ planes we can realize 6d superconformal field theories hitherto unobtainable in F-theory, such as those with hypermultiplets in the symmetric representation of su. We also examine a couple of compact models. These reproduce some famous perturbative models, and in some cases enhance their gauge symmetries non-perturbatively.

1 That the Chan-Paton indices can carry u, so and sp indices was originally pointed out by Schwarz in [4] and that they were the only possibilities was soon showed by Marcus and Sagnotti in [5], both in 1982; see also Sec. 1.3 of Schwarz's review [6]. That the choice of so and sp is reflected in the sign of the RR-charge of the O9-plane was already essentially noticed in the seminal paper by Green and Schwarz on the anomaly cancellation in Type I superstring theory [7] in 1984. That one can have a consistent T 2 /Z 2 compactification of type IIB theory with three O7 − s and one O7 + was originally noted in [8] in 1991. realized by conventional F-theory constructions.
A typical feature of these cases is that their massive IIA brane construction involves O8 + s. By a T-duality, this is mapped to a IIB brane construction involving O7 + s. This motivated us to look at F-theory compactifications to six-dimensions in the presence of O7 + s.
At this point, it is natural to worry if there could be frozen singularities other than O7 + -planes which have not been studied in conventional F-theory. This question was settled, at least for supersymmetric seven-branes, in a recent re-analysis of 7branes in F-theory [21] which concluded that the O7 + is in fact the only type of frozen singularity in F-theory. 2 Therefore, the only ingredient missing in conventional F-theory compactifications to six-dimensions is the inclusion of O7 + -planes, and indeed including them we find F-theory realizations of 'missing' 6d SCFTs, as we will see later in the paper. 3 Once we are convinced that O7 + -planes can be included in the F-theory construction, we realize that we need to revisit every part of the standard F-theory machinery, such as the assignment of the gauge algebras and of the matter content to the components of the discriminant and to their intersections, and the way the 6d anomalies cancel via the Green-Schwarz-West-Sagnotti effect [25,26], derived geometrically for F-theory by Sadov in [27]. This paper is the authors' first attempt to provide such generalizations.
One unexpected consequence of the introduction of O7 + -planes is the following. To appreciate it, let us first recall the situation without O7 + -planes. In a conventional F-theory compactification without O7 + -plane, once one is given the geometry of the elliptically-fibered Calabi-Yau, there is a standard method to assign a unique set of gauge algebras and matter content to the geometry. In particular, under this standard assignment, each simple factor in the gauge algebra is associated to a single component of the discriminant divisor, and each component has at most one simple factor of gauge algebras associated to it. This choice corresponds to having zero holonomies of the gauge fields on these divisors themselves. We have the option of turning on the nontrivial gauge configurations, including the effects often called the T-branes [28], but we also have the standard option of not turning them on at all.
With O7 + -planes, however, we will often be forced to have at least some nontrivial 2 There are various other less-studied types of higher-codimension singularities one can incorporate in F-theory, such as the ones used by García-Etxebarria and Regalado [22] to construct 4d N =3 SCFTs. Frozen versions of singularities also occur in M-theory [10,11], where they play an important role in M5-brane fractionation [11,23,24]. 3 We will find F-theory realizations for certain examples, but we defer a general treatment of the classification problem formulated in [12,13] to future work. gauge configurations on some of the components. More precisely, we even lose the concept of a unique, standard assignment of gauge algebras and matter content, since we do not even have a natural origin in the space of the all possible holonomies. Because of this, we often have multiple simple factors of gauge algebras on a single component of the discriminant locus, and also a single simple factor of gauge algebra shared across multiple components, as we will see later.
Unfortunately, at present, we do not have any algorithmic method to find consistent assignments given an elliptic Calabi-Yau and a specification of where the O7 + -planes are; we do not even have a method to tell if there are any consistent assignments at all. Therefore we are forced to rely on consistency checks via anomaly cancellation and dualities to backgrounds that are better understood.
The rest of the paper is organized as follows. In Sec. 2, we study the properties of O7 + -planes in the context of F-theory, using string theory and M-theory dualities. This will let us figure out how to assign gauge algebras and matter content. In Sec. 3, we study the anomaly cancellation of F-theory models with O7 + -planes. We will see that the analysis of Sadov [27] can be naturally generalized by introducing a divisor which represents where O7 + -planes lie. Then in Sec. 4, we discuss some 6d SCFTs which can be realized only with O7 + -planes in F-theory construction, and in Sec. 5, we study the massless spectrum of a couple of compact six-dimensional models with O7 + -planes.
In Appendix A, we review the 8d compactifications with O7 + -planes, which is simpler than the 6d examples discussed in the main text. Finally in Appendix B, we give an alternative derivation, using intersecting brane models, of the spectrum of some compact models discussed in Sec. 5.

Frozen seven-branes and their properties
In this section, we use perturbative string techniques to obtain some properties of frozen singularities.
We start in section 2.1 with a lightning review of O-planes. We then discuss the basics of O7 + -planes in F-theory in Sec. 2.2, and in Sec. 2.3 we study the physics at individual intersection points of O7 + -planes and other seven-branes. To prepare ourselves for the analysis of an O7 + -plane which intersects with more than one sevenbrane, we then need to have short digressions, on the T-duals of NS5-and D6-branes in Sec. 2.4 and on the phenomenon of shared gauge algebras in Sec. 2.5. We then come back to the case with O7 + -planes in Sec. 2.6. In the final subsection 2.7, we see that with O7 + -planes a shrunken divisor does not necessarily signify any singularity in the low energy physics.

Basics of orientifold planes
Let us start by a quick review of the basics of the orientifolds. 4 Action on the closed strings: An orientifold is usually defined as a Z 2 symmetry Π that includes world-sheet parity Ω. It can also include a spacetime involution σ. It is often necessary to also include an extra factor (−) F L (where F L is the left-moving spacetime fermion number) so that Π 2 acts as the identity. If locally σ is the reflection of 9−p coordinates, so that the orientifold plane Op (the fixed locus of σ) 5 has (spatial) dimension p, one needs to include (−) F L if p = 2, 3 mod 4. 6 To summarize, locally the orientifold action is O9 O8 O7 O6 O5 · · · Ω ΩR 9 ΩR 8 R 9 (−1) F L ΩR 7 R 8 R 9 (−1) F L ΩR 6 R 7 R 8 R 9 · · · , (2.1) where R p denotes a reflection of the p-th coordinate. This specifies the orientifold's action on closed strings. In this paper, we will be interested in particular in O7s, with O6s and O8s making occasional appearances.
Action on the open strings: In presence of open strings, one also needs to decide its action on the Chan-Paton matrix λ, which appears in a superposition i,j λ ij |ij of the states |ij , that in turn can be interpreted as going from the i-th to the j-th brane in a stack (omitting other quantum numbers). Since the world-sheet parity Ω reverses orientation, it acts by transposing Λ, but it may also mix the states with a change of basis M : namely, λ → M λ t M −1 . Imposing that this action is an involution leads to the condition that M −1 M t = ∓1 .
(2.2) 4 A good review of the basics can also be found in [29]. More detailed and rigorous analysis of perturbative orientifolds were done e.g. in [30,31], but we stick to the traditional, ad hoc approach in this paper. The name orientifold itself was introduced in [32] by Dai, Leigh and Polchinski. The concept of the orientifold goes back further in history, see e.g. [33,34] and references therein. 5 We will also consider actions that include translations and thus have no fixed locus as in (2.5); the conclusions in (2.1) below also apply. 6 To check this, one first uses the fact that a reflection R I of the I-th spatial coordinate acts by Γ I on the 10d Majorana spinor, which satisfies (Γ I ) 2 = +1. Therefore, R 2 I1···Ip = 1 or (−1) F L +F R depending on whether p = 0, 1 or 2, 3 mod 4, respectively. Then one compensates this (−1) F L +F R by the fact that Ω(−1) F L Ω = (−1) F R and therefore (Ω(−1) This sign choice leads to two different types of O-plane, which we call Op ± . 7 The RR-charge: The RR charge can be computed through a one-loop computation, which contains − tr M −1 M t in its Möbius strip contribution (see for example the reviews [29,35]). In the end one concludes that the charge is ±2 p−5 that of a full Dp-brane 8 : explicitly, Thus, the Op − has negative charge and the Op + has positive charge, as the name implies.
The gauge group: The gauge group is also influenced by the sign (2.2). If a stack of N Dp-branes is parallel to the Op-plane but not on top of it, the action will relate the strings ending on them to strings ending on an image stack in a different locus; the gauge group will be the usual U(N ). On the other hand, if the stack is on top of the Opplane, the action will relate the open string states to themselves, projecting out some of them. To read off the gauge group, we can consider the gauge field states λ ij α µ −1/2 |0; ij . Since Ωα −1/2 Ω = −α −1/2 , the surviving states will be those with Chan-Paton factors λ such that λ = −M λ t M −1 . If the sign in (2.2) is −1, M is antisymmetric; by a change of basis (λ → C −1 λC, M → CM C t ) it can be chosen to be of the form J ≡ ( 0 1 N −1 N 0 ), and thus λ will be in the sp N algebra. 9 If on the other hand the sign in (2.2) is +1, then M can be chosen to be 1 2N , and λ ∈ so 2N . Summarizing, the choice (2.2) leads to two different orientifolds: • Op − , with so 2N gauge algebra and charge −2 p−5 , and • Op + , with sp N gauge algebra and charge +2 p−5 .
Dq-branes intersecting Op-planes: More generally, if we also have a stack of Dqbranes which intersect our Op, there are subtle signs [37] coming from the fact that the strings from the Op-to the Dq-branes needed to be expanded to both integer and half-integer modes. In flat space (and vanishing B field), the number #ND of Neumann-Dirichlet directions (the number of directions transverse to the Dp and parallel to the 7 In [10] and other older papers, Op ± -planes are called planes of type O ∓ , with the opposite sign. We stick to the more modern conventions which are now standard. 8 Naively the fractional charge of the Op-plane for p ≤ 4 contradicts the Dirac quantization. For a resolution, see [36]. 9 We follow the standard convention that sp 1 = su 2 . Dq, or vice versa) has to be a multiple of 4, for unbroken supersymmetry. The result for the gauge algebra on the Dq-branes is then as follows: 10 T-duality: Let us next discuss the T-duality of orientifolds, since we often need to perform T-duality of the setup on S 1 /Z 2 where two fixed points support Op-planes, possibly of different types. Two most straightforward cases are when both fixed points have Op − or both fixed points have Op + . The T-dual is then simply O(p + 1) − or O(p + 1) + wrapped around S 1 . When one fixed point has Op − and the other fixed point has Op + , the T-dual is known to be a shift-orientifold, namely an orientifold whose spacetime action σ not only flips the coordinates transverse to the orientifold, but also translates a circle by half its radius The derivation of this fact can be found e.g. in [10, p. 41] or [38]. A rough argument goes as follows. We start from the shift-orientifold background (2.5), and T-dualize the x p+1 direction. Its T-dual should be a compactification on S 1 /Z 2 . Therefore this 10 The fields on the Dq stack get mapped to fields on another point of the stack, unless the Dq stack is completely embedded in the Op-plane. A priori this only restricts the behavior as a function of the coordinates of the gauge field, which would then locally remain of u(2m) type. However, in situations where the divisor wrapped by the stack is compact, in most applications we want to keep only the zero-modes of the gauge field under its equation of motion, and this restricts the gauge group as in (2.4).
should result in a combination of two Op-planes at two fixed points. The original shiftorientifold background did not have any D(p + 1)-charge. Therefore, in the T-dual, we should have zero Dp-plane charge. This is only possible if one fixed point is Op − and the other is Op + .
Another intuitive argument is as follows. The shift operator s : x p+1 → x p+1 + R/2 can be thought of as e i R 2 p , where p is the momentum operator. Its T-dual iss = e i 1 2R w , where w is the "winding operator", which measures the length of the string.s gives 1 on strings of total length zero, such as those that begin and end on the same Op, but it gives −1 on the strings that begin and end on different Op's, signaling the fact that the two have different signs.
Other types of orientifolds: It is also known that there areÕp ± -planes when p ≤ 6, distinguished from the more ordinary Op ± -planes by the RR-torsion flux. As we will not use them heavily, we will not discuss them further.

Frozen divisors in F-theory
Our main interest lies in seven-branes in Type IIB theory and F-theory. An ordinary O7 − without any D7-branes on top is known to lift to two I 1 divisors, due to quantum effects [9]. Similarly, with n < 4 D7-branes on top, the F-theory realization is given by (n + 2) I 1 divisors. With at least 4 D7-branes, it is interpreted in F-theory as an I * n−4 divisor (where n is the number of D7-branes). Since string theory also has O7 + -planes, it is natural to ask how these are described in F-theory.
First of all, from (2.3) we see that O7 ± have charge equal to that of ±4 full D7branes. So an O7 + has the same charge and tension as an O7 − with 8 full D7-branes on top. In F-theory, they will give rise to the same monodromy [10,39]; we expect both to be described by an I * 4 divisor. However, the O7 − with 8 D7 gives rise to an so 16 gauge algebra, while the O7 + gives rise to none. A related difference is that the O7 − with 8 D7 can be deformed by pulling the D7s away (which corresponds in F-theory to a complex structure deformation), while the O7 + cannot. Thus an O7 + is described by a I * 4 singularity which for some reason cannot be deformed; we will call this a frozen singularity, and denote it by I * 4 . More generally, an O7 − with n D7s has the same charge and tension as an O7 + with (n − 8) D7s; both are described by an I * n−4 singularity, but in the latter case the gauge algebra is sp n−4 rather than so 2n , and the deformations are correspondingly reduced. In this case too we say that the singularity is frozen, and we denote by I * n−4 .
To be more expicit, an F-theory vacuum is typically described by the "Weierstrass coefficients" f and g which are sections of the line bundles O B (−4K B ) and O B (−6K B ) on the F-theory base B, and which lead to the equation for the total space of the elliptic fibration. Along a divisor D with a I * n−4 singularity, f vanishes to order 2, g vanishes to order 3, and the equation 4f 3 +27g 2 of the discriminant locus vanishes to order (n − 8) + 10, for a configuration with n − 8 D7-branes on top of an O7 + . Although the "freezing" mechanism is not understood, it must prevent any deformation which lowers the order of vanishing of either f or g at all, or which lowers the order of vanishing of 4f 3 + 27g 2 below 10.
Note that the Weierstrass coefficients are accompanied by periods of type IIB twoforms over appropriate two-cycles in B; for compactifications to 6d, the complex moduli provided by Weierstrass coefficients are paired with these periods of two-cycles to provide the two complex scalars in a hypermultiplet. In particular, by activating a vev represented by one of these two-form periods we may disturb the gauge group assigned to a divisor without changing the geometry of the divisor (which would have required a change of complex modulus). Such deformations are often described in the language of T-branes [28], for which a number of geometric tools have been developed [40][41][42].
As an exercise in using the rule (2.4), let us consider D3-branes embedded in the worldvolume of O7 ± . Since #ND = 4, the gauge group on the embedded D3-branes is so for O7 + and sp for O7 − . In particular, the smallest gauge algebra allowed is so 1 and sp 1 , with one and two Chan-Paton indices, respectively. A bulk D3-brane has two Chan-Paton indices. Therefore, a bulk D3-brane can fractionate into two separate objects on O7 + but not on O7 − . These D3-branes can be considered as point-like instantons of the gauge fields on O7 ± , and therefore the D3-charges of the minimal-charge instanton on O7 ± differ by a factor of 2. This fact becomes important in the anomaly analysis in Sec. 3.1.

Intersections: perturbative analysis
As mentioned in the introduction, O7 + s are the only frozen F-theory singularities [21]. As our main interest lies in the compactification to 6d, we now want to understand their behavior when they intersect other singularities, namely, how they modify the gauge algebras of neighboring divisors and the matter representations at intersections with them. We will do so by using perturbative string techniques, and dualities. Some readers might want to study the simpler situation in 8d summarized in Appendix A, before considering the more interesting but complicated examples of 6d compactifications discussed here.

I * -I intersection
Let us now start working out what happens when the frozen divisors intersect ordinary divisors. We will begin with the intersections of frozen I * n with I m divisors. Let us first recall what this intersection gives in the unfrozen case, i.e. an I * -I intersection. The intersection with the I * induces on the I a so-called "Tate" monodromy, a nontrivial automorphism of the gauge algebra that reduces it [43]. 11 This is expressed by saying that the divisor is non-split, and denoted by a superscript ns . Its effect on the gauge algebra is that it reduces from u 2m to sp m . We summarize this situation by writing so 2n+8 sp m I * n I ns 2m . (2.7) As a warm-up, let us also see how it is reproduced by orientifolds. Consider an intersection of an O7 − + (n + 4) D7 along directions 01256789 with m full D7s along directions 03456789. From (2.4) we see again that the gauge algebra on the m D7s is reduced to sp m ; see also footnote 10. We thus recover (2.7). Notice that the spacetime action of the orientifold projection can be interpreted as the Tate monodromy we mentioned above.
We can similarly work out what happens if the I * divisor is replaced by its frozen I * counterpart: the configuration now involves an O7 + + (n − 4) D7s, and 2m transverse D7s (see Fig. 2.2, where only directions 6789 are depicted). Looking again at (2.4), we see that the gauge algebra on the m D7s is reduced this time to so 2m . We conclude sp n−4 so 2m I * n I ns 2m . (2.8) Thus, an I ns divisor intersecting a frozen divisor has an so gauge algebra, rather than an sp gauge algebra. In both cases (2.7) and (2.8) there is a bifundamental at the intersection, due to the strings from one set of branes to the other.  We will now consider intersections between two I * divisors, both frozen and unfrozen.
We will see that using perturbative O7s we will have only partial success in understanding the full possibilities. This will lead us in Sec. 2.6 to consider T-dual configurations.
I * -I * intersection: Let us again start by recalling what F-theory gives in the ordinary unfrozen case. The intersection of two I * divisors actually falls outside Kodaira's classification. To cure this, one can blow-up the base; this reveals a new divisor that touches both I * 's, and that behaves like in (2.7): where we assumed k + to be even, and the • denotes the bad singularity that we blew up. Physically, it signals a six-dimensional superconformal sector which is sometimes called D k+4 -D +4 conformal matter ; 12 the blow-up represents moving along its tensor branch, namely the part of its moduli space where we give a vev to the scalar in the tensor multiplet. Let us now try to engineer an I * -I * intersection using O7s. The most natural generalization of Fig. 2.2 consists of two O7s that intersect transversally. This can be 12 In fact this superconformal theory depends only on k + and has so(2k + 2 + 16) flavor symmetry. Thus we will simply call it D k+ +8 conformal matter in what follows. We use the blackboard letter D since the notation D i denotes an i-th divisor in this paper. One can also define D 2n as the 6d superconformal theory which has a one-dimensional tensor branch on which it becomes an sp n−4 theory with 4n fundamentals with at least so 4n flavor symmetry. For example, then, the D 8 theory is the E-string theory.
achieved by an orientifold projection in flat space that has more than one generator of the type we recalled in (2.1). For an intersection of two O7s, locally one takes the two generators We can see that in this situation there is an O7 on the locus x 6 = x 7 = 0, and another on the locus x 8 = x 9 = 0. (Notice that one is then also quotienting by their product R 6 R 7 R 8 R 9 , so that at the intersection between the O7s there is in fact also a Z 2 orbifold singularity.) Choosing the ± type of these two orientifold planes affects their charge and their action on Chan-Paton indices in the way we reviewed earlier; we will see shortly what their combined effect amounts to. Another ingredient is that the projection on the closed Z 2 -twisted sector is reversed if two orientifolds of different type intersect [44]. This comes about by considering the exchange of closed strings between two crosscaps, one from one O7 and another from another O7. The sign of this diagram is reversed when two orientifolds are of different type, and the modular transformation of this diagram determines the orientifolding projection on the closed string Z 2 twisted sector. In the end, one finds that an O7 − -O7 + intersection has a six-dimensional tensor multiplet, while O7 − -O7 − or O7 + -O7 + intersection has a hypermultiplet: (2.11) As we mentioned, if D-branes are present, they will now feel the effect of both projections. Consider for example choosing both planes to be O7 − , with k + 4 and + 4 D7s present on the x 6 = x 7 = 0 and x 8 = x 9 = 0 loci respectively. The first set of D7s, say, would be projected to so 2k+8 by the O7 − parallel to it; but, recalling (2.4), it would also be projected to sp k+4 by the O7 − transverse to it. This means that it actually gets projected to the intersection of the two, u k+4 . In the language of F-theory branes, this gives u k+4 u +4 where the · now represents the hypermultiplet found in (2.11). 13 This hypermultiplet is 13 A warning is in order. The orientifold projection leaves the gauge algebra u on I * , but the u 1 part usually gets Higgsed and becomes massive by the Green-Schwarz mechanism, each u 1 eating a neutral hypermultiplet. This point was carefully analyzed in [45,Sec. 2]. In our case, the diagonal u 1 of u k+4 and u +4 will be gone. In a compact model, we usually expect every u 1 part to be eliminated in this neutral under u k+4 ⊕ u +4 . The presence of this neutral hypermultiplet signals that the configuration (2.12) is obtained by moving along a particular direction in the Higgs branch of D k+4 -D +4 conformal matter whose tensor branch was depicted in (2.9). This particular direction in the Higgs branch is parametrized by vevs of the neutral hypermultiplet in (2.12). Another well-known direction in the Higgs branch, distinct from the one represented by (2.12), is provided by brane recombination, where the two I * divisors merge.
I * -I * intersection: For an O7 + -O7 + projection, for the same reason we get In analogy with our discussion below (2.12), it is natural to think that this is the Higgsing of a "frozen conformal matter" and that upon blowing up (moving along the tensor branch) an I ns k+ with so(k + ) gauge algebra would be created, which would behave as in (2.8). We will see later that this expectation is borne out. I * -I * intersection: For an O7 + -O7 − intersection, on each set of D7s the two projections will be of the same type. For example, on the D7s on the O7 − , we have We can make M 1 = 1 as in section 2.1; with the residual freedom in change of basis we can diagonalize M 2 , but a priori it could have any number of positive and negative eigenvalues. If we also impose that the D7s can move off the O7 − , we obtain that M 2 = 1 +4 0 0 −1 +4 , and the gauge symmetry is so +4 ⊕ so +4 . Similar considerations apply to the O7 + + (k − 4)D7s; hence we get where we assumed k to be even. Notice that in this case there is no neutral hypermultiplet at the origin, according to (2.11); we have included the symbol • to mark this.
manner, agreeing with the usual expectation that only the su algebras are realized on the 7-branes, not the u algebras.
So in this case we do not expect this configuration to be a Higgsing of a conformal one. This might look surprising, but it will become clearer in section 2.7 below, where we will see an alternative realization of the same setup (in the case k = is even).

NS5-and D6-branes
To go beyond the results in section 2.3.2, we will need to consider configurations which are dual to IIA in presence of NS5-branes. To set the stage, in this subsection we will discuss a situation without orientifolds. We consider IIA on R 9 × S 1 ; let us say the S 1 corresponds to direction 4, and has periodicity R. Let us have a single NS5 whose worldvolume is in directions 056789, localized at x α = x 4 = 0, α = 1, 2, 3. T-dualizing it along direction 4 turns it into an Euclidean Taub-NUT geometry. The space transverse to the NS5 is R 3 × S 1 ; T-duality turns the H flux of the NS5 into a Chern class that signals the S 1 is now Hopf-fibred over the S 2 s at x α x α = r 2 . The inverse images of these S 2 s are thus copies of S 3 . These shrink smoothly at x α = 0, so that locally around this point the fibration is S 1 → R 3 → R 4 . One way to realize this fibration in coordinates is where σ α are the Pauli matrices. So If we have several NS5s localized at several positions in the 3 direction ( , T-duality turns the geometry into a multi-Taub-NUT geometry where the S 1 shrinks at the x 3 = x 3 i . The inverse image under the S 1 fibration of a path between two of these points is an S 2 . We represent this in Fig. 2

.3.
Let us now suppose some D6s are also stretched along the 0356789 directions. First let us imagine there are n D6s stretched along the entire 3 axis. Under T-duality along direction 4, they will turn into n D7s. More precisely, as Fig. 2.3 suggests, they will turn into a sequence of D7s wrapping the various S 2 on the Taub-NUT with multiplicity n. What the picture does not show is that these S 2 s are holomorphic cycles. Locally around an NS5 at x α = 0, for example, we see from (2.18) that the locus x 3 = 0 is turned into zw = 0, which is the union of the curve z = 0 and of w = 0. In F-theory terms, this is a chain of intersecting I n curves.
In the presence of a Romans mass, parameterized conventionally by an integer 2πF 0 ≡ n 0 = 0, one can also have situations where D6s are ending on NS5s. Focusing on an NS5 on which a D6 ends from the right, we see again from (2.18) that T-duality turns it into the single curve z = 0. This would be one of the S 2 s in Fig. 2.3. The number of D6s ending on an NS5 from the left minus the number of D6s from the right is proportional to F 0 . We then have a chain of intersecting curves supporting I n , I n+n 0 , I n+2n 0 , . . . .
Another possible generalization is to move the D6s in the x 4 direction, so that there is now a stack of n j D6s at x 4 = x 4 j . On the IIB side, this corresponds to Wilson lines for the gauge field on the D7s.

Shared gauge algebras
From the setup of Fig. 2.3, we can also wonder what happens if we move only some of the D6s away from the NS5s in direction 4; say from an initial stack of n D6s we move m to the position x 4 = x 4 0 . These D6s recombine: they no longer end on the NS5s. In field theory, this corresponds to a partial Higgsing where the su m at the middle is the diagonal subalgebra of two copies of su m ⊂ su n .
Since the displacement has happened along the 4 direction, it is not immediately apparent on the IIB side: the T-dual still consists of two stacks of m + n D7-branes meeting at a point, as in section 2.4. The only consequence of the displacement is the presence of a Wilson line: there is a worldvolume gauge field with non-zero holonomy, a = x 4 0 l 2 s diag(0, . . . , 0, 1, . . . , 1)dx 4 . Since direction4 shrinks at the intersection point, on both D7s there is a worldvolume da = f field strength proportional to a δ-function supported on the intersection point.
By comparing with the IIA picture, we conclude that a Wilson line can partially break the gauge algebra on two intersecting D7s, as in (2.19): part of the gauge algebra can recombine. The su m algebra is now shared between the two intersecting divisors; this is summarized in Fig. 2  On the IIB side, this corresponds to a gauge algebra su m that is shared between two curves meeting at a point. We denote this with a double-sided arrow.
If we move all the D6s off the NS5 (i.e. if n = m), only the shared gauge algebra is present. In this case, one might be puzzled by the fact that on the IIB side the Wilson line is now proportional to the identity. This would not seem to cause a Higgsing, while from the IIA picture it is clear that it does, since the D6s are away from the NS5.
To clarify this point, we need to identify the T-dual of the NS5 position in IIB. Since the NS5 position in IIA is shifted by a diffeomorphism in the x 4 position, its T-dual should be shifted in IIB by a gauge transformation for the NS-NS two-form field, namely B → B + dΛ, for Λ a one-form. In fact this one-form was identified in [46,Sec. 2.2] explicitly. Thus more generally we conclude that, in the intersection between two curves C 1 , C 2 , there is a shared gauge algebra if on either curve there is an eigenvalue a i of the Wilson line α on the curves that does not match with the pullback of Λ at large distance from the intersection: (2.20) In F-theory language, we could consider a deformation of the Weierstrass coefficients which "recombined" two branes, i.e., smoothed the two divisors out into a single divisor. If instead of this deformation, the corresponding periods of two-forms are activated, the gauge theory will recombine without any change in the geometry.

Intersections: via T-duality
Having made a detour in the last two subsections, we now reintroduce O-planes in our story.
First we need to review the behavior of NS5s in presence of orientifolds. Like any other brane, any NS5 must come with a mirror image under the orientifold action. Each copy is usually called a half-brane to emphasize that it can become full if the two copies are brought to the O-plane. It turns out [47] that when this is done the two half-NS5s can be separated again: this time along the O-plane worldvolume, while staying on it. When this happens, the orientifold type changes between the two half-NS5s.
The situation relevant for our purposes consists in having an O6 defined by a reflection inverting directions 124, and for example two half-NS5s at two values of x 3 . (Thus the O6-plane and the half-NS5s are stretched along the same directions as the D6 and NS5 in the previous subsection.) If the O6 is taken to be an O6 − outside the two half-NS5s, its type will change to O6 + inside. This leads to a sequence of gauge algebras so 2n+8 , sp n , so 2n+8 .
Actually, since direction 4 is compact, a reflection involving 124 will have a fixed point both at x 4 = 0 and at x 4 = R/2, the opposite locus on the circle. The O6-plane on that locus can be of both O6 − and O6 + type. We show both those cases in Fig. 2 Upon T-duality, we again find a chain of curves. To see what type of curves we have, we need to use the rules reviewed in section 2.1; see in particular Fig. 2.1. We learn from there that an orientifold with O6 ± -planes on both sides of a circle gets T-dualized to an orientifold with an O7 ± -plane, while a circle which has an O6 + on one side and an O6 − on the other gets T-dualized to a shift-orientifold. This is another realization n 6 (n + 4) 6 (n + 4) 6 I n I n I 2n n 6 (n + 4) 6 (n + 4) 6 6 + 6 6 6 6 + 6 6 so 2n+8 so 2n+8 sp n so 2n+8 so 2n+8 sp n 7, 8, 9 6 6, 7 n 6 (n + 4) 6 (n + 4) 6 I n I n I 2n n 6 (n + 4) 6 (n + 4) 6 6 + 6 6 6 6 + 6 6 so 2n+8 so 2n+8 sp n so 2n+8 so 2n+8 sp n 7, 8, 9 6 6, 7 of Tate monodromy, which we discussed at the beginning of section 2.3.1.
Thus, in the case of Fig. 2.5(a), after T-duality we end up with a curve I ns 2n between two ordinary I * n curves. This is familiar from (2.9) with m = n, and is in agreement with the sequence of gauge algebras (2.21) we found in IIA.
In the case of Fig. 2.5(b), we have a frozen I * n+4 curve touching two I ns 2n+8 ones. The presence of the frozen singularity alters the usual F-theory rules: from the IIA picture, we see that as expected an I * n+4 curve supports an sp n gauge algebra; moreover, we also see that an I ns 2m touching a frozen curve supports an so 2m . This can be generalized to . This is the theory on the tensor branch of (2.14), thus realizing the expectation discussed there. If we put the half-NS5s back on top of each other, we recover a full NS5. We can now split it again by moving the two halves along the periodic 4 direction, together with some of the D6s, or by moving them in another direction, so that the degeneration induced by T-dualizing the NS5s no longer happens on the O6-D6 system. These two new configurations represent respectively the Higgsing in (2.12), and the one mentioned below it involving brane recombination. These two possibilities are depicted in Fig. 2.6.
The setup of this section can also be decorated by adding m D6-branes at the bottom orientifold plane; this would add a gauge algebra so 2m to line, and to a gauge algebra that is shared among the three curves, in the language of section 2.5. Again, this can be realized through the T-brane-like phenomena of activating the two-form-period partner of a geometric deformation.

Smooth transitions
In the chains of curves considered so far, shrinking one or more of the curves leads to some strongly coupled physics. This is clear from the IIA picture, where it corresponds to making two or more NS5-branes coincide. In an effective field theory description, this often manifests itself in a gauge coupling becoming infinite. The positions of the NS5s parameterize the tensor branch of a six-dimensional effective theory; these situations correspond to non-generic loci of the tensor branch. For example, in the situations depicted in figures 2.3 and 2.5, there is a onedimensional tensor branch, parameterized by a 6d tensor multiplet whose scalar φ corresponds to the distance between the two NS5s, and which in the 6d theory also plays the role of the inverse square of the gauge coupling. At the origin φ = 0, the gauge coupling diverges. At this strong coupling point it is expected that a CFT arises, describing two coincident NS5s on top of a D6 stack.
However, on the IIA side we can also consider placing the NS5s at different values of x 9 (the compact direction). In this case, bringing the NS5s at the same value of x 9 does not actually put them on top of each other; now we do not expect strong coupling physics at the origin φ = 0 of the tensor branch. A first example not involving orientifolds is shown in Fig. 2.7. In this case without frozen seven-branes, we can of course put all NS5-branes on the same stack of D6-branes so that this smooth transition does not happen. When we start involving orientifolds, we can engineer more interesting situations. The example in Fig. 2.8 has a non-split I ns 2n touching both a frozen and a non-frozen I * . In this case there is no way to put all NS5-branes on the same side of the O6-planes. Note also that in both sides of the figure the overall gauge algebra remains the same, but the roles of localized and shared simple subalgebras are exchanged.  When the two NS5s are aligned, for m = n we are in fact in the situation of (2.15), with k = = 2n + 4. This is in agreement with our observation made there (motivated by the absence of a hypermultiplet) that there is no conformal point at that intersection; in this case the transition is completely smooth, and there is no special point on the tensor branch.
In 6d compactifications of F-theory, we are accustomed to getting conformal theories when a divisor shrinks. One reason for this is that one can engineer string states from D3-branes, and these strings become tensionless when we shrink a curve. In the situations of Fig. 2.7 and 2.8, in fact we cannot wrap a D3-brane on the middle curve: this is made clear by T-dualizing back to IIA, where it would become a D2-brane, which can terminate on either one or the other half-NS5, but not on both.
The situation in Fig. 2.8 is a simple illustration of the fact mentioned in the introduction that in the presence of O7 + we lose the notion of a canonical assignment of gauge algebras and matter content. In this situation, this happens for two reasons. First, we can only take m D6-branes from bottom to top of the cylinder. After doing that, we are still left with 4 D6-branes ending on half-NS5-brane. This implies that there is no canonical 'zero' for the Wilson lines. Second, the half-NS5s are stuck at fixed values of x 4 . This implies that there are fixed non-zero periods of NS-NS 2-form potential on the curves.

Tangential intersections and O8-planes
The discussion of I * -I and I * -I intersections in section 2.3.1 has an interesting exception, that occurs when the intersection is tangential. We discuss it now because T-duality helps in the analysis, as we will now see.
We start by considering O7s and D7s that again share the directions 056789, but which are extended in the remaining directions in a more complicated fashion than in section 2.3.1. Define z = x 1 + ix 2 , w = x 3 + ix 4 , and let the orientifold act on the spacetime by σ : z ↔ w. The O7 ∓ will then be on the locus z = w; place again n ± 4 D7s on top of it. Now also place m half-D7s on the locus z = 0; their m images will be on the locus w = 0. In this case, the gauge fields on the D7s on z = 0 will have a U(m) gauge field, which the O7 maps to a gauge field on the D7s on w = 0. To see why this is related to a tangential intersection, consider the invariant coordinates v = z + w, u = zw. The configuration we are considering is then mapped to an O7 ∓ + (n ± 4) D7s on the locus v 2 = 4u, and m D7s on u = 0. These two loci intersect tangentially. We can summarize this as follows: where we have used || to denote tangency as in [48]. This coordinate change is illustrated in the top part of Fig. 2.9, in the O7 + case. An additional subtlety concerns the matter content in (2.23). One can in principle work this out directly in the original setup on the left of Fig. 2.9, but it is instructive to do it instead in a dual frame. First of all we change coordinates, using again (2.18); only this time we take z = x 1 + ix 2 , w = x 3 + ix 4 introduced earlier, and define new coordinatesx 1 + ix 2 = zw,x 3 = |z| 2 − |w| 2 , with a fourth periodic coordinate e ix 4 = zw zw . We are once again rewriting R 4 as a fibration S 1 → R 4 → R 3 . The orientifold is now defined by the involution σ :x 3 → −x 3 ,x 4 → −x 4 ; the O7-plane then sits at x 3 =x 4 = 0, while the D7s are on the locusx 1 =x 2 = 0. (Notice that thex 4 circle shrinks atx 3 = 0.) If we now T-dualize along direction 4, we end up with an O8 at x 3 = 0 with a half-NS5 stuck on it, and with D6s crossing it.
All this is depicted on the lower part of Fig. 2.9, again for the O8 + case. At this point we can read off the matter content from a perturbative string computation similar to the one leading to (2.4), as already done in [16,17]; the result is that in the tangential intersection (2.23) the u m has a hypermultiplet in the antisymmetric in the unfrozen case, and in the symmetric in the frozen case.
We can deform a tangential intersection into two transverse intersections. This corresponds to giving a vev to the hypermultiplet in the antisymmetric or symmetric representation, and breaks the gauge algebra to sp or so respectively. We will study an explicit example in Sec. 4.2.

Anomaly analysis
In this section we discuss the cancellation of one-loop anomalies and the Green-Schwarz contributions in 6d compactifications with frozen seven-branes.

Anomaly cancellation with frozen singularities
A compactification of F-theory on an elliptically fibered Calabi-Yau threefold gives rise to an effective 6d gauge theory with N = (1, 0) supersymmetry at low energies. When there are no frozen singularities present, it is possible to turn off the holonomies of gauge fields on stacks of seven-branes, and the periods of 2-form NS-NS and R-R potentials. Then, each simple summand g i of the 6d gauge algebra is associated to a single irreducible component D i of the discriminant locus of the elliptic fibration, and can be determined from the knowledge of the type of singular fiber over D i along with the data of the monodromy of the elliptic fiber around D i [3,43,49]. The matter content [50,51] and the coupling of tensor multiplets [27] is encoded in the intersection numbers of various divisors in the base of the elliptic fibration. These data allow us to compute both the 1-loop contribution I 8 1-loop to the anomaly polynomial, as well as the Green-Schwarz contribution I 8 GS to the anomaly polynomial. Combining these two, one finds that I 8 = I 8 1-loop + I 8 GS vanishes for any smooth elliptically fibered Calabi-Yau threefold [52,51]. Now let us include frozen singularities in the geometry. In this situation, it is not always possible to tune the above mentioned holonomies to zero. We do not have any canonical nonzero choice either. Because of the nonzero holonomies, one is forced to consider situations in which simple summands of the 6d gauge algebra are realized on divisors which are positive linear combinations of irreducible components of discriminant locus. We will call the divisors associated to simple summands of gauge algebra as gauge divisors.
In this paper, we will not be able to list down all the possible 6d spectra that could result from a geometry, as that will require a systematic understanding of holonomies and fluxes in F-theory compactifications, which we do not have at present. Therefore, we suppose that an assignment of gauge algebras on the components of the discriminant is given, and study the Green-Schwarz contribution to the anomaly. We follow the work of Sadov [27] but we include the effects from the frozen singularities.
The 6d tensor multiplets descend from Kaluza-Klein reduction of the chiral 4-form C (4) of type IIB string theory. To determine the coupling of 6d tensor multiplets, we need to look at two couplings of C (4) in ten-dimensional type IIB string theory, namely the coupling to the gauge theory living on seven-branes and the coupling to gravity in the bulk.
Gauge Green-Schwarz terms: We start with the coupling of the gauge fields to the RR 4-form field C (4) . When there are no O7 + -planes, the stack of seven-branes on D a has a ten-dimensional coupling given by where F a is the field strength valued in the "Kodaira" 8d gauge algebra k a on the D a component of the discriminant, and ν(F a ) is the instanton number density 14 , normalized so that it integrates to one on the standard BPST instanton embedded into k a with embedding index 1. This normalization reflects the familiar fact that an instanton in the worldvolume of a seven-brane has D3-charge 1.
When the component D a carries an I * n+4 singularity, i.e. when it corresponds to an O7 + -plane with n D7-branes on top, the local 8d gauge algebra is k a = sp n , and the ten-dimensional coupling is Note a factor-of-two difference in the coefficient between (3.1) and (3.2). This is due to the fact that a bulk D3-brane can fractionate into two on O7 + , as reviewed in Sec. 2.2, and the gauge instanton in the standard normalization corresponds to the D3-brane of minimal possible charge. Let us now write the 6d gauge algebra in the form ⊕ i g i where g i is simple. Each g i is shared on some of the D a ; we let µ i,a = 1 or 0 depending on whether g i is on D a or not. An embedding ρ i,a : g i → k a must exist whenever µ i,a = 1, and otherwise we let ρ i,a be the zero map. These embeddings have the properties 14 In the literature many different conventions have been used; tr F 2 is defined variously as the trace in the smallest nontrivial representation (e.g. [20]), or in the adjoint representation divided by the dual Coxeter number (e.g. [36]), or by twice the dual Coxeter number (e.g. [52,51]), with or without (2π) 4 in the denominator implicitly included. We follow the physical convention introduced by Intriligator in [53], where the notation c 2 (F ) was used. This choice has the virtue that the coefficient in the resulting anomaly polynomial of the term ν(F a )ν(F b ) have a direct physical meaning, i.e. the Dirac pairing of two instanton-strings.

1.
i ρ i,a (g i ) ⊂ k a , and 2. g i is the diagonal in a ρ i,a (g i ).
The Green-Schwarz coupling for the gauge fields is given in terms of the field strenghs F i valued in g i by where we defined the i-th gauge divisor to be and o i,a is the embedding index 15 of g i ⊂ k a , multiplied by 1/2 when k a = sp n is supported on a frozen singularity. Note that even when there is no "sharing" (so the gauge divisors are Σ a = D a ) and no O7 + -planes, g a could still be different from k a , due to the "Tate monodromy" phenomenon [43].
Before proceeding, we point out here that the inverse square of the gauge coupling of g i is given by a µ i,a o i,a A a where A a is the area of D a . This follows from the fact that the scalar A a and the 2-form Da C (4) are the bosonic components of a single supermultiplet, and therefore Green-Schwarz coupling C (4) a µ i,a o i,a ν(F i )D a comes with the coupling a A a µ i,a o i,a tr F i ∧ * F i . This means in particular that when the gauge algebra g i is shared on multiple components, the gauge theory does not become singular when a single component D a involved in the gauge divisor shrinks to zero size.
Gravitational Green-Schwarz terms: We turn our attention to the gravitational coupling. When there are no O7 + s, the stack of seven-branes on D a has a ten-dimensional 15 The embedding indices we often encounter in this paper can be summarized in the following diagram: u 2n sp n so 2n u n where N a is the order of vanishing of discriminant ∆ on D a , p 1 (T ) is the Pontryagin class of the tangent bundle of the worldvolume. We also slightly abuse notation and use D a within the integral to represent the two-form determined by the divisor. 16 In particular, a D7-brane contributes N a = 1 and an O7 − -plane contributes N a = 2. Now, the contribution of O7 + to this gravitational coupling is opposite to that of O7 − ; the "effective N a " is −2. Since an I * n singularity corresponds to O7 + + (n − 4)D7branes, its "effective N a " is −2 + (n − 4) = n − 6. In comparison, N a of I * n is n + 6. Hence, in the presence of O7 + we need a correction term to the coupling, which be written as where s a = 1 when the curve D a carries an O7 + and s a = 0 when it does not.

The cancellation: Combining (3.3) and (3.7), the full six-dimensional coupling relevant for the Green-Schwarz mechanism is
where C (4) has two legs on the base B and F = a s a D a (3.9) is the frozen divisor, signifying the divisor along which we find the frozen singularities. We have also used the condition for unbroken supersymmetry (the Calabi-Yau condition) to substitute the canonical divisor K in place of − 1 12 N a D a . The contribution to anomaly polynomial is then a square of the coefficient of C (4) , with a factor of 1/2 in front, to take into account that the RR 4-form field is self-dual: (3.10) It is a standard result (see e.g. [55-57, 45, 27]) that the one-loop anomaly of the 6d system is given by 17 where n T is the number of tensor multiplets, and N i , M ij are some numerical coefficients, assuming that the coefficient of tr R 4 vanishes, i.e.
At the end of this subsection, we comment on how to obtain the numerical values N i and M ij . We see that cancellation of gauge and gauge-gravity anomalies requires the following: Here, K · Σ i = K · ( µ i,a o i,a D a ) can be computed from the adjunction formula 2(g a − 1) = (K + D a ) · D a , where g a is the genus of the curve D a . If the 6d theory contains dynamical gravity and satisfies (3.12), then we obtain the following condition as well, from the vanishing of the coefficient of (tr R 2 ) 2 : 9 − n T = (K + F ) 2 (3.14) This condition (3.14) follows just from geometry, as we now demonstrate. If D a carries a frozen singularity, then the singular fiber over D a has Kodaira type I * n≥4 . For these Kodaira fibers, it is known that D a · (−2K − D a ) = 0 [51]. Moreover, any two distinct components D a and D b of F must not intersect each other because two I * n≥4 singularities cannot intersect each other(in the absence of conformal matter). From 17 Again there are various different normalizations in the literature. We follow the convention that 2πiI 8 ours yields the anomalous phase variation via the descent formalism; in particular I 8 should have rational coefficients when expressed in terms of geometrically-defined characteristic classes. The early paper by Erler [55] used I 8 Erler = 2πiI 8 ours . Another common convention during the early years of the second revolution, apparently introduced by Schwarz [56], was to normalize I 8 to contain (tr R 2 ) 2 with coefficient 1, for a model with one tensor multiplet. We have I 8 Schwarz = 16(2π) 4 I 8 ours .
these two facts, it follows that 15) and the equality K 2 = 9 − n T . 18 By now, the cancellation of the Green-Schwarz anomaly and of the one-loop anomaly in the conventional F-theory compactification without O7 + is well-established. This allows us to read off N i and M ij for almost all the cases. First, for i = j, we have M ij = 1 for a bifundamental of su-su or a half-bifundamental of so-sp. To read off N i and M ii (without suming over i), let us say that the given algebra is g i and the total set of hypermultiplets for g i is ρ. One then looks up the pair of (g i , ρ) e.g. in Eqs. (2.10)-(2.14) of [58], to find a conventional F-theory realization of the 6d gauge theory on a sphere of self-intersection −n. Then M ii = −n and N i = n − 2. Essentially the only case not covered by this procedure is when g i = su(n), with one sym and n − 8 fundamentals. For this, one first Higgses the hypers in sym, to give so(n) with n − 8 fundamentals. This has a well-known anomaly polynomial, which can be determined in the method just described above. Then one can convert it back to the anomaly polynomial of the original su(n) theory by using ν(so(n)) = 2ν(su(n)).

Matter content with frozen singularities
Transversal intersections: In the situation when there are no frozen singularities and each simple factor of gauge algebra g a is associated to a single irreducible component of discriminant locus D a , Grassi and Morrison [51] wrote down the matter content charged under g a in terms of intersection numbers of combinations of D a and K, assuming that every intersection among D a and D b is transversal. The geometry underlying the derivation of those formulas, analyzed in the M-theory dual (and therefore on the Coulomb/tensor branch of the theories), consists of finding the curves in the total space upon which M2-branes can be wrapped, and finding the intersection numbers of those curves with the divisors which represent the Cartan subgroup of the original nonabelian gauge group, since those intersection numbers specify gauge charges. This was carried out in a number of works [43,50,59,49,60] which [51] relied upon. Now we would like to understand the matter content in the presence of the frozen g a ρ Number of hypers in ρ su 2 adj Table 3.1: Number of hypermultiplets for each relevant representation of each simple gauge algebra when frozen singularities are present. This includes the contribution of vector multiplet as a −1 hypermultiplet in adjoint. By definition, K = K + F . spin * denotes the sum of number of hypers in two irreducible spinors spin ± for so even , and the number of hypers in the unique irreducible spinor for so odd . For so 8 , number of hypers in spin + must equal number of hypers in spin − . The two different proposals for so 7 coincide when (−2K − Σ i ) · Σ i = 0. For so n≥15 we have a further constraint that Σ i · (−2K − Σ i ) = 0, and for e 8 we have a further constraint that (6K + 5Σ i ) · Σ i = 0. singularities. We do not have a geometric derivation for our proposed answer, since the M-theory geometry of frozen singularities is not well understood. However, as we have seen in detail, the effect of the frozen singularity in the anomaly contribution from the Green-Schwarz effect is summarized by the replacement of individual components D a by the gauge divisor Σ i , and of the canonical class K by K = K + F . The one-loop contribution should then be able to exactly cancel this contribution. We thus propose that the corect answer for the matter content is to perform the same replacement in the results of [51]. We tabulate the results of this replacement, i.e., of our precise proposal for matter content, in is the number of hypermultiplets charged in the adjoint representation. The −1 incorporates the contribution to the anomaly of a vector multiplet, which indeed comes with the opposite sign with respect to an adjoint hypermultiplet.
• For so even , the number of hypers in spin * denotes the combined sum of the number of hypers in the two irreducible spinor representations spin ± . For so odd , the number of hypers in spin * denotes the number of hypers in the unique irreducible spinor representation.
• For a generic so even we can choose the number of hypers in spin + and spin − arbitrarily as long as their sum equals the number of hypers required in spin * . However, for so 8 , the number of hypers in spin + must equal the number of hypers in spin − , because there are two linearly-independent Casimirs of degree 4. See [51] for more details on this requirement.
• The entry for so 7 in our table contains a refinement over [51], in which only the spinor representation was considered. But the coefficient of the spinor representation is negative whenever (−2K − Σ i ) · Σ i < 0. In this case, a different representation with the same contribution to the anomaly needs to be used. One finds sym 2 irr does the job. 19 • For g a = so n≥15 , we have the additional constraint Σ i · (−2K − Σ i ) = 0. The physical meaning of this constraint is that the intersection points of Σ i and −2K − Σ i carry spinor representations, but it is impossible to satisfy anomaly cancellation for so n≥15 if we have hypermultiplets transforming as spinors. There is a similar constraint for e 8 which states that (6K + 5Σ i ) · Σ i = 0.
Tangential intersections: We know that this simple replacement cannot be the full story. We saw at the end of Sec. 2.8 that if a curve carrying frozen singularities intersects a curve carrying I n singularity tangentially, then it traps a hypermultiplet in the twoindex symmetric representation of su n . In light of this, for g i = su n we define t a to be the number of tangential intersections of F with D a . Let t i = a µ i,a t a , in terms of which we write our modified proposal for su n as This still satisfies anomaly cancellation because σ = −adj + asym 2 + sym 2 has the property that tr σ F 2 and tr σ F 4 are both zero. This proposal gives correct predictions for models which have a perturbative dual for which the spectrum can be determined by other methods.

Noncompact models
Now let us analyze how the anomaly cancellation works out in a few examples. We are particularly interested in 6d SCFTs which supplement the lists given in [12,13]. As in [12,13], we expect to be able to realize the tensor branch of a 6d SCFT by means of a contractible collection of curves in the F-theory base, with the difference that we will now allow frozen branes as well.
non-negative and the number of adjoints are ≥ −1, we obtain: Combining the first and third inequalities, we find that Σ 2 i ≥ 0. Combining the second and third inequalities, we find that (8 − n)Σ 2 i ≥ 32. These two together imply that n < 8.

so-sp chains
We first come back to the setup discussed in Sec. 2.6. In the type IIA frame, we consider the following chain: where the first, the second, the third row shows the gauge algebra, the negative of the self-intersection number, and the singularity type, respectively. Denoting the two CP 1 's in the middle by D 1 and D 2 , the Green-Schwarz contribution to the anomaly is In the second case we obtain a setup with frozen singularities: Note that the gauge group, matter content, and flavor symmetry group of (4.5) are identical to those of (4.3): only the F-theory realization is different.
Denoting the two middle CP 1 's byD 1 andD 2 this time, and the canonical class by K to distinguish it from the case above, the Green-Schwarz contribution is now (4.6) where the factor 1/2 in front ofD 1 is due to the fractionation of D3-branes on O7 + , and the factor 2 in front ofD 2 is due to the embedding index 2 of so 2n+8 ⊂ su 2n+8 . The frozen divisorF isD 1 +D 3 , whereD 3 is the noncompact divisor on the far right. The terms with tr F 2 sp and tr F 2 so in the two expressions (4.4) and (4.6) should agree, since they cancel the same 1-loop anomalies. Indeed, we can easily check that In addition, as observed earlier, K 2 = (K +F ) 2 .

su-su chains
Let us next consider the IIA configurations shown in Fig. 4.1. The top row and the bottom row are distinguished by the type of the O8-plane; we add 16 D8-branes for the top row to have the same Romans mass for the both rows. The configurations on the left column contain tangential intersections of the type discussed in section 2.8. The configurations on the right column are obtained by moving the half-NS5-brane at the intersection of the 6-branes and the 8-branes away from the intersection. Gauge theoretically, this operation corresponds to giving a vev to hypermultiplets. Using the discussion in section 2.8 and following [16,17], we find that these configurations realize 6d quivers whose structures are summarized in Fig. 4.2. (We did not explicitly write in that figure the standard bifundamental matter hypermultiplets between two consecutive gauge factors.) The type of the O8-plane is correlated to the type of the two-index tensor representation of the su n gauge algebra. Higgsing is done by giving a vev to the hypermultiplet in the antisymmetric or symmetric two-index tensor representations of su n , breaking it to sp n/2 or so n . Here for simplicity n is assumed to be even in the former case; if n is odd, the gauge algebra is sp n/2 and one needs to add a flavor to su n−8 .
We can discuss the F-theory duals by T-dualizing the original IIA configurations along the lines of section 2.8; the results are shown in Fig. 4 4 or I * 4 is tangent to the divisor with I n singularity. To go to the right column, we deform the divisors so that the tangent point is split to two transversal intersection points. This operation in turn changes the singularity type from I n to I ns n . The two models on the bottom row realizes 6d quiver gauge theories (the tensor branches of 6d SCFTs) which were not previously possible in an ordinary F-theory compactification without frozen singularities.
Let us name the four divisors in each model as C 1 , D 1 , D 2 , C 2 from the left to the right; C 1,2 are non-compact and D 1,2 are compact. The Green-Schwarz contribution to the anomaly can be written down as follows.
For the top row with the non-frozen I * 4 singularity, we have both before and after the Higgsing. For the bottom row with the frozen I * 4 singularity, we have where we used the fact that the frozen divisor is C 1 . After the Higgsing, the Green-Schwarz contribution is where the factor in front of D 1 is due to the embedding index of so n ⊂ su n . 20 . It is a straightforward exercise to show that these Green-Schwarz contributions correctly cancel the gauge squared and the gauge-gravity part of the one-loop anomalies. The construction discussed here gives a first indication of how the classification results of [12,13] need to be modified in order to include frozen branes. We leave a thorough consideration of the effect of frozen branes on this classification to future work.

Compact models
In this section we discuss some compact models with O7 + -planes in F-theory language. They are obtained from very classic F-theory models by flipping some of the O7 − to O7 + . Our current understanding of the compact models is rather incomplete. In this paper we will be content with presenting some of the consistent assignments of gauge algebras and hypermultiplet matter content, leaving systematic studies in the future.

The F −4 model and its flip
Without frozen 7-brane: Aspinwall and Gross considered the following model [62]: the F-theory base is the Hirzebruch surface F −4 , which is a CP 1 bundle over CP 1 such that the base is a −4 curve. We have the I * 12 singularity along the −4 curve C and a fiber Φ hosts an I ns 48 singularity; see This model has the following massless matter content: • so 32 on C and sp 24 on Φ, • a half-hypermultiplet in 32 ⊗ 48, • a hypermultiplet in ∧ 2 48, together with • one supergravity multiplet, one tensor multiplet and 20 neutral hypermultiplets.
Let us remind ourselves how this spectrum can be understood in a dual frame. We start from the heterotic or type I so 32 on a K3, realized as an elliptic fibration over CP 1 . The Green-Schwarz mechanism in ten dimensions requires that the instanton number of the gauge bundle over K3 is 24. To keep the whole so 32 gauge algebra unbroken, we use 24 point-like instantons. We then collapse the whole 24 instantons to a point. This is known to generate sp 24 on the heterotic side [63]. The spectrum as written above can be found perturbatively on the type I side.
Assuming that the elliptic fiber has small area, we perform fibre-wise the duality between heterotic on T 2 and F-theory on an elliptically-fibered K3. This converts the whole to an elliptically-fibered K3 fibered over CP 1 . The so 32 gauge algebra is now realized on the base C as the I * 12 singularity, and the point-like instanton is on the fiber Φ as the I ns 48 singularity.
Here and below, we mean by the sentence "a gauge algebra g on D" that the gauge divisor associated to g is D, in the language of Sec. 3. Let us give a derivation of these spectra, using the same duality as in the unfrozen case shown above. We again start from the heterotic or type I so 32 on a K3, realized as an elliptic fibration over CP 1 , but with the generalized Stiefel-Whitney class of Spin(32)/Z 2 being nonzero along the fiber, destroying the vector structure [10]. The maximal possible gauge algebra is now sp 8 . We now need a gauge configuration of instanton number 12 on the K3 surface, since the embedding index of sp 8 ⊂ so 32 is two. We choose to put all 12 point-like instantons at the same place.
The spectrum can be determined perturbatively using the type I description. 21 We find that when the point-like instanton is on a generic point, the spectrum is as in the Higgsed case above, while when it is on a singularity of the form C 2 /Z 2 , the spectrum is the one before the Higgsing.
To go to the F-theory frame, we perform the fiber-wise duality as before. This time we use the frozen version reviewed in Appendix A, which relates heterotic or type I so 32 on T 2 without vector structure to F-theory on K3 with one frozen singularity. We now have the I * 12 singularity on C and the I ns 48 singularity on Φ. The Higgsing distinguishing the two versions is related to how the residual part of the discriminant with the I 1 type singularity intersects with the fiber Φ.

The unfrozen CP
The next compact model we consider was first considered at a perturbative level by Bianchi and Sagnotti [64] before the second superstring revolution, during which these models were revisited by many others, including by Gimon and Polchinski [37]. In this subsection we will consider its F-theory realization in the case where all O7-planes are O7 − ; in section 5.3 we will consider what happens by changing one or both of them to O7 + .
The model is obtained by considering the T 2 /Z 2 × T 2 /Z 2 compactification in type IIB theory, with O7 − at each Z 2 singularity, together with 16 mobile D7-branes along the first T 2 /Z 2 and another 16 mobile D7-branes along the second T 2 /Z 2 . We give the perturbative derivation of the spectrum of these models in Appendix B. The aim here is to understand the spectrum from the F-theory point of view. 22 Conformal matter point: Since T 2 /Z 2 CP 1 , we take the F-theory base to be CP 1 × CP 1 . We pick divisors C and D wrapping each of the CP 1 above. We let each divisor host an I * 12 singularity. At the intersection we expect to have the conformal matter theory (see footnote 12) D 32 , where so 32 × so 32 ⊂ so 64 is gauged.
Let us see this in more detail. We choose coordinates ([s, t], [u, v]) on CP 1 × CP 1 , and consider bihomogenous polynomials. We want to engineer I * 12 along t = 0 and also along v = 0. Doing so is quite constrained. The equation defining the elliptic fibration 21 An analysis after a T-dual along one direction in the T 2 without vector structure is given around equation (B.3) of Appendix B.2. 22 Analyses of unfrozen compact models with conformal matter will also be provided in [65] by other authors, where a detailed analysis of the CP 1 × CP 1 model with four D7-branes per O7 − -plane is given, among others. was derived in [66] but we follow the notation of [67,Eq. (42)]: where p 3,3 is bihomogeneous of degree (3,3). (We shall usually suppress the variables in writing polynomials such as p 3,3 .) This equation is not in Weierstrass form. By completing the cube, we find By the Kodaira vanishing criteria, we indeed see I * 12 along t = 0 and v = 0. Therefore we have D 32 conformal matter at t = v = 0.
The discriminant has components t = 0 and v = 0 along which I * 12 fibers are located, as well as two components which comprise the "residual discriminant" (the part with no gauge algebra or type II enhancement). We denote these by ∆ ± , and note that the defining equation of each has bidegree (3,3). Both of these components intersect with t = 0 at the three points t = p 3,3 = 0, and similarly intersect with v = 0 at the three points v = p 3,3 = 0.
The multiplicities of f , g, and ∆ at all six intersection points are (4,6,20) so there is conformal matter at those points as well. We can roughly see how this F-theory setup corresponds to the perturbative model reviewed in App. B.1. The I * 12 curves are the counterpart of the O7 − -planes with 16 D7s on top. The residual discriminant corresponds to the O7 − -planes without D7s. As is customary, such planes are realized in F-theory by a pair of I 1 curves.
We illustrate this initial model, which is at its transition point between tensor and Higgs branches, in Fig. 5.2. We have marked the seven CFT points with red dots. The curves ∆ + and ∆ − have a third order contact with each other at each point of intersection, and also pass transversally through one of the I * 12 curves (labeled C and D) at each such point. Key mathematical features not found in the illustration include the intersection data: C 2 = D 2 = 0, C · D = 1, K · C = K · D = −2, ∆ 2 + = ∆ 2 − = ∆ + · ∆ − = 18, and K · ∆ + = K · ∆ − = −12. It follows that ∆ + and ∆ − each have genus 4. The matter content is then • so 32 on C and so 32 on D, • the conformal matter D 32 gauged by so 32 × so 32 ⊂ so 64 • three copies of D 16 gauged by so 32 , • three copies of D 16 gauged by so 32 , • one supergravity multiplet, one tensor multiplet and 13 neutral hypermultiplets.
Higgsed model: To obtain the standard perturbative massless spectrum of the model, we can Higgs the conformal matter theories. From (2.12) and Fig. 2.6(a), we see that at the collision point of two I * curves there is a hypermultiplet which, when activated, breaks the global symmetry from 23 so 2n+8 ⊕ so 2n+8 to su n+4 ⊕ su n+4 .
Doing this for the conformal matter theories in the model of Fig. 5.4(b), one reproduces the perturbative spectrum: • u 16 on C and u 16 on D, • a hypermultiplet in 16 ⊗ 16 , 23 The u 1 part of both u 16 are known to get Higgsed by the Green-Schwarz mechanism, eating one neutral hypermultiplet each, and becoming massive [45,Sec. 2]. Here we follow the older perturbative string terminology.
• two hypermultiplets in ∧ 2 16, • two hypermultiplets in ∧ 2 16 , • one supergravity multiplet, one tensor multiplet and 20 neutral hypermultiplets, as can be found in the original papers [64,37,45], and reviewed in App. B.1. The Ftheory interpretation of this Higgsed spectrum was given in [9,68]; this study eventually led to a refined understanding of the relation between F-theory and O7 − [69]. 24 So far we used the process of Fig. 2.6(a), which is non-geometric in F-theory, to realize the Higgsed spectrum. We also expect that giving vevs to other scalars in the same hypermultiplet would have the same effect. We thus seek a geometric deformation of the original equation (5.1) in which the Kodaira fibers I * 12 become Kodaira fibers I s 16 . The deformation involves a new polynomial q 2,2 of bidegree (2, 2) and takes the form When we complete the square and then complete the cube, we find the data for Weierstrass form: This is Kodaira type I s 16 on each curve; it is split because (g/f )| t=0 = − 1 18 εq 2 2,2 | t=0 is a perfect square, and likewise for v = 0.
This time, the intersection of t = 0 with the residual discriminant is at two points t = q 2,2 , each of which has multiplicities of (f, g, ∆) being (2,3,18). Such an intersection point is associated to a matter representation Λ 2 rather than to conformal matter, so the corresponding points should not be blown up. The same is true of the two points v = q 2,2 = 0.
Similarly, the intersection of t = 0 with u = 0 is ordinary bifundamental matter, and this point should not be blown up either. The geometry is illustrated in Fig. 5.3. This model reproduces the perturbative spectrum again, but this time with a geometrical Higgsing.
Tensor deformation: We will now consider the tensor branch deformation of the model with conformal matter in Fig. 5.2.
The blowup of the collision point t = v = 0 is straightforward and produces an exceptional curve E along which the Kodaira type is I ns 24 . Let us study the intersection points of p 3,3 = 0 with t = 0 in more detail. By a change of coordinates, we may locate one of the intersection points at t = u = 0. In that case, we can write p 3,3 = u p 3,2 + tp 2,3 . Multiplicities of f , g, and ∆ at t = u = 0 are easily seen to be 4, 6, and 20 so we have a conformal fixed point and we need to blow up. To perform the blowup, we work in the affine coordinate chart v = s = 1. In one coordinate chart of the blowup, we have t 1 = t, u 1 = u/t, and the Weierstrass coefficients and discriminant become The exceptional divisor t 1 = 0 supports an I 8 fiber, since the orders of vanishing are (0, 0, 8), and there is monodromy: the usual branch divisor (g 1 /f 1 )| t 1 = 0 vanishes at u 1 = t 1 = 0 in this chart and has a single order of vanishing. Thus, this is I ns 8 and the gauge algebra is sp 4 . No matter is visible in this chart. Note that this branch point is the point at which the residual discriminant meets the exceptional divisor. The multiplicities at this point are 2, 3, 10 which is consistent with the enhancement from A 7 to D 8 which is expected at such a point. In the other coordinate chart of the blowup, we have t 2 = t/u, u 2 = u. The Weierstrass coefficients and discriminant become 14) and we indeed see the exceptional divisor u 2 = 0 meeting the original I * 12 at t 2 = 0. This intersection point also provides the second branch point defining the monodromy.
This same analysis applies at all six points t = p 3,3 = 0 and v = p 3,3 = 0 so six additional blowups need to be done. All in all, we have blown up CP 1 × CP 1 at seven points, and we obtain a model with no conformal matter and with eight tensor multiplets.
This model is illustrated in Fig. 5.4(a) and Fig. 5.4(b). The curves ∆ + and ∆ − are now simply tangent at each of their points of intersection, which occur at a point of one of the new exceptional divisors C j or D j . The intersection data this time are: Note that because of the tangencies we now have ∆ + · ∆ − = 12.
The massless matter content is: • so 32 on C, sp 12 on E, so 32 on D, and a copy of sp 4 on each C j and on each D j , • a half-hypermultiplet in 32 ⊗ 24, a half-hypermultiplet in 24 ⊗ 32 , three halfhypermultiplets in 32⊗8 (corresponding to (C, C j )) and three half-hypermultiplets in 8 ⊗ 32 (corresponding to (D j , D)), together with • one supergravity multiplet, eight tensor multiplets and 13 neutral hypermultiplets.

Frozen CP 1 × CP 1 models
We will now consider what happens in the CP 1 × CP 1 model of section 5.2 when one changes the type of one or both O7 − to O7 + .
With two frozen seven-branes: Let us first consider what happens when one changes both O7 − to O7 + . In the original geometry without blowups in Fig. 5.2, the two I * 12 curves are now changed into I * 12 . Since the residual discriminant represents O7 − -planes, we do not expect conformal matter at the its intersection with the I * 12 curves, which represent O7 + -planes. We have not analyzed this situation before, but we expect it to be similar to the one in Fig. 2.8 and (2.15), thus with no conformal matter and a tensor.
We do not venture to guess the field theory content at this point. It is easier to follow a tensor deformation by blowing up all curves. The result lives again on the geometry illustrated in Fig. 5.4, I * 12 → I * 12 . A possible choice of gauge divisors that cancels all gauge anomalies gives the following model: • (sp 2 ) 1,2,3,4 , supported on C +C 1 +C 2 +E, C +C 1 +C 2 +2C 3 +E, D +D 1 +D 2 +E, • so 8 supported on 2E, • hypermultiplets in 4 i ⊗ 4 j for i < j, • one supergravity multiplet, 8 tensor multiplets, and 13 neutral hypermultiplets.
The sp groups living on the I * 12 curves have been shared with the I ns 8 curves, just like in Fig. 2.8, where O7-planes of different types meet. The so 8 has appeared on the E curve just like in a usual tensor-Higgs transition. Indeed the field content is related to the one for the perturbative model in App. B.3 by such a transition.
With one frozen seven-brane: We now consider what happens if only one of the I * 12 is changed to I * 12 (say D). As in the previous case, we don't try to write the field content at the singular locus; we instead follow the tensor deformations. Here we encountered a problem: we have not found a credible model that cancels all anomalies after blowing up all those singular points, perhaps because of some global constraint.
What we were able to achieve is the following. We consider the geometry in a situation intermediate between Fig. 5.3 and Fig. 5.4(a), namely, we tune the Kodaira type on D to be I * 12 , but we deform the Kodaira type on C to be I 16 . Then, the intersections of D with others are still conformal points and need to be blown up, but the intersections of C with others are smooth. We then only need to blow up the intersection of C and D, and the intersections of the residual discriminant and D.
This is exactly the spectrum of the perturbative model described in appendix B.2. Note that we chose to blow up the intersections of D with other discriminant loci, while we decided to deform other intersections. In other words, we chose to go to the tensor branch side for the conformal points on D whereas we went to the Higgs branch side for the conformal points on C. This is in accord with our analysis in Sec. 2.3.2, since the perturbative construction naturally gives a tensor at an O7 + -O7 − intersection whereas it gives a hyper at an O7 − -O7 − intersection. As discussed in the main text, we were not able to find a consistent assignment when we have I * 12 on C instead.

A Dimension eight
There are three families [10] of vacua with 16 supercharges in dimension eight. The standard one has gauge algebra of rank 20, the next one has gauge algebra of rank 12, and the final one has gauge algebra of rank 4. The rank-12 case was found in the perturbative type I frame by Bianchi, Prasidi and Sagnotti in [8] in 1992 and then in the context of heterotic string by Chaudhuri, Hockney and Lykken in [73] in 1995; the latter construction is known under the name of the CHL string. An easy generalization of either construction leads to the rank-4 case. The moduli space of these systems and the possible enhancements of gauge algebras are studied in detail in [74].
In this appendix, we give an F-theory description of three cases: they are models on elliptically-fibered K3 with 0, 1, or 2 frozen seven-branes.

A.1 IIB with seven-branes
Let us start by the perturbative IIB setup on the orientifold T 2 /Z 2 . We can either have zero O7 + , one O7 + or two O7 + : with 16, 8 or 0 D7-branes, respectively. The first one, under T-duality, maps to 2 O8 − in type IIA, and then 1 O9 − in type IIB. The last one, under T-duality, maps to O8 − and O8 + , or to a shift-orientifold of type IIA, and then a shift-orientifold of type IIB, without any D9-brane. The second one is more peculiar. One T-duality should combine a pair of two O7 − s to O8 − , while the other pair of O7 − and O7 + to a shift orientifold. The resulting geometry is shown below: Namely, we consider a T 2 whose complex structure modulus is of the form τ ∈ 1 2 +iR, and take the Z 2 flip along the horizontal axis. Then we have just one O8 − locus and a shift-orientifold locus. Another T-duality leads to the Spin(32)/Z 2 bundle without vector structure [10]. equation for these models (in Tate form) is y 2 = x 3 + u 2 (z, w)zwx 2 + v 0 (z, w)z 4 w 4 x (A. 6) with frozen brane-locus δ = zw. Note that v 0 (z, w) is constant, and the x 0 term in the equation vanishes due to degree considerations. This implies that (x, y) = (0, 0) is a section which has order 2 in the Mordell-Weil group, and suggests a subtle modification of the F-theory gauge group. 27 B CP 1 × CP 1 model and its flips via branes

B.1 Unflipped case
The original model considered by Bianchi-Sagnotti and Gimon-Polchinski was given in terms of Type I on T 4 /Z 2 . It has O9 − with 16 D9s in the bulk, with 16 O5 − at the Z 2 fixed points, and 16 D5s. 28 Let us determine its massless spectrum. From the bulk closed string modes, we have one supergravity multiplet, one tensor, and four neutral hypermultiplets. From the Z 2 twisted closed strings, one neutral hypermultiplet arises from each Z 2 singularity.
As for the open strings, O9 − wants to make the gauge algebra on D9 orthogonal. Therefore the bulk of the 9-brane has Spin(32)/Z 2 as the gauge group. But O5 − wants to make the gauge algebra on D9 symplectic. This gives a localized Spin(32)/Z 2 holonomy around the intersection point, and the massless gauge algebra on D9 that can remain is u 16 , the intersection of sp 16 and so 32 . This will keep charged hypermultiplets in 2 · ∧ 2 16.
One can do the same analysis on the D5-branes, and get the same answer, when all the D5s are on a single O5 − . Finally, the 5-9 strings give hypermultiplets in 16 × 16. The spectrum is then • gauge algebras u 16 × u 16 , • charged hypermultiplets in 2 · ∧ 2 16 ⊕ 16 × 16 ⊕ 2 · ∧ 2 16 , • one supergravity multiplet, one tensor multiplet, and 20 neutral hypermultiplets. 27 We are assuming here that the torsion in the Mordell-Weil group is calculated for frozen F-theory models in the same way it is calculated for conventional F-theory models [78]. We leave detailed investigations of this for the future. 28 Here the number of D-branes is counted in terms of Type IIB or Type IIA RR-charge, in a way invariant under T-duality. In simple orientifold models, this number equals the number of mobile D-branes or the rank of the gauge groups, but in more complicated models such as those discussed in this note, they can be different.
We can take T-duality along two directions and bring this model to the type IIB T 4 /Z 2 orientifolds with seven-branes, with the structure below: To deduce the open string spectrum on the 8 D7s perpendicular to the second T 2 , we just T-dualize one direction of the first T 2 and apply the rules of [18]. When the 8D7s are on a generic point, one gets sp 4 with a full antisymmetric tensor (both the traceless part and a singlet), and with 16 fundamentals. If they are on O7 − , it gets enhanced to u 8 with 2 · ∧ 2 8, and if they are on O7 + , it gets enhanced to sp 4 × sp 4 with a bifundamental.
For the 16 D7s on the first T 2 , we can take the T-dual of the second T 2 : This T-duality was derived from the worldsheet point of view in [79]. When 16 D7s are on a single generic point on T 2 , the T-dual is just 8 D6s suspended between two D8s that are in fact the same due to the funny geometry. This is sp 4 with one asym and 16 flavors. When they are all on an O7 − , this gets enhanced to u 8 with 2 · ∧ 2 8. Although we started from 32 Chan-Paton indices but we got just u 8 . We give two other explanations to this somewhat unexpected fact: • If we T-dualize the second torus twice, this describes instantons (or 5-branes) in the Spin(32)/Z 2 gauge fields on T 2 without vector structure. As discussed in [10], a minimal flat Spin(32)/Z 2 configuration without vector structure is in SU(2) embedded in so 32 as sp 1 × sp 8 . Then the instanton needs to be embedded into this sp 8 ; a single such instanton counts as two instantons in the original Spin(32)/Z 2 . In other words, two small instantons of Spin(32)/Z 2 need to move together.
• In the original 7-brane description, there are four intersections with transverse O7s; one is with O7 + and three are with O7 − . The former has a monodromy that squares to −1 and the latter has a monodromy that squares to 1. But one cannot embed them into O(1): they are not consistent, since the four monodromies need to multiply to one. To compensate this, one needs an additional flat SO (3) background on the 7-brane.
Summarizing, when 16 D7s perpendicular to the first T 2 are on a single O7 − and 8 D7s perpendicular to the second T 2 are on a single O7 + , the spectrum is • charged hypermultiplets in 2 · ∧ 2 8 ⊕ ( i=1,2 8 ⊗ 8 i ) ⊕ 8 1 ⊗ 8 2 , • one supergravity multiplet, 5 tensor multiplets, and 16 neutral hypermultiplets. with 8 D7-branes on each T 2 . Using the analysis as in case II, we see that when 8 D7s are on a single O7 + , the gauge algebra is sp 2 × sp 2 . Considering D7s on both T 2 , we have (sp 2 ) 4 in total. The matter spectrum can be worked out as before:
The anomaly cancels; although there are 8 additional tensors, they do not participate in the gauge anomaly cancellation. This is as it should be, since they are localized on the intersections of O7 − and O7 + , and bifundamentals are supported away from them.