$E_6$ Yukawa couplings in F-theory as D-brane instanton effects

In a weak coupling limit the neighborhood of $E_6$ Yukawa points in $SU(5)$ GUT F-theory models is described by a non-resolvable orientifold of the conifold. We explicitly show, first directly in IIB and then via a mirror symmetry argument, that in this limit the $E_6$ Yukawa coupling is better described as coming from the non-perturbative contribution of a euclidean D1-brane wrapping the non-resolvable cycle. We also discuss how the M-theory description interpolates between the weak and strong coupling viewpoints.


Introduction
In recent years, starting with [1][2][3][4][5], F-theory [6] has emerged as a rich and powerful framework in which to do string model building, and more specifically GUT model building. The main desirable feature that distinguishes F-theory from ordinary type IIB model building at weak coupling is the natural appearance of the 10 10 5 Yukawa coupling in SU (5) GUT F-theory models, which in weakly coupled IIB models can only realized via D-brane instanton effects, i.e. nonperturbatively, and is thus expected to be fairly suppressed. Since phenomenological constraints require the 10 10 5 coupling associated to the top quark multiplet to be of order one, a non-perturbative suppression factor makes realistic model building in IIB a challenge. The situation in F-theory seems to be better, with computations of the physical Yukawa couplings in toy models yielding promising results [7][8][9][10]. To a large extent this feature of F-theory justifies the considerable effort spent in developing the theoretical tools behind F-theory model building in the last few years, trying to address the significantly increased technical difficulties involved in computing important physical aspects of the backgrounds (when compared to the weakly coupled IIB approach), such as quantum corrections to Kähler potentials [11][12][13], non-chiral spectra [14,15], D3/M5-instanton effects [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34], or fluxes on 7-branes [14,33, What we will show in this paper is that, despite superficial appearances to the contrary, there is no qualitative distinction between the F-theory and IIB approaches when it comes to the generation of the 10 10 5 coupling: when we take the F-theory models generating the 10 10 5 Yukawa to small coupling we reach a complementary weakly coupled description in which the same 10 10 5 coupling arises from a D-brane instanton effect.
We will provide strong evidence for this assertion from various dual viewpoints via a careful technical analysis, but there are a priori reasons to expect this connection to exist.
The key observation is that, as explained in [56,57], in a specific weakly coupled limit the neighborhood of the E 6 point 1 in SU (5) GUT models becomes an orientifold of the conifold. This orientifold is peculiar in that it projects out the small resolution mode of the conifold, so attempts to study this configuration using ordinary singularity resolution techniques in algebraic geometry are not applicable.
Nevertheless, the non-resolvability of the conifold singularity, in itself, does not obstruct the existence of a perfectly sensible weakly coupled description of the system. Holomorphicity of the Yukawa couplings in the superpotential, together with the fact that the weakly coupled limit is nothing but motion in complex structure moduli space of the fourfold, suggests then that one should equally well be able to compute the 10 10 5 coupling at the E 6 point using purely weakly coupled language. Since in the IIB description the 10 10 5 coupling is forbidden in perturbation theory, it must be generated non-perturbatively by D-brane instantons. These 1 As is conventional we will often refer to the point where the 10 10 5 coupling is generated as the E6 point.
As discussed in detail in [51,57], and reviewed in §B.1.3, the terminology here is somewhat misleading, since the resolved fiber over the Yukawa point is not of IV * type, but we will stick to the usual nomenclature henceforth.
considerations lend support to the idea that the 10 10 5 coupling should be generated by D1-brane instantons at weak coupling (as Donagi and Wijnholt already suggested in [56]). The goal of this paper will be to show that this conclusion is indeed correct, and to initiate the study of some of its implications.
We have organized this work as follows. In §2 we review how to take the relevant weak coupling limit for a neighborhood of the E 6 point. The core of our paper is §3, where we compute, in the weakly coupled IIB description, the instanton contribution to the superpotential, showing that as expected it generates a 10 10 5 coupling. In particular, the actual computation of the superpotential coupling by integration of instanton zero modes is in §3.5. We then rederive the same result in the mirror IIA description in §4. In §5 we analyze the weak coupling limit from the geometric M-theory viewpoint, and reproduce some of the features of the weakly coupled analysis directly in this language. Appendix A contains a technical result we use in the text, and appendix B reviews the M-theory description of the 10 10 5 coupling away from weak coupling.

IIB description of the E 6 point
In this section we briefly review the derivation in [56] for the convenience of the reader, and to set notation. We will focus on a SU (5) model with a 5 and a 10 multiplet which couple via a 10 10 5 coupling. Such couplings can be naturally engineered in the context of local F-theory models (in the small angle limit) by an unfolding of an exceptional singularity. In particular, the 5 and 10 curves should intersect over a point where the enhancement is of E 6 type. 2 This can be easily achieved by writing the local structure of the fibration in Tate form [61,62] y 2 + a 1 xyz + a 3 yz 3 = x 3 + a 2 x 2 z 2 + a 4 xz 4 + a 6 z 6 . (2.1) We choose to denote the transverse coordinate to the SU (5) stack by σ, so we impose a 1 = −b 5 , a 2 = σb 4 , a 3 = −σ 2 b 3 , a 4 = σ 3 b 2 , a 6 = σ 5 b 0 (2.2) with the b i generic polynomials in σ nonvanishing at σ = 0. Following [61] we introduce We are interested in taking this configuration to weak coupling. As discussed in [56] we can achieve this by replacing a 3 → a 3 , a 4 → a 4 , a 6 → 2 a 6 (2.6) and taking the → 0 limit. In this limit and the string coupling goes to zero almost everywhere. The b 2 = 0 component of the discriminant was identified in [63,64] as the location of the O7 − plane at weak coupling, and the b 2 b 6 − b 2 4 = 0 component as the location of D7 branes. For the particular ansatz (2.2) we have ) . (2.9) In this last line we identify the σ 5 component as the SU (5) stack, and the rest of the expression as the flavor brane responsible for the existence of the 5 representation.
The Calabi-Yau threefold where the IIB theory is formulated is given by the double cover branched at the orientifold locus. We are particularly interested in the neighborhood of the E 6 Yukawa coupling point, which is located at [56,62] If we introduce the variables u = b 5 and w = 4b 4 we arrive to the conifold equation [56,57] ξ 2 = (u ) 2 + σw . (2.12) Although this describes a CY threefold with standard conifold singularities, any attempt at a small resolution will be projected out by the orientifold involution. Concretely, there would be two small resolutions, constructed by taking as an ambient space the product and as a subvariety either one of the following 14) The involution ξ → −ξ maps one small resolution into the other. Therefore, neither one yields an orientifold-invariant smooth threefold. One way around this problem is to change the way one takes the weak coupling limit in F-theory. This strategy was studied in generality in [65], and more specifically for this setup in [66]. Although in this way one manages to have a model with a smooth threefold, the nature of the model changes significantly, letting the sought-for Yukawa coupling elude us.
Our strategy will be to deal with the singular space directly, using the language of noncommutative crepant resolutions, which physically entails studying the quiver gauge theory on D-branes probing this conifold singularity.
For our purposes it will be convenient to define a slightly different set of coordinates in which the flavor branes have a simpler expression. We introduce (2.16) so that we have where we have ignored cubic and higher terms in (u, σ, w), which do not affect the singular behavior close to the singular point at ξ = u = σ = w = 0. The virtue of these coordinates is that they will make manifest the algebraic structure of the SU (5) and flavor stacks close to the conifold singularity. In particular, notice that the SU (5) stack is at σ = 0, while the flavor stack is at w = 0. The algebraic structure of these divisors is best understood in terms of the GLSM for the conifold In the first case, the homogeneous coordinates α i represent the coordinates of the exceptional P 1 , which is located at the locus β i = 0. This exceptional P 1 has volume given by |t|.
The correspondence between the toric and algebraic descriptions is explicit under the following map: In these variables the orientifold involution acts as follows: We summarize the toric data for the resulting geometry in figure 1. Note, that this involution is not an automorphism of the resolved conifold. Instead, it is a map from a resolved conifold with Kähler modulus t to a flopped conifold with modulus −t. Only for t = 0 can we regard this as an involution that maps the conifold into itself. As a consequence, the orientifold involution projects out the resolution mode of the conifold. On the other hand, the integral of the B-field over the exceptional P 1 survives, since B has an intrinsic minus sign under the action of (−1) F L Ω.
Note that the SU (5) stack factorizes without any approximation necessary, while factorization for the flavor stack at w = 0 only happens as we zoom into the conifold singularity, since in (2.17) we dropped higher order terms. This distinction is not particularly important for the analysis in the rest of the paper. The subleading terms will affect the precise form of the effective action, but not the existence of the instanton contribution to the superpotential that we find, since the instanton lives at the singular locus.

IIB instanton computation via noncommutative crepant resolution
Our goal in this paper will be to understand the behavior of D-brane instantons living at the singularity appearing at weak coupling. 3 The main difficulty in doing this is that, as we have just seen in the previous section, having F-theory SU (5) models with a 10 10 5 coupling 3 One may worry about the fact that gs formally diverges close to the O7 plane, which is precisely where we want to do our computation. This effect is not incompatible with the existence of a weakly coupled description of the system at any given energy scale. For instance, consider a D3 probe of a O7 − [67]. The divergence of gs on the O7 signals that the probe theory confines [68], but the dynamical scale of this theory can be made arbitrarily small by tuning the ambient string coupling. The conifold can be obtained by partially smoothing an orbifold of this configuration, so it can also be made arbitrarily weakly coupled. of order one means that, in the weak coupling limit, one is forced to deal with a conifold singularity whose resolution mode is projected out by the orientifolding. 4 Indeed, this has been the main obstruction to studying these setups in perturbative string theory, since elementary algebro-geometric methods are not reliable on a singular space. Techniques for analyzing such systems, based largely on mirror symmetry, are known [73][74][75][76][77][78], and the result of applying such techniques will be briefly reviewed in §4 below. Nevertheless the application of these techniques involves a certain amount of heuristics (at least at the level that they are currently developed), so in the interest of making our derivation as assumptionfree as possible, we have opted to give a first principles derivation of the physics at the singularity using the powerful technology of non-commutative crepant resolutions [79,80] (NCCRs in what follows). In practice, the NCCR approach gives us a concrete way of defining D-branes on the singular space, without referring to a small resolution. We will see that computing open string spectra is surprisingly easy in this language.
Armed with the knowledge of the instanton zero-modes present in our system, and how these couple to the background D7/D7-strings, we will be able, in §3.5, to reproduce very straightforwardly the 10 10 5 interaction. We encourage the impatient reader to skip ahead to the derivation of the 10 10 5 coupling in §3.5, and then return here for the systematic justification of the basic ingredients going into the computation.

NCCRs
Intuitively, the NCCR construction can be understood as a replacement of the coordinate ring 5 R = C[u, ξ, σ, w]/(−ξ 2 + u 2 + σw), describing the singular space, with the ring A of open string modes of probe branes. This new ring A, which is also an algebra over R, can be thought of as the path algebra of the quiver. It is non-commutative, since paths cannot be composed in arbitrary order. The fact that the ring R is singular implies that one cannot describe fractional branes easily, as these correspond to modules with infinitely long resolutions. On the other hand, the noncommutative ring A is such that any module will admit a finite resolution. This is the essence of the noncommutative resolution. One further demands that A be Cohen-Macaulay. This is the ring-theoretic analog of requiring a trivial canonical bundle, leading to an noncommutative crepant resolution (NCCR).
We define the conifold threefold algebraically by the equation This variety has a coordinate ring R = C[ξ, u, σ, w]/(u 2 − ξ 2 + σw), and admits a so-called 4 Note that the resolution mode had to be projected out in order for the D1 instanton to have a chance of contributing to the superpotential. Otherwise, by resolving the conifold we could misalign the central charge of the D1 with respect to that of the N = 1 background, and this would imply [69,70] that we could at best generate a higher F-term [71,72], instead of a superpotential contribution. 5 In what follows we will use various elementary notions in category theory freely. For introductions for physicists see [81,82]. matrix factorization, a pair of square matrices 6 (φ, ψ) such that φ·ψ = ψ·φ = (u 2 −ξ 2 +σw)·1. From these two matrices, we can define two so-called maximal Cohen-Macaulay (MCM) modules over R. Essentially these are R-modules defined as the cokernels of the matrices We define For the conifold, these are all the non-trivial irreducible MCM modules up to isomorphism. They are akin to line-bundles over the conifold 7 , except that they fail to be locally free at the singularity. This can be understood by noticing that the matrices ψ, φ have rank one on the conifold, leaving a one-dimensional cokernel over every non-singular point. At the singularity, the matrices vanish, and the cokernels jump to dimension two. So we can think of these as Calabi-Yau filling branes with an added point-like brane at the origin. These modules M andM over R can be fit into exact complexes as follows: We can think of M andM as the cokernels of the maps ψ and φ, respectively. Henceforth, we will replace M orM by their resolution complex as follows: For our complexes, we use cohomological degree staring at zero on the right, and increasing as we move left. We will underline the zeroth position for clarity. The isomorphisms here state that our modules are simply the H 0 modules for these complexes. Note, that these resolutions are semi-infinite, a hallmark of singular spaces. The goal of an NCCR is to replace R with a ring such that all modules admit finite resolutions.
The NCCR for the singular ring R is defined by picking one of these two modules, say M , and defining the endomorphism algebra A = End(R ⊕ M ). A turns out to be a noncommutative ring that will serve as our NCCR. It can be decomposed into four pieces and can be encoded as the path algebra of the following quiver with relations: Here, the arrows are morphisms that can be represented as cochain maps between resolution complexes. Take two complexes K • , C • . Concretely, an element in Hom(K • , C • ) corresponds to a collection of vertical maps f • such that each square commutes, i.e. f i • k i+1 = c i+1 • f i+1 , and modulo homotopies, i.e.
for some collection of the diagonal dotted g i . These notions are reviewed in many physics papers, see [81] for a general account, and [15,83] for concrete examples.
Back to the conifold, the α 1,2 : R −→ M can be written as follows and for β 1,2 : Finally, e 0 ∈ Hom(R, R) ∼ = R and e 1 ∈ Hom(M, M ) ∼ = R are the multiplicative identities (actually idempotents) of the endomorphism ring of each node. These morphisms satisfy the relations: where composition is defined from right to left. For instance, we see that The two column vectors differ by an element of the image of ψ: Such a morphism is discarded in the homotopy category as being gauge-equivalent to zero, and it actually corresponds to the zero morphism at the level of the cohomology. The relations can be encoded in the superpotential W = α 1 β 1 α 2 β 2 − α 1 β 2 α 2 β 1 which we recognize as the superpotential of the Klebanov-Witten theory [84].
The noncommutative crepant resolution is then simply defined as the ring A, which can be identified with the path algebra of the quiver modulo the relations derived from the superpotential. The product of the ring is the concatenation of arrows, and the sum is simply taking complex linear combinations of arrows. D-branes are described in this formulation as complexes of right A-modules. More precisely, the bounded derived category D b (mod−A) comprises complexes of A-modules modulo certain equivalences known as quasi-isomorphism. The interest in the NCCR stems from the fact that this category D b (mod−A), which is defined entirely in the singular geometry, is known to be derived equivalent to the bounded derived category of coherent sheaves of the resolved conifold in both resolution phases. By studying D-branes through the NCCR, we sit in the middle between the two flopped phases, and are never forced to break our orientifold involution.

Quiver representations
A-modules are equivalent to representations of the quiver. We will be interested in two kinds of modules: 1. Modules that correspond to D7-branes. These can be regarded as infinite-dimensional representations.
2. Modules corresponding to fractional D(−1)-branes, which means D1 and anti-D1-branes wrapping the vanishing P 1 of the conifold. These correspond to finite-dimensional representations.
We now explain how to construct these modules.

Recovering the conifold
As a warmup, we will study the finite-dimensional representation corresponding to a complete (non-fractional) D(−1)-brane. Physically, we expect the moduli space of this brane to be its transverse space, i.e. the conifold. By switching on FI terms in the quiver gauge theory, we can reach both flopped phases. A (finite-dimensional) quiver representation consists in assigning a vector space to each node of the quiver, and promoting the arrows to linear maps between the vector spaces it connects. The idempotents e 0 and e 1 are set to one and left out of the diagram. In our case, we are interested in the following representation: Denoting a representation by its dimension vector d, in this case we have d = (1, 1). We can interpret this as a single D(−1)-brane splitting up into one fractional brane on each node, leading to an abelian quiver theory. Here, the α i and β i are simply complex numbers that transform in the (anti-)bifundamental of the relative U (1) ⊂ U (1) × U (1). In order to study the moduli space of d = (1, 1) representations, we quotient out by the action of the relative U (1). This leads us to a toric variety We must also impose a D-term constraint The resolved phases correspond to t > 0 and t < 0. In our case, the orientifold involution (2.21) acts as t → −t, so it is not possible to respect it for non-zero t.

Non-compact branes
There are two basic infinite-dimensional representations P 0 and P 1 , which are the projective right A-modules, defined as follows: These can be thought of as CY-filling branes with line bundles on them. From these two modules, we can build all branes of interest through complexes. Let us now define how the orientifold involution σ acts on a complex. Take a complex K • of the form where the cohomological degree increases from left to right. Then, the orientifold image of this complex is another complexK • defined through the mapping Σ which maps to objects of a complex as follows: where depending on the choice of orientifold involution σ * (K 1−i ) may further act on the object K 1−i (we will determine the proper action below), and it acts on the maps as follows: Concretely, on a complex K • , we have the following action: Let us now define the non-compact D7-branes of interest, which are supported over divisors. We will introduce a stack of brane/image-brane pairs called G 0 and G 1 , where our SU (5) theory will live: We will also need a 'flavor' brane/image-brane pair: We can easily see that Σ : G 0 → G 1 , and Σ : F 0 → F 1 by applying the rules in (3.20) and (3.21), if we assume that σ * (P 0,1 ) = P 0,1 , which is the choice compatible with the geometric action of our orientifold (as will become clear momentarily, when we map these complexes to sheaves).
How can we interpret these modules? One way is to study what happens when we promote the maps in the complexes to coordinates in the moduli space of the d = (1, 1) representation. As we saw in the previous section, the α i and β i become toric homogeneous coordinates for the conifold. Hence, we clearly see by inspection that, for instance, G 0 will correspond to a D-brane supported on the divisor β 1 = 0. The other piece of information we need is how to transport the projective modules to P 0 and P 1 to sheaves on the moduli space. In this case, one choice is to send Hence, we conclude that our modules are mapped to sheaves as follows:

Fractional branes from the (1, 0) and (0, 1)
The fractional branes, which in our case will be fractional D(−1)-instantons, correspond to the two simple (i.e. not admitting subrepresentations), one-dimensional representations: S 0 with d = (1, 0), and S 1 with d = (0, 1). As A-modules, these are simply defined as the modules generated by e 0 and e 1 , respectively: As quiver representations, each consists of a node with a C, and the self-arrow e 0 = 1. Hence, these have no moduli. This is a reflection of the fact that these are fractional branes wrapping a rigid vanishing curve. These fractional branes admit the following projective resolutions: In appendix A we show that these transport to D1 and anti-D1 branes wrapping the vanishing P 1 .
We can compute the Ext 1 's between these complexes by brute force. However, there is a much quicker way to do this, which exploits the fact that these are finite-dimensional representations. Suppose we want to compute Ext 1 (S 1 , S 0 ). This is equivalent to the so-called Yoneda-Ext group that parametrizes all possible non-trivial extensions of S 0 by S 1 , i.e., exact sequences of the form such that E ∼ = S 0 ⊕ S 1 . Clearly, the middle object E, which will depend on the choice of maps (a, b), must always have dimensions d = (1, 1). Hence, we can draw an exact sequence of quiver representations as follows: In this sequence, we have omitted the zeroes corresponding to empty quiver nodes. However, these are important, because arrows going to and from empty nodes are the zero map. A sequence of quiver representations means all squares must commute. In the upper (0, 1) quiver representation, the arrows going to the left are zero. Hence, we must impose that the composition β i • a = 0 for i = 1, 2, which implies β i = 0. For each choice of the pair α i we have an exact sequence of quiver representations. Hence, . For the zero element of this group, i.e. α = 0, we see indeed that the middle element is isomorphic to S 0 ⊕S 1 . We can take this one step further by modding out these maps by the C * that acts homogeneously on these two maps, and excluding the trivial map. This yields a P 1 . We can think of this as the exceptional curve after resolving, since it corresponds to the moduli space for the recombination tachyon between an D1 and and anti-D1 with a net flux difference of O(1). In other words, we could think of this P 1 as the moduli space of the induced D(−1)-brane trapped on this pair. Alternatively, in field theory terms we have that the fact that we have β i = 0 implies that all mesons vanish, so we are stuck at the exceptional locus of the resolution.
Similarly, we deduce that Ext 1 (S 0 , S 1 ) ∼ = C 2 . Finally, we will define our fractional branes as follows: Where we have shifted our complexes one position to the right (in the conventions of [81]) to take into account that the D1 brane mutually supersymmetric with respect to a D7 brane wrapping an arbitrary cycle is, from the perspective of sheaves, the anti-brane of the skyscraper sheaf [85]. The spectrum between these two will not change under a simultaneous shift. However, the spectrum with the non-compact branes will.

Ext's
Let us now compute the Ext groups involving the flavor branes. We have already established in the previous section that: Let us now calculate the spectra involving the non-compact branes. We will do a few sample calculations and then list the results for all sectors. Let us start with the spectrum between G 0 and I 0 . It is given by the Ext i (G 0 , I 0 ) groups which correspond to cochain maps, whereby the degree i instructs us to shift the second complex to the right as follows: The fact that this is computed in the derived category of modules means that we have to quotient out by quasi-isomorphisms. Fortunately, it is known fact that if a ring has enough projectives, as it is the case here, it is sufficient to consider projective resolutions and quotient out by cochain homotopies (explained around formula (3.8)).
Let us then compute Ext 1 (G 0 , I 0 ). Since I 0 = S 0 [−1], we see that Ext 1 (G 0 , I 0 ) ∼ = Ext 0 (G 0 , S 0 ). So the task is to find the vertical maps (f 0 , f 1 ) in the following diagram: Let us start with f 0 . For consistency, it is necessarily of the form with z a complex number. The elided parts can be arbitrarily complex, but since they are composed with β i on the right they always give contributions to f 0 which are homotopy equivalent to zero, so we can simply forget about these and set f 0 = z. Then, in order for the square to commute, the only solution is f 1 = (z, 0) T , which is also non-trivial w.r.t. homotopy. In conclusion, Ext 1 (G 0 , I 0 ) ∼ = C, where C is parametrized by the number z. The rest of the D1/D7 Ext groups are computed in the same manner. We list the Ext groups between fractional and non-compact branes here: Since the spectrum between two non-compact branes is infinite-dimensional, the calculation is qualitatively different. Let us perform a sample calculation, say Ext 1 (G 0 , G 1 ). It consists in the set of vertical maps f 1 ∼ f 1 + g 0 • β 1 − α 1 • g 1 modulo homotopies given by (g 0 , g 1 ) Here, we see that any loop from node '1' to itself not made of β 1 at its start or α 1 at its end is allowed. In other words, we can take any linear combination of any power of β 2 • α 2 . Hence, we conclude that our Ext group is a polynomial ring Ext . This is nothing other than the ring of polynomials on the complex plane. This reflects the fact that our D7-branes intersect over a complex plane, and the low energy bifundamental strings are simply fields defined on it. In this non-compact setup, this spectrum is an infinite-dimensional vector space. However, upon compactifying the intersection curve of the two D7-branes, the Figure 2: Quiver representation for the theory discussed in the text, before taking the orientifold involution (described by a reflection with respect to a horizontal line through the quiver nodes, as described in §3.4). We have indicated the matter between the flavor branes by the blue dashed line, the actual matter content coupling to the quiver degrees of freedom will depend on choices of flux away from the singularity.
spectrum will become finite-dimensional, and upon switching on an appropriate flux, it will be chiral. Note that this information is input from the point of view of the theory at the conifold singularity. Below we will choose this input to be that appearing in F-theory GUT models.
The other D7/D7 spectra are computed in the same manner, with the results: The resulting quiver is shown in figure 2.

Orientifold projection
We would now like to understand the nature of the orientifold projection acting on the singular conifold. I.e. which projection should we take on the quiver found in figure 2. Recall from §2 that, for a conifold defined as the orientifold involution acted geometrically as ξ → −ξ, leaving the other coordinates invariant. If we rewrite this conifold as xy = zw, with (x, y, z, w) For the (1, 1) representation of the quiver, these geometric coordinates are in one-to-one correspondence with vevs of the elementary mesons of the quiver theory. The conifold theory admits various involutions, but it is easy to see (from the classification in [77], for instance) that the only class of involutions compatible with the action (3.47) on the mesons is the action shown in figure 3(a). In principle the projection associated to the two fixed lines can (a) Action on the dimer model.  [73][74][75] is given by a line orientifold, in the terminology of [77]. The choice of equal signs for the two fixed lines is associated with having an O7 plane instead of an O3 plane. (b) Resulting quiver gauge theory. We will determine in the text that G = USp(2N ).
be independently chosen, but as discussed in detail in [86] the choice with opposite relative signs corresponds to an O3 plane, instead of an O7 plane, so we need to choose the two fixed lines to have the same sign. The resulting theory is depicted in figure 3. We still have the ambiguity of whether the gauge group on the nodes is of type SO(N ) or USp(2N ). 8 One can easily see that the right choice is G = USp(2N ) by a probe argument, as follows. In our setup we have an O7 − plane coming out of the conifold point. We know that in flat space a single D3 brane on top of an O7 − gives rise to a four dimensional N = 2 USp(2) theory (it would be a SO(2) algebra for a O7 + ). This will also be true at low energies for a D3 on the orientifolded conifold background, as long as the D3 is on top of the O7 − plane but away from the singular point. This implies that there should be a submanifold on the moduli space of the G × G theory for N = 1 where at low energies one recovers the N = 2 USp(2) theory. A single probe brane on the orientifolded conifold background will have gauge group SO(2) × SO (2) or USp(2) × USp(2), since it corresponds to two D3 branes on the covering space. Out of these two choices, the only one that can Higgs to a USp(2) subgroup anywhere in the moduli space is USp(2) × USp (2), so we conclude that the right projection is USp (2N ) × USp(2N ).
The fact that we obtain a USp(2N ) × USp(2N ) projection on the quiver will be essential in order to be able to understand the E 6 Yukawa coupling in the superpotential as coming from an instanton: in the F-theory background we have no gauge dynamics localized at the Yukawa point, so we need to set N = 0, and USp(0) nodes in empty quivers are precisely those which can give rise to stringy instanton effects [87][88][89][90]. The microscopic reason for this is that the projection on spacetime filling branes and the corresponding instantonic branes is reversed [91][92][93][94] (see also [95] for a review), so a USp projection on fractional D3 branes implies the existence of an orthogonal projection on the fractional D(−1) branes. This is precisely the projection necessary for eliminating the neutral τ modes on D-brane instantons, and allowing for the existence of a superpotential contribution. 9 The previous argument is somewhat indirect, and it may be illuminating to see explicitly the enhancement in moduli space. In the rest of this section we do this exercise. (The reader not interested in this derivation should feel free to skip ahead to the next section.) In addition to the quiver in figure 3(b), there is a superpotential of the form [96] .  (4) rotations. Note that these rotations act independently on the real an imaginary parts of a µ . Let us go to a gauge in which (a (1) µ ) = (a 0 , 0, 0, a 3 ) with (a 0 , a 3 ) = (0, 0). (This will be the generic case for either the real or imaginary part of one of the A i fields, we choose to talk about the real part of A 1 for concreteness. It may be the case that a 0 = 0 or a 3 = 0 for all A i at certain points in moduli space, we will analyze these points momentarily.) Non-abelian D-terms for this theory are given by for any of the three Pauli matrices σ k . With the gauge choice above, these imply that we can write We see that the USp(2) × USp(2) gauge symmetry is broken down to a U (1) × U (1) There is an overall U (1), obtained when α = β, which acts trivially on A i . The nontrivial O(2) = U (1) Z 2 symmetry can be understood in the representation of the A i as four-vectors 9 It is also clear that the fractional D(−1) has no deformation modes, so the only zero modes to worry about, in order to make sure that one has a bona fide superpotential contribution (as opposed to a higher F-term [71,72]), are the charged zero modes. We will deal with these in the next section. 10 This can be argued as follows. For any 2 × 2 matrix Ai, we have that σ2A t i σ2 = Tr(Ai) · I − Ai. By the Cayley-Hamilton theorem this implies that Ai · σ2A t i σ2 = σ2A t i σ2 · Ai = det(Ai) · I. Direct substitution into (3.48) then shows that W vanishes.
as the rotations and reflection on the 2-plane of non-vanishing directions, associated to σ 0 and σ 3 . Introduce the coordinates u ± i = z 0 i ± iz 3 i . It is easy to see that (3.51) acts on these coordinates as u ± i → e ±i(β−α) u ± i . Furthermore, the non-abelian D-terms (3.49) become in these coordinates the single equation We recognize this moduli space as the usual GLSM construction of the conifold given above with u + i = α i and u − i = β i , together with a decoupled U (1) factor, associated with the U (1) symmetry of a D3 in flat space.
The geometric involution we are interested in appears here as the remnant discrete Z 2 action (3.52), which acts on the u ± i coordinates as u ± i ↔ u ∓ i . The fixed point locus is obtained whenever u ± i = u ∓ i (in some gauge). This implies z 3 i = 0 (in this gauge, or equivalently z 0 i = 0 in a different gauge), and assuming z 0 1 z 0 2 = 0 on this locus we have an enhanced USp(2) symmetry, in agreement with the expected behavior of the D3 on the O7 − plane.

Instanton effects in the orientifolded background
Let us now focus on the specific choice of ranks that will arise in GUT models, as described in §2. We take in particular multiplicity 5 for the G i stacks (so they will be associated with the GUT stack) and multiplicity 1 for the F i stacks (associated with the flavor brane). For simplicity, we choose the spectrum between the G i branes to be given by a single chiral bifundamental, which upon orientifolding will give rise to the 10 representation. For phenomenological purposes it is often more interesting to take three copies of the 10 representation, we generalize the analysis below. We also choose to have a single 5 between the G i and F i stacks, with the same chirality as the 10. With these choices, the theory after orientifolding is that described in figure 4.
The instanton effects come from euclidean D1 branes wrapping the (empty) I i nodes. We will consider the case of a single fractional instanton wrapping either I 0 or I 1 . (Contributions due to multi-instantons may be interesting, but their analysis is significantly more subtle so we leave the analysis of these to the interested reader.) Note that due to our choice of chirality for the 10 and 5 representations the two nodes behave rather differently. Let us start with I 1 . In our conventions, the I 1 node admits terms in the instanton action coupling the zero modes λ 1 , ν 1 to matter fields of the form for some c, d ∈ C. We have indicated the U (5) indices explicitly. This instanton action, upon integration of the fermionic zero modes λ 1 , ν 1 , will generate an effective 10 10 5 coupling in the low-energy theory, 11 in agreement with the F-theory expectation. On the other hand, the euclidean D-brane on I 0 behaves very differently. Gauge invariance forbids any couplings analogous to (3.54), so due to the unsaturated zero modes the instanton will generate a higher F-term, instead of a superpotential contribution [71,72].
Let us now argue that the couplings in (3.54) do in fact appear in the instanton action. In both terms, the matter fields correspond to recombination modes of the flavor stacks. In the first case, G 0 and G 1 are Weyl divisors that cannot leave the singularity. Indeed, if the 10 field is turned off, the charged zero-modes are always massless. Recombination of the G 0 and G 1 divisors in order to obtain a Cartier divisor not intersecting the singularity will manifest itself, from the point of view of the instanton, in giving masses to the λī 1 modes. The mass term transforms in the 10 representation of SU (5), and can indeed be seen as the expectation value of the corresponding GUT matter field (which is, from the instanton worldvolume point of view, a background field). A similar argument holds for the second term in (3.54).

Yukawa rank in the multiple family case
For simplicity we have considered the case in which there is a single 10 representation involved in the Yukawa coupling of interest. In realistic models one would like to have three generations, often all living on the same curve. The most general instanton action is now given by with I the family index. The resulting Yukawa matrix is proportional to which, as already remarked in [97], is of rank one. This is in agreement with the general expectation from the F-theory analysis [98].

Figure 5:
Mirror for the conifold, as a fibration over C, parameterized by a coordinate W . The generic fiber is C * times P 1 punctured at four points. The C * fiber degenerates at W = 0, and the punctured P 1 at W ∈ {w 0 , w 1 }. The mirror of the fractional branes on the conifold wrap the three-spheres S 3 0 and S 3 1 , obtained by fibering the product of the degenerating 1-cycles in the fiber over the segment in the base.

Mirror description
The previous results can also be justified, to some extent, from the point of view of the IIA mirror description of the conifold using the techniques in [75,99]. In particular, the structure of the mirror manifold in the presence of the orientifold involution of interest to us has been discussed in [77]. One additional ingredient with respect to the discussion in [77] is the presence of flavor branes. How to include these in the present context was discussed in [76,78]. Note that the discussion in these works involves a nontrivial amount of guesswork, which is why we have chosen to give a more principled analysis above. Nevertheless the mirror picture is technically simpler to analyze, so we will briefly describe it here in order to provide some additional intuition for the reader.
As described in [75,99], for the purposes of computing chiral information one can take the mirror of the conifold to be described by a C * × Σ fibration over C given by the equations uv = W (4.1) with u, v, W ∈ C, x, y ∈ C * . The complex plane is parameterized by W . The first equation gives a C * fiber over the W plane, with a degeneration at W = 0. The second equation defines a Riemann surface fiber with four punctures. We pick coordinates and a framing [100] such that P (x, y) = q + y + xy − xy 2 . Here q encodes the complex structure of the mirror, or equivalently the complexified Kähler modulus of the original conifold. The resulting Riemann surface is singular for W = q and W = q + 1; we denote these points w 0 , w 1 respectively. Consider the T 2 over any point in the W plane formed by the S 1 ⊂ C * collapsing at W = 0 times the one-cycle in Σ associated with the degenerations at w i . The total space of this T 2 over a segment in the W plane connecting W = 0 and w i has the topology of S 3 , and the two fractional branes of the conifold are obtained by wrapping D6 branes on these cycles. The resulting geometry is depicted in figure 5. These two fractional branes intersect over W = 0. We will focus on the structure of the Σ fiber over this point, which is described by {q + y + xy − xy 2 } ⊂ (C * ) 2 . The structure of the fiber at this point is summarized in figure 6. As we see from this figure, in this mirror description various nontrivial aspects of the IIB analysis are manifest. The structure of zero modes and chiral multiplets arises in a simple way from brane intersections (perhaps at infinity along a puncture, as for the 5 and 10 matter fields), and the couplings in the instanton effective action (3.54) arise naturally from worldsheet instanton effects.

M-theory description
In this section we would like to reproduce some of the effects we have observed in the weakly coupled IIB and IIA descriptions directly in the more usual (in the F-theory model building literature) geometric M-theory language. We will focus on two interrelated effects. The first effect we would like to understand is simply how the usual computation of Yukawa couplings  in M-theory language, that is, in terms of M2 branes wrapping appropriate chains with boundaries on the matter states [51][52][53], connects to the instantonic picture.
At a heuristic level it is not hard to argue that there is a connection, as follows. The IIA configuration T-dual to the fractional D(−1) on the conifold is given by D2 brane wrapping the T-duality cycle, and the exceptional P 1 of the conifold. At the edges of the T-duality interval we have O6 planes, and the P 1 of the conifold necessarily collapses there. On a generic point along the interval the P 1 is not necessarily of zero size. The D2 instanton wraps the total space of this P 1 fibration over the interval, which defines a non-trivial three-cycle Σ 3 .
Consider now in the IIA picture for a coupling mediated by D2-brane instantons. This is depicted in figure 7(a). We have that various open string states meeting at a given spacetime point, where the euclidean D2-brane lives, giving rise to the effective vertex in the low energy action. In the M-theory picture both the D2 and the F1 lift to M2 branes, so the lift of the IIA configuration will be given by a recombined smooth M2, shown in figure 7(b). This M2 wraps the three-chain Σ 3 , with boundaries given by P 1 cycles associated with the states appearing in the coupling. This is precisely the usual description for how perturbative couplings (such as the 10 10 5 coupling of interest to us) appear in the M-theory description [51][52][53].
We can understand the transition from figure 7(a) to 7(b) as follows. Focus on a junction where an open F1-worldsheet stretched between two D6-branes ends on an interval on the Euclidean D2-brane, say the upper junction in 7(a). Now take Euclidean τ time to run horizontally from left to right in the diagrams. Then, at each slice with constant τ we have a semi-infinite line from the F1 ending on the D2. From the point of view of the D2worldvolume, the F1 is an electric source, and a backreacted solution creates a funnel-type geometry, whereby the F1 is replaced by a smooth spike made entirely of the D2-brane [101]. At any horizontal slice, the boundary of this surface is a circle, whose radius grows with g s . This system lifts straightforwardly to a smooth M2 brane configuration in M-theory. This  recombination process is shown in figure 8.
Now take a family of these funnel geometries parametrized by τ , covering the whole interval. At the two extremities g s vanishes, since there are D6-branes there. Hence, the boundary circles of the M2 funnels shrink to zero size at the extremities of the interval. In the end, the full boundary of this family is a circle fibration over the interval collapsing over two points, i.e. an S 2 . Now we have replaced the F1/D2-junction with a smooth D2 with boundaries, which trivially uplifts to M-theory to an M2 with boundaries, as depicted in 7(b).
In what follows we will present evidence supporting this heuristic picture, by explicitly tracking the M-theory three-chain Σ 3 to the weak coupling limit, and seeing that it survives the limit, getting localized at the E 6 point. A full proof of the connection requires an explicit identification of Σ 3 with the uplift of the D2 brane. It would be very interesting to work this out in detail, but we will not do this here.
Along the way we will find that in the strict weak coupling limit (g s = 0), the geometry develops a non-flat fiber at the E 6 point. This effect only appears at g s = 0, and disappears as long as g s is finite, no matter how small. We conjecture that the physical origin for this effect has to do with the light strings appearing for vanishing B-field on the collapsed cycle.
This conjecture is based on the following observations. Notice first that in the weakly coupled description this B-field period can be arbitrarily tuned while staying on the quiver locus (contrary to what is stated in [56]), so it is possible to enhance the contribution of the instanton by tuning the B-field. More in detail, the instanton contribution to the superpotential goes as gs P 1 B 2 + P 1 C 2 10 10 5 where we have omitted the (unknown) dependence on the complex structure moduli, and schematically indicated by P 1 B 2 and P 1 C 2 the periods of the B 2 and C 2 fields on the non-resolvable P 1 . (This is schematic since the P 1 is never present in the geometry as a proper algebraic cycle, but the corresponding periods are still well defined.) We see that taking P 1 B 2 → 0 we enhance the magnitude of the non-perturbative effect. Nevertheless, we cannot simply take P 1 B 2 to vanish. This is because in order to have a standard field theory description of the conifold background one needs the B field on the collapsing P 1 not to vanish [102,103]. Otherwise one has massless strings coming from D3 branes wrapping the collapsing P 1 . The mass of these states is, in fact, given by the same expression as the instanton action.
Our conjecture is that these D3 branes dualize to M5 branes wrapping the non-flat fiber.
Assuming that this identification is correct, the IIB analogue of the disappearance of the non-flat fiber as we take g s nonzero is presumably related to some dynamical effect on the D3 making it massive. Also, the D3 branes on the type IIB side get a mass by turning on a B-field background, which should imply that the M5 branes on the non-flat fiber get a mass upon turning on an appropriate C 3 background. It would be very interesting to verify (or refute) our conjectured identification, and to understand the mechanisms behind both mass mechanisms, but we will not attempt to do so here.

The weak coupling limit
We will start by reviewing the techniques in [104,105], which provide an efficient way of taking a weak coupling limit in F-theory such that one can more easily recognize the IIB brane system encoded by an elliptic fibration, elaborating on Sen's original proposal [63,64]. We will first describe the procedure in [104,105] in general, and then apply it to the SU (5) model. Let us start with the following particular form for an elliptic fibration: Following Sen [63,64], we introduce a small parameter and rescale (b 4 , b 6 ) → ( b 4 , 2 b 6 ), then we find that the j-function of this model blows up everywhere except at b 2 = 0. In this way, the discriminant has the following leading term: As explained in [63,64], there is an O7-plane at b 2 = 0 and a D7-brane at b 2 4 − b 2 b 6 = 0. The problem with this approach to the weak coupling limit stems from the fact that taking → 0 does not commute with computing the discriminant of the elliptic fibration. In order to see Sen's result, one must first keep arbitrary, compute the discriminant, and only then expand. If, on the other hand one first expands the Tate equation in and then takes the limit, then the model washes away all the D7-brane data, giving too crude a rendition of the situation.
One can improve on this by adopting the philosophy of [104,105], and promoting the parameter to a coordinate. So we study the fivefold X 5 , which is a family of CY fourfolds, of which the limiting hypersurface = 0 would be the weakly coupled F-theory model. X 5 is then given by the hypersurface equation: One first notices that this fivefold is singular at the ideal (y, x, ). Then, one proceeds to blow-up the fivefold at this ideal. The result is the following ambient spacê with the following irrelevant ideals (x,ŷ, z), (x,ŷ,t), (z, v), and the following ideal describing the proper transformX 5 of the fivefold: From now on, we will drop the 'hats', hoping not to cause confusion. The fourfold Y 0 over the central fiber = tv = 0 breaks into two components Y 0 = Y pert ∩ Y rem . The first component inside the ambient space x y t 1 1 1 with irrelevant ideal (x, y, t), reveals the perturbative IIB data. For instance, its discriminant is simply i.e., the locus of the perturbative D7-brane. It is a (quadric) P 1 -fibration over the base B 3 . The second 'remaining' component, Y rem , is given by inside the ambient space Since x = 0 ⇒ y = 0, which would be forbidden, we can fix x → 1. Now we end up with a linear equation in v, allowing us to eliminate it. Therefore, we are left with a purely toric space given by y z 1 1 which is a constant P 1 -bundle over B 3 . The two sphere fibrations meet at the following ideal: in the ambient space define by the complex y-plane. This is a double cover of B 3 branched over the locus b 2 = 0, which we recognize as the O7-plane. Put differently Y pert ∩ Y rem is isomorphic to the perturbative IIB CY threefold target space.

Constructing the SU (5) model from the bottom up
Armed with this technology, we can easily construct the F-theory fourfold Y 4 corresponding to any given perturbative IIB setup; we simply need to run this machinery backwards. We start with a CY threefold X 3 that admits a description of the form with orientifold involution ξ → −ξ. We give as input an D7-brane hypersurface In [106], it was shown that this is the most general admissible form for a D7-brane consistent with an O7-involution, even if the brane is reducible and non-reduced. In our case, we would like to have ∆ pert = σ 5 (w + σP ) ≈ σ 5 w near the GUT brane , (5.13) where σ and w are coordinates in B 3 , and P is some polynomial. This ensures a stack of 5 branes at σ = 0, and that the intersection between that stack and the remainder is described by a simple equation, (σ, w).
We also know that, in order to have an SU (5) gauge group as opposed to Sp(5), we need the divisor σ to be reducible into a brane/image-brane pair. This is achieved by requiring X 3 to have the following conifold structure where u is another base coordinate. In other words, we are defining b 2 ≡ u 2 + σw. Now the SU (5) stack and its image are given by the ideals which are swapped by the involution ξ → −ξ. Now all we need to do is fix the forms of b 4 and b 6 by requiring This leads us to the following ansatz for the Tate equation of the CY fourfold: The full ideal for Y can therefore be written as follows:

Yukawa interactions at weak coupling
Having reviewed what creates the 10 10 5 Yukawa coupling in F-theory, we now want to see this process through the weak coupling limit. Since the point of this paper is that such a coupling exists in perturbative IIB, then we should be able to see it as Y degenerates into Y 0 = Y rem ∪ Y pert . (The analysis in the = 0 case is well understood by now [43,53], we review it in appendix B). Let us take our central fiber, i.e. = 0 in (5.19), and focus on the 'perturbative' Y pert branch defined by v = 0. Explicitly, we take inside the following toric ambient space: x y t σ u w 1 1 1 0 0 0 with irrelevant ideal (x, y, t). This space is singular at the ideal (x, y, σ). Since the physics that interests us is happening at the singularity, we will focus on the patch t = 0 and gauge fix that coordinate to one. In this way we have reduced the problem to studying the hypersurface To gain some intuition, notice that, away from the orientifold locus given by b 2 ∼ = u 2 +σw = 0, we can rewrite this as So, in a neighborhood where b 2 is constant, we can interpret this equation as a C * fibration that degenerates over ∆ pert = σ 5 w = 0, which matches our expectation about our perturbative D7-brane setup. Before we can start performing resolutions, we will make the convenient redefinition Y ∼ = y + ux + σ 2 , such that our fourfold is now given by Y (−Y + 2ux + 2σ 2 ) + σwx 2 . (5.23) Now we can resolve the singularity at (Y, x, σ). It turns out that we will need two blow-ups and two further small resolutions. There are several possible resolution phases for this setup, as explored in [57]. We will pick one, leading to the following toric ambient space in the following ambient toric fivefold: The fully resolved fourfold is described by the hypersurface: and blow-down map where the underlined coordinates are coordinates of the blown-down singular space.
Let us now study the fiber structure of this resolved space, and the various degenerations it undergoes as we restrict to special loci of increasing codimension.

Codimension one
We begin by studying the fiber over σ ≡ σv 1 v 2 v 3 v 4 = 0. Intersecting the various factors with Y pert , we obtain a pattern of curves. Two words on our notation: 1. When we write a sum of ideals, we really mean the homological sum of the curves associated to the ideals, e.g. (eq 1 ) = (eq 2 ) + (eq 3 ) (5. 27) really means that the lhs corresponds to a curve that decomposes into the two curves on the rhs. Technically we should say that it is the intersection of the two ideals.
2. We will gauge-fix coordinates whenever possible without explicitly saying so. For instance, in the ideal (v 1 , −v 4 +2ux), x cannot vanish. For this would require v 4 to vanish, even though (x, v 4 ) is ruled out. So we can use a projective action to fix x → 1 and simply write the ideal as (v 1 , −v 4 + 2u).
Let us now proceed: The intersection pattern of the full fiber has the following graph: The blue part showcases a non-affine A 4 diagram. The extremities, C L,R , are non-compact curves. We can think of this as a five-centered Taub-NUT space, such that, upon projecting onto a C-plane, we have a C * -fibration that collapses over the D7-branes. The two extremal pieces are just the fiber expanding as we move away from the branes.

Codimension two
Over the loci Σ 5 = (σ, w) and Σ 10 = (σ, u), we expect to see enhancements to A 5 and D 5 , respectively. Now let us examine the fiber over these loci.
We clearly recognize an A 5 Dynkin diagram, sandwiched between the two complex planes, as expected.
The arrangement has the following shape: The blue part is a D 5 Dynkin diagram. This can be understood as an orbifold of an A 5 Dynkin diagram: The two external pairs plus the C L and C R get identified. The middle node gets itself orbifolded with two fixed points, which lead to two singularities, which after resolution give the D 5 (4) and D 5 (5) nodes.

Codimension three
We will now study the fate of the fiber at the Yukawa 'E 6 '-point. This is where things differ drastically from the more conventional = 0 case reviewed in appendix B. What we are about to see is that our fibration is non-flat, meaning that the fiber dimension will jump. In this case, the fiber will decompose into three curves plus a surface! Tensionless strings By taking (5.25) and setting (σ, u, w) = 0, we get the following degenerate ideal:Ŷ Clearly, this fiber has a component given by v 4 = 0, which is a toric divisor described by the following data An M5-brane wrapping this divisor will become an effective tensionless string upon blowingdown, unless (as discussed in the introduction to the section) the C 3 field gives it a non-zero mass.
The configuration of the full fiber has the following shape: As one might expect, the weak coupling limit avoids creating the non-affine E 6 -diagram that develops in the strongly coupled case. It is interesting to see that the way this is avoided is by growing a vanishing four-cycle.
"E 6 " enhancement: Σ 5 → Yuk E 6 Now let us approach the Yukawa point (σ, u, w), where all D-brane stacks meet each other and the O7-plane, coming from the Σ 5 -curve. We find the following splittings Yukawa Σ 10 → E 6 Now let us approach the Yukawa point from the 10-matter curve: The aim now is to demonstrate that there is a homological relation of curves: just as in the non-perturbative situation. Here, we can easily see the relation as follows. By consulting the table (5.3.3) which describes the toric divisor v 4 = 0, we see that the curves correspond to homogeneous coordinates that have the following weights under the three C * actions: This implies that the homology classes add up as expected, thereby confirming the existence of a 3-chain connecting the three curves as in the non-perturbative case. In this situation, one can construct the following 3-chain Σ 3 We speculate that the D1-instanton discussed in §3 uplifts to an open M2-brane instanton wrapped on this 3-chain. Note, that this 3-chain is entirely contained in the vanishing four-cycle given by v 4 = 0.

Equivalence of M2-instanton effects
In the previous two sections, we saw that there is a 3-chain mediating the transition whereby an M2-state in the 10 representation transitions two a sum of another state in the 10 and one in the 5 representations. A euclidean M2-brane that wraps such a 3-chain will indeed mediate such a transition. What we showed in the previous sections and in appendix B is that such a 3-chain exists both in Y with = 0, and for Y pert . However, the possibility still exists that these 3-chains are inequivalent. Let us now show that this is not the case, by constructing a homotopy relating the 3-chains as is taken to zero.
Let us first understand the 3-chain on the non-perturbative side. The curves of interest are Let us restrict to the Σ 10 -curve by setting u = 0. Now we are looking at three curves inside the divisor v 4 = 0. So, our space is now The three curves are now As we saw before, as w → 0, we see that 3) . Since these three curves are co-bordant, there is a 3-chain connecting them. Let us define for each fixed the 3-chain Σ via the following ideal: (5.69) If we now take → 0, the equation (5.68) defining the 3-chain becomes w cos θ, σ sin θ + v 2 v 3 cos θ = cos θ, σ + w, σ sin θ + v 2 v 3 cos θ .

Conclusions
We have shown that the 10 10 5 Yukawa coupling in SU (5) GUT F-theory models is generated by a D1-instanton effect in the weak coupling description of the system. We have directly argued for this result from a couple of different viewpoints, namely weakly coupled IIB string theory and its weakly coupled IIA mirror. We have also presented evidence from the geometric M-theory viewpoint that further supports our conclusions. We find this result interesting in that it demystifies somewhat the nature of the coupling, and brings F-theory closer to the much better understood weakly coupled IIB model building. A particularly interesting take-away from our result is that there is a second way of generating 10 10 5 couplings in weakly coupled IIB models, in addition to the known process mediated by euclidean D3 branes (see for instance [107] for concrete examples). The euclidean D1 contribution analyzed in this paper is an independent effect, which may also be a useful ingredient in the IIB model builder's toolbox: the instanton contribution requires the mild condition of the Calabi-Yau background having a conifold singularity admitting an appropriate involution, which is a condition that is not too constraining, and can be imposed early when constructing the model (along the lines of [86], for instance).
Our observation also raises some interesting questions in itself, particularly in connection to the usual M-theory description of the E 6 coupling.
As explained in §5, at weak coupling there is tension between having a significant contribution to the superpotential and keeping perturbative control of the theory, due to the presence of light strings. We expect this tension to relax as we go away from the weak coupling limit, but it would sill be interesting to follow the fate of the light string states at P 1 B 2 = 0 as we go away from weak coupling. We have started this analysis in §5, but more work is needed to properly understand the effect of these states in ordinary F-theory compactifications away from weak coupling.
A second point concerns the transmutation of instantons into classical couplings as we go towards strong coupling. As we argued in §5, the fact that D1-instantons can describe some classical couplings in F-theory is not too surprising, since both effects lift in M-theory to M2-branes wrapping appropriate chains [51][52][53]. But it would clearly be interesting to elucidate this point further via an explicit dualization of the D1 instantons into M2 branes, and an explicit matching of moduli in both pictures.
Finally, it would be interesting to understand whether we can describe what happens at a Yukawa E 6 point in F-theory without resorting to fourfold resolutions, such as those performed in §5. The NCCR approach we have used here to treat the conifold singularity in perturbative IIB string theory begs for a counterpart on the F-theory fourfold. Perhaps an approach along the lines of [15] might be fruitful.
admitting two matrix factorizations (φ, ψ) and (ψ, φ), with such that φ · ψ = ψ · φ = (−ξ 2 + u 2 + σw) · 1. From these matrices, two modules can be constructed as cokernels: These are what are known as irreducible maximal Cohen-Macaulay modules. In this case, the conifold admits only these two up to isomorphism. In order to perform a noncommutative crepant resolution (NCCR), we are instructed to pick one of these two, say M , and construct the endomorphism algebra This quiver has a path algebra encoded by the quiver with superpotential W = α 1 β 1 α 2 β 2 − α 1 β 2 α 2 β 1 We define two projective right A-modules P 0 = e 0 · A , P 1 = e 1 · A , (A. 6) of paths ending in the node in the label. In order to obtain a resolution, we construct the following representation of the quiver Each node has a C * -action that redefines the basis of each C vector space. The arrows are complex numbers that transform under the relative C * action as the toric coordinates of the conifold: We must also impose a D-term constraint The two resolved phasesX ± correspond to t > 0 and t < 0. It is known [79,108] that there is a derived correspondence Indeed, we see heuristically that 12 Hom(P 0 , P 1 ) ∼ = Hom(O, O(1)) = α 1 , α 2 , (A.10) Now we can study how our various branes are mapped from the singular to the X + phase. For the non-compact branes, we can now readily confirm the mappings in (3.26). The fractional branes, on the other hand, require more work. Let us define the brane S 0 through its projective resolution (3.29) This is lifted to the following complex in the resolved X + This object S 0 ∈ D b (X + ) can be rewritten after a basis transformation as follows: We have underlined the zero on the rhs to indicate that that is the starting position, i.e. the degree zero object in the complex. This can be understood as a so-called mapping cone between two complexes. This essentially means that we can regard this object as a bound state between the lower complex and the upper complex via tachyon condensation. We will not go into this here, but refer instead to [109] for a general introduction to these notions, and to [83] for a concise introduction in the string theory context. Suffice it to say that in this complex, the lower part has trivial cohomology at every position. In fact, it corresponds to the skyscraper sheaf over the deleted point (α 1 , α 2 ) = (0, 0).
By taking the cohomology of the full complex, we notice that the portion containing O(2) 1 → O(2) drops out entirely. There exists a so-called quasi-isomorphism that simply maps it to the upper complex: where C is the resolution P 1 . To summarize, we can say that corresponds to a D-brane wrapping the resolution curve. In our case, this will be a Euclidean D1. Now let us run through the calculation for the other simple representation S 1 : It transports in the resolved phase to the following complex of sheaves: which can be rewritten as Now the upper half corresponds to a trivial object (i.e. one that has no cohomology). After a quasi-isomorphism to eliminate it, we are left with the following complex: This shifted object corresponds to an anti -D1-brane with flux of Chern number minus one wrapped on the resolution P 1 : Indeed, now we see that these two fractional branes have net D1-charge zero, and net D(−1)charge one, since the negative flux on an anti-D1-brane induces positive D(−1)-charge.

B Resolving the generic fourfold
We will now study the resolution of the model written in the previous section. We will do it for a generic fiber Y of the family of fourfolds, with = 0. We start with the full ideal for Y : −y 2 + x 3 v + (uxz + σ 2 tz 3 ) 2 + σwx 2 z 2 , tv = . (B.1) In order to perform the resolution as economically as possible, we introduce an auxiliary coordinate Y , with the following relation Now, our model is written as the following ideal: The fully resolved fourfoldỸ is then given by the ideal in the following ambient space The blow-down map for this resolved space is the following (x, Y, z, t, v, σ, v 1 , v 2 , v 3 , v 4 ) (B.6) where the underlined coordinates are those on the blown-down space.

B.1 Yukawa interactions at strong coupling
We will now study how representations and Yukawa couplings appear in F-theory, following the logic of [43] and [53].

B.1.1 Codimension one
Let us study first the fiber over a generic point 13 in σ ≡ σv 1 v 2 v 3 v 4 = 0 for a generic Y . In this case the fiber is comprised of the following divisors: We have used the irrelevant ideals to 'gauge-fix' coordinates to '1' whenever they are forbidden from vanishing. It can be shown that all of these divisors are P 1 's, and that they have the intersection pattern of the affine Dynkin diagram for A 4 , as expected: A 4 (4)

C
The blue part corresponds to the non-affine Dynkin diagram.

B.1.2 Codimension two
Σ 5 matter curve: The locus (σ, w) is the curve where the SU (5) stack meets the extra 'flavor' brane. Hence, we expect an A 5 enhancement. Indeed, the resolution shows that the A 4 3 curve splits, leading to the following map: The arrangement takes the following form 13 Here, by 'generic' we mean a point such that u, w = 0. C Σ 10 matter curve: Let us proceed to the D 5 enhancement at (σ, u). This is the locus where the GUT stack meets its image.  The arrangement takes the following shape: Here, implicitly C = E 6 (1) . We stress that this is the non-affine Dynkin diagram of E 6 .
This peculiarity was discovered in [57]. Common lore would have us expect an affine-Dynkin diagram. This paradox was solved in [60], where the authors showed that the standard Weierstrass/Tate model for SU (5) actually has a T-brane built into it. In other words, a non-diagonalizable vev of the E 6 Higgs field taking values in the SU (2) × U (1), breaking it to SU (5) is responsible for this strange fiber geometry. By switching off this vev and following the consequences for the elliptic fibration, one recovers the expected affine Dynkin fiber. As explained in [58,59] introducing this T-brane vev is crucial in order to have the top quark much heavier than the other two generations.
"E 6 " enhancement: Σ 10 → Yuk E 6 Let us now approach the Yukawa point from the Σ 10 -curve. with the same arrangement, since the curves are the same as before.

B.1.4 Yukawa interactions
Let us now study a possible Yukawa interaction. The various elements of the 5 and 10 representations of SU (5) can be formal linear combinations of the P 1 's at the respective enhancement loci. However, some elements are straightforward curves. We will focus on the following examples, Here, we have labeled curves following the conventions of [43], by the order of appearance of an element in the construction of a representation, starting from the highest weight at number one. This confirms our expectation that these curves sit in the aforementioned representations. Therefore, the relation translates to This relation of homology classes implies the existence of a 3-chain with these three curves as boundaries. If this 3-chain is wrapped by an open M2-brane, it mediates a C 10 6 → −C 10 5 − C 5 3 transition, which implies the existence of the 10 10 5 coupling.