Vector-Like Pairs and Brill--Noether Theory

How likely is it that there are particles in a vector-like pair of representations in low-energy spectrum, when neither symmetry nor anomaly consideration motivates their presence? We address this question in the context of supersymmetric and geometric phase compactification of F-theory and Heterotic dual. Quantisation of the number of generations (or net chiralities in more general term) is also discussed along the way. Self-dual nature of the fourth cohomology of Calabi--Yau fourfolds is essential for the latter issue, while we employ Brill--Noether theory to set upper bounds on the number $\ell$ of vector-like pairs of chiral multiplets in the SU(5) 5+5bar representations. For typical topological choices of geometry for F-theory compactification for SU(5) unification, the range of $0 \leq \ell \lesssim 4$ for perturbative unification is not in immediate conflict with what is already understood about F-theory compactification at this moment.


Introduction
"Who ordered that?" The Standard Model of particle physics contains three generations of quarks and leptons. Particle theorists have long been wondering what can be read out from the number of generations, N gen = 3. If the Standard Model as a low-energy effective theory is obtained as a consequence of compactification of a high-energy theory in higher dimensional space-time, N gen is often determined by index theorem (or an equivalent topological formula) on some internal geometry. Historically, it was first considered to be χ(Z; T * Z) = χ(Z) top /2, the Euler characteristic of the cotangent bundle of a Calabi-Yau threefold Z, in a (2,2) compactification of Heterotic string theory [1]. Its generalisation in Heterotic string (0,2) compactifications is χ(Z; V ), where V is a vector bundle on Z. In Type IIB / F-theory language, N gen is given by χ(Σ; K 1/2 Σ ⊗ L) = c 1 (L), where L is a line bundle on a holomorphic curve Σ in a complex threefold M int . In any one of those implementations, the fact that N gen = 3 only means that one number characterising topology of compactification data happens to be 3.
Study of string phenomenology in the last three decades provides a dictionary of translation between the data of effective theory models and those for compactifications. An important question, then, is whether such a dictionary is useful. 1 The former group of data have direct connection with experiments, while we need to be lucky to have an experimental access to the latter in a near future; this means that the dictionary may not be testable. Compactification data may still provide correlations/constraints through the dictionary among various pieces of information in the effective theory model data-that is the remaining hope. From this perspective, it is crucial which observable parameter constrains compactification data more. This letter shows, in section 2, that the value of N gen brings virtually no constraint on the topology of the curve Σ, threefold M int or Z; this is due to the self-dual nature of the middle dimensional cohomology group of Calabi-Yau fourfolds, in F-theory language. This is a good news for those who seek for existence proof of appropriate compactifications, and a bad news for those who seek for profound meaning in N gen = 3.
In section 3, we focus on the number of matter fields in a vector-like pair of representations, as in the title of this article. It has often been adopted as a rule of game in bottom-up model building that vector-like pairs of matter fields are absent unless their mass terms are forbidden by some symmetry. Papers from string phenomenology community, on the other hand, often end up with such vector-like pairs in low-energy spectrum; difficulty of removing them from the spectrum is reflected the best in the heroic effort the U. Penn group had to undertake until they find a Heterotic compactification with just one pair of Higgs doublets. We will see, in section 3, that there is no reason to trust the bottom up principle based on the current understanding of F-theory/Heterotic string compactification; in the meanwhile, there is a good reason to believe (cf [5]) that generic vacua of F-theory compactification (and Heterotic dual) will predict smaller number of vector-like pairs than in papers (such as [2,3]) that have been written. 2 Brill-Noether theory sets upper bounds on the number of vector-like pairs ℓ for a given genus g of a relevant curve Σ; given the typical range O(10)-O(100) for g(Σ) for the matter fields in the SU(5) GUT -(5 +5) representations, the range of 0 ≤ ℓ 4 for perturbative unification are not in immediate conflict with most of internal geometry for F-theory / Heterotic string compactifications.
Discussions in section 2 and section 3 are mutually almost independent. Despite many math jargons, logic of section 3 will be simple enough for non-experts to follow. Observations in both sections will have been known to stringpheno experts already to some extent (e.g. section 7 of [5]), but have not been written down as clearly and in simple terms as in this article, to the knowledge of the author. So, there will be a non-zero value in writing up an article like this.
Language of supersymmetric and geometric phase F-theory compactification is used in most of discussions in this article. Heterotic string compactification on elliptic fibred Calabi-Yau threefolds is also covered by the same discussion, due to the Heterotic-F-theory duality. It is worth noting that large fraction of Calabi-Yau threefolds admit elliptic fibration [6]. 3 2 In this article, we are concerned about vector-like pairs in string compactification that are not associated in any way with symmetry or anomaly (and its flow). In compactifications that have an extra U(1) symmetry (which may be broken spontaneously or at non-perturbative level), low-energy spectrum tends to be richer, partially due to the 6D box anomaly cancellation of U(1) (cf [4]). This article is concerned about more conservative set-ups, where there may or may not be an extra U(1) symmetry; matter parity is enough for SUSY phenomenology.
3 M-theory compactification on G 2 -holonomy manifolds is not discussed here, because the author is not a big fan of it. It is difficult to obtain realistic flavour pattern in SU(5) GUT in that framework [7], and a solution to this problem has not been known so far. If SU(5) unification is not used as a motivation, however, almost all kinds of string vacua (including IIA, IIB, Type I and those in non-geometric phase) will be just as interesting.

Self-dual Lattice
Let X be a compact real 2n-dimensional oriented manifold. Combination of the Poincare duality and the universal coefficient theorem implies that the middle dimensional homology group [H n (X; Z)] free forms a self-dual lattice; 4 the intersection pairing matrix in is symmetric and integer-valued, and its determinant is ±1.

F-theory Applications
Warming-up We begin with the simplest example imaginable. Consider using the sextic fourfold X = (6) ⊂ P 5 for M-theory compactification. We have an effective theory of 2+1dimensions then.
For a generic complex structure of X, algebraic two-cycles (real four-cycles) generate a rank-1 sublattice M := Z H 2 | X of L := H 4 (X; Z); the generator 5 is H 2 | X , where H is the hyperplane divisor of P 5 , and (H 2 , H 2 ) = 6. Let M ′ := [M ⊥ ⊂ L] be the orthogonal complement of M. Since the dimension of the primary horizontal and primary vertical components of H 2,2 (X; R)-h 2,2 H (X) and h 2,2 V (X)-add up to be h 2,2 (X) in this example, M ′ ⊗ R ⊂ L ⊗ R corresponds to the primary horizontal component of X = (6) ⊂ P 4 . M ′ must be a lattice of rank-(b 4 (X) − 1) whose intersection form is given by a matrix with the determinant 6. Due to the property of self-dual lattices stated earlier,

When a fourform is restricted within a class
it is guaranteed to be purely of (2,2) Hodge component for any complex structure of the sextic fourfold. Its integral over the algebraic cycle H 2 | X can take a value in the value is quantised in units of 6, and cannot be 0, 1, 2, 4 or 5 modulo 6. When we allow the flux to be in [9] G ∈ c 2 (T X) however, the self-dual nature of the lattice L = H 4 (X; Z) indicates that the integral H 2 | X G = (H 2 | X , G) can take any integer value. Such a flux G is not purely of (2,2) Hodge component in an arbitrary complex structure of X, but the Gukov-Vafa-Witten superpotential drives the complex structure of X to an F-term minimum, where the (1,3) and (3,1) Hodge components of the flux G vanish (see also a comment later) 6 SU(5) GUT models: Let us consider F-theory compactification on a fourfold X 4 so that there is a stack of 7-branes along a divisor S in B 3 . This means that there is an elliptic fibration π : X 4 −→ B 3 , there is a section σ : B 3 −→ X 4 , and X 4 has a locus of codimension-2 A 4 singularity in π −1 (S). LetX 4 be a non-singular Calabi-Yau fourfold obtained by resolving singularities of X 4 (see [10,16] for conditions to impose onX 4 ).
For concreteness of presentation, we choose the base threefold to be a P 1 -fibration over P 2 , and the remaining five These 9 cycles generate a rank-9 sublattice M vert of a self-dual lattice L = H 4 (X 4 ; Z). The intersection form is given by in the basis of those 9 cycles; the determinant of this 9 × 9 matrix is discr(M vert ) = (3 + n)(18 + n), which does not vanish in the range −3 < n < 3 of our interest.
It is not obvious whether the lattice M vert generated by the nine elements above is a primitive sublattice of L; since L is not necessarily an even lattice, we have a limited set of tools to address this question. When it is not, however, we just have to replace the nine generators appropriately, so that M vert becomes the primitive sublattice of L. Arguments in the following needs to be modified accordingly, but not in an essential way. discr(M vert ) may not be the same as (3 + n)(18 + n) after the replacement, but the sublattice M vert still remains non-degenerate.
Let M ′ be the orthogonal complement, [M ⊥ ⊂ L], in the lattice L.
In the examples considered here, M ′ corresponds to the horizontal components, M horz , because M ⊗ Q = M vert ⊗ Q and the non-vertical non-horizontal component is empty [5]. The quotient is a finite group isomorphic to M ∨ /M = M ∨ vert /M vert . For a flux G to preserve the SO(3,1) and SU(5) symmetry, it has to satisfy all of [11] (G, When we choose a fourform flux G from c 2 (TX 4 )/2 + M, the conditions above leave as the only possible choice. This flux is always of pure (2,2) Hodge component for any complex structure ofX 4 , and hence defines a supersymmetric vacuum. This is the flux constructed in [12]; see [13,14,15,16]. Within this class of choice of the fourform flux, the number of generations is quantised as follows [17]: although λ FMW can change its value by ±1, N gen cannot change by ±1. This would serve as a tight constraint in search of a geometry with "right topology" for the real world; the value of |λ FMW (3 + n)(18 + n)| would never be as small as 3 for the choice of (B 3 , S) we made here.
In fact, we do not have to choose the flux from c 2 (TX 4 )/2 + M. The condition of [9] does not rule out choice of flux from a broader class c 2 (TX 4 )/2 + L. Because of the self-dual nature of L, the homomorphism L −→ M ∨ is surjective. This means that we can change the flux by ∆G ∈ L whose image in M ∨ is anything one likes. In particular, there exists a change ∆G ∈ L so that (∆G, x) = 0 for all the eight generators in (10), while N gen is changed by (∆G, E 2 · E 4 ) = ±1. Therefore, the flux G can be chosen within c 2 (TX 4 )/2 + L so that N gen = 3, and the SO(3,1) and SU(5) symmetry is preserved. Certainly such a flux is not purely of (2,2) Hodge component for generic complex structure ofX 4 , but the complex structure ofX 4 is driven to an F-term minimum of the Gukov-Vafa-Witten superpotential, where the (1,3) + (3,1) Hodge component of the flux is absent automatically, and the moduli are stabilised (cautionary remark follows shortly, however).
To put it from a slightly different perspective, the surjectivity of the homomorphism L −→ M ∨ means that we can choose the M ∨ ⊂ M ⊗ Q component of the flux in L ⊗ Q arbitrarily, to suit the need from phenomenology (such as symmetry preservation and choosing N gen ); this is, in effect, to relax the condition λ FMW ∈ (1/2) + Z and allow the overall coefficient (denoted λ instead of λ FMW ) to take any value in [1/(3 + n)(18 + n)] × Z. Once the M ∨ component is chosen, then one can always find some element in (M ′ ) ∨ so that their sum fits within L ⊂ (M ∨ ⊕ (M ′ ) ∨ ). Depending upon phenomenological input, such as N gen = 3, we may not be able to choose the flux so that the (M ′ ) ∨ component vanishes, but that is an advantage rather than a problem, since complex structure moduli ofX 4 tend to be stabilised then.
One can see that the M ∨ -component of the flux, (11) with a relaxed quantisation in λ, satisfies the primitiveness condition J ∧ G = (t S S + t P 2 H P 2 ) · G = 0, where J is the Kähler form on B 3 . This is enough to conclude that the primitiveness condition is satisfied, because the non-vertical component does not contribute to J ∧ G.
A cautionary remark is in order here. First, the (M ′ ) ∨ = M ∨ horz component of the flux G horz needs to be chosen so that (G horz ) 2 > 0, or otherwise there is no chance of finding a supersymmetric vacuum. This condition is not hard to satisfy, because we can change G horz freely by +M horz without changing the value of N gen or breaking the SO(1,3) and SU(5) symmetry, and the lattice M horz is not negative definite. An open question is, for a given [G horz ] ∈ M ∨ horz /M horz , how one can find out whether there is a choice of Hodge structure ofX 4 so that there exists G horz ∈ M ∨ horz with the vanishing negative component; note that a choice of Hodge structure introduces a decomposition of M hor ⊗ R into (2h 4,0 + h 2,2 H )-dimensional positive definite directions and 2h 3,1 -dimensional negative definite directions. 7 Due to the absence of a convenient Torelli theorem for general Calabi-Yau fourfolds, the author does not have a good idea how to address this problem.
Generalisation: The argument above can be used in set-ups where more phenomenological requirements are implemented. One can impose an extra U(1) symmetry (for spontaneous R-parity violation scenario instead of Z 2 parity), and a flux for SU(5) → SU(3) C × SU(2) L × U(1) Y symmetry breaking can be introduced in the non-vertical non-horizontal component of H 4 (X 4 ) [19]. One just has to take the lattice M ⊂ L = H 4 (X 4 ; Z) so it contains all the cycles relevant to symmetry (symmetry breaking) and the net chiralities of various matter representations in the low-energy spectrum. The self-dual nature of H 4 (X 4 ; Z) is the only essential ingredient in the argument above, and hence the same argument applies to more general cases. 8

Heterotic Dual
The same story should hold true, when the argument above in F-theory language is translated into the language of Heterotic string. N gen can be chosen as we want it to be, by choosing the value of λ FMW characterising the vector bundle for Heterotic compactification not necessarily in (1/2) + Z. Supersymmetry can still be preserved, presumably by choosing the complex structure of a Calabi-Yau threefold Z and vector bundle moduli appropriately and introducing a threeform flux and non-Kählerity of the metric on Z. It is hard to verify this statement directly in Heterotic string language, but that must be true, if we believe that 7 Even when such G horz ∈ M ∨ horz and an appropriate Hodge structure is present, too large a positive value of (G horz ) 2 would violate the D3-tadpole condition. So, this is another physics condition to be imposed. 8 The algebraic cycles S to be used in χ = S G to determine net chiralities need to be primitive elements of the primitive sublattice M ⊂ L, for the argument to apply. If some cycle S were an integer multiple of another topological cycle, mS ′ for some m ∈ Z, then the net chirality on S is always divisible by m, no matter how we choose a flux. The Madrid quiver [18]-fractional D3-branes at C 3 /Z 3 singularity-is the best known example of that kind. The matter curve is effectively the canonical divisor of the vanishing cycle there is one-to-one dual correspondence (even at the level of flux compactification) between elliptic fibred Calabi-Yau threefold compactification of Heterotic string and elliptic fibred K3-fibred Calabi-Yau fourfold compactification of F-theory. 9

Number of Vector-Like Pair Multiplets
We often encounter in supersymmetric string compactification with SU(5) GUT unification that there are multiple pairs of chiral multiplets in the SU(5) GUT -5 + 5 representations left in the low-energy spectrum and no perturbation in moduli can provide large masses to those vector-like multiplets. A good example is the one in [2], where the low-energy spectrum has 34 + N ′ chiral multiplets in the 5 representation and 34 + N ′ + N gen of those in the5 representation. 10 The N ′ > 0 copies of chiral multiplets in the 5 +5 representations have ∆W = φ ·5 · 5 coupling with moduli fields φ, but 34 other vector-like pairs remain in the low-energy spectrum (at least without supersymmetry breaking) in the example studied in [2]. It is likely that those 34 vector-like pairs have nothing to do with some symmetry in the 4D effective theory.
Symmetry has been one of the most important guiding principles in bottom-up effective theory model building for more than three decades. It has often been assumed in model building papers that matter fields in a vector-like pair of representations are absent in lowenergy spectrum, unless their mass terms are forbidden by some symmetry principle. Does the bottom-up guiding principle overlook something in string theory, or is there something yet to be understood in string phenomenology?
This guiding principle in bottom-up model building corresponds to the following statement in mathematics. Let us first note 11 that the number of SU(5) GUT -5 and5 chiral multiplets are given by and respectively, for some holomorphic curve Σ and a line bundle O(D) on Σ, quite often in supersymmetric and geometric phase compactifications of F-theory for SU(5) unification models [12,17,2,20,14,15]. We assume that the flux (i.e., O(D)) is chosen to realise the appropriate net chirality (cf discussion in section 2) Thus, this general statement in math is in line with the bottom-up principle. The gap between the bottom-up guiding principle and the predictions of multiple vector-like pairs as in [2,3] must be due to non-genericity of the complex structure of the holomorphic curve, of the flux configuration, or of both, in the math moduli space M g and Pic χ+g−1 (Σ g ). Most of papers for spectrum computation in F-theory or Heterotic string compactification so far employed the flux (11) or something similar. With more general type of flux configuration (as discussed in section 2), however, more general elements of O(D) ∈ Pic χ+g−1 (Σ g ) can be realised than, for example, in [2,3]. Smaller number of vector-like pairs may be predicted in F-theory and elliptic fibred Heterotic string compactifications then ( [5]).
The question is how general τ ∈ M g and O(D) ∈ Pic χ+g−1 (Σ) can be in such string compactifications. It is easy to see that the complex structure of the holomorphic curve Σ for the 5 +5 matter cannot be fully generic. Let us take the example (5) for illustration purpose. The genus g of Σ is given by [21,15] 2g − 2 = (3n + 24)(3n + 21) − 2(3 + n)(9 + n) = 7n 2 + 111n + 450, (16) and the dimension of M g is 3g − 3. On the other hand, the defining equation of the curve Σ involves 5 + n 2 + 8 + n 2 + 11 + n 2 + 14 + n 2 + 21 + n 2 − 9 = 5n 2 + 113n + 770 2 (17) complex parameters; the first five terms correspond to h 0 (P 2 ; L) for line bundles L = O(3+n), O(6 + n), O(9 + n), O(12 + n) and O(18 + n); the last term accounts for the isometry of P 2 and the overall scaling of the defining equation. The freedom (17) available for the complex structure of Σ in F-theory compactification remains to be smaller than the 3g − 3 dimensions of the math moduli sapce M g , as long as −3 ≤ n, which allows SU(5) GUT models. The condition (a) necessary for the general math statement ℓ = 0 (and absence of vector-like pairs) is not satisfied in string compactifications. 12 We will also find more direct evidence for this in footnote 14.
To summarise, predictions of multiple vector-like pairs in string compactifications, such as those in [2,3], do not have to be taken at face value, because only purely vertical flux was considered in those works; more generic choice (that involves horizontal components) would predict smaller number of vector-like pairs. But, the bottom-up guiding principle does not have to be trusted too seriously either, because the holomorphic curve Σ for SU(5) GUT -5 +5 matter fields is not expected to have a generic complex structure.
Brill-Noether theory [22] 13 tells us a little more than the general math statement quoted above. Let Σ be a genus g curve and O(D) a line bundle on Σ whose degree is d = χ + g − 1.
First of all, When 0 ≤ d ≤ g − 1, there are soft upper bound and hard upper bound. Clifford's theorem provides the hard upper bound, 12 An intuitive (but not rigorous) alternative explanation is this. In Heterotic string, with a gauge field background in SU(5) str (which breaks E 8 symmetry down to SU(5) GUT = [SU(5) ⊥ str ⊂ E 8 ]), the5 GUT matter fields are determined by the Dirac equation in the 10 = ∧ 2 5 str representation of SU(5) str . Despite the 10 components participating in this Dirac equation, the structure group remains SU(5) str , not SU (10). 13 The phenomenon that the values of ℓ and (ℓ − χ) jump up and down over the math moduli space M g and Pic χ+g−1 is a math translation of the coupling ∆W = z · 5 ·5. The remaining question, which is partially discussed with (17) vs (3g − 3), is how much of the math moduli space is covered by the physical moduli space (fields) of compactification. In other words, it is to study z(φ, G), where φ denotes physical moduli and G the flux. which holds for any complex structure of smooth curve Σ. When the complex structure of Σ is not non-generic, there is a stronger upper bound, 14 because the Brill-Noether number ρ := g − ℓ(ℓ − χ) becomes negative for ℓ beyond this upper bound. Due to the Serre duality, it is enough to focus on the cases with d ≤ g − 1.
In the case of SU(5) GUT -10 + 10 matter fields, string compactification often ends up with g ≤ −χ = N gen = 3 (though not always), and hence the d < 0 case applies. The vector like pair of 10 + 10 is absent then. In the case of SU(5) GUT -5 +5 matter fields, however, g often takes a much larger value (as in the example (16)), and hence the ℓ = 0 result does not apply. Typical values of g listed in Table 1  For such large values of g, d = χ + g − 1 is close to g − 1 for χ = −N gen = −3 or χ = 0. The upper bounds (19,20) for those g and d have no conflict with vector-like pairs in the range of 0 ≤ ℓ 4 for perturbative gauge coupling unification. 15 It requires much more dedicated study to go beyond. One could try to characterise what the physically realised subspace-one with the dimension given in (17)-in M g would be like, or to work out the image of not necessarily purely vertical fluxes mapped into Pic χ+g−1 (Σ); the cautionary remark in page 6 also needs to be taken care of along the way. They are way too beyond the scope of this article, however. It is also worth studying how discussion in this article needs to be modified, when spontaneous R-parity violation scenario is at work (where an off-diagonal 4D scalar field breaking a U(1) symmetry to absorb a non-zero Fayet-Iliopoulos parameter (cf section 5 of [12] and [25,26,27,28,29,30])).
14 This upper bound is not always satisfied (hence this is a soft upper bound), when the complex structure of Σ is somewhat special. A good example is found in [3]. There, a flux is chosen as in (11), including the quantisation condition on λ FMW , so that χ = −N gen = −17. In addition to this net chirality in the SU(5) GUT -5 +5 sector, non-removable ℓ = 11 vector-like pairs are predicted in that example. In this case, g = 174, and hence d = 156. The hard upper bound ℓ ≤ d/2 + 1 = 79 is satisfied, but the stronger upper bound for Σ with a generic complex structure, ℓ ≤ 7.15, is not satisfied. So, this computation is a direct evidence that the curve Σ for the 5 +5 matter in F-theory does have a special complex structure (even after choosing the complex structure ofX 4 completely generic). The dimension counting argument using (17) is not the only evidence for the non-genericity of τ ∈ M g . It will be possible to carry out similar study for the examples in [23]. 15 The H 2,1 moduli of F-theory compactification (and also presumably their Heterotic dual) do not receive large supersymmetric mass terms from the Gukov-Vafa-Witten superpotential, and are likely to change O(D) = K 1/2 Σ ⊗ L in Pic χ+g−1 (Σ g ). So they are good candidates of a singlet field S that have a coupling ∆W = S · 5 ·5; some of the H 3,1 moduli may also remain unstabilized supersymmetrically (i.e., in the low-energy spectrum) and play the same role. There is nothing new in that observation, but there will be some value to leave such a footnote in this article as a reminder.