Upper bound on the Atiyah-Singer index from tadpole cancellation

We propose an upper bound on the Atiyah-Singer index in the effective action of string theory. For $E_8 \times E_8^\prime$ and $SO(32)$ heterotic string theories on smooth Calabi-Yau threefolds with line bundles, we find that the tadpole cancellation and supersymmetry conditions lead to an upper bound on the generation number of quarks and leptons as well as Higgs doublets. By taking into account the observed value of four-dimensional gauge couplings and the supergravity approximation, we explicitly evaluate the bound on favorable complete intersection Calabi-Yau threefolds. The bound can be extended to Calabi-Yau threefolds in the Kreuzer-Skarke database. We also put the upper bound on the Atiyah-Singer index in Type IIB/F-theory compactifications.


Introduction
The origin of fermion generations is one of the unsolved problems in the Standard Model (SM).Higher-dimensional theories have been utilized to understand the generation structure of chiral fermions.Indeed, background fluxes and curvatures in the extra-dimensional space will lead to degenerate chiral fermions in the four-dimensional (4D) low-energy effective action through the Kaluza-Klein compactifications [1].The degeneracy of chiral zero modes will be counted by the Atiyah-Singer index theorem, but an arbitrary value of background fluxes is allowed in general.Thus, there is no guiding principle to fix the chiral index, and it will be a challenging problem to answer the generation problem of quarks and leptons.
In string theory, brane charges induced by the background fluxes and curvatures should be canceled in compact spaces, the so-called tadpole cancellation conditions.It suggests that the possibility of background fluxes is finite through the cancellation of brane charges, and the Atiyah-Singer index determining the number of chiral zero modes is bounded by some topological quantities.There was an attempt to put the bound on the chiral index in the context of heterotic E 8 × E ′ 8 string theory on Calabi-Yau (CY) threefolds with line bundles [2][3][4] under the assumption that the effective action is described by the 4D N = 1 supergravity and the four-dimensional coupling constants are finite.This bound was derived by the requirement of the supersymmetry and anomaly cancellation.It motivates us to explore the maximal number of quarks and leptons in other corners of string theories.
The purpose of this paper is to derive a rigorous bound on the Atiyah-Singer index in the effective action of string theory.We focus on three classes of string effective action: (i) E 8 × E ′ 8 heterotic line bundle models incorporating the 4D SU (5) grand unified theory (GUT), (ii) SO (32) heterotic line bundle models incorporating the 4D Pati-Salam model or SM-like model, (iii) Type IIB magnetized D7-brane models lift to the F-theory vacua on compact CY fourfolds.In these models, Abelian fluxes lead to degenerate chiral zero modes counted by the Atiyah-Singer index theorem, but the degeneracy will be correlated with the brane charges due to the tadpole cancellation conditions.Thus, the tadpole charge puts a severe bound on the chiral index.Our findings about the bound on the index of chiral matters are summarized as follows: • For E 8 ×E ′ 8 heterotic line bundle models, we find that the index of quarks and leptons is bounded from above by the tadpole cancellation condition: where α is the O(1) constant, m max is the largest U (1) flux quanta and ∥c 2 (T M)∥ denotes the Euclidean norm of the second Chern number of CY threefolds.Note that m max is also bounded from above by CY topological data through the supersymmetry, K-theory and tadpole cancellation conditions.When we move to SO(32) heterotic line bundle models, we arrive at a similar bound on the index of Higgs doublets and vector-like quarks under the SM gauge group.
• We explicitly check the upper bound on the chiral index on favorable complete intersection CY threefolds [5,6], taking into account the finite value of 4D coupling constants.The analysis can be extended to the CY threefolds in the Kreuzer-Skarke database [7].
• We explore the bound of the chiral index in the context of Type IIB/F-theory compactifications.Since the index of chiral zero modes arising from the intersection between two stacks of magnetized D7-branes is determined by world-volume fluxes on D7-branes, the chiral index is bounded by the brane charges through the tadpole cancellation condition of D3-branes.
This paper is organized as follows.First, we revisit the E 8 × E ′ 8 heterotic line bundle models.After summarizing the consistency conditions in the heterotic model building, we derive the upper bound on the chiral index in E 8 × E ′ 8 heterotic line bundle models in Sec. 2. For SO (32) heterotic line bundle models, we consider two scenarios: (i) 4D Pati-Salam models in Sec.3.1 and (ii) 4D SM-like models realized by the so-called hypercharge flux breaking.In addition to the chiral index of quarks/leptons, the index of vector-like quarks and Higgs doublets is also bounded from above by geometric quantities of CY threefolds.In Sec. 4, we analyze the chiral index on magnetized D7-branes in Type IIB/F-theory compactifications.Sec. 5 is devoted to the conclusions.

E 8 × E ′ 8 heterotic string theory
We start from E 8 × E ′ 8 heterotic string theory on CY threefolds with line bundles. 1 The bosonic part of low-energy effective action up to the order of α ′ is described by with consisting of the Kalb-Ramond two-form B (2) (whose hodge dual is B (6) ) and the gauge and gravitational Chern-Simons three-forms, w YM and w L .Here, ϕ 10 denotes the dilaton, and the 10D gravitational coupling κ 10 and gauge coupling g 10 are normalized as 2κ 2 10 = (2π) 7 (α ′ ) 4 and g 2 10 = 2(2π) 7 (α ′ ) 3 , respectively.The trace of 10D gauge field strength F is taken in the fundamental representation of E 8 .Furthermore, we introduce N s stacks of heterotic five-branes wrapping the holomorphic two-cycles γ s whose Poincaré dual four-form is represented by δ(γ s ), and the tension of five-branes is given by T 5 = ((2π) 5 To realize the semi-realistic spectra, we introduce the internal gauge bundle consisting of multiple line bundles l a , each with a structure group U (1) ⊂ E 8 , that is, (2.3) These line bundles play a role not only in breaking the E 8 gauge group to G × Π a U (1) a but also in generating the chiral fermions from the adjoint representation of E 8 : under G and Π a U (1) a , respectively.Indeed, the index of chiral massless fermions (chiral superfields if the supersymmetry is preserved) in the representation R p is counted by the Atiyah-Singer index: where ch 3 (C p ) and c 2 (T M) denote the third Chern character of the internal bundle of each C p and the second Chern class of the tangent bundle of CY threefolds M, respectively.However, the line bundles should satisfy four consistency conditions, as will be discussed below.

Tadpole cancellation condition
From the Bianchi identity of the Kalb-Ramond field B (6) : d(e 2ϕ 10 * dB (6) with F and R being the internal background field strengths, the line bundles satisfy the following inequality in cohomology: Here and in what follows, the trace of R is taken in the fundamental representation of SO(1, 9).

K-theory condition
The first Chern class of the total bundle W is constrained as to cancel the anomalies in the two-dimensional non-linear sigma model [10,11].This condition will be regarded as the K-theory condition in Type I string theory [12,13].
In what follows, n a is normalized as one in the analysis of E 8 × E ′ 8 heterotic string theory, but it depends on the embedding of line bundles, as shown in Sec. 3 for SO(32) heterotic string theory.Throughout this paper, we focus on the case with c 1 (W ) = 0, corresponding to a structure group S(U (1) n ).

Supersymmetry condition
The internal gauge field strength F has to be holomorphic, that is, due to the supersymmetry condition.The so-called Hermitian Yang-Mills equation can be solved when at the leading order.Here, l s denotes the string length, and J denotes the Kähler form of CY threefolds.

Bound on the 4D gauge coupling
From the dimensional reduction of Einstein-Hilbert term on a six-dimensional internal manifold, the 4D Planck mass M Pl is extracted as with g s = e ⟨ϕ 10 ⟩ being the string coupling.Here, the CY volume is measured in units of string length, Vol(M) = V l 6 s .Since the 4D gauge coupling is derived from the dimensional reduction of 10D gauge field strength the internal volume is bounded from above by where we adopt the 4D gauge coupling at the GUT scale, GeV.We recall that the CY volume is described by with Here, t i and d ijk denote the Kähler moduli and the triple intersection numbers of CY manifolds, that is, d ijk = M w i ∧ w j ∧ w k , respectively.Note that the Kähler moduli should reside in t i > 1 in units of l s , otherwise the supergravity description is not valid.Thus, the bound on the CY volume (2.14) restricts the sum of intersection number as follows: Here, we assume non-negative triple intersection numbers as in the favorable complete intersection Calabi-Yaus.
In the following, we restrict the index of chiral zero modes by using the above consistency conditions in E 8 × E ′ 8 heterotic string theory on CY threefolds with line bundles.In particular, we focus on the SU (5) × S(U (1) 5 ) GUT as a branch of E 8 . 2 The net number of chiral zero modes is then described by where the first Chern classes are of the form in the basis Here, m i a denotes the quantized integer, and the number of U (1) is taken as an arbitrary integer n, although n = 5 in the case of SU (5) × S(U (1) 5 ) GUT.Following Ref. [3], we present the bound of line bundle vector m a .Since the moduli metric: is positive definite in the interior of the Kähler cone, one can derive the following inequality: where we use the supersymmetry condition (2.10) in the first step and the tadpole cancellation condition (2.7) in the last step.Here, c 2i (W ) = − 1 2 a,j,k d ijk m j a m k a and ∥ • ∥ denotes the Euclidean norm.It suggests the following redefinition of the moduli metric: from which the inequality is rewritten as Furthermore, when we define Eigen( G ij ) min = λ min , the flux vector m a is bounded from above by as derived in Ref. [3].For instance, in the case of single Kähler modulus, i.e., h 1,1 = 1, the modulus metric is given by Thus, the flux quanta are bounded from above as Let us derive the maximal value of the flux quanta m i a in a different approach of Ref. [3].By using the so-called K-theory condition (2.8) which is rewritten in terms of m i a3 : for all i, it turns out that where we use the Cauchy-Schwarz inequality.Note that a similar inequality holds on the other a, that is, for all a.Thus, when we define m max satisfying |m i a | ≤ max a,i (m i a ) =: |m max | for all a, i, we arrive at where we assume m i a ̸ = 0 for all i, but it is possible to derive m i a = 0 for some i.In this case, the term h 1,1 − 1 in the above equation will be modified.When we evaluate the index of chiral modes, we adopt the above conservative bound of the flux quanta throughout this paper.
We are now ready to derive the upper bound on the Atiyah-Singer index of chiral zero modes.Since the index of SU (5) matter multiplets is determined by Eq. (2.16), we find the following inequality: where we use the tadpole cancellation in the last step.Thus, the index is bounded from above by the maximal value of the flux quanta m max and the second Chern number of the tangent bundle of CY.The authors of Ref. [3] derived the index bounded by the CY volume itself, but we check that our finding bound is stronger than the known bound for all favorable complete intersection CY threefolds (CICYs).
Here and in what follows, we focus on CICYs defined in the ambient space The CICYs are specified by the m × R configuration matrix [5,6]: where the positive integers q I r (r = 1, • • • , R) denote the multi-degree of R homogeneous polynomials on the ambient space P n 1 × • • • × P nm with the homogeneous coordinates of P n I being x I α for I = 1, • • • , m and α = 0, • • • , n I .Since the first Chern class of the tangent bundle can be zero under R r=1 q I r = n I + 1 for all I, the CICYs are defined on the common zero locus of these R polynomials. 4In particular, we focus on all "favorable" CICYs among complete intersection CYs, where the "favorable" means that the second cohomology of CY descends from that of ambient space. 5o calculate the upper bound on the index, we have to calculate λ min which depends on the values of moduli t i .In our analysis, we adopt the universal value of moduli fields, t := t i for all i whose value is fixed by a given CY volume V = 12 or V = 25.By using the explicit values of CY topological data, we plot the index in Fig. 1, where the CY volume is fixed as V = 12 in the left panel and V = 25 in the right panel, respectively.We find that there are no viable models on the large h 1,1 due to the fact that the volume of favorable CICYs is bounded from above by the phenomenological requirement (2.14).So far, we have focused on the so-called "upstairs" CICYs.However, it was known that some of the CICYs admit a freely-acting discrete symmetry group Γ; one can define the quotient CICYs M = M/Γ [5,6] on which one can turn on Wilson lines (see, Ref. [17] for the classification of Γ).Remarkably, such discrete quotients decrease the index: although the topological quantities of quotient CICYs are different from before.Here, we consider a Γ-equivariant structure for the vector bundle V .For instance, the CY volume also reduces to V Γ := V/|Γ|.For more details, see, e.g., Ref. [18].In Fig. 2  It is interesting to apply our methods for finding a bound on the index to other classes of CY models.We study the CY threefolds in the Kreuzer-Skarke database [7], whose topological data was statistically analyzed in Ref. [19].Note that we focus on favorable CY threefolds in the Kreuzer-Skarke database as in Ref. [19].As seen in figure 1 of Ref. [19] with the large h 1,1 region, the nonvanishing triple intersection number behaves as for h 1,1 ≳ 25 6 .By introducing a normalized vector t i , we rewrite the Kähler moduli with ∥t∥ = 1, which scales as t i ≃ (h 1,1 ) −1/2 [19,20].Together with Eq. (2.33), the CY volume in the geometrical regime scales as which will be consistent with the bound on CY volume (2.14) required in the GUT.
Furthermore, one can derive the non-trivial bound on the CY topological quantities from the tadpole cancellation.By utilizing the scaling of κ i ≃ (h 1,1 ) −1 ∥t∥ 2 and the mostly diagonal κ ij [19,20], i.e., κ ii ≃ (h 1,1 ) −1/2 ∥t∥, the sum of the flux vector is bounded from below by with h 1,1 ≫ 1. Combining with Eq. (2.20), we arrive at (2.37) Thus, the second Chern number of CY will be bounded by the hodge number due to the tadpole cancellation.Note that we have focused on favorable CYs, but it will be interesting to figure out the behavior of the chiral index on non-favorable CYs, which is left for future work.

SO(32) heterotic string
In this section, we discuss the bound on the chiral index in the framework of SO(32) heterotic string theory on smooth CY threefolds with line bundles. 7In particular, we focus on the Pati-Salam group in Sec.3.1 and a direct flux breaking, so-called hypercharge flux breaking in Sec.3.2.
To avoid the existence of chiral exotic modes, we require The tadpole cancellation condition is given by with By identifying left-handed quarks {Q 1 , Q 2 }, right-handed quarks {u c 4 R 2 , u c 5 R 2 } and vectorlike quarks under the SM gauge group d c R 1 , their generation numbers are counted by respectively.Note that the index of each field is calculated by using the Atiyah-Singer index theorem: with Thus, the generation numbers are rewritten as where we used Eq.(3.4).The index of left-and right-handed quarks should be equal, that is, N In the following, we will derive the upper bound on these Atiyah-Singer indices.
From the equality:
Note that the hypercharge gauge boson will couple to string axions through the Stückelberg couplings.Taking into account the Stückelberg couplings of Kähler axions and dilaton axion, the hypercharge gauge boson U (1) Y = a f a U (1) a becomes massless under10 a,b,c,d where f a is chosen as in Eq. (3.21).Furthermore, flux quanta are constrained by the

Type II string
In this section, we move to the Type II orientifold compactifications.In particular, we focus on Type IIB orientifold compactifications with O3-and O7-planes, but the analysis can be extended to the T-dual Type IIA intersecting D-brane models.
On N a stacks of magnetized D7-branes wrapping holomorphic divisors Γ D7 of Calabi-Yau threefolds M12 , world-volume fluxes induce the degenerate number of chiral zero modes.Similar to the previous sections, it is possible to embed U (1) gauge fluxes into U (N a ) on the D7-branes that do not lie on the O7-planes or SO(2N a )/Sp(2N a ) on the D7branes that lie on the O7-planes.Specifically, we focus on U (N a ) magnetized D7-branes.Following Ref. [29], we choose the background gauge invariant field strength: where ι * denotes the pull-back from the CY threefolds M to Γ D7 .Here, T 0 and T i denote the generators of diagonal U (1) a and the other traceless Abelian elements of U (N a ), respectively.An orientifold projection Ω(−1) F L σ for B 2 and F is chosen as and the holomorphic involution σ acts on the Kähler form and NS-NS two-form B 2 on M: Thus, the following internal gauge invariant field strength will be relevant to the realization of chiral zero modes: Note that such background fluxes F are assumed to be holomorphic and preserve the supersymmetry: on the divisor Γ D7 , where J denotes the Kähler form of M. Furthermore, such magnetized D7-branes as well as O7-planes induce the D3-brane charge: where χ(D a ) and χ(D O7 ) denote the Euler characteristic of divisors D a and D O7 , respectively.Then, the D3-brane tadpole cancellation conditions become where Here, we suppose an uplifting of Type IIB orientifolds to the F-theory compactifications on Calabi-Yau fourfolds Y 4 with Euler characteristics χ(Y 4 ).
As discussed in Secs. 2 and 3, we derive the upper bound on the magnetic flux quanta determining the number of chiral zero modes.Now that the gauge fluxes are rewritten by with c 1 (l a ) = m a [D a ] and it is bounded from above by using the tadpole cancellation condition: Here, we use N flux ≥ 0 due to the imaginary self-dual condition of three-form fluxes.The above expression is similar to Eq. (2.28) discussed in the heterotic line bundle models.When we define m max satisfying |m a | ≤ max a (m a ) := |m max | for all a, we find that where we assume m i a ̸ = 0 for all a, but it is possible to derive m a = 0 for some a.Although the term h 1,1 (D a ) − 1 in the above equation will be modified in this case, we adopt the above conservative bound of the flux quanta in the following analysis.
Let us discuss the chiral zero modes arising from intersections of two different stacks of magnetized D7-branes carrying holomorphic line bundles l a and l b , respectively.In particular, we focus on matter fields in the bifundamental representation (N a , Nb ), as classified in the following two classes13 :

Matter divisors
If divisors of two D7-brane coincide with each other, i.e., D := D a = D b , the index of bifundamental matter fields is counted by the extension group [30].The chiral index is determined by . Thus, the chiral index is bounded from above by where we use Eq.(4.11) in the last inequality.

Matter curves
If divisors of two D7-brane are different but intersect over a curve C of genus g, the chiral index is determined by . By using Eq.(4.11), we arrive at the upper bound on the chiral index: For illustrative purposes, let us focus on toroidal orientifolds.The distribution of flux quanta was explicitly discussed in Ref. [31] in the context of Type IIB string theory on T 6 /(Z 2 × Z ′ 2 ) orientifolds with N a stacks of magnetized D(3 + 2n)-branes and O-planes.When we turn on U (1) a magnetic fluxes on (T 2 ) i : the wrapping numbers m i a and magnetic fluxes n i a lead to the net number of chiral zero modes between two stacks a and b of D-branes: where we consider 64 O3-planes.As discussed in Ref. [32], one can realize the supersymmetric brane configurations with the SM-like spectra:

Conclusions
We studied the index of chiral fermions in the effective action of E 8 × E ′ 8 and SO(32) heterotic line bundle models, and magnetized D7-branes in Type IIB/F-theory compactifications.We derived an analytical expression for the upper bound of the chiral index, which depends on the topological quantities of CY manifolds.It results in a finite number of consistent line bundle models subject to phenomenological constraints on favorable CYs.We focused on a limited class of consistent string models, but it is interesting to extend our analysis into other corners of the string landscape in the future.
In E 8 × E ′ 8 and SO(32) heterotic line bundle models, we evaluated the upper bound on complete intersection CY threefolds.Since the CY volume is bounded from above by V ≤ 25 to realize the observed value of 4D gauge couplings, semi-realistic models are realized on some of the complete intersection CY threefolds.On quotient CY threefolds, the index of chiral fermions decreases due to the freely-acting discrete symmetry group.It turned out that the chiral index satisfies N gen ≤ 100 on quotient CY threefolds we analyzed.We also commented on the favorable CY threefolds with the large number of Kähler moduli in the Kreuzer-Skarke database.Note that for SO(32) heterotic string theory, we found the bound on the chiral index of Higgs doublets and vector-like quarks in addition to quarks and leptons.Our finding bound on the chiral index is correlated with the second Chern number of the tangent bundle of CY threefolds.Such a mutual relation between the chiral index and the second Chern number was also pointed out in the context of a machine learning study [33], and it would be interesting to figure out a physical background of this suggestive behavior.
In Type IIB/F-theory compactifications, we focused on the index of chiral fermions arising in the intersection between two stacks of magnetized D7-branes.Since the worldvolume fluxes on D7-branes play a role not only in determining the chiral index but also in inducing the D3-brane charges, the D3-brane tadpole cancellation condition restricts the maximal value of flux quanta.It turned out that the chiral index on matter divisors or matter curves is restricted by the number of D7-brane and topological quantities of CY manifolds.For concreteness, we analyzed the generation number of quarks and leptons on T 6 /(Z 2 × Z ′ 2 ) orientifolds.The index is indeed bounded by the orientifold contributions.Thus, the small tadpole charge leads to a few generations of chiral fermions.

Figure 1 .
Figure 1.The upper bound on the chiral index in SU (5) GUT on upstairs CICYs M.
, we plot the chiral index on 195 quotient CICYs satisfying the volume constraint(2.15).Similarly, the moduli values are chosen as t := t i for all i with the CY volume V Γ = 12 in the left panel and V Γ = 25 in the right panel of Fig.2, respectively.It turns out that the chiral index suppressed by |Γ| satisfies N gen ≤ 100 on quotient CY threefolds.

Figure 2 .
Figure 2. The upper bound on the chiral index in SU (5) GUT on quotient CICYs M.

. 17 )N a n 1 a n 2 a n 3
Furthermore, magnetized D-branes carry low-dimensional D-brane charges.In particular, the D3-brane tadpole charges are satisfied when a