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Yukawa couplings in intersecting D-brane models

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D. Cremades, L.E. Ibanez, F. Marchesano

High Energy Physics - Theory · Submitted 14 Feb 2003 · 63 references

Pith's verdict: The theta-function Yukawa formula is a real, checkable result; the non-coprime ansatz is a genuine but narrower gap than it first looks, and the abstract oversells the Fukaya connection.

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arXiv IDhep-th/0302105 (version 2)
Pith IDEXEHNGV6, Pith's permanent record of this paper and its review
CitationD. Cremades, F. Marchesano, L.E. Ibanez. "Yukawa couplings in intersecting D-brane models." arXiv:hep-th/0302105 (2003). https://pith.science/paper/EXEHNGV6
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The paper

The paper claims

Yukawa couplings in intersecting D6-brane models arise from worldsheet disc instantons and evaluate to products of complex Jacobi theta functions with characteristics, depending on Kahler and open string moduli but not on complex structure.

The reading

One route to realistic particle physics from string theory wraps D-branes around small cycles inside a compact six-dimensional space. Where two branes cross, a chiral particle sits. Three such crossings can form a triangle, and a string worldsheet can stretch across it like a soap film; summing over all such films generates the Yukawa coupling that gives the particle its mass. This paper performs that sum exactly for the simplest case, flat tori. The answer is a product of Jacobi theta functions, one factor per two-torus, with arguments set by the intersection numbers, the brane positions, and the Wilson lines. The couplings depend on the Kahler moduli and on brane positions but not on the shape (complex structure) of the torus. The authors then apply the formula to a brane configuration whose chiral spectrum is that of the MSSM and find that only the third generation gets a mass, with the CKM matrix equal to the identity. Finally they show the same theta-function machinery applies to a non-compact Calabi-Yau that is mirror to D3-branes at a Z3 singularity, and observe that the disc sums they compute are precisely the composition maps of Fukaya's category.
Extended reading notes

Core claim

The Yukawa coupling for three chiral fields at the intersections of factorizable n-cycles on a T^2n is given by Eq. (3.49), Y_ijk = h_qu sigma_abc times a product over r of theta functions with characteristics, with parameters fixed by intersection numbers, brane positions, Wilson lines, and the complexified Kahler structure. If correct, this makes Yukawa couplings in any toroidal intersecting brane model computable from topological intersection numbers plus moduli, with no dependence on complex structure and no explicit dependence on the wrapping numbers themselves.

Load-bearing premise

The non-coprime case d = gcd(I_ab, I_bc, I_ca) > 1, treated in Sec. 3.1.3. The paper states: 'Although no explicit formula is given, we propose an ansatz that has been checked in plenty of models.' The ansatz supplies the theta parameters (3.30)-(3.32) and the selection rule i+j+k = 0 mod d, but no derivation and no cited or appended check is given. This matters because the paper itself calls this case 'the most interesting from the phenomenological point of view', and the MSSM example of Sec. 4 has I = 3 on each torus, so d = 3: the flagship application rests on the unproved ansatz rather than on the derived formula.

Original abstract

We compute the Yukawa couplings among chiral fields in toroidal Type II compactifications with wrapping D6-branes intersecting at angles. Those models can yield realistic standard model spectrum living at the intersections. The Yukawa couplings depend both on the Kahler and open string moduli but not on the complex structure. They arise from worldsheet instanton corrections and are found to be given by products of complex Jacobi theta functions with characteristics. The Yukawa couplings for a particular intersecting brane configuration yielding the chiral spectrum of the MSSM are computed as an example. We also show how our methods can be extended to compute Yukawa couplings on certain classes of elliptically fibered CY manifolds which are mirror to complex cones over del Pezzo surfaces. We find that the Yukawa couplings in intersecting D6-brane models have a mathematical interpretation in the context of homological mirror symmetry. In particular, the computation of such Yukawa couplings is related to the construction of Fukaya's category in a generic symplectic manifold.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,202 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.23 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.07 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We compute the Yukawa couplings among chiral fields in toroidal Type II compactifications with wrapping D6-branes intersecting at angles. Those models can yield realistic standard model spectrum living at the intersections. The Yukawa couplings depend both on the Kahler and open string moduli but not on the complex structure. They arise from worldsheet instanton corrections and are found to be given by products of complex Jacobi theta functions with characteristics. The Yukawa couplings for a particular intersecting brane configuration yielding the chiral spectrum of the MSSM are computed as an example. We also show how our methods can be extended to compute Yukawa couplings on certain classes of elliptically fibered CY manifolds which are mirror to complex cones over del Pezzo surfaces. We find that the Yukawa couplings in intersecting D6-brane models have a mathematical interpretation in the context of homological mirror symmetry. In particular, the computation of such Yukawa couplings is related to the construction of Fukaya's category in a generic symplectic manifold.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We compute the Yukawa couplings among chiral fields in toroidal Type II compactifications with wrapping D6-branes intersecting at angles. contribution 0.99 0.96 0.10 asserted 1.00
s1 Those models can yield realistic standard model spectrum living at the intersections. background 0.94 0.13 0.06 hedged possibility 0.99
s2 The Yukawa couplings depend both on the Kahler and open string moduli but not on the complex structure. result 1.00 0.84 0.20 asserted 1.00
s3 They arise from worldsheet instanton corrections and are found to be given by products of complex Jacobi theta functions with characteristics. result 0.90 0.94 0.19 asserted 1.00
s4 The Yukawa couplings for a particular intersecting brane configuration yielding the chiral spectrum of the MSSM are computed as an example. contribution 0.72 0.95 0.10 asserted 1.00
s5 We also show how our methods can be extended to compute Yukawa couplings on certain classes of elliptically fibered CY manifolds which are mirror to complex cones over del Pezzo surfaces. contribution 0.88 0.92 0.08 asserted 0.95
s6 We find that the Yukawa couplings in intersecting D6-brane models have a mathematical interpretation in the context of homological mirror symmetry. interpretation 0.98 0.92 0.37 asserted 1.00
s7 In particular, the computation of such Yukawa couplings is related to the construction of Fukaya's category in a generic symplectic manifold. interpretation 0.99 0.72 0.19 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 0.95.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Yukawa coupling between two quarks of opposite chirality and a Higgs boson. in figure 1, where two quarks of oppo￾site chirality couple to a Higgs boson. The worldsheet connecting the corre￾sponding intersection has the topology of a disc, and involves three different boundary states. These are, in a target space perspective, branes a, b and c, whose intersections localize the matter particles. Given such boundary con￾ditions, there will exist an infinity of worldsheets satisfying them. In order to compute the instanton correction to our effective theory, however, we must concentr
Figure 2. The Standard Model at intersecting D-branes. a) Four stacks of branes, baryonic, left, right and leptonic are needed to get all quark and leptons at the intersections. b) The SM branes may be wrapping cycles on a, e.g., CY manifold, with appropriate intersections so as to yield the SM chiral spectrum. Notice that considering a full D6-brane configuration instead of one single brane makes the supersymmetry discussion more involved. Although each of the components of the 6This picture may be slightly changed in orientifold models, see e.g., section 4 below. – 9 –
Figure 3. Yukawa couplings as euclidean maps from the worldsheet. area. In the specific setup discussed in [4], the geometry of intersecting brane worlds is reduced to each stack wrapping (linear) 1-cycles on a T 2 . It is thus easy to see that in this case the target space worldsheet instanton has a planar triangular shape. This, however, will not be the general shape for a holomorphic curve, even in the familiar case of higher dimensional tori with a flat metric (see appendix A). More concretely, we expect the Yukawa couplings between the fields Qi L , q j R and Hk to be roughly of the
Figure 4. Picture of the recombination. (1) Before the recombination, the worldsheet instantons connecting the QL, QR and Higgs multiplets corresponds to holomorphic discs with their boundaries embedded in three different branes. (2) After giving a v.e.v. to H, stacks b and c have recombined into a third one e. If the recombination is soft enough, the number of chiral fermions at the intersections will not vary. However, they will get mass terms by holomorphic discs that connect fermions of opposite chirality, having its boundary on stacks a and e. Let us then consider the recombination of…
Figure 5. Relevant intersections and triangles for the three 1-cycles (na, ma) = (1, 0), (nb, mb) = (1, 2) and (nc, mc) = (1, −3). In the left figure we have depicted these 1-cycles intersecting on the covering space C. The dashed lines represent various images of the brane c under torus translations. The indexing of the bc intersections by the integer k coincides with expressions (3.3) and (3.5). In the right figure we have depicted a single fundamental region of the torus, and have indexed every intersection. Notice that we have chosen that all branes intersect at the origin, and the…
Figure 6. Relevant intersections and triangles for the three 1-cycles (na, ma) = (1, 0), (nb, mb) = (1, 2) and (nc, mc) = (1, −4). Notice that the fundamental region of the torus has two identical regions, exactly matching with d = g.c.d.(Iab, Ibc, Ica) = 2. Also notice that a triangle exists connecting the vertices (i, j, k) if and only if i + j + k = even. feature of these configurations, which is that the fundamental region of the torus divides in d identical copies. That is, the intersection pattern of any of these regions exactly matches the others. This is a direct consequence from the…
Figure 7. Scheme of the model discussed in the text. Moving brane c on top of its mirror c* one gets an enhanced SU(2)R symmetry. If in addition brane d is located on top of brane a one gets an enhanced SU(4) Pati-Salam symmetry. Let us now give a specific realization of such abstract construction. For simplicity, we will consider a plain orientifold of type IIA compactified on a T 6 = T 2 × T 2 × T 2 , with R : zr 7→ z¯r r = 1, 2, 3 being a simultaneous reflection on each complex plane. Our D6-branes will wrap factorizable cycles [Πα] ≡ [(n 1 α , m1 α )] ⊗ [(n 2 α , m2 α )] ⊗ [(n 3 α , m3 α…
Figure 8. Brane configuration corresponding to the MSSM-like model described in the text, for the choice ρ = 1. For simplicity, we have not depicted the leptonic stack nor the mirror a* stack. Considering the leptonic sector involves triplets dbc and dbc*. Now, since the stack d is similar to the a, the above discussion also apply to this case, and the only change that we have to make is considering new variables (ǫ (2) l , ǫ (3) l ; θ (2) l , θ(3) l ) instead of (ǫ (2), ǫ(3); θ (2), θ(3)). Notice that the difference of this two sets of variables parametrizes the breaking SU(4) → SU(
Figure 9. D-brane configuration in (5.1), restricted to the elliptic fibre at z = 0 in the base space. Due to the quantum Z3 symmetry of this configuration, each stack of branes must be related to the others by a e ±2πi/3 rotation. Hence, the complex structure of the elliptic fibre is frozen. Notice that a triangle between vertices i, j, k closes only if the condition i + j + k ≡ 0 mod 3 is satisfied. not coprime, so the Yukawa couplings between intersections i, j, k will be given by a theta function with characteristics (3.30), (3.31) and (3.32), with d = 3. Given the specific choice of…
Figure 10. Final D-brane configuration, with the brane positions fixed by (5.6). Again we restrict to the elliptic fibre at z = 0 in the base space. 6. Yukawa versus Fukaya In the previous section we have shown how, combined with mirror symmetry, the computa￾tion of worldsheet instantons between chiral fields in intersecting D-brane models can yield a powerful tool to compute Yukawa couplings in more general setups as, e.g., D-branes at singularities. The purpose of the present section is to note that computation of Yukawas and other disc worldsheet instantons is not only a tool, but lies at…
Figure 11. Construction of higher dimensional holomorphic discs. into itself, which is given by Conf(D) = {φ τ ω (z) = τ · z − ω 1 − ωz¯ |ω ∈ Int(D), τ ∈ ∂D}. (A.2) A several statement holding for antianalytic functions. If we also require the boundary of such regions to match under f, that is, if we require f(∂∆abc) = ∂D following the second point above, then we will select one unique map from the whole family φ τ ω ◦ f, since Con(D) ∼= P SL(2, Z), and this latter group can be fixed by specifying the location of ωab, ωbc and ωca in ∂D. Moreover, the map f will be either analytic or anti

Figures reproduced from arXiv:hep-th/0302105 by the authors.

References
How later papers used it

83 later papers cite it

Evidence: substantial. 71 of the citing papers share no author with it, and 11 of those carry a confident label from Pith's citation classifier.

Pith read 23 of the 151 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 2 papers
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How the literature uses it

78 arXiv papers cite this in 146 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 99 sentences without a confident label

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Uses its data or values 1 paper
Compares with it 10 papers
Cites it as background 26 papers

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