Heterotic Compactification, An Algorithmic Approach

Andre Lukas; Lara B. Anderson; Yang-Hui He

REVIEW

Heterotic Compactification, An Algorithmic Approach

Lara B. Anderson, Yang-Hui He, Andre Lukas

High Energy Physics - Theory · Submitted 27 Feb 2007 · 45 references

Pith's verdict: A solid methods paper whose main gap is an acknowledged sketch of the rank 5 stability proof for four of seven bundles — and the stress-test's arithmetic objection does not survive checking.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on November 11, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 2 major objections · 6 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDhep-th/0702210 (version 2)

Pith ID3MS2LHGN, Pith's permanent record of this paper and its review

CitationAndre Lukas, Lara B. Anderson, Yang-Hui He. "Heterotic Compactification, An Algorithmic Approach." arXiv:hep-th/0702210 (2007). https://pith.science/paper/3MS2LHGN

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Short version: the stress-test's arithmetic objection is wrong, and the paper is in better shape than the CONDITIONAL verdict implies — but the rank 5 stability argument really is a sketch for four of the seven bundles, and that is the honest gap to press on. First, the numbers. The stress-test recomputes Eq. (53) for the first row of Table 10 and gets 264 against the tabulated 277. The slip is in C*⊗C. With c = (4,4), C*⊗C is four copies of O(0), so h^0 = 4, not 17 — 17 is what you get for c = (4,2). Redo it: 340 − 60 − 4 + 1 = 277, exactly as printed. I spot-checked a couple of other rows and they reproduce too. So Eq. (53) and Table 10 are consistent, and the claim in Eq. (58) that only n10 moves for the non-generic map stands. Don't propagate that concern. What is actually here: a genuine methods paper. The classification is small — 37 bundles — but the constraints are stated up front and the enumeration is finite and checkable. The rank 3 and rank 4 stability arguments are clean: Kodaira vanishing plus the exterior-power sequences. The singlet formula (53) is derived rank-independently, which is the reusable piece. The 'no anti-generations' result is a derivation from stability plus the long exact sequence, not a fit, and the moduli-dependence claim is backed by an explicit non-generic map with a worked Macaulay2 computation. The appendix tutorial is the kind of thing that actually gets picked up and used. Soft spots, in order. The rank 5 stability proof for [5|2 4], [5|3 3] and [6|2 2 3] is an overview by the authors' own admission, and the [6|2 2 3] case leans on an embedding trick that is also summarized. That is four of seven rank 5 bundles, and it is the gap between the abstract's 'we prove stability for all such bundles' and what is on the page. Appendix B.2 has the same shape: H^0(X,V)=0 is proved for the quintic and asserted analogous elsewhere. Neither looks likely to fail, but a referee should ask for the bookkeeping. Minor: the B.3 title says n1 where it means n10. Also worth flagging to a physics audience — n10 = 0 generically means no Higgs, which the authors handle honestly via the jumping phenomenon but which limits the phenomenological payoff. Who should read it: anyone doing monad model building, and anyone who wants a template for pushing cohomology computations into Macaulay2. It deserves referees.

Referee report

Significance

If correct, the paper is a genuine methodological advance: it shows that stability and complete spectra can be handled algorithmically for a nontrivial class of heterotic monad bundles, rather than case by case. The explicit Macaulay2 code in Appendix A.4, the reproducible spectrum tables, and the concrete moduli-dependence example are real strengths and are exactly the kind of artifacts that make the results checkable. The no-anti-generations result follows cleanly from stability plus the long exact sequence, and the honest discussion of the 'jumping phenomenon' in Section 4.3.1 is a credit to the paper. The main caveat is that the rank 5 stability proof for the codimension-2 and codimension-3 manifolds is presented as an overview rather than exhibited in full.

Major objections (2)

[Section 4.4.2] The stability proof for the four rank 5 bundles on [5|2 4], [5|3 3], and [6|2 2 3] is given only as an overview: 'In the interests of space, we will only give an overview of it here.' The abstract claims 'we prove stability for all such bundles,' and this is load-bearing for the 'all 37' claim. The argument structure is clear and the key maximal-rank step is stated, but the explicit cohomology dimensions that close it (e.g., h^4(A, ∧^2N*⊗S^3C*) versus h^4(A, ∧^2N*⊗S^2C*⊗B*), and the analogous dimensions for the [6|2 2 3] case) are not given, so the reader cannot verify the vanishing of ker(h') without redoing the calculation. Please supply these dimensions, for instance in a short table or an appendix, or state explicitly that they were obtained by Macaulay2 and give the output.
[Section 3.3 and Appendix B.2] There is an internal inconsistency about whether the rank 3 stability proof uses H^0(X,V)=0. Section 3.3 states that for p = n−1 one needs H^0(X,V)=0, that this is proved in Appendix B.2, and that 'this completes the stability proof for the rank 3 bundles.' Appendix B.2, however, states that 'for the rank 3 bundles in our list we cannot apply these methods, nor do we make use of h0(X,V)=0 in those stability proofs.' These two statements cannot both be right as written. Since ∧^2V* ≅ V for SU(3), H^0(X,∧^2V*)=0 is exactly H^0(X,V)=0, so the analytic proof does appear to need it. Please reconcile the two passages and clarify whether the rank 3 stability rests on the analytic argument, on the Macaulay2 computation, or on both.

Minor objections (6)

[Appendix B.3 (title) and Section 4.3.1] The title of Appendix B.3 reads 'Proof that n1 = h1(X, ∧2V*) = 0 for the SO(10) Models,' but in Table 2 and Section 4.3.1 n1 denotes h^1(V⊗V*) while h^1(∧2V*) is n10. The title should refer to n10. In addition, Section 4.3.1 says the proof of n10 = 0 'can be found in Appendix B.2,' but the relevant appendix is B.3; B.2 is the proof of H^0(X,V)=0.
[Section 4.4.3] The sentence 'As for rank 4 and 5 bundles, we have h0(X,V) = h3(X,V) = 0 from stability and h2(X,V) = 2 from the sequence (30)' contains two slips: the comparison should be to rank 3 and 4 bundles, and h^2(X,V) should be 0, not 2, as follows from the same long exact sequence used throughout.
[Appendix A.4] In the Macaulay2 tutorial the map is defined as 'gmap = map(C, B, random(C, B));' but the subsequent line uses 'Vdual = sheaf coker transpose fmap;'. The variable should be 'gmap', not 'fmap'; as written the code will not run as a self-contained example, which undercuts the reproducibility value of the appendix.
[Abstract] The abstract says 'we classify positive monad bundles,' but the classification is of positive monad bundles satisfying the additional anomaly and index-divisibility constraints of Section 4.1. Qualifying the phrase (e.g., 'positive monad bundles subject to the physical constraints below') would match the body of the paper.
[Section 4.4.1] The chain 'h1(A, N*⊗∧3V*) = h3(A, K1) = ker(h′)' uses equality where isomorphism or dimension equality is meant; writing '≅ ker(h′)' or '= dim ker(h′)' would be clearer.
[Eq. (42)] The middle expression in Eq. (42) is the coefficient of J^2 in c2(V), i.e., c2(V)/J^2; labelling it explicitly would make the anomaly constraint easier to read against Eq. (26).

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

The paper argues that heterotic string model building on five Calabi–Yau threefolds can be turned into an algorithm: enumerate the positive monad bundles, prove each is stable, and read off the full particle spectrum.

The reading

Heterotic string compactification needs a Calabi–Yau threefold $X$ plus a holomorphic vector bundle $V$ on it whose structure group breaks $E_8$ down to a grand unified group. The hard part has always been stability: a bundle is only admissible if it is (semi-)stable, and checking that condition directly is usually a research project per bundle. This paper asks whether the whole job can be made mechanical for a restricted but well-defined class — bundles written as monads, $0→V→⨁_i O_X(b_i)→⨁_j O_X(c_j)→0$, on the five Calabi–Yau manifolds that are complete intersections in a single projective space. Imposing vanishing first Chern class, positivity of all the line-bundle degrees, an anomaly condition satisfiable with a five-brane, and an index that is a nonzero multiple of three, the authors enumerate 37 bundles: 20 of rank 3, 10 of rank 4, and 7 of rank 5. They then argue that all 37 are stable and compute their complete spectra, singlets included, using a mixture of analytic vanishing arguments and the computer algebra package Macaulay2. If the argument holds, a slice of heterotic model building becomes a screening procedure rather than a craft, and the tabulated spectra become a starting point for more detailed phenomenology.

What carries the argument

The monad short exact sequence $0→V→⨁_i O_X(b_i)→⨁_j O_X(c_j)→0$ converts every question about $V$ into a question about sums of line bundles, whose cohomology on these five manifolds is known in closed form from the index theorem. Hoppe's criterion then turns stability into a finite list of vanishing statements for $H^0(X,∧^pV^*)$, which are dispatched by the exterior-power sequences induced by the dual monad together with Kodaira vanishing; the rank-5 case needs Koszul resolutions to move the calculation to the ambient projective space, where the maximal-rank property of the defining matrix propagates to the induced maps on cohomology. Singlet counts come from $h^1(X,V⊗V^*)$, computed by a web of six interlocking exact sequences and reducible to a closed formula, $h^1(X,V^*⊗V)=h^0(X,B^*⊗C)-h^0(X,B^*⊗B)-h^0(X,C^*⊗C)+1$, valid for all ranks. Every step is cross-checked by explicit sheaf cohomology and Ext computations in Macaulay2.

Extended reading notes

Core claim

The central claim is that for every one of the 37 positive monad bundles in the classification, stability can be established and the full low-energy particle spectrum computed. Stability is reduced to Hoppe's criterion — on a cyclic manifold with $c_1(V)=0$, vanishing of $H^0(X,∧^pV)$ for $p=1,…,rk(V)-1$ implies stability — and the required vanishings are obtained from long exact sequences in cohomology, Kodaira vanishing for the negative bundles $B^*$ and $C^*$, and, for rank 5, Koszul resolutions that lift the computation to the ambient projective space. Two structural results follow. First, the number of anti-generations vanishes for all 37 bundles: $n_{27}=0$ for $E_6$, $n_{16}=0$ for $SO(10)$, $n_{10}=0$ for $SU(5)$; only the generation number survives, and it is read off from the index. Second, the spectrum is not rigid: the authors exhibit an explicit non-generic map $g$ for an $SU(4)$ bundle on the quintic for which $n_{10}$ jumps from 0 to 13 while $n_{16}=90$ and $n_1=277$ stay fixed, so the Higgs count in particular is a property of a point in moduli space, not of the bundle's topology alone.

Load-bearing premise

The load-bearing premise is that the defining matrix of each monad is generic enough that it has maximal rank and that this maximal rank propagates to the induced maps on ambient-space cohomology; the stability proof for the four rank-5 bundles on the codimension-2 and codimension-3 manifolds is given only as an overview, the authors writing that in the interests of space they will only sketch it, so those cases rest on an argument outline rather than a reproduced computation.

Editorial extensions

If this is right

All 37 bundles in the classification are admissible supersymmetric heterotic backgrounds, so their spectra can be tabulated rather than derived case by case; the paper lists generation and singlet counts for every entry.
Because no anti-generations appear, the chiral asymmetry of each model equals its total generation number, which is fixed by the index and is a nonzero multiple of three by construction — the prerequisite for three-family models after quotienting by a discrete symmetry.
The singlet counts are large and explicit (for example 141, 231, 277 for various quintic models), giving concrete targets for Wilson-line breaking and for checking whether unwanted light fields can be lifted.
The demonstrated jump in $n_{10}$ from 0 to 13 on a special locus shows that the spectrum depends on the complex structure moduli through the map $g$, so statements about Higgs content must be made region by region in moduli space.
The method extends to complete intersections in products of projective spaces and in weighted projective spaces, which the authors identify as the next target.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the generic vanishing of $n_{10}$ and $n_5$ is really a property of the maximal-rank locus, then the physically interesting Higgs spectra live on special divisors in moduli space; a systematic scan over degenerate maps, rather than random ones, is the natural next search strategy.
The stability argument for the four rank-5 bundles on the codimension-2 and codimension-3 manifolds is presented as an overview rather than a full bookkeeping, so an independent Koszul computation for those cases would be the cheapest way to test the paper's strongest claim.
The same machinery should apply to monads whose constituent line bundles are not all positive, provided a substitute for Kodaira vanishing exists; relaxing positivity would enlarge the classification well beyond 37 and might reintroduce anti-generations.
Since singlet counts are computable but Yukawa couplings are not addressed here, the natural follow-on test of these models is whether the same computer-algebra pipeline can extract triple products of cohomology classes.

Original abstract

We approach string phenomenology from the perspective of computational algebraic geometry, by providing new and efficient techniques for proving stability and calculating particle spectra in heterotic compactifications. This is done in the context of complete intersection Calabi-Yau manifolds in a single projective space where we classify positive monad bundles. Using a combination of analytic methods and computer algebra we prove stability for all such bundles and compute the complete particle spectrum, including gauge singlets. In particular, we find that the number of anti-generations vanishes for all our bundles and that the spectrum is manifestly moduli-dependent.

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, 3,616 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.43 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.59 · compares with prior results 0.10 · mentions code or data availability 0.03.

The abstract, sentence by sentence

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

We approach string phenomenology from the perspective of computational algebraic geometry, by providing new and efficient techniques for proving stability and calculating particle spectra in heterotic compactifications. This is done in the context of complete intersection Calabi-Yau manifolds in a single projective space where we classify positive monad bundles. Using a combination of analytic methods and computer algebra we prove stability for all such bundles and compute the complete particle spectrum, including gauge singlets. In particular, we find that the number of anti-generations vanishes for all our bundles and that the spectrum is manifestly moduli-dependent.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We approach string phenomenology from the perspective of computational algebraic geometry, by providing new and efficient techniques for proving stability and calculating particle spectra in heterotic compactifications.	contribution 0.97	0.87	0.13	asserted 0.99
s1	This is done in the context of complete intersection Calabi-Yau manifolds in a single projective space where we classify positive monad bundles.	method 0.48	0.72	0.06	asserted 0.99
s2	Using a combination of analytic methods and computer algebra we prove stability for all such bundles and compute the complete particle spectrum, including gauge singlets.	method 0.62	0.96	0.23	asserted 1.00
s3	In particular, we find that the number of anti-generations vanishes for all our bundles and that the spectrum is manifestly moduli-dependent.	result 1.00	0.96	0.48	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.98 · evidence from proof or derivation 0.92.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

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

References

Works cited (45)

[1]

P. Candelas, G. T. Horowitz, A. Strominger and E. Witten, “Vacuum Conﬁgura- tions For Superstrings,” Nucl. Phys. B 258, 46 (1985)

View on Pith 1985
[2]

B. Greene, K. H. Kirklin, P. J. Miron and G. G. Ross, A Superstring Inspired Standard Model , Phys. Lett. B 180 (1986) 69; A Three Generation Superstring Model. 1. Compactiﬁcation and Discrete Symmetries , Nucl. Phys. B 278 (1986) 667; A Three Generation Superstring Model. 2. Symmetry Breaking a nd the low- Energy Theory, Nucl. Phys. B 292 (1987) 602

View on Pith 1986
[3]

V. Braun, Y. H. He, B. A. Ovrut and T. Pantev, “A heterotic s tandard model,” Phys. Lett. B 618, 252 (2005) [arXiv:hep-th/0501070]. V. Braun, Y. H. He, B. A. Ovrut and T. Pantev, “A standard model from the E(8) x E(8 ) heterotic superstring,” JHEP 0506, 039 (2005) [arXiv:hep-th/0502155]

View on Pith Pith review arXiv 2005
[4]

V. Braun, Y. H. He, B. A. Ovrut and T. Pantev, “The exact MSS M spectrum from string theory,” JHEP 0605, 043 (2006) [arXiv:hep-th/0512177]

View on Pith Pith review arXiv 2006
[5]

V. Bouchard and R. Donagi, “An SU(5) heterotic standard m odel,” Phys. Lett. B 633, 783 (2006) [arXiv:hep-th/0512149]

View on Pith Pith review arXiv 2006
[6]

R. Donagi, Y. H. He, B. A. Ovrut and R. Reinbacher, “The par ticle spectrum of heterotic compactiﬁcations,” JHEP 0412, 054 (2004) [arXiv:hep-th/0405014]

View on Pith Pith review arXiv 2004
[7]

R. Donagi, Y. H. He, B. A. Ovrut and R. Reinbacher, “Higgs d oublets, split mul- tiplets and heterotic SU(3)C x SU(2)L x U(1)Y spectra,” Phys . Lett. B 618, 259 (2005) [arXiv:hep-th/0409291]. R. Donagi, Y. H. He, B. A. Ov rut and R. Rein- bacher, “The spectra of heterotic standard model vacua,” JH EP 0506, 070 (2005) [arXiv:hep-th/0411156]

View on Pith Pith review arXiv 2005
[8]

V. Braun, Y. H. He, B. A. Ovrut and T. Pantev, “Vector bundl e extensions, sheaf cohomology, and the heterotic standard model,” [arXiv:hep -th/0505041]

View on Pith
[9]

L. B. Anderson, Y. H. He and A. Lukas, In preparation

View on Pith
[10]

E. Witten, “New Issues In Manifolds Of SU(3) Holonomy,” Nucl. Phys. B 268 (1986) 79

View on Pith 1986
[11]

J. Distler and B. R. Greene, “ASPECTS OF (2,0) STRING COM PACTIFICA- TIONS,” Nucl. Phys. B 304 (1988) 1

View on Pith 1988
[12]

R. Donagi, “Principal bundles on elliptic ﬁbrations,” Asian J. Math. Vol. 1 (1997), 214-223, [alg-geom/9702002]. R. Friedman, J. Morgan, E. Wi tten, “Vector Bundles And F Theory,” Commun.Math.Phys. 187 (1997) 679-743 R. Friedman, J. Morgan, E. Witten “Vector Bundles over Elliptic Fibrations,” [alg- geom/9709029]

View on Pith Pith review arXiv 1997
[13]

R. P. Thomas, Examples of Bundles on Calabi-Yau 3-Folds for String Theory Com - pactiﬁcations, Adv. Theor. Math. Phys. 4, 231 (2000) [math.ag/9912179]

View on Pith arXiv 2000
[14]

T. L. Gomez, S. Lukic and I. Sols, ”Constraining the K¨ ahler Moduli in the Heterotic Standard Model, [arXiv:hep-th/0512205]. 34

View on Pith Pith review arXiv
[15]

V. Braun, Y. H. He and B. A. Ovrut, “Stability of the minim al heterotic standard model bundle,” JHEP 0606, 032 (2006) [arXiv:hep-th/0602073]

View on Pith Pith review arXiv 2006
[16]

R. Blumenhagen, S. Moster and T. Weigand, “Heterotic GU T and standard model vacua from simply connected Calabi-Yau manifolds,” Nucl. P hys. B 751, 186 (2006) [arXiv:hep-th/0603015]. R. Blumenhagen, S. Moster, R. Reinbacher and T. Weigand, “Ma ssless spectra of three generation U(N) heterotic string vacua,” [arXiv:hep -th/0612039]

View on Pith Pith review arXiv 2006
[17]

S. Kachru, “Some Three Generation (0,2) Calabi-Yau Mod els,” Phys. Lett. B 349, 76 (1995) [arXiv:hep-th/9501131]

View on Pith Pith review arXiv 1995
[18]

R. Blumenhagen, “Target space duality for (0,2) compac tiﬁcations,” Nucl. Phys. B 513, 573 (1998) [arXiv:hep-th/9707198]. R. Blumenhagen, R. Schimmrigk and A. Wisskirchen, “(0,2) mi rror symmetry,” Nucl. Phys. B 486, 598 (1997) [arXiv:hep-th/9609167]

View on Pith Pith review arXiv 1998
[19]

M. R. Douglas and C. G. Zhou, “Chirality change in string theory,” JHEP 0406, 014 (2004) [arXiv:hep-th/0403018]

View on Pith Pith review arXiv 2004
[20]

J. Distler and E. Sharpe, “Heterotic compactiﬁcations with principal bundles for general groups and general levels,” arXiv:hep-th/0701244

View on Pith Pith review arXiv
[21]

H. Hoppe, “Generischer spaltungstypumun zweite Chern klassestabiler Vektorraum- b¨ undel vom rang 4 auf P4,” Math. Z., 187 (1984), 345-360

View on Pith 1984
[22]

Maria Chiara Brambilla, “Semistability of certain bun dles on a quintic Calabi-Yau threefold,” [math.AG/0509599]

View on Pith arXiv
[23]

D. Grayson and M. Stillman, “Macaulay 2, a software syst em for research in alge- braic geometry.” Available at http://www.math.uiuc.edu/Macaulay2/

View on Pith
[24]

R. Donagi, Y. H. He, B. A. Ovrut and R. Reinbacher, “Modul i dependent spectra of heterotic compactiﬁcations,” Phys. Lett. B 598, 279 (2004) [arXiv:hep-th/0403291]

View on Pith Pith review arXiv 2004
[25]

S. Hosono, A. Klemm, S. Theisen and S. T. Yau, “Mirror sym metry, mirror map and applications to complete intersection Calabi-Yau spac es,” Nucl. Phys. B 433 (1995) 501 [arXiv:hep-th/9406055]

View on Pith Pith review arXiv 1995
[26]

J. Gray, Y. H. He, V. Jejjala and B. D. Nelson, “Exploring the vacuum geometry of N = 1 gauge theories,” Nucl. Phys. B 750, 1 (2006) [arXiv:hep-th/0604208]. J. Gray, Y. H. He and A. Lukas, “Algorithmic algebraic geomet ry and ﬂux vacua,” JHEP 0609, 031 (2006) [arXiv:hep-th/0606122]

View on Pith Pith review arXiv 2006
[27]

J. Gray, A. Lukas and B. Ovrut, “Perturbative anti-bran e potentials in heterotic M-theory,” [arXiv:hep-th/0701025]

View on Pith Pith review arXiv
[28]

M. Green, J. Schwarz and E. Witten, “Superstring theory ,” CUP, 1990

View on Pith 1990
[29]

K. Uhlenbeck and S.-T. Yau, “On the existence of Hermiti an Yang-Mills connec- tions in stable bundles,” Comm. Pure App. Math. 39 257 (1986) , 42 703 (1986). S. 35 Donaldson, “Anti Self-Dual Yang-Mills Connections over Co mplex Algebraic Sur- faces and Stable Vector Bundles,” Proc. London Math. Soc. 3 1 (1985)

View on Pith 1986
[30]

W. Barth and K. Hulek, “Monads and Moduli of Vector Bundl es,” Manuscripta Math. 25, 323-347 (1978)

View on Pith 1978
[31]

T. Hubsch “Calabi-Yau Manifolds — A Bestiary for Physic ists,” World Scientiﬁc, Singapore, 1994

View on Pith 1994
[32]

Marcos Jardim, “Stable bundles on 3-fold hypersurface s,” [math.AG/0606658]. Marcos Jardim and Rosa M. Mir-Roig, “On the semistability of instanton sheaves over certain projective varieties,” [math.AG/0603246]

View on Pith arXiv
[33]

G. Horrocks and D. Mumford, “A rank 2 vector bundle on P4 with 15000 symme- tries,” Topology, 12:63-81, (1973)

View on Pith 1973
[34]

A. Beilinson, “Coherent sheaves on Pn and problems in linear algebra,” Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68-69

View on Pith 1978
[35]

M. Maruyama,” Moduli of stable sheaves, II ,” J. Math. Ky oto Univ. 18-3, 1978, 557-614

View on Pith 1978
[36]

C. Okonek, M. Schneider, H. Spindler, “Vector Bundles o n Complex Projective Spaces,” Birkhauser Verlag, 1988

View on Pith 1988
[37]

D. Huybrechts and M. Lehn, “The Geometry of Moduli Space s of Sheaves,” Aspects of Mathematics E 31, Vieweg (1997)

View on Pith 1997
[38]

H. Schenck, “Computational Algebraic Geometry,” LMS 5 8, CUP, 2003

View on Pith 2003
[39]

D. Eisenbud, D. Grayson, M. Stillman, and B. Sturmfels, Ed, “Computations in algebraic geometry with Macaulay 2,” Springer-Verlag, 2 001. Available in http://www.math.uiuc.edu/Macaulay2/Book/

View on Pith
[40]

R. Hartshorne, “Algebraic Geometry, Springer,” GTM 52 , Springer-Verlag, 1977. P. Griﬃth, J. Harris, “Principles of algebraic geometry,” 1 978

View on Pith 1977
[41]

W. Fulton and J Harris, “Representation Theory. A ﬁrst c ourse,” Graduate Texts in Mathematics. Readings in Mathematics 129. Springer Verl ag (1991)

View on Pith 1991
[42]

W. Fulton, “Intersection Theory,” Springer-Verlag (1 998). pg. 430-440

View on Pith
[43]

V. Mu˜ noz and F. Presas, “Semipositive Bundles and Bril l-Noether Theory,” Bull. London Math. Soc. 35, 179-190 (2003)

View on Pith 2003
[44]

F. Laytimi and D. Nagaraj, “Maximal Degeneracy Loci and the Secant Vector Bundle,” Journ. of Math. Sci., Vol 94., No.1, 1068-1072 (199 9)

View on Pith
[45]

W. Fulton and R. Lazarsfeld, “On the connectedness of de generacy loci and special divisors,” Acta Math. 146, 271-283 (1981). 36

View on Pith 1981

Forward citations

Cited by 31 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. OpenAlex reports about 122 citations worldwide. Full citation record

Exploring Line Bundle Standard Models with Transformers
hep-th 2026-06 unverdicted novelty 7.0 of 10

A Transformer trained by reinforcement learning generates heterotic line-bundle sums that satisfy anomaly-cancellation, stability, and chirality constraints, and its policy transfers usefully across Calabi-Yau geometries.
Hilbert Functions and Line Bundle Cohomology on CICY Threefolds
hep-th 2026-06 unverdicted novelty 7.0 of 10

Hilbert functions of Koszul maps turn empirical chamber-wise polynomial formulae for line bundle cohomology on CICY threefolds into explicit analytic or finite-box certified statements.
Revisiting the Quantum Geometry of Torus-fibered Calabi-Yau Threefolds
hep-th 2025-10 conditional novelty 7.0 of 10

The conjectured Jacobi modularity of topological string amplitudes on torus-fibered Calabi-Yau threefolds is derived conditionally from the wave-function property under the relative conifold monodromy, which also maps...
Deep-Learning the Landscape
hep-th 2017-06 conditional novelty 7.0 of 10

Multi-layer neural networks trained on pixelated configuration matrices of Calabi-Yau manifolds, vector bundles, and quivers predict geometric invariants on held-out data, establishing machine learning as a practical ...
A New Construction of Calabi-Yau Manifolds: Generalized CICYs
hep-th 2015-07 conditional novelty 7.0 of 10

Allowing negative entries in CICY configuration matrices, with the defining relations imposed sequentially, yields a larger class of Calabi-Yau threefolds, including new Hodge number pairs and new manifolds.

Show 26 more citing papers

The MSSM Spectrum from (0,2)-Deformations of the Heterotic Standard Embedding
hep-th 2011-12 conditional novelty 7.0 of 10

Eight branches of the moduli space of (0,2) deformations of the heterotic standard embedding on a Z12 Calabi-Yau quotient yield exactly the massless MSSM spectrum.
Stability Walls in Heterotic Theories
hep-th 2009-05 accept novelty 7.0 of 10

The paper derives the anomalous U(1) D-term at a heterotic stability wall and shows its Fayet-Iliopoulos term is proportional to the bundle slope, reproducing slope stability from the four-dimensional effective theory.
Yukawa Couplings in Heterotic Compactification
hep-th 2009-04 conditional novelty 7.0 of 10

A polynomial-algebra algorithm computes Yukawa couplings for heterotic compactifications on Calabi-Yau threefolds with non-standard embeddings, reducing the calculation to multiplying and reducing polynomials.
Stabilizing the Complex Structure in Heterotic Calabi-Yau Vacua
hep-th 2010-10 conditional novelty 6.5 of 10

A holomorphic gauge bundle in a heterotic Calabi-Yau vacuum stabilizes complex structure moduli, because shape deformations that break bundle holomorphy generate a positive F-term potential; a worked example fixes 17 ...
Supersymmetric Protection and the Swampland
hep-th 2020-03 conditional novelty 6.0 of 10

Exact vanishing of supersymmetric protected quantities in quantum gravity occurs only when the theory is an orbifold of a higher-supersymmetry theory, motivating a new Swampland conjecture.
Machine Learning Line Bundle Cohomology
hep-th 2019-06 conditional novelty 6.0 of 10

A region-detecting neural network recovers known piecewise-polynomial formulas for line bundle cohomology and generates new conjectured formulas for several surfaces and three-folds.
Fibrations in CICY Threefolds
hep-th 2017-08 accept novelty 6.0 of 10

Complete classification of genus-one fibrations for 4,957 of the 7,890 CICY threefolds, plus 139,597 obvious fibrations across the full set and an explicit infinite family on the Schoen manifold.
T-Branes and Geometry
hep-th 2013-10 conditional novelty 6.0 of 10

T-brane data in six-dimensional F-theory is the singular limit of three-form potential periods in the intermediate Jacobian, and that limit is the fiber of a Hitchin system with defects.
The Atiyah Class and Complex Structure Stabilization in Heterotic Calabi-Yau Compactifications
hep-th 2011-07 conditional novelty 6.0 of 10

Three computations of how a holomorphic vector bundle constrains the complex structure moduli of a Calabi-Yau threefold are argued to agree, and explicit bundles that freeze 80 of 82 moduli are exhibited.
Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds
hep-th 2011-06 conditional novelty 6.0 of 10

A systematic scan of line-bundle sums on complete-intersection Calabi-Yau threefolds with up to five Kahler parameters yields 208 heterotic models with the MSSM matter spectrum, the Standard Model gauge group, and no exotics.
Stabilizing All Geometric Moduli in Heterotic Calabi-Yau Vacua
hep-th 2011-01 conditional novelty 6.0 of 10

All geometric moduli of a smooth heterotic Calabi-Yau compactification can be stabilized without NS flux by combining bundle holomorphy, anomalous U(1) D-terms, and non-perturbative effects, with an explicit AdS example.
Yukawa Textures From Heterotic Stability Walls
hep-th 2010-01 accept novelty 6.0 of 10

Extra U(1) symmetries that appear on heterotic stability walls constrain Yukawa couplings and vector-like masses throughout the adjacent stable region, producing persistent hierarchical textures.
The Edge Of Supersymmetry: Stability Walls in Heterotic Theory
hep-th 2009-03 conditional novelty 6.0 of 10

An anomalous U(1) D-term in the four-dimensional effective theory vanishes exactly where the heterotic gauge bundle is slope-stable and is positive where it is unstable, giving a field-theoretic description of bundle ...
Torsional Heterotic Geometries
hep-th 2009-03 accept novelty 6.0 of 10

Duality with orientifold flux compactifications generates new torsional heterotic backgrounds and a perturbative solution to the heterotic Bianchi identity, plus an exact NS5-brane metric.
Monad Bundles in Heterotic String Compactifications
hep-th 2008-05 conditional novelty 6.0 of 10

Positive monad bundles on favourable complete intersection Calabi-Yau manifolds form a finite class of 7118 examples on 36 manifolds, all with vanishing anti-families and no vector-like family pairs.
Triadophilia: A Special Corner in the Landscape
hep-th 2007-06 conditional novelty 6.0 of 10

Calabi-Yau manifolds with both Hodge numbers small are rare, the (3,3) MSSM manifold and the Tian-Yau three-generation manifold are linked by conifold transitions, and heterotic bundles may be carried across those tra...
Rational points in the 6d supergravity landscape and simple current extensions
hep-th 2026-03 unverdicted novelty 5.0 of 10

Rationality of the 2d SCFT on BPS strings provides a consistency criterion that uniquely determines the elliptic genus and global gauge group in a class of 6d supergravity models without tensor multiplets.
Multiple Fibrations in Calabi-Yau Geometry and String Dualities
hep-th 2016-08 conditional novelty 5.0 of 10

Different elliptic fibrations of a single Calabi-Yau threefold give rise to distinct six-dimensional F-theory vacua that all reduce to the same five-dimensional M-theory theory on a circle, and multiple K3 fibrations ...
Non-supersymmetric heterotic model building
hep-th 2014-07 conditional novelty 5.0 of 10

Compactifications of the non-supersymmetric SO(16)xSO(16) heterotic string on Calabi-Yau backgrounds are argued to be tachyon-free at large volume, and a three-generation Standard Model-like orbifold model with one Hi...
A Comprehensive Scan for Heterotic SU(5) GUT models
hep-th 2013-07 conditional novelty 5.0 of 10

An exhaustive-by-saturation scan over roughly 10^40 sums of five line bundles on 68 favourable complete-intersection Calabi-Yau manifolds yields 34,989 SU(5) GUT models with three-family chiral asymmetry, no 10-anti-f...
Heterotic Line Bundle Standard Models
hep-th 2012-02 conditional novelty 5.0 of 10

About 400 explicit heterotic line bundle compactifications reproduce the Standard Model matter spectrum with three families, one to three Higgs pairs, and no exotic charged states, catalogued in a public database.
Exploring Positive Monad Bundles And A New Heterotic Standard Model
hep-th 2009-11 conditional novelty 5.0 of 10

An exhaustive scan rules out all positive two-term monad bundles on complete intersection Calabi-Yau manifolds as sources of the exact Standard Model spectrum, and a semi-positive monad on the bi-cubic is shown to yield it.
Towards Realistic String Vacua From Branes At Singularities
hep-th 2008-10 conditional novelty 5.0 of 10

Fractional D-branes at a del Pezzo 1 singularity give three-family Standard Model-like matter with hierarchical Yukawa couplings, and inside a LARGE volume compactification all leading-order soft supersymmetry-breakin...
Machine Learning CICY Threefolds
hep-th 2018-06 conditional novelty 4.0 of 10

Neural networks and SVMs predict Hodge numbers, favourability, and discrete symmetries of complete intersection Calabi-Yau threefolds, reaching 88% accuracy on h^{1,1} but only weak performance on symmetry prediction.
STRINGVACUA: A Mathematica Package for Studying Vacuum Configurations in String Phenomenology
hep-th 2008-01 accept novelty 4.0 of 10

STRINGVACUA provides a Mathematica front end to Singular for decomposing and solving the polynomial equations that define vacua of four-dimensional N=1 supergravities.
The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning
hep-th 2018-12 conditional novelty 2.0 of 10

The draft is a pedagogical survey connecting Calabi-Yau geometry, string-theory model building, computational algebraic geometry, and machine-learning applications, with code examples and dataset pointers for beginners.

How the literature uses it

52 arXiv papers cite this in 102 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 · 51 sentences without a confident label

Builds on it 2 papers

These conditions are a generalization of Hoppe's criterion [ref] , used in [ref] to prove stability in the case of cyclic Calabi-Yau manifolds.

Monad Bundles in Heterotic String Compactifications · Stability · 0.89
In this respect, it would be interesting to generalize the results of [ref] to vector bundles over generic toric varieties, which are not products of projective spaces.

Landscape Study of Target Space Duality of (0,2) Heterotic String Models · Introduction · 0.86

Uses its method or tool 10 papers

A break-through in the method of attack of the programme in [ref] is the extensive use of the recent rapid advances in computer algebra, especially in algorithmic algebraic geometry [ref] .

An Algorithmic Approach to Heterotic String Phenomenology · Monad Bundles: A Large Class · 0.91
Secondly, efficient methods to compute line bundle cohomology on CICYs have been described and implemented by the authors in Ref. [ref] .

Heterotic Line Bundle Standard Models · Heterotic line bundle models on CICYs · 0.89
Instead, our starting point is an explicit formal construction of a rank $n$ holomorphic vector bundle (for example a sum of line bundles, or a bundle constructed via a monad [ref] , or by extension [ref] ).

A Comprehensive Scan for Heterotic SU(5) GUT models · Principal Bundles vs. Vector Bundles · 0.81

Uses its data or values 2 papers

We provide here a concrete proof-of-concept of this idea using explicit data, viz., what was referred to as “landscape data” in [ref] We will focus on the so-called complete intersection Calabi-Yau (CICY) manifolds in complex dimension three [ref] and four [ref] , which are chosen for their deep relevance to foundational problems in algebraic geometry and string theory [ref] as well as the proven effectiveness of AI/ML methods in analyzing these datasets [ref] .

The World in a Grain of Sand: Condensing the String Vacuum Degeneracy · Introduction and Summary · 0.86
From Table 4 in [ref] we see that there exist two $SU(3)$ monad bundles with second Chern classes of 3 and 7 (relative to the square of the overall Kähler form).

Supersymmetric Protection and the Swampland · Non-standard embeddings · 1.00

Compares with it 2 papers

Firstly, such configurations are relatively simple to deal with from a computational point of view and, as a result, vastly greater numbers of models can be considered as compared to other approaches, such as [ref] .

Heterotic String Models on Smooth Calabi-Yau Threefolds · Introduction · 0.93
Firstly, such configurations are relatively simple to deal with from a computational point of view, and as a result, vastly greater numbers of models can be considered as compared to other approaches, such as [ref] .

A Comprehensive Scan for Heterotic SU(5) GUT models · Introduction and Summary · 0.90

Cites it as support 2 papers

As was shown in Refs. [ref] , for a generic choice of the morphism, $f$, this bundle has $h^2(X,∧^2 V)=0$ and, hence, cannot produce $5$-multiplets in its low energy spectrum.

Exploring Positive Monad Bundles And A New Heterotic Standard Model · The $SU(5)$ model · 0.97
For example, according to [ref] , most bundles built via the monad construction (see also [ref] ) will yield a degenerate spectral cover.

T-Branes and Geometry · The Spectral Cover Construction · 0.96

Cites it as background 22 papers

In recent years, the focus has moved to a wider class of models based on general holomorphic vector bundles on the internal space which can lead to any of the three GUT groups mentioned above [ref] .

Exploring Positive Monad Bundles And A New Heterotic Standard Model · Introduction · 0.94
Several approaches have been proposed and used over the years: smooth Calabi-Yau compactifications based on the standard embedding [ref] , non-standard embedding models [ref] , models based on orbifolds [ref] , free fermionic strings [ref] and Gepner models [ref] .

A Comprehensive Scan for Heterotic SU(5) GUT models · Introduction and Summary · 1.00
The supersymmetric heterotic theories [ref] , and in particular the E$_8×$E$_8$ theory with non-Abelian bundles [ref] or line bundles on orbifold resolutions [ref] and Complete Intersection Calabi-Yau manifolds (CICYs) [ref] , have been well-studied since the seminal paper [ref] .

Calabi-Yau compactifications of non-supersymmetric heterotic string theory · Introduction and summary · 0.92

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

P. Candelas, G. T. Horowitz, A. Strominger and E. Witten, “Vacuum Conﬁgura- tions For Superstrings,” Nucl. Phys. B 258, 46 (1985)

View on Pith 1985

[2] [2]

B. Greene, K. H. Kirklin, P. J. Miron and G. G. Ross, A Superstring Inspired Standard Model , Phys. Lett. B 180 (1986) 69; A Three Generation Superstring Model. 1. Compactiﬁcation and Discrete Symmetries , Nucl. Phys. B 278 (1986) 667; A Three Generation Superstring Model. 2. Symmetry Breaking a nd the low- Energy Theory, Nucl. Phys. B 292 (1987) 602

View on Pith 1986

[3] [3]

V. Braun, Y. H. He, B. A. Ovrut and T. Pantev, “A heterotic s tandard model,” Phys. Lett. B 618, 252 (2005) [arXiv:hep-th/0501070]. V. Braun, Y. H. He, B. A. Ovrut and T. Pantev, “A standard model from the E(8) x E(8 ) heterotic superstring,” JHEP 0506, 039 (2005) [arXiv:hep-th/0502155]

View on Pith Pith review arXiv 2005

[4] [4]

V. Braun, Y. H. He, B. A. Ovrut and T. Pantev, “The exact MSS M spectrum from string theory,” JHEP 0605, 043 (2006) [arXiv:hep-th/0512177]

View on Pith Pith review arXiv 2006

[5] [5]

V. Bouchard and R. Donagi, “An SU(5) heterotic standard m odel,” Phys. Lett. B 633, 783 (2006) [arXiv:hep-th/0512149]

View on Pith Pith review arXiv 2006

[6] [6]

R. Donagi, Y. H. He, B. A. Ovrut and R. Reinbacher, “The par ticle spectrum of heterotic compactiﬁcations,” JHEP 0412, 054 (2004) [arXiv:hep-th/0405014]

View on Pith Pith review arXiv 2004

[7] [7]

R. Donagi, Y. H. He, B. A. Ovrut and R. Reinbacher, “Higgs d oublets, split mul- tiplets and heterotic SU(3)C x SU(2)L x U(1)Y spectra,” Phys . Lett. B 618, 259 (2005) [arXiv:hep-th/0409291]. R. Donagi, Y. H. He, B. A. Ov rut and R. Rein- bacher, “The spectra of heterotic standard model vacua,” JH EP 0506, 070 (2005) [arXiv:hep-th/0411156]

View on Pith Pith review arXiv 2005

[8] [8]

V. Braun, Y. H. He, B. A. Ovrut and T. Pantev, “Vector bundl e extensions, sheaf cohomology, and the heterotic standard model,” [arXiv:hep -th/0505041]

View on Pith

[9] [9]

L. B. Anderson, Y. H. He and A. Lukas, In preparation

View on Pith

[10] [10]

E. Witten, “New Issues In Manifolds Of SU(3) Holonomy,” Nucl. Phys. B 268 (1986) 79

View on Pith 1986

[11] [11]

J. Distler and B. R. Greene, “ASPECTS OF (2,0) STRING COM PACTIFICA- TIONS,” Nucl. Phys. B 304 (1988) 1

View on Pith 1988

[12] [12]

R. Donagi, “Principal bundles on elliptic ﬁbrations,” Asian J. Math. Vol. 1 (1997), 214-223, [alg-geom/9702002]. R. Friedman, J. Morgan, E. Wi tten, “Vector Bundles And F Theory,” Commun.Math.Phys. 187 (1997) 679-743 R. Friedman, J. Morgan, E. Witten “Vector Bundles over Elliptic Fibrations,” [alg- geom/9709029]

View on Pith Pith review arXiv 1997

[13] [13]

R. P. Thomas, Examples of Bundles on Calabi-Yau 3-Folds for String Theory Com - pactiﬁcations, Adv. Theor. Math. Phys. 4, 231 (2000) [math.ag/9912179]

View on Pith arXiv 2000

[14] [14]

T. L. Gomez, S. Lukic and I. Sols, ”Constraining the K¨ ahler Moduli in the Heterotic Standard Model, [arXiv:hep-th/0512205]. 34

View on Pith Pith review arXiv

[15] [15]

V. Braun, Y. H. He and B. A. Ovrut, “Stability of the minim al heterotic standard model bundle,” JHEP 0606, 032 (2006) [arXiv:hep-th/0602073]

View on Pith Pith review arXiv 2006

[16] [16]

R. Blumenhagen, S. Moster and T. Weigand, “Heterotic GU T and standard model vacua from simply connected Calabi-Yau manifolds,” Nucl. P hys. B 751, 186 (2006) [arXiv:hep-th/0603015]. R. Blumenhagen, S. Moster, R. Reinbacher and T. Weigand, “Ma ssless spectra of three generation U(N) heterotic string vacua,” [arXiv:hep -th/0612039]

View on Pith Pith review arXiv 2006

[17] [17]

S. Kachru, “Some Three Generation (0,2) Calabi-Yau Mod els,” Phys. Lett. B 349, 76 (1995) [arXiv:hep-th/9501131]

View on Pith Pith review arXiv 1995

[18] [18]

R. Blumenhagen, “Target space duality for (0,2) compac tiﬁcations,” Nucl. Phys. B 513, 573 (1998) [arXiv:hep-th/9707198]. R. Blumenhagen, R. Schimmrigk and A. Wisskirchen, “(0,2) mi rror symmetry,” Nucl. Phys. B 486, 598 (1997) [arXiv:hep-th/9609167]

View on Pith Pith review arXiv 1998

[19] [19]

M. R. Douglas and C. G. Zhou, “Chirality change in string theory,” JHEP 0406, 014 (2004) [arXiv:hep-th/0403018]

View on Pith Pith review arXiv 2004

[20] [20]

J. Distler and E. Sharpe, “Heterotic compactiﬁcations with principal bundles for general groups and general levels,” arXiv:hep-th/0701244

View on Pith Pith review arXiv

[21] [21]

H. Hoppe, “Generischer spaltungstypumun zweite Chern klassestabiler Vektorraum- b¨ undel vom rang 4 auf P4,” Math. Z., 187 (1984), 345-360

View on Pith 1984

[22] [22]

Maria Chiara Brambilla, “Semistability of certain bun dles on a quintic Calabi-Yau threefold,” [math.AG/0509599]

View on Pith arXiv

[23] [23]

D. Grayson and M. Stillman, “Macaulay 2, a software syst em for research in alge- braic geometry.” Available at http://www.math.uiuc.edu/Macaulay2/

View on Pith

[24] [24]

R. Donagi, Y. H. He, B. A. Ovrut and R. Reinbacher, “Modul i dependent spectra of heterotic compactiﬁcations,” Phys. Lett. B 598, 279 (2004) [arXiv:hep-th/0403291]

View on Pith Pith review arXiv 2004

[25] [25]

S. Hosono, A. Klemm, S. Theisen and S. T. Yau, “Mirror sym metry, mirror map and applications to complete intersection Calabi-Yau spac es,” Nucl. Phys. B 433 (1995) 501 [arXiv:hep-th/9406055]

View on Pith Pith review arXiv 1995

[26] [26]

J. Gray, Y. H. He, V. Jejjala and B. D. Nelson, “Exploring the vacuum geometry of N = 1 gauge theories,” Nucl. Phys. B 750, 1 (2006) [arXiv:hep-th/0604208]. J. Gray, Y. H. He and A. Lukas, “Algorithmic algebraic geomet ry and ﬂux vacua,” JHEP 0609, 031 (2006) [arXiv:hep-th/0606122]

View on Pith Pith review arXiv 2006

[27] [27]

J. Gray, A. Lukas and B. Ovrut, “Perturbative anti-bran e potentials in heterotic M-theory,” [arXiv:hep-th/0701025]

View on Pith Pith review arXiv

[28] [28]

M. Green, J. Schwarz and E. Witten, “Superstring theory ,” CUP, 1990

View on Pith 1990

[29] [29]

K. Uhlenbeck and S.-T. Yau, “On the existence of Hermiti an Yang-Mills connec- tions in stable bundles,” Comm. Pure App. Math. 39 257 (1986) , 42 703 (1986). S. 35 Donaldson, “Anti Self-Dual Yang-Mills Connections over Co mplex Algebraic Sur- faces and Stable Vector Bundles,” Proc. London Math. Soc. 3 1 (1985)

View on Pith 1986

[30] [30]

W. Barth and K. Hulek, “Monads and Moduli of Vector Bundl es,” Manuscripta Math. 25, 323-347 (1978)

View on Pith 1978

[31] [31]

T. Hubsch “Calabi-Yau Manifolds — A Bestiary for Physic ists,” World Scientiﬁc, Singapore, 1994

View on Pith 1994

[32] [32]

Marcos Jardim, “Stable bundles on 3-fold hypersurface s,” [math.AG/0606658]. Marcos Jardim and Rosa M. Mir-Roig, “On the semistability of instanton sheaves over certain projective varieties,” [math.AG/0603246]

View on Pith arXiv

[33] [33]

G. Horrocks and D. Mumford, “A rank 2 vector bundle on P4 with 15000 symme- tries,” Topology, 12:63-81, (1973)

View on Pith 1973

[34] [34]

A. Beilinson, “Coherent sheaves on Pn and problems in linear algebra,” Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68-69

View on Pith 1978

[35] [35]

M. Maruyama,” Moduli of stable sheaves, II ,” J. Math. Ky oto Univ. 18-3, 1978, 557-614

View on Pith 1978

[36] [36]

C. Okonek, M. Schneider, H. Spindler, “Vector Bundles o n Complex Projective Spaces,” Birkhauser Verlag, 1988

View on Pith 1988

[37] [37]

D. Huybrechts and M. Lehn, “The Geometry of Moduli Space s of Sheaves,” Aspects of Mathematics E 31, Vieweg (1997)

View on Pith 1997

[38] [38]

H. Schenck, “Computational Algebraic Geometry,” LMS 5 8, CUP, 2003

View on Pith 2003

[39] [39]

D. Eisenbud, D. Grayson, M. Stillman, and B. Sturmfels, Ed, “Computations in algebraic geometry with Macaulay 2,” Springer-Verlag, 2 001. Available in http://www.math.uiuc.edu/Macaulay2/Book/

View on Pith

[40] [40]

R. Hartshorne, “Algebraic Geometry, Springer,” GTM 52 , Springer-Verlag, 1977. P. Griﬃth, J. Harris, “Principles of algebraic geometry,” 1 978

View on Pith 1977

[41] [41]

W. Fulton and J Harris, “Representation Theory. A ﬁrst c ourse,” Graduate Texts in Mathematics. Readings in Mathematics 129. Springer Verl ag (1991)

View on Pith 1991

[42] [42]

W. Fulton, “Intersection Theory,” Springer-Verlag (1 998). pg. 430-440

View on Pith

[43] [43]

V. Mu˜ noz and F. Presas, “Semipositive Bundles and Bril l-Noether Theory,” Bull. London Math. Soc. 35, 179-190 (2003)

View on Pith 2003

[44] [44]

F. Laytimi and D. Nagaraj, “Maximal Degeneracy Loci and the Secant Vector Bundle,” Journ. of Math. Sci., Vol 94., No.1, 1068-1072 (199 9)

View on Pith

[45] [45]

W. Fulton and R. Lazarsfeld, “On the connectedness of de generacy loci and special divisors,” Acta Math. 146, 271-283 (1981). 36

View on Pith 1981