Mirror Symmetry, Mirror Map and Applications to Calabi-Yau Hypersurfaces

A. Klemm (2); Harvard University; (2) Sektion Physik der Universit\"at M\"unchen); S. Hosono (1); S. Theisen (2); S.-T. Yau (1) ((1) Dept. of Mathematics

REVIEW

Mirror Symmetry, Mirror Map and Applications to Calabi-Yau Hypersurfaces

S. Hosono (1), A. Klemm (2), S. Theisen (2), S.-T. Yau (1) ((1) Dept. of Mathematics, Harvard University; (2) Sektion Physik der Universität München)

High Energy Physics - Theory · Submitted 26 Aug 1993 · 48 references

Pith's verdict: A real landmark in the multi-parameter mirror program, with one load-bearing identification in §2.3 that is asserted rather than proved and never carried through the pipeline.

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CitationA. Klemm (2), Harvard University; (2) Sektion Physik der Universit\"at M\"unchen), S. Hosono (1), S. Theisen (2), S.-T. Yau (1) ((1) Dept. of Mathematics. "Mirror Symmetry, Mirror Map and Applications to Calabi-Yau Hypersurfaces." arXiv:hep-th/9308122 (1993). https://pith.science/paper/T24RR5QE

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This is the paper that took the Candelas–de la Ossa–Green–Parkes machinery off the one-parameter reservation. The durable contribution is the appendices: per-model Mori cone bases, Picard–Fuchs operators, discriminants, Yukawa couplings, m matrices, Kähler cones, and instanton tables for two- and three-moduli hypersurfaces. That is a lot of explicit, checkable computation, and people used it. The other real contribution is importing Batyrev's reflexive-polyhedron construction into the physicists' toolkit and showing it reaches weighted projective hypersurfaces where the orbifold recipes of refs [8]–[11] found nothing. The IP4(3,9,17,22,24) example in §2.3 is the concrete instance, and it is a genuinely new mirror at the time. The §3.3 extension of the hypergeometric system by the automorphism operators Z'_k is a sensible idea and the X14(7,2,2,2,1) worked example is convincing on its own terms. Credit where due: the paper is candid about what it does not have. It flags the negative instanton numbers as unexpected and unexplained, and it flags the incompleteness of the type III treatment rather than papering over it. On the soft spots, the stress-test note is right and I would not soften it. The §2.3 identification of X-hat(w) with Z-hat_{f,Delta(w)} for non-Gorenstein ambient spaces rests on one example, Hodge-number agreement over several thousand classified models, and a proof deferred to a paper listed as forthcoming. Hodge numbers are necessary, not sufficient, and the deferred proof is unpublished, so that citation does not carry weight yet. The check against the classification is more than a sample, which helps, but it does not establish the isomorphism. More to the point, no non-Gorenstein case is ever run through the PF/mirror-map/instanton machinery, so the abstract's claim at [abstract:s1] is an existence argument, not worked physics. Second, the completeness criterion (3.47) is said to be verified for all models in Appendix A but the verification is not shown. Third, m_ij is fixed by matching the leading topological couplings plus integrality of instanton numbers, so the leading term of each expansion is an input. That is not circular — the intersection numbers come independently from toric geometry — but it should be read as calibration, not prediction. Who should read it: anyone doing multi-parameter mirror computations, and anyone who wants the explicit PF data. It deserves referees.

Referee report

Significance

If the construction holds, this is a substantial advance: it converts mirror symmetry for weighted-projective hypersurfaces from a case-by-case orbifold art into a systematic toric algorithm, and it supplies the first multi-modulus Picard-Fuchs operators, discriminants, mirror maps and instanton tables that others can check and use. The explicit per-model data in Appendix A (Mori cone bases l(k), operators D_k, discriminant components dis_1(X*), Yukawa couplings, matrices m_ij, Kähler cones) are a genuine resource, and the cross-checks reported in §6 are meaningful: the summed invariants Sum_{n_D} N(n_J,n_D) reproduce the one-modulus results of refs. [16],[18],[19], and the degree-(1,0) and (0,1) counts for the bicubic agree with the independent computation of ref. [35]. The paper is also candid about what it has not settled, flagging both the deferred combinatorial proof and the unexpected negative instanton numbers rather than concealing them.

Major objections (3)

[§2.3] The identification X-hat(w) = Z-hat_{f,Delta(w)} for non-Gorenstein IP(w) is stated as a claim, not derived. The stated support is (a) one worked example, IP^4(3,9,17,22,24), where the Newton polyhedra are isomorphic and the Hodge numbers from eq. (2.4) coincide, (b) an empirical survey of several thousand models from refs. [6],[11], and (c) a general combinatorial proof deferred to a publication listed only as 'will be published elsewhere'. Hodge-number agreement is necessary but not sufficient for an isomorphism of varieties, and the deferred proof is not available to the reader. Since this identification is what underwrites the abstract's claim [abstract:s1] that mirror symmetry is established for spaces whose mirrors were previously unavailable, the manuscript should either label the identification explicitly as a conjecture supported by the evidence listed, or supply one further independent check. A natural check would be to carry the IP(3,9,17,22,24) example through the Picard-Fuchs and mirror-map machinery of §§3-5 and show that the resulting Yukawa couplings and low-degree instanton numbers are geometrically sensible; as it stands, no non-Gorenstein case is taken through the pipeline, so the headline claim rests on existence arguments rather than worked physics.
[§6 and Appendix B] Several models (X12(4,3,2,2,1), X14(7,2,2,2,1), X18(9,6,1,1,1), X24(12,8,2,1,1), X12(3,3,3,2,1)) yield negative integers N({n_i}), and the manuscript itself calls this 'a priori unexpected'. The proposed explanation, that the numbers count covers of degenerate or non-isolated rational curves and that Batyrev has pointed out there are no rational curves on the bicubic for n > 3, is asserted rather than demonstrated. Because the paper's language throughout is that these are 'topological invariants' extracted from an 'instanton expansion', the geometric meaning of a negative entry is load-bearing for the interpretation, even though the Yukawa couplings themselves are unaffected. At minimum the terminology should be hedged where the geometric reading is unsupported, and ideally one negative entry should be checked against an independent computation or a direct curve count at that degree.
[§2.4 and §6] The topological triple couplings K^0 collected in eqs. (2.15)-(2.17) are the independent input that fixes the mirror-map matrix m_ij and the overall Yukawa normalization in §6 (see eq. (6.1) and the surrounding discussion). The rules (B) and (C) of §2.4 from which these couplings are computed are attributed to refs. [31], [37] and [30], of which [37] is listed as 'to appear' and [30] as 'in preparation'. Since the normalization of every instanton table in Appendix B traces back to these intersection numbers, at least one of the multi-modulus K^0 expressions should be cross-checked against a published computation, or the derivation of rules (B) and (C) sketched sufficiently that a reader can verify them without access to the unpublished reference.

Minor objections (8)

[§4.1, following eq. (4.6)] The text says the indicial equation at x1, x2 -> 0 has 'six (= dim H^{2,1}(X*))' solutions. For this model h^{2,1}(X*) = 2, so the parenthetical should read dim H^3(X*) = 2(h^{2,1}+1) = 6.
[§6, paragraph on X12(6,3,1,1,1)] The text quotes the intersections as 'K^0 = 18J^2 + 18(E1+E2)^3', whereas eq. (2.16) gives K^0 = 18J^3 + 9E1^3 + 9E2^3. The exponent on J and the factor in front of the E combination are both inconsistent with (2.16); please reconcile, since this normalization feeds the quoted m = ((1,1),(0,-3)).
[§3.3] The treatment of type III models is demonstrated on the single example X12(4,3,2,2,1), via the auxiliary period Pi'_i of eq. (3.37) and the total-derivative identity that follows it. It would help to state explicitly whether this device is algorithmic for arbitrary type III models or whether each case requires an ad hoc choice of charge-two invariants tau_i, as the phrase 'though the algebraic relations ... are not unique, we may choose' suggests.
[Appendix A, X12(6,3,1,1,1)] The note that this model has h^{2,1} = 3 but only two monomial deformations, so that the Picard-Fuchs system has two variables, is honest and useful. The subsequent identification of t-tilde_1 and t-tilde_2 with J and the symmetric combination E1+E2 is, however, heuristic; a sentence acknowledging that the third modulus is a twisted-sector state not captured by the differential system would prevent misreading.
[Appendix B, X18(9,6,1,1,1) table] The entries '(1,1) 540, (2,2) 540, ... ..., (6,6) 540' use an ellipsis in place of data. Either spell out the intermediate rows or state the pattern explicitly, so the table is self-contained.
[Appendix B, X8(2,2,2,1,1) table] The stated range is p+q <= 10, but rows such as (8,0), (8,2), (9,1), (9,3), (9,5) and (10,0) are absent. If these vanish, a sentence saying so would make the table complete relative to its stated range.
[§5.1] The illustrative example is introduced as 'the torus X = X3(1,1,1)'. Since X3(1,1,1) is an elliptic curve and the surrounding discussion concerns threefold Kähler cones, a word of clarification that this is a toy model for the cone construction would avoid confusion.
[Throughout] There are a number of typographical errors that should be corrected in proof, among them 'accompagnie' (§1), 'neccesary' and 'construnctiong' (§3.1), 'Topolocical Invariants' (Appendix B title), and 'perevious' (Appendix B, X12(6,2,2,1,1)).

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The paper

The paper claims

Using reflexive polyhedra from toric geometry, the authors construct mirror partners for Calabi-Yau hypersurfaces in weighted projective spaces where orbifold methods found none, and compute the Picard-Fuchs equations, mirror maps and Yukaw

The reading

Mirror symmetry says two geometrically different Calabi-Yau spaces can describe the same string theory, with their Hodge numbers exchanged. Finding the partner of a given space had been an art: one looked for a discrete symmetry of the defining polynomial and took a quotient, which worked for some families and failed for others. The authors adopt a construction in which a Calabi-Yau hypersurface is encoded by a lattice polyhedron that is 'reflexive' (its dual is again an integral polyhedron), and the dual polyhedron automatically supplies the mirror. Because this encoding exists for every weighted-projective hypersurface, the authors claim the missing mirrors can now be written down, and they demonstrate it on a hypersurface in $IP^4(3,9,17,22,24)$ whose ambient space is not Gorenstein, a case the earlier methods could not reach. The paper then does the practical work that makes the claim useful: it derives the Picard-Fuchs differential equations for two- and three-modulus models from a hypergeometric system attached to the polyhedron, extracts the mirror map and the Yukawa couplings, and reads off integer invariants that count rational curves, some of which turn out to be negative.

What carries the argument

Three pieces carry the argument. First, the reflexive polyhedron: a Calabi-Yau hypersurface is encoded by an integral polyhedron $Delta$ containing the origin, and the mirror family lives on the toric variety of the dual polyhedron $Delta^*$; the Hodge numbers of the pair are computed by counting lattice points on faces of $Delta$ and $Delta^*$, with the last term in each formula counting dual-face pairs that correspond to twisted sectors. Second, the monomial-divisor mirror map, which sends an integral point of $Delta^*$ to a monomial in the homogeneous coordinates of the original weighted projective space, so that divisors on one side correspond to complex-structure deformations on the other. Third, the $Delta^*$-hypergeometric system of Gel'fand-Kapranov-Zelevinsky: for each lattice relation $l$ among the lifted points one writes a differential operator $D_l$, and the algebraic relations among the invariant monomials become the Picard-Fuchs equations after the constraint operators are solved and trivial factors removed. The authors classify their models into three types according to whether extra symmetries or higher-degree invariants must be adjoined, and give a recipe for extending the hypergeometric system in each case. The mirror map is then $t_k(x)=partial_{rho_k}w(x;0)/w(x;0)$, and the Yukawa couplings follow from the Picard-Fuchs operators up to one integration constant, which is fixed by matching the topological triple intersection numbers.

Extended reading notes

Core claim

The central claim is that Batyrev's reflexive-polyhedron construction systematically produces mirror partners for Calabi-Yau hypersurfaces in weighted projective spaces, including cases where the mirror had been unavailable in previous constructions, and that the same toric data determines the Picard-Fuchs equations, the mirror map and the Yukawa couplings in closed form. The load-bearing example is the hypersurface $z_1^{25}+z_2^8 z_1+z_3^3 z_5+z_4^3 z_2+z_5^5 z_1+z_2^5 z_3^2=0$ in $IP^4(3,9,17,22,24)$, whose ambient space is not Gorenstein: the authors show that the associated polyhedron $Delta(vec{w})$ is still reflexive, that its Newton polyhedron matches the one cut out by the constraint, and that the Hodge numbers agree, so that the mirror is a hypersurface in the dual toric variety $IP_{Delta^*}$. They report checking several thousand models from the existing classification lists, including ones for which no mirror had been found even after all abelian orbifolds, and finding reflexive polyhedra in every case. On top of the existence statement, the paper gives explicit Picard-Fuchs operators, logarithmic solutions at the point of maximally unipotent monodromy, mirror maps, discriminant surfaces and Yukawa couplings for a set of two- and three-moduli models, and converts them into tables of instanton numbers.

Load-bearing premise

For the non-Gorenstein ambient spaces, everything rests on identifying the resolved hypersurface with the toric hypersurface of the reflexive polyhedron, an identification supported here by one worked example plus matching Newton polyhedra and Hodge numbers, with the general combinatorial proof deferred to a forthcoming publication.

Editorial extensions

If this is right

Every Calabi-Yau hypersurface in a weighted projective space acquires a constructible mirror, so the mirror-symmetry prediction of rational-curve counts becomes available for families where it previously could not be made.
For two- and three-moduli models, the Yukawa couplings and hence the instanton-corrected topological couplings are given explicitly as rational functions divided by a component of the discriminant surface.
The instanton numbers extracted for these models are integers but not always positive, which forces the interpretation of these numbers as counts of rational curves to be extended to singular and non-isolated curves.
Summing the two-modulus invariants over one degree reproduces the known one-modulus results for related complete intersections, tying the new construction to earlier computations.
The Kähler cone of each model is read off from the same toric data, giving the region of moduli space where the large-radius expansion and the instanton sum are under control.

Reading between the lines

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

If the polyhedron encoding really is available for every weighted-projective hypersurface, the classification of Calabi-Yau hypersurfaces becomes a classification problem for reflexive polyhedra, which is a finite combinatorial question rather than a geometric search for discrete symmetries.
The negative invariants are a warning that 'number of rational curves' is the wrong name for these coefficients once singular curves are present; a separate enumerative theory for multiple covers of degenerate curves would be needed to give them a direct geometric meaning.
The paper's method of fixing the mirror map by requiring integrality of the instanton numbers plus agreement with the topological intersection numbers is a consistency check that could be turned around: a model where no integer matrix satisfies both would falsify the identification of the Mori cone basis with the large-radius coordinates.

Original abstract

Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.

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Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry. It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions. Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.

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s0	Mirror Symmetry, Picard-Fuchs equations and instanton corrected Yukawa couplings are discussed within the framework of toric geometry.	contribution 0.61	0.55	0.14	asserted 0.93
s1	It allows to establish mirror symmetry of Calabi-Yau spaces for which the mirror manifold had been unavailable in previous constructions.	contribution 0.38	0.77	0.53	asserted 0.95
s2	Mirror maps and Yukawa couplings are explicitly given for several examples with two and three moduli.	result 0.84	0.89	0.17	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.72 · evidence from proof or derivation 1.00.

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References

Works cited (48)

[8]

S.-S. Roan, Int. Jour. Math., Vol. 2, No. 4 (1991) 439; Topological properties of Calabi- Yau mirror manifolds , preprint 1992

View on Pith 1991
[11]

M. Kreuzer and H. Skarke, Nucl. Phys. B405 (1993) 305; M. Kreuzer, Phys. Lett. 314B (1993) 31

View on Pith 1993
[10]

B. Greene and M. Plesser, Nucl. Phys. B338 (1990) 15

View on Pith 1990
[16]

A. Klemm and S. Theisen, Nucl. Phys. B389 (1993) 153

View on Pith 1993
[18]

A. Libgober and J. Teitelbaum, Duke Math. Jour., Int. Math. Res. Notices 1 (1993)29

View on Pith 1993
[19]

A. Klemm and S. Theisen, Mirror Maps and Instanton sums for Complete Intersec- tions in Weighted Projective space , preprint LMU-TPW 93-08

View on Pith
[35]

V. Batyrev and D. van Straten, Generalized Hypergeometric functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varie ties, Essen preprint, 1993

View on Pith 1993
[6]

A. Klemm and R. Schimmrigk, Landau-Ginzburg String Vacua , CERN-TH-6459/92, to appear in Nucl. Phys. B; M. Kreuzer and H. Skarke, Nucl. Phy s. B388 (1993) 113

View on Pith 1993
[31]

T. Oda, Convex Bodies and Algebraic Geometry, (An Introduction to t he Theory of Toric Varieties), Ergebnisse der Mathematik und ihrer Grenzgebiete , 3.Folge, Bd. 15, Springer-Verlag (1988); V.I. Danilov, Russian Math. Surve ys, 33 (1978) 97

View on Pith 1988
[37]

S.-S. Roan: Topological Couplings of Calabi-Yau Orbifolds , Max-Planck-Institut Se- ries, No. 22 (1992), to appear in J. of Group Theory in Physics

View on Pith 1992
[30]

D. Dais and A. Klemm: On the Invariants of Minimal Desingularizations of 3- dimensional Calabi-Yau Weighted Complete Intersection Va rieties, in preparation

View on Pith
[1]

For reviews, see Essays on Mirror Manifolds (ed. S.-T. Yau), Int. Press, Hong Kong, 1992

View on Pith 1992
[2]

L. Dixon, in Superstrings, Uniﬁed Theories and Cosmology 1987 , G. Furlan et al, eds., World Scientiﬁc 1988

View on Pith 1987
[3]

W. Lerche, C. Vafa and N. Warner, Nucl. Phys. B324 (1989) 4 27

View on Pith 1989
[4]

P. Candelas, A. M. Dale, C. A. L¨ utken, R. Schimmrigk : Complete Intersection Calabi- Yau Manifolds I, II , Nucl. Phys. B298 (1988) 493, Nucl. Phys. B306 (1988) 113

View on Pith 1988
[5]

L. Dixon, J. A. Harvey, C. Vafa and E. Witten : Nucl. Phys. B 261 (1985) 678 and Nucl. Phys. B274 (1986) 285; D. G. Markushevich, M. A. Olshan etsky and A. M. Perelomov, Commun. Math. Phys. 111 (1987) 247; J. Erler, A. K lemm, Commun. Math. Phys. 153 (1993) 579

View on Pith 1985
[7]

C. Vafa and N. Warner, Phys. Lett. 218B (1989) 51; C. Vafa, Mod. Phys. Lett. A 4 (1989) 1169

View on Pith 1989
[9]

P. Berglund and T. H¨ ubsch, Nucl. Phys. 393B (1993) 377

View on Pith 1993
[12]

E. Witten, Nucl. Phys. B403 (1993) 159

View on Pith 1993
[13]

V. Batyrev, Dual Polyhedra and the Mirror Symmetry for Calabi-Yau Hyper surfaces in Toric Varieties , Univ. Essen Preprint (1992), to appear in Journal of Alg. Ge om. and Duke Math. Journal, Vol. 69, No 2, (1993) 349

View on Pith 1992
[14]

P. Candelas, X. De la Ossa, P. Green and L. Parkes, Nucl. P hys. B359 (1991) 21

View on Pith 1991
[15]

D. Morrison, Picard-Fuchs Equations and Mirror Maps for Hypersurfaces , in ref.[1]

View on Pith
[17]

A. Font, Nucl. Phys. B391 (1993) 358

View on Pith 1993
[20]

M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Nucl. P hys. B405 (1993) 279

View on Pith 1993
[21]

Yu. I. Manin, Report on the Mirror Conjecture , Talk at the Mathematische Arbeit- stagung, Bonn 1993

View on Pith 1993
[22]

P. Candelas, E. Derrick and L. Parkes, Nucl. Phys. B407 ( 1993) 115 56

View on Pith 1993
[23]

T. Eguchi and S.K. Yang, Mod. Phys. Lett. A5 (1990) 1693; R. Dijkgraaf, H. Verlinde and E. Verlinde, Nucl. Phys. B352 (1991) 59; E. Witten : Mirror Manifolds And Topological Field Theory , in Essays on Mirror Manifolds (ed. S.-T. Yau), Int. Press, Hong Kong, 1992 120-158

View on Pith 1990
[24]

J. Distler and B. Greene, Nucl. Phys. B309 (1988) 295

View on Pith 1988
[25]

L. Dixon, D. Friedan, E. Martinec and S, Shenker, Nucl. P hys. B282 (1987) 13; S. Hamidi and C. Vafa, Nucl. Phys. B279 (1987) 465; J. Lauer, J . Mas and H. P. Nilles Nucl. Phys. B351 (1991) 353; S. Stieberger, D. Jungni ckel, J. Lauer and M. Spalinski, Mod. Phys. Lett. A7 (1992) 3859; J. Erler, D. Jung nickel, M. Spalinski and S. Stieberger, Nucl. Phys. B...

View on Pith 1987
[26]

B. Blok and A. Varchenko, Int. J. Mod. Phys. A7 (1992) 146 7; W. Lerche, D.J. Smit and N. Warner, Nucl. Phys. B372 (1992) 87

View on Pith 1992
[27]

A.C. Cadavid and S. Ferrara, Phys. Lett. 267B (1991) 193 ; S. Ferrara and J. Louis, Phys. Lett. 278B (1992) 240; A. Ceresole, R. D’Auria, S. Ferr ara, W. Lerche and J. Louis, Int. J. Mod. Phys. A8 (1993) 79

View on Pith 1991
[28]

A. Klemm, S. Theisen and M. G. Schmidt, Int. J. Mod. Phys. A7 (1992) 6215

View on Pith 1992
[29]

L. Dixon, V. Kaplunovsky and J. Louis, Nucl. Phys. B329 ( 1990) 27; S. Cecotti, S. Ferrara and L. Girardello, Int. J. Mod. Phys. A4 (1989) 2465

View on Pith 1989
[32]

M. Beltrametti and L. Robbiano, Expo. Math. 4 (1986) 11

View on Pith 1986
[33]

D.Aspinwall, B.Greene and D.Morrison, Phys.Lett. 303 B (1993) 249

View on Pith 1993
[34]

P. Aspinwall, A. L¨ utken and G. Ross, Phys. Lett. 241B (1 990) 373; P. Aspinwall and A. L¨ utken,A New Geometry from Superstring Theory , in [1]

View on Pith
[36]

T. Yonemura, Tohoku Math. J. 42 (1990) 351

View on Pith 1990
[38]

D. Markushevich, M. Olshanetsky and A. Perelomov, Comm un. Math. Phys. 111 (1987) 247

View on Pith 1987
[39]

I. Dolgachev, Weighted Projective Varieties. In: Group Actions and Vector Fields , Proc. Polish - North Amer. Sem. Vancouver (1981) LNM Vol. 965, Springer-Verlag (1982) 34

View on Pith 1981
[40]

P. Griﬃths, Ann. of Math. 90 (1969) 460

View on Pith 1969
[41]

A. Aleksandrov, Math. USSR Sbornik, 45 (1983) 1 57

View on Pith 1983
[42]

I.M.Gel’fand, A.V.Zelevinsky and M.M.Kapranov, Func . Anal. Appl. 28 (1989) 12 and Adv. Math. 84 (1990) 255

View on Pith 1989
[43]

D. Morrison, Amer. Math. Soc. 6 (1993) 223; Compactiﬁcations of Moduli spaces inspired by mirror symmetry DUK-M-93-06, alg-geom/9304007

View on Pith Pith review arXiv 1993
[44]

P. Candelas and X. De la Ossa, Nucl. Phys. B355 (1991) 455

View on Pith 1991
[45]

T. Oda and H.S. Park, Tohoku Math. J. 43 (1991) 375

View on Pith 1991
[46]

V.V. Batyrev, Quantum cohomology rings of toric manifolds , preprint (1993)

View on Pith 1993
[47]

P. Berglund, P. Candelas, X. de la Ossa, A. Font, T. H¨ ubsch, D. Jancic and F. Quevedo, Periods for Calabi-Yau and Landau-Ginzburg Vacua , preprint CERN-TH.6865/93

View on Pith
[48]

P. Candelas, X. de la Ossa, A. Font, S. Katz and D. Morriso n, Mirror Symmetry for Two-Parameter Models, preprint CERN-TH.6884/93 58

View on Pith

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How later papers used it

131 later papers cite it

Evidence: substantial. 113 of the citing papers share no author with it, and 43 of those carry a confident label from Pith's citation classifier.

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1 later paper by other authors is listed below: 1 built on it. 1 more paper by the same authors is listed: 1 used its method or tool.

Builds on it 1 of 4 papers with this label

Generalized this paper's approach to threefolds with non-simplicial Mori cones and presented an efficient algorithm for carrying out the mirror map in this setting.

Computational Mirror Symmetry (2023) · 0.99 · paraphrased: the licence does not allow quoting

Uses its method or tool 1 of 47 papers with this label · 1 by the same authors

By the same authors

Obtained BPS numbers with the methods explained in this paper, implementing the explicit calculations in the program Instanton using toric data as input.

U(1) Mediation of Flux Supersymmetry Breaking (2008) (same authors) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

127 arXiv papers cite this in 292 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 · 151 sentences without a confident label

Builds on it 4 papers

In the present work, we overcome the two remaining obstacles: we generalize the approach of [ref] to threefolds with non-simplicial Mori cones, and we present an efficient algorithm for carrying out the mirror map in this setting.

Computational Mirror Symmetry · Introduction · 0.99
As was also discussed in [ref] , generalizing the well known results for $N=1$ from [ref] , the prepotential $F(t)$ takes the form [equation] Here $κ$ is the triple intersection number of the smooth deformation $X$ of $X$ and the coefficient of the constant term takes the form [equation] Here we define $n_q$ for $q=1,…,⌊N/2⌋$ as [equation]

In search of almost generic Calabi-Yau 3-folds · The B-model at genus zero · 0.86
Our method is an extension of the procedure which is studied for the period integrals without boundaries [ref] .

A Note on Computations of D-brane Superpotential · An alternative technique to Griffiths' reduction · 0.97

Uses its method or tool 47 papers

We will obtain closed forms for the periods using four ingredients: * the known formulae [ref] for the periods around the LCS point, which express the periods in terms of parameter derivatives of hypergeometric functions; * Clausen's identity, reducing the problem to the solutions of second order operators; * a relation between the periods originating from supersymmetry [ref] , which allows expressing the periods in terms of the first parameter derivatives; * the so-called $ϵ$-expansion of a family of hypergeometric functions to first order [ref] .

Analytic Periods via Twisted Symmetric Squares · Computation of the periods · 0.80
Notably, the series coefficients of the relevant power series for $ϖ(ψ),c^a(ψ)$ and $ĉ^ab(ψ)$ are all rational.[Formulas for the $c_n$ and $ĉ^ab_n$ are given in Appendix [ref], following, with some minor simplifications, Appendix A of [ref] .]

Computational Mirror Symmetry · Enumerative invariants · 0.86
We descend from the lofty heights of all genus results, as all we require are genus 0 invariants (though in the case of E-strings, these in fact uniquely specify all invariants [ref] ), which we obtain via mirror symmetry computations [ref] .[We refer to these invariants as “genus 0 Gromov-Witten invariants” throughout the paper to emphasize that the only tool involved in their computation is mirror symmetry.

Determining F-theory matter via Gromov-Witten invariants · Introduction · 0.89

Uses its data or values 1 paper

The extended set of integral points in its toric geometric construction is given in [ref] as [equation] and after the embedding into $ℝ^5$ leads to a lattice of dependencies ([ref]) from which we extract the Mori cone basis[Notice a typing error in [ref] affecting $l_1$ and $𝒟_3$.] [equation] These give through [ref] the appropriate LCS coordinates [equation] The Picard-Fuchs operators in these coordinates read[footnote] [equation] where we have employed the logarithmic derivatives $Θ_x_i = x_i ∂_x_i$.

Small Flux Superpotentials for Type IIB Flux Vacua Close to a Conifold · A (not so) special example P(1,1,2,8,12)[24] · 0.81

Compares with it 5 papers

In Section [ref], we found that the technique suggested by Hori and Vafa [ref] for determining the Picard-Fuchs equations satisfied by $Π_M_G$ when the Calabi-Yau condition ([ref]) holds yields results which agree with those obtained in [ref] working in the Batyrev-Borisov [ref] framework.

Hori-Vafa mirror periods, Picard-Fuchs equations, and Berglund-H\"{u}bsch-Krawitz duality · Discussion · 0.95
The resulting numbers of $^1$ curves in each homology class has been compared with the Gromov-Witten invariants obtained using [ref] , and a perfect match has been found in all cases.

Heterotic Instanton Superpotentials from Complete Intersection Calabi-Yau Manifolds · Example of type 2 · 0.96
The function $F_nm(t)=F_mn(t)$ is given by [equation] Finally, we relate the residue expansions ([ref]) with the geometric formula proposed in [ref] .

B-brane Transport and Grade Restriction Rule for Determinantal Varieties · Determinantal quintic in $^4$ · 0.86

Cites it as background 45 papers

The classic paper of Hosono, Klemm, Theisen, and Yau [ref] laid out a procedure that is valid for Calabi-Yau threefold hypersurfaces in which the Mori cone is simplicial.

Computational Mirror Symmetry · Conclusions · 0.96
Such cases are discussed in [ref] .] $T$ determine the set of generators $l^(k)$ of $L$ and the Stanley-Reisner ideal $Z_T$.

The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods · Calabi-Yau hypersurfaces in toric ambient spaces · 0.92
Their quantum geometry has been analyzed in great detail using mirror symmetry in [ref] .

Analytic Periods via Twisted Symmetric Squares · Examples · 0.89

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] [8]

S.-S. Roan, Int. Jour. Math., Vol. 2, No. 4 (1991) 439; Topological properties of Calabi- Yau mirror manifolds , preprint 1992

View on Pith 1991

[2] [11]

M. Kreuzer and H. Skarke, Nucl. Phys. B405 (1993) 305; M. Kreuzer, Phys. Lett. 314B (1993) 31

View on Pith 1993

[3] [10]

B. Greene and M. Plesser, Nucl. Phys. B338 (1990) 15

View on Pith 1990

[4] [16]

A. Klemm and S. Theisen, Nucl. Phys. B389 (1993) 153

View on Pith 1993

[5] [18]

A. Libgober and J. Teitelbaum, Duke Math. Jour., Int. Math. Res. Notices 1 (1993)29

View on Pith 1993

[6] [19]

A. Klemm and S. Theisen, Mirror Maps and Instanton sums for Complete Intersec- tions in Weighted Projective space , preprint LMU-TPW 93-08

View on Pith

[7] [35]

V. Batyrev and D. van Straten, Generalized Hypergeometric functions and Rational Curves on Calabi-Yau Complete Intersections in Toric Varie ties, Essen preprint, 1993

View on Pith 1993

[8] [6]

A. Klemm and R. Schimmrigk, Landau-Ginzburg String Vacua , CERN-TH-6459/92, to appear in Nucl. Phys. B; M. Kreuzer and H. Skarke, Nucl. Phy s. B388 (1993) 113

View on Pith 1993

[9] [31]

T. Oda, Convex Bodies and Algebraic Geometry, (An Introduction to t he Theory of Toric Varieties), Ergebnisse der Mathematik und ihrer Grenzgebiete , 3.Folge, Bd. 15, Springer-Verlag (1988); V.I. Danilov, Russian Math. Surve ys, 33 (1978) 97

View on Pith 1988

[10] [37]

S.-S. Roan: Topological Couplings of Calabi-Yau Orbifolds , Max-Planck-Institut Se- ries, No. 22 (1992), to appear in J. of Group Theory in Physics

View on Pith 1992

[11] [30]

D. Dais and A. Klemm: On the Invariants of Minimal Desingularizations of 3- dimensional Calabi-Yau Weighted Complete Intersection Va rieties, in preparation

View on Pith

[12] [1]

For reviews, see Essays on Mirror Manifolds (ed. S.-T. Yau), Int. Press, Hong Kong, 1992

View on Pith 1992

[13] [2]

L. Dixon, in Superstrings, Uniﬁed Theories and Cosmology 1987 , G. Furlan et al, eds., World Scientiﬁc 1988

View on Pith 1987

[14] [3]

W. Lerche, C. Vafa and N. Warner, Nucl. Phys. B324 (1989) 4 27

View on Pith 1989

[15] [4]

P. Candelas, A. M. Dale, C. A. L¨ utken, R. Schimmrigk : Complete Intersection Calabi- Yau Manifolds I, II , Nucl. Phys. B298 (1988) 493, Nucl. Phys. B306 (1988) 113

View on Pith 1988

[16] [5]

L. Dixon, J. A. Harvey, C. Vafa and E. Witten : Nucl. Phys. B 261 (1985) 678 and Nucl. Phys. B274 (1986) 285; D. G. Markushevich, M. A. Olshan etsky and A. M. Perelomov, Commun. Math. Phys. 111 (1987) 247; J. Erler, A. K lemm, Commun. Math. Phys. 153 (1993) 579

View on Pith 1985

[17] [7]

C. Vafa and N. Warner, Phys. Lett. 218B (1989) 51; C. Vafa, Mod. Phys. Lett. A 4 (1989) 1169

View on Pith 1989

[18] [9]

P. Berglund and T. H¨ ubsch, Nucl. Phys. 393B (1993) 377

View on Pith 1993

[19] [12]

E. Witten, Nucl. Phys. B403 (1993) 159

View on Pith 1993

[20] [13]

V. Batyrev, Dual Polyhedra and the Mirror Symmetry for Calabi-Yau Hyper surfaces in Toric Varieties , Univ. Essen Preprint (1992), to appear in Journal of Alg. Ge om. and Duke Math. Journal, Vol. 69, No 2, (1993) 349

View on Pith 1992

[21] [14]

P. Candelas, X. De la Ossa, P. Green and L. Parkes, Nucl. P hys. B359 (1991) 21

View on Pith 1991

[22] [15]

D. Morrison, Picard-Fuchs Equations and Mirror Maps for Hypersurfaces , in ref.[1]

View on Pith

[23] [17]

A. Font, Nucl. Phys. B391 (1993) 358

View on Pith 1993

[24] [20]

M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Nucl. P hys. B405 (1993) 279

View on Pith 1993

[25] [21]

Yu. I. Manin, Report on the Mirror Conjecture , Talk at the Mathematische Arbeit- stagung, Bonn 1993

View on Pith 1993

[26] [22]

P. Candelas, E. Derrick and L. Parkes, Nucl. Phys. B407 ( 1993) 115 56

View on Pith 1993

[27] [23]

T. Eguchi and S.K. Yang, Mod. Phys. Lett. A5 (1990) 1693; R. Dijkgraaf, H. Verlinde and E. Verlinde, Nucl. Phys. B352 (1991) 59; E. Witten : Mirror Manifolds And Topological Field Theory , in Essays on Mirror Manifolds (ed. S.-T. Yau), Int. Press, Hong Kong, 1992 120-158

View on Pith 1990

[28] [24]

J. Distler and B. Greene, Nucl. Phys. B309 (1988) 295

View on Pith 1988

[29] [25]

L. Dixon, D. Friedan, E. Martinec and S, Shenker, Nucl. P hys. B282 (1987) 13; S. Hamidi and C. Vafa, Nucl. Phys. B279 (1987) 465; J. Lauer, J . Mas and H. P. Nilles Nucl. Phys. B351 (1991) 353; S. Stieberger, D. Jungni ckel, J. Lauer and M. Spalinski, Mod. Phys. Lett. A7 (1992) 3859; J. Erler, D. Jung nickel, M. Spalinski and S. Stieberger, Nucl. Phys. B...

View on Pith 1987

[30] [26]

B. Blok and A. Varchenko, Int. J. Mod. Phys. A7 (1992) 146 7; W. Lerche, D.J. Smit and N. Warner, Nucl. Phys. B372 (1992) 87

View on Pith 1992

[31] [27]

A.C. Cadavid and S. Ferrara, Phys. Lett. 267B (1991) 193 ; S. Ferrara and J. Louis, Phys. Lett. 278B (1992) 240; A. Ceresole, R. D’Auria, S. Ferr ara, W. Lerche and J. Louis, Int. J. Mod. Phys. A8 (1993) 79

View on Pith 1991

[32] [28]

A. Klemm, S. Theisen and M. G. Schmidt, Int. J. Mod. Phys. A7 (1992) 6215

View on Pith 1992

[33] [29]

L. Dixon, V. Kaplunovsky and J. Louis, Nucl. Phys. B329 ( 1990) 27; S. Cecotti, S. Ferrara and L. Girardello, Int. J. Mod. Phys. A4 (1989) 2465

View on Pith 1989

[34] [32]

M. Beltrametti and L. Robbiano, Expo. Math. 4 (1986) 11

View on Pith 1986

[35] [33]

D.Aspinwall, B.Greene and D.Morrison, Phys.Lett. 303 B (1993) 249

View on Pith 1993

[36] [34]

P. Aspinwall, A. L¨ utken and G. Ross, Phys. Lett. 241B (1 990) 373; P. Aspinwall and A. L¨ utken,A New Geometry from Superstring Theory , in [1]

View on Pith

[37] [36]

T. Yonemura, Tohoku Math. J. 42 (1990) 351

View on Pith 1990

[38] [38]

D. Markushevich, M. Olshanetsky and A. Perelomov, Comm un. Math. Phys. 111 (1987) 247

View on Pith 1987

[39] [39]

I. Dolgachev, Weighted Projective Varieties. In: Group Actions and Vector Fields , Proc. Polish - North Amer. Sem. Vancouver (1981) LNM Vol. 965, Springer-Verlag (1982) 34

View on Pith 1981

[40] [40]

P. Griﬃths, Ann. of Math. 90 (1969) 460

View on Pith 1969

[41] [41]

A. Aleksandrov, Math. USSR Sbornik, 45 (1983) 1 57

View on Pith 1983

[42] [42]

I.M.Gel’fand, A.V.Zelevinsky and M.M.Kapranov, Func . Anal. Appl. 28 (1989) 12 and Adv. Math. 84 (1990) 255

View on Pith 1989

[43] [43]

D. Morrison, Amer. Math. Soc. 6 (1993) 223; Compactiﬁcations of Moduli spaces inspired by mirror symmetry DUK-M-93-06, alg-geom/9304007

View on Pith Pith review arXiv 1993

[44] [44]

P. Candelas and X. De la Ossa, Nucl. Phys. B355 (1991) 455

View on Pith 1991

[45] [45]

T. Oda and H.S. Park, Tohoku Math. J. 43 (1991) 375

View on Pith 1991

[46] [46]

V.V. Batyrev, Quantum cohomology rings of toric manifolds , preprint (1993)

View on Pith 1993

[47] [47]

P. Berglund, P. Candelas, X. de la Ossa, A. Font, T. H¨ ubsch, D. Jancic and F. Quevedo, Periods for Calabi-Yau and Landau-Ginzburg Vacua , preprint CERN-TH.6865/93

View on Pith

[48] [48]

P. Candelas, X. de la Ossa, A. Font, S. Katz and D. Morriso n, Mirror Symmetry for Two-Parameter Models, preprint CERN-TH.6884/93 58

View on Pith