Holomorphic anomaly equations and the Igusa cusp form conjecture

Oberdieck, Georg; Pixton, Aaron

doi:10.1007/s00222-018-0794-0

Cited by 51 publications

(116 citation statements)

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“…Note added: after this article appeared on the arxiv, Georg Oberdieck pointed out that our results in section 4.5 match with his and Aaron Pixton's conjectured holomorphic anomaly equation on Calabi-Yau n-folds appearing in [23,24]. Moreover, he explained to us the explicit form of the generalized holomorphic anomaly equation for the Gromov-Witten potentials on Calabi-Yau fourfolds, which we include now in appendix B.…”

Section: Jhep01(2018)086supporting

confidence: 70%

“…For the non π-vertical cycles the conjectured anomaly equation agrees with our results for M E 8 P 3 and we also checked it for the Gromov-Witten potentials of M E 8 P 1 ×P 2 . Our results on the modular structure can therefore be seen as a partial derivation and non-trivial check of the holomorphic anomaly equation conjectured in [23,24] for Calabi-Yau fourfolds. We will now briefly describe the general form of the holomorphic anomaly equations for genus zero Gromov-Witten potentials.…”

Section: Jhep01(2018)086mentioning

confidence: 53%

“…After releasing a pre-print of this paper it was brought to our attention by Georg Oberdieck that there is a conjectured modular anomaly equation for elliptic Calabi-Yau n-folds in [23,24]. For n = 4 the conjecture implies the modular anomaly equations for the GromovWitten potentials associated to π-vertical cycles and for genus one free energies that we derived in this paper.…”

Section: Jhep01(2018)086mentioning

confidence: 82%

“…On the other hand, conjecture B of [23,24] implies a general modular anomaly equation for F (g) γ 1 ,...,γm . Following the discussion of sections 3.1 and 4.4, we can make a generalization of the pure modular basis by taking the 4-cycles,…”

Section: Jhep01(2018)086mentioning

confidence: 99%

“…Then for a given γ ∈ H 2,2 (M, C) Georg Oberdieck pointed out to us that conjecture B of [23,24] implies a modular anomaly equation for F (0) γ , which in the modular basis (B.3) reads…”

Section: Jhep01(2018)086mentioning

confidence: 99%

See 4 more Smart Citations

Modular amplitudes and flux-superpotentials on elliptic Calabi-Yau fourfolds

Cota¹,

Klemm

Schimannek³

2018

J. High Energ. Phys.

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We discuss the period geometry and the topological string amplitudes on elliptically fibered Calabi-Yau fourfolds in toric ambient spaces. In particular, we describe a general procedure to fix integral periods. Using some elementary facts from homological mirror symmetry we then obtain Bridgelands involution and its monodromy action on the integral basis for non-singular elliptically fibered fourfolds. The full monodromy group contains a subgroup that acts as PSL(2,Z) on the Kähler modulus of the fiber and we analyze the consequences of this modularity for the genus zero and genus one amplitudes as well as the associated geometric invariants. We find holomorphic anomaly equations for the amplitudes, reflecting precisely the failure of exact PSL(2,Z) invariance that relates them to quasi-modular forms. Finally we use the integral basis of periods to study the horizontal flux superpotential and the leading order Kähler potential for the moduli fields in F-theory compactifications globally on the complex structure moduli space. For a particular example we verify attractor behaviour at the generic conifold given an aligned choice of flux which we expect to be universal. Furthermore we analyze the superpotential at the orbifold points but find no stable vacua.

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Section: Jhep01(2018)086supporting

confidence: 70%

Section: Jhep01(2018)086mentioning

confidence: 53%

Section: Jhep01(2018)086mentioning

confidence: 82%

Section: Jhep01(2018)086mentioning

confidence: 99%

Section: Jhep01(2018)086mentioning

confidence: 99%

See 3 more Smart Citations

Modular amplitudes and flux-superpotentials on elliptic Calabi-Yau fourfolds

Cota¹,

Klemm

Schimannek³

2018

J. High Energ. Phys.

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The Donaldson-Thomas Theory of K3 × E via the Topological Vertex

Bryan

2018

Geometry of Moduli

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Oberdieck and Pandharipande conjectured [9] that the partition function of the Gromov-Witten/Donaldson-Thomas invariants of S × E, the product of a K3 surface and an elliptic curve, is given by minus the reciprocal of the Igusa cusp form of weight 10. For a fixed primitive curve class in S of square 2h − 2, their conjecture predicts that the corresponding partition functions are given by meromorphic Jacobi forms of weight −10 and index h − 1. We calculate the Donaldson-Thomas partition function for primitive classes of square -2 and of square 0, proving strong evidence for their conjecture.Our computation uses reduced Donaldson-Thomas invariants which are defined as the (Behrend function weighted) Euler characteristics of the quotient of the Hilbert scheme of curves in S × E by the action of E. Our technique is a mixture of motivic and toric methods (developed with Kool in [5]) which allows us to express the partition functions in terms of the topological vertex and subsequently in terms of Jacobi forms. We compute both versions of the invariants: unweighted and Behrend function weighted Euler characteristics. Our Behrend function weighted computation requires us to assume Conjecture 18 in [5].Date: November 22, 2017. 1 Since this paper was originally written in 2015, Oberdieck and Pixton have recently given a complete proof of this conjecture using methods (different from ours) from both Donaldson-Thomas theory and Gromov-Witten theory [11]. 1 2 JIM BRYANOur general computational strategy is the following. Donaldson-Thomas invariants are given by weighted Euler characteristics of Hilbert schemes. We stratify the Hilbert scheme using the geometric support of the curves and we compute Euler characteristics of strata separately. Many of the strata acquire actions of E or C * (that were not present globally) and we restrict to the fixed point loci. We are able to further stratify the fixed point loci and those strata sometimes acquire further actions. Iterating this strategy, we reduce the computation to subschemes which are formally locally given by monomial ideals. These are counted using the topological vertex. New identities for the topological vertex lead to closed formulas. To incorporate the Behrend function into this strategy, we must assume the conjecture formulated in [5, Conj. 18]. This is a general conjecture regarding the behavior of the Behrend function at subschemes given locally by monomial ideals.Acknowledgements. I'd like to thank George Oberdieck, Rahul Pandharipande, and Yin Qizheng for invaluable discussions. I've also benefited with discussions with Tom Coates, Sheldon Katz, Martijn Kool, Davesh Maulik, Tony Pantev, Balazs Szendroi, Andras Szenes, and Richard Thomas. The computational technique employed in this paper was developed in collaboration with Martijn Kool whom I owe a debt of gratitude. I would also like to thank the Institute for Mathematical Research (FIM) at ETH for hosting my visit to Zürich, and for Matematisk Institutt, UiO and Jan Christophersen for organizing the 2017 Abel S...

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Lost chapters in CHL black holes: untwisted quarter-BPS dyons in the ℤ2 model

Fischbach¹,

Klemm

Nega³

2021

J. High Energ. Phys.

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Motivated by recent advances in Donaldson-Thomas theory, four-dimensional $$ \mathcal{N} $$ N = 4 string-string duality is examined in a reduced rank theory on a less studied BPS sector. In particular we identify candidate partition functions of “untwisted” quarter-BPS dyons in the heterotic ℤ2 CHL model by studying the associated chiral genus two partition function, based on the M-theory lift of string webs argument by Dabholkar and Gaiotto. This yields meromorphic Siegel modular forms for the Iwahori subgroup B(2) ⊂ Sp4(ℤ) which generate BPS indices for dyons with untwisted sector electric charge, in contrast to twisted sector dyons counted by a multiplicative lift of twisted-twining elliptic genera known from Mathieu moonshine. The new partition functions are shown to satisfy the expected constraints coming from wall-crossing and S-duality symmetry as well as the black hole entropy based on the Gauss-Bonnet term in the effective action. In these aspects our analysis confirms and extends work of Banerjee, Sen and Srivastava, which only addressed a subset of the untwisted sector dyons considered here. Our results are also compared with recently conjectured formulae of Bryan and Oberdieck for the partition functions of primitive DT invariants of the CHL orbifold X = (K3 × T2)/ℤ2, as suggested by string duality with type IIA theory on X.

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Holomorphic anomaly equations and the Igusa cusp form conjecture

Cited by 51 publications

References 49 publications

Modular amplitudes and flux-superpotentials on elliptic Calabi-Yau fourfolds

Modular amplitudes and flux-superpotentials on elliptic Calabi-Yau fourfolds

The Donaldson-Thomas Theory of K3 × E via the Topological Vertex

Lost chapters in CHL black holes: untwisted quarter-BPS dyons in the ℤ2 model

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