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      CHL Calabi-Yau threefolds: Curve counting, Mathieu moonshine and Siegel modular forms

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          Abstract

          A CHL model is the quotient of \(\mathrm{K3} \times E\) by an order \(N\) automorphism which acts symplectically on the K3 surface and acts by shifting by an \(N\)-torsion point on the elliptic curve \(E\). We conjecture that the primitive Donaldson-Thomas partition function of elliptic CHL models is a Siegel modular form, namely the Borcherds lift of the corresponding twisted-twined elliptic genera which appear in Mathieu moonshine. The conjecture matches predictions of string theory by David, Jatkar and Sen. We use the topological vertex to prove several base cases of the conjecture. Via a degeneration to \(\mathrm{K3} \times \mathbb{P}^1\) we also express the DT partition functions as a twisted trace of an operator on Fock space. This yields further computational evidence. An extension of the conjecture to non-geometric CHL models is discussed. We consider CHL models of order \(N=2\) in detail. We conjecture a formula for the Donaldson-Thomas invariants of all order two CHL models in all curve classes. The conjecture is formulated in terms of two Siegel modular forms. One of them, a Siegel form for the Iwahori subgroup, has to our knowledge not yet appeared in physics. This discrepancy is discussed in an appendix with Sheldon Katz.

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          Most cited references23

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          Some Theorems on Actions of Algebraic Groups

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            Curve counting via stable pairs in the derived category

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            For a nonsingular projective 3-fold \(X\), we define integer invariants virtually enumerating pairs \((C,D)\) where \(C\subset X\) is an embedded curve and \(D\subset C\) is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of \(X\). The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of \(X\). For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described. The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We prove that our integrality predictions for Gromov-Witten invariants agree with the BPS integrality. Conversely, the BPS geometry imposes strong conditions on the enumeration of pairs.
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              Finite groups of automorphisms of K3 surfaces and the Mathieur group

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                Author and article information

                Journal
                14 November 2018
                Article
                1811.06102
                d182ac97-85ef-4128-b6ae-58f3d2d8c23d

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                73 pages, including a 10 page Appendix by Sheldon Katz and the second author. Comments welcome
                math.AG hep-th

                High energy & Particle physics,Geometry & Topology
                High energy & Particle physics, Geometry & Topology

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