Vertex Operator Algebras and Weak Jacobi Forms

Krauel, Matthew; Mason, Geoffrey

doi:10.1142/s0129167x11007677

Cited by 17 publications

(16 citation statements)

References 18 publications

(9 reference statements)

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“…(In particular, it actually predates Borcherds' introduction of the notion of vertex algebra in [9].) More general results were established recently in [82], by applying earlier work [89] of Miyamoto. Cf.…”

Section: K3 Surfaces and Jacobi Formsmentioning

confidence: 96%

Derived equivalences of K3 surfaces and twined elliptic genera

Duncan

Mack-Crane

2016

Res Math Sci

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We use the unique canonically twisted module over a certain distinguished super vertex operator algebra-the moonshine module for Conway's group-to attach a weak Jacobi form of weight zero and index one to any symplectic derived equivalence of a projective complex K3 surface that fixes a stability condition in the distinguished space identified by Bridgeland. According to work of Huybrechts, following Gaberdiel-Hohenegger-Volpato, any such derived equivalence determines a conjugacy class in Conway's group, the automorphism group of the Leech lattice. Conway's group acts naturally on the module we consider. In physics, the data of a projective complex K3 surface together with a suitable stability condition determines a supersymmetric non-linear sigma model, and supersymmetry-preserving automorphisms of such an object may be used to define twinings of the K3 elliptic genus. Our construction recovers the K3 sigma model twining genera precisely in all available examples. In particular, the identity symmetry recovers the usual K3 elliptic genus, and this signals a connection to Mathieu moonshine. A generalization of our construction recovers a number of Jacobi forms arising in umbral moonshine. We demonstrate a concrete connection to supersymmetric non-linear K3 sigma models by establishing an isomorphism between the twisted module we consider and the vector space underlying a particular sigma model attached to a certain distinguished K3 surface.

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Section: K3 Surfaces and Jacobi Formsmentioning

confidence: 96%

Derived equivalences of K3 surfaces and twined elliptic genera

Duncan

Mack-Crane

2016

Res Math Sci

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“…-indeed, this simply reduces to the statement that E 2 (τ) is the logarithmic derivative of ∆(τ). Therefore any solution to (19) is a scalar multiple of ∆ w (τ). Now, fix γ ∈ Γ .…”

Section: Existence Of Vvmfmentioning

confidence: 99%

The Theory of Vector-Valued Modular Forms for the Modular Group

Gannon

2014

Contributions in Mathematical and Computational Sciences

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We explain the basic ideas, describe with proofs the main results, and demonstrate the effectiveness, of an evolving theory of vector-valued modular forms (vvmf). To keep the exposition concrete, we restrict here to the special case of the modular group. Among other things, we construct vvmf for arbitrary multipliers, solve the Mittag-Leffler problem here, establish Serre duality and find a dimension formula for holomorphic vvmf, all in far greater generality than has been done elsewhere. More important, the new ideas involved are sufficiently simple and robust that this entire theory extends directly to any genus-0 Fuchsian group.

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“…Motivated to express transformation properties of trace functions with automorphisms using only ordinary modules, Miyamoto [14] studied a deviation of characters for the modules of a VOA and developed their transformation laws with respect to the group SL 2 (Z). By exploiting these transformation laws, a number of works have been developed pertaining to elliptic genera [6], Jacobi forms [9,10], and modular-invariance relations between shifted VOAs and orbifold theory [16], among others. It is the aim of this paper to develop a 1-point analogue of Miyamoto's theta functions and deduce their transformation laws with respect to SL 2 (Z).…”

Section: Introductionmentioning

confidence: 99%