Modified Elliptic Genus

Gritsenko, Valery

doi:10.1007/978-3-030-42400-8_2

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2022

2023

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Cited by 3 publications

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“…It has nice automorphic property only if c 1 (M ) = 0. If c 1 (M ) = 0 then one can define an appropriate quasi-modular correction of χ(M ; τ, z) in order to obtain a generalized elliptic genus χ(M, E; τ, z) for any vector bundle over M (see [9] and [10]). This function is a meromorphic function of Jacobi type.…”

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confidence: 99%

Elliptic genus and modular differential equations

Adler¹,

Gritsenko²

2022

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We study modular differential equations for the basic weak Jacobi forms in one abelian variable with applications to the elliptic genus of Calabi-Yau varieties. We show that the elliptic genus of any CY3 satisfies a differential equation of degree one with respect to the heat operator. For a K3 surface or any CY5 the degree of the differential equation is 3. We prove that for a general CY4 its elliptic genus satisfies a modular differential equation of degree 5. We give examples of differential equations of degree two with respect to the heat operator similar to the Kaneko-Zagier equation for modular forms in one variable. We find modular differential equations of Kaneko-Zagier type of degree 2 or 3 for the second, third and fourth powers of the Jacobi theta-series.

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confidence: 99%