Elliptic genus of Calabi-Yau manifolds and Jacobi and Siegel modular forms. (English) Zbl 1101.14308
St. Petersbg. Math. J. 11, No. 5, 781-804 (2000); and Algebra Anal. 11, No. 5, 100-125 (2000).
Summary: We study two types of relations: a one is between the elliptic genus of Calabi-Yau manifolds and Jacobi modular forms, another one is between the second quantized elliptic genus, Siegel modular forms and Lorentzian Kac-Moody Lie algebras. We also determine the structure of the graded ring of the weak Jacobi forms with integral Fourier coefficients. It gives us a number of applications to the theory of elliptic genus and of the second quantized elliptic genus.
MSC:
14G35 | Modular and Shimura varieties |
11F23 | Relations with algebraic geometry and topology |
11F50 | Jacobi forms |
14J28 | \(K3\) surfaces and Enriques surfaces |
17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
58J26 | Elliptic genera |
14J15 | Moduli, classification: analytic theory; relations with modular forms |
11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |
14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |