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Seminários do IMPA

Folheações Holomorfas

Título |
Calabi-Yau modular forms, $sl_2$ Lie algebra and Rankin-Cohen Bracket |

Expositor |
Younes Nikdelan
Universidade do Estado do Rio de Janeiro (UERJ) |

Data |
Quinta-feira, 8 de fevereiro de 2018, 15:30 |

Local |
Sala 224 |

Resumo |

The idea of this talk is to introduce a certain type of $q$-expansions with integer coefficients and give some evidence which convince us to consider them as a generalization of (quasi)-modular forms. For doing this, we work on the moduli space $\sf{T}$ of Calabi-Yau $n$-folds arising from Dwork family enhanced with differential forms. There exists a unique vector field $\sf{R}$ in $\sf{T}$ with certain properties with respect to the underlying Gauss-Manin connection, calling modular vector field, whose solutions can be written in terms of the desired $q$-expansions. In particular, for $n=1,2$ we see that a solution of $\sf{R}$ can be written in terms of classic (quasi-)modular forms. We call this $q$-expansions as Calabi-Yau modular forms.

An algebraic group $\sf{G}$ acts from right on $\sf{T}$. The Lie algebra $\rm{Lie}(\sf{G})$ of $\sf{G}$ together with a modular vector field $\sf{R}$ generates another Lie algebra $\mathfrak{G}$, called AMSY-Lie algebra, such that $\rm{dim} (\mathfrak{G})=\rm{dim} (\sf{T})$. We find $\mathfrak{sl}_2(\mathbb{C})$ as a Lie subalgebra of $\mathfrak{G}$ that contains $\sf{R}$.

Finally, we endow the space $\mathcal{M}_n$ generated by Calabi-Yau modular forms with the Rankin-Cohen algebra structure.