Chiral De Rham complex on the upper half plane and modular forms

Abstract

In this thesis, we introduce an SL(2,R)-action on the chiral de Rham complex on the upper half plane, and study the vertex subalgebra Ω^ch(H,Γ) of Γ-invariant global sections which are holomorphic at all the cusps, for any congruence subgroup Γ. The thesis can be divided into three parts. The first part includes a recollection on the theory of vertex algebras and modular forms. The second part consists of a brief review of the construction of chiral de Rham complex by Malikov, Schechtman and Vaintrob, which will be applied to the upper half plane. We consider the vertex algebra Ω^ch(H,Γ) associated to an arbitrary congruence subgroup Γ, and compute its character formula. We also give an explicit formula for the lifting of modular forms to Ω^ch(H, Γ), and the lifting formula is essentially unique and universal. The last part discusses the relations between the modified Rankin-Cohen bracket and the elements in Ω^ch(H,Γ), and some further properties about Ω^ch(H,Γ).

Date of Award	2020
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology