Gromov–Witten theory and cycle-valued modular forms

Abstract

Abstract In this paper, we review Teleman’s work on lifting Givental’s quantization of ℒ + ( 2 ) ⁢ GL ⁢ ( H ) {\mathcal{L}^{(2)}_{+}{\rm GL}(H)} action for semisimple formal Gromov–Witten potential into cohomological field theory level. We apply this to obtain a global cohomological field theory for simple elliptic singularities. The extension of those cohomological field theories over large complex structure limit are mirror to cohomological field theories from elliptic orbifold projective lines of weight ( 3 , 3 , 3 ) (3,3,3) , ( 2 , 4 , 4 ) (2,4,4) , ( 2 , 3 , 6 ) (2,3,6) . Via mirror symmetry, we prove generating functions of Gromov–Witten cycles for those orbifolds are cycle-valued (quasi)-modular forms.