Theta functions, broken lines and 2-marked log Gromov-Witten invariants

Theta functions were defined for varieties with effective anticanonical divisor [11] and are related to certain punctured Gromov-Witten invariants [2]. We show that in the case of a log Calabi-Yau surface (<i>X</i>, <i>D</i>) with smooth very ample anticanonical divisor we can relate theta functions and their multiplicative structure to certain 2-marked log Gromov-Witten invariants. This is a natural extension of the correspondence between wall functions and 1-marked log Gromov-Witten invariants [8]. It gives an enumerative interpretation for the intrinsic mirror construction of [17] and will be related to the open mirror map of outer Aganagic-Vafa branes in [9].