^There's this belief that lovers are connected with a red thread on their pinkies. The red thread symbolizing destiny. (Ever seen Otsuka Ai's "POCKET" PV? It shares the same symbolism.) Therefore ayu sings in the next line that she thought destiny was something one could grasp.
And I thought "end of an era" in "Duty" refers to the end of her career?
PS. You can read more on the meaning of the red thread
here. ^^