There are texts that everyone knows are permitted to learn.
https://www.ou.org/holidays/the_story_of_the_churban_the_destruction/
However, certain portions of the Torah are permitted to be studied even on Tisha B’Av, the saddest day of the Jewish year. They include the Book of Job, which is a dramatic probe into the question of seemingly undeserved suffering, “Megilat Eichah,” the Book of Lamentations of the Prophet Yirmiyahu over the fate of Yerushalayim, the Laws of Mourning and certain portions of the Talmud.
The permitted Talmudical portions are in the “Aggadic” part of the Talmud; that is, the non-legal, but rather, historical or poetic, but with a moral lesson, portions. In particular, those portions of the Aggadah which are permitted to be studied are those dealing with the “Churban,” or Destruction, of Yerushalayim and of the Temple, the destruction of Beitar at the end of the Bar Kochba Revolt with its river, literally, of Jewish blood, and of certain similarly tragic episodes in Jewish History. Presumably, books about the Holocaust perpetrated by the Nazis, may their name be erased, are also acceptable reading matter on Tisha B’Av.
The major location in which these Aggadic portions are found is in “Masechet Gittin,” the Volume of the Talmud the legal portion of which deals mainly with “Gittin,” or Divorces (pages 55b – 58a). This is most appropriate, since the destruction of Yerushalayim and the Temple about two thousand years ago represented a “temporary” divorce, separating G-d and the Jewish People.
In addition Rabbi Friefeld told me in the name of Rav Hutner that it is also permitted to learn sefarim that help correct the sins that led to Tisha B'Av
