You need to create a composite index on tableA:
CREATE INDEX ix_tablea_t1id_t2id ON table_A (t1id, t2id)
Indexes in MySQL are considered a part of a table: they are updated automatically, and used automatically whenever the optimizer decides it's a good move to use them.
MySQL does not use the term index table.
This term is used by Oracle to refer to what other databases call CLUSTERED INDEX: a kind of table where the records themselves are arranged according to the value of a column (or a set of columns).
In MySQL:
When you use MyISAM storage, an index is created as a separate file that has .MYI extension.
The contents of this file represent a B-Tree, each leaf containing the index key and a pointer to the offset in .MYD file which contains the data.
The size of the pointer is determined by the server setting called myisam_data_pointer_size, which can vary from 2 to 7 bytes, and defaults to 6 since MySQL 5.0.6.
This allows creating MyISAM tables up to 2 ^ (8 * 6) bytes = 256 TB
In InnoDB, all tables are inherently ordered by the PRIMARY KEY, it does not support heap-organized tables.
Each index, therefore, in fact is just a plain InnoDB table consisting of a single PRIMARY KEY of N+M records: N records being an indexed value, and M records being a PRIMARY KEY of the main table record which holds the indexed data.