Fraser, who recently appeared in The Pajama Game at Chichester Festival Theatre, has portrayed both Marius and Javert in Les Miserables in the West End.
Francesca Annis will play Garry's mother, Clara. Recently seen on television in Cranford, Annis' film credits include Revolver, The Libertine and Milk.
Doctor Hsu, the "wunderkind engineer" behind chess machine Deep Blue, will be played by Kenneth Lee, who starred in US comedy drama Delocated.
The cast also includes: Lorenzo Allchurch, Antonia Bernath, Cornelius Booth, Nicholas Croucher, Daniel Harries, Rob Lines, David Mumeni, Phil Nichol, John Ramm, Lucille Sharp, Brian Sills and Trevor White.
Donmar Warehouse artistic director [Josie Rourke] directs the play, which is based on Kasparov's headline-grabbing 1997 New York chess tournament against Deep Blue, a super-computer developed by technology giant IBM.
The Machine runs at Campfield Market Hall in Manchester from 12-21 July (previews from 10 July).