Answer by taking a quote from the following article:

Robert James Fischer (March 9, 1943 - January 17, 2008) was an American chess grandmaster and the eleventh World Chess Champion. Many consider him to be the greatest chess player of all time. Bobby Fischer showed great skill in chess from an early age; at 13, he won a brilliancy known as "The Game of the Century". At age 14, he became the US Chess Champion, and at 15, he became both the youngest grandmaster up to that time and the youngest candidate for the World Championship.

Sources implying that Paul Nemenyi, a Hungarian-Jewish mathematician and physicist and an expert in fluid and applied mechanics, was Fischer's biological father were first made public in a 2002 investigation by Peter Nicholas and Clea Benson of The Philadelphia Inquirer. Throughout the 1950s, the FBI investigated Regina and her circle for her alleged communist sympathies, as well as her previous life in Moscow.  FBI files identify Paul Nemenyi as Bobby Fischer's biological father, showing that Hans-Gerhardt Fischer never entered the United States, having been refused admission by US immigration officials due to his alleged Communist sympathies. Not only were Regina and Nemenyi reported to have had an affair in 1942, but Nemenyi made monthly child support payments to Regina and paid for Bobby's schooling until his own death in 1952. Nemenyi had lodged complaints with social workers, saying he was concerned about the way that Regina was raising Bobby, to the point that, on at least one occasion, Nemenyi broke down in tears. Later on Bobby told the Hungarian chess player Zita Rajcsanyi that Paul Nemenyi would sometimes show up at the family's Brooklyn apartment and take him on outings.  After Paul Nemenyi died in 1952, Regina Fischer wrote a letter to Nemenyi's first son, Peter, asking if Paul had left money for Bobby in his will:  Bobby was sick 2 days with fever and sore throat and of course a doctor or medicine was out of the question. I don't think Paul would have wanted to leave Bobby this way and would ask you most urgently to let me know if Paul left anything for Bobby.  On one occasion, Regina told a social worker that the last time she had ever seen Hans-Gerhardt Fischer was in 1939, four years before Bobby was born. On another occasion, she told the same social worker she had traveled to Mexico to see Hans-Gerhardt in June 1942 and that Bobby was conceived during that meeting. According to Bobby Fischer's brother-in-law, Russell Targ (who was married to Joan), Regina concealed the fact that Nemenyi was Bobby's father because she wanted to avoid the stigma of an out-of-wedlock birth.

Who is Bobby's mother?
Regina