I must remember: are SBB a theory or fact?
Since both "screened bottom board" and "solid bottom board" can be abbreviated "SBB", I'll assume you mean "screened". It is theory, which is supported by some research, but research also indicates that the issue is not simple. Examples:http://www.countryrubes.com/images/AV-BOTTOM_BOARD.pdf
This study shows that screened boards reduce varroa only if they are "closed" (i.e. if the bottom board is a flat box with a screen at the top, and not simply a hole in the bottom of the hive with a screen across it). The study also cites three other studies which concluded some stuff.http://scientificbeekeeping.com/managing-varoa-part-1-ipm-realities/
In this repost, Roy Hendrickson states that he's phasing out screened boards in his 500 hives, but he doesn't say whether he had actually tested them against non-screened boards in a controlled study or whether he just got the impression that they're useless for him.