How do you mean 'written conversation' ? Do you mean texting or email ? We need to know the nature of the falsity in it, what the lie is.
As 'written' it could be a libel.The person who wrote the libel could be said to have 'published', as lawyers put it, to the person reading it.The trouble will be getting disclosure of the libellous document, getting sight of it in its written form. You have no right to see any and every 'written conversation'. What evidence do you have of its existence and what it says? All that's needed for libel is that your reputation would lowered in the estimation of right thinking people if they read it.
A secondary problem is that the courts are a bit slow to condemn idle gossip, which few, if any, would take seriously or statements merely an abusive expression of some opinion ("She's a ***"),the latter being rarely actionable at all as libel.
The third problem is cost. Defamation, libel, actions are very expensive to pursue; they can run into hundreds of thousands in costs in extreme cases;and costs of both parties are paid by the loser. There are claimants who manage to get solicitors to agree 'no win, no fee' arrangements (which annoy the defendant, because the claimant's recoverable costs are much higher) but there is a reason why you don't see TV ads advertising such a service for people who think they've been defamed, but only for personal injury and mispaid insurance cases! Legal Aid was never available in defamation cases.