Thanks for your enquiry.
There is no set law as to which boundary fences a party is responsible for.
First, you will need to check your Deeds to see if there is any mention as to which boundaries you are responsible for. Please note that less than 50% of house deeds actually say anything about boundary responsibilities, so your Deeds may not assist.. You can obtain a copy of your Deeds from the land Registry website at a cost of £6 (called "Official Copies").
If your Deeds do not mention anything about the boundary responsibilities, then the law is that you and the respective next door neighbour are jointly responsible for maintaining the fence.
As an alternative, you may just want to ask the neighbour to provide evidence to you that you are responsible for the left boundary-as they have just moved in, their Solicitor will no doubt have advised them about the boundary responsibilities, and may have confirmed this in writing to them.
I hope this assists, but please let me know if you require any further clarification.