Beat me to it. Note that you're also interested in the coproduct in the category of ringed spaces (so the "gluing" is the topological gluing that you expect). The paper shows that if R is a closed subscheme of one of the others then all three concepts coincide.

Replace Witten with Wiles here, and you get the general idea: abstrusegoose.com/272

Oh, ok! Well, the exponential map is also invariant under integration, and is also a solution here if you replace your lower limit of integration with minus infinity.