I have been using 120V line power and a dimmer switch for a long time with .032 stainless safety wire. Cuts beautifully and you don't have to worry about putting tension on it. I bought a pound spool of the stuff like 20 years ago and still haven't gone through it.The voltage of 70V is a bit exaggerated for nichrome wire unless your wire is thick and long. The impedance of the wire (inner resistance) varies with the tension of the wire too, thus a variable voltage power source would be a better solution to have clean foam cuts.
I see people use nichrome wire quite a lot but there is a better wire for foam cutting if smooth surfaces are needed. Nichrome has an inner specific resistance of 1.08 Ohm while Kanthal wire has 1.45 Ohm per mm2 per meter of length. The composition of nichrome wire is nickel and chrome while the Kanthal wire is iron, chrome, aluminum and cobalt. The aluminum part in the Kanthal wire allows to have smoother cuts because molten foam will not stick to the wire surface. This wire is less strong than the nichrome wire but it is intended for heating so has a better resistance to heat. Thus a Kanthal wire if the calculated resistance and voltage are met will almost never break and will cut smoothy. The thinner the wire the more resistance it has, so an optimum is to balance between resistance and voltage to get the right wire temperature. Tensioning the wire (be it nichrome or kanthal) impacts the thin structure of the walls and adds resistance to the wire. Basically, the higher the resistance the electricity 'does not want to go through' so you have to increase the voltage. As you increase the voltage in a high resistance wire the heat goes up. With a good Kanthal wire and a variable voltage power source you could make a foam cutter which could set an ideal temperature for the foam you intend to cut.
The 0.5mm kanthal wire has an impedance of 6 Ohm while the 0.6mm wire has 0.4 Ohm. So you see the thinner you get the more voltage you will need. When tensioning with a spring the 0.6mm wire extends with the heat and it's resistance increases almost to the one of the 0.5mm wire. So, every spring that is put in a system might influence the heat of the foam cutter wire temperature and has to be taken into consideration.