# Is there any GENERAL symmetry behind holography and dualities?

Holography tells that a theory with gravity in D-dimensions is equivalent to a field theory in D-1 on the boundary. Similarly, after the second string revolution, we know there are symmetries called dualities, pervading every superstring or M-theory. What is the symmetry principle and/or general invariance under these maps? And moreover, what would be the conserved quantity or Noether identity associated to these? Addendum: in classical mechanics, when you pass from a sistem with N d.o.f., to N-1 d.o.f. you get a momentum conservation from cyclic coordinate. What is the cyclic "coordinate" under a holography or duality map? If any...I mean, is there any known consequence after dimensional reduction. Of course, I suppose the trick is in the counting of the fields.

• This is actually a deep question but it may still be too deep for the present day. You can see preliminary ideas like arxiv.org/abs/1705.02500 – Mitchell Porter Apr 22 at 5:10
• Noether's theorem relates continuous symmetries to conserved currents. Dualities are non-trivial alternate descriptions of a theory. They are not continuous and they are not symmetries in the sense of Noether's theorem – octonion Apr 22 at 18:14
• Electromagnetic duality DOES have conservation laws related to zilches and helicity...Perhaps we should enlarge our notion of symmetry to include dualities somehow... – riemannium Apr 22 at 18:41
• @riemannium, In that case the duality is a map between field configurations of the same theory (so it can be thought of as a symmetry), and it can be extended to a continuous symmetry. By the way, I don't see your comment unless you link me with @ – octonion Apr 22 at 20:48
• @octonion How do you extend em duality to a continuous symmetry? I only know how to extend to $SL(2,\mathbb{Z})$, which is discrete. – Ryan Thorngren Apr 22 at 21:08