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The dualities described the theories under certain transformations to be equivalent, i.e. $T$-duality described the equivalence between theories with $R$ and $\frac{1}{R}$. However, this looked very much like a symmetry, and should thereby have conserved charges.

On the other hand, the dualities were equivalence between the theories, not just the same one, so there does seem to be some distinct features.

Are dualities symmetries? If so what are their conserved charges?

Qmechanic
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ShoutOutAndCalculate
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    Symmetries are transformations acting on the fields of a theory without changing the theory. So, you transform the fields, but the theory is the same. Dualities are maps between two different theories, i.e., they change the theory so that you actually change your action. – Gold Nov 28 '22 at 16:00
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    Possible duplicates: https://physics.stackexchange.com/q/162027/2451 and https://physics.stackexchange.com/q/8518/2451 – Qmechanic Nov 28 '22 at 16:27
  • @Gold Yes, I read something about the type I and type II theories so it's probably not the same in that case. But if just in the CFT sigma model, it's really the exchange of the momentum and winding modes. In the rational radius it looked like a discrete symmetry, and in irrational radius there's infinite number of fields. – ShoutOutAndCalculate Nov 28 '22 at 16:36
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    One way to think about dualities is that the quantum theory has multiple classical limits. There's not necessarily a continuous transformation that links one to the other, which is needed for Noether's theorem. – Andrew Nov 28 '22 at 16:48

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I don't think you've gotten a full answer yet. John is right that dualities, trialities, etc would not give rise to conserved charges even if they were symmetries because $\mathbb{Z}_2$, $\mathbb{Z}_3$, etc are discrete. But there are continuous "infinite-alities" and a simple example ($\phi \mapsto \phi + \lambda \phi^2$ for a free scalar) is given in problem 10.5 of Srednicki. This changes the action, allowing us to see that it's not a symmetry. And the more general fact that dualities are never symmetries can be explained without referring to actions or classical limits.

Various definitions of symmetry have been given in recent years but they all have something to do with automorphisms of the theory's Hilbert space. In short, you can stare at an abstract Hilbert space long enough and then notice that its states can be arranged into faithful unitary representations of some group $G$ which becomes a symmetry group of your theory. Unlike $G$ however, dualities do not act on the theory's Hilbert space (or other fundamental data of it) at all. They leave the theory untouched and instead relate two human-made constructions of it. These could be actions (what some people imprecisely refer to as theories), IR limits of flows away from simpler theories (these dualities are usually conjectural) or various geometric engineering procedures.

Connor Behan
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If I remember it right, Noether's theorem is about continuous symmetries. Duality is in general an integer one, so no conserved charges are expected, at least at the level of most literal reading of Noether's statement.

John
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Symmetries DO NOT act on coupling constants OR on background fields. Duality transformations do ($S$-transform maps $g \to 1/g$). Consequently, they are not a symmetry. If you take the viewpoint that different values of coupling constants are different theories, then dualities map you between theories.

Note, however, that at the self-dual point ($g=1$ in the $S$-transform example above) the duality map does indeed have the potential of becoming a true symmetry since the coupling constant in this case remains invariant as required for a symmetry.

Prahar
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