Intuitive

A duality is like a translation scheme between languages. The same thing, say a chair, can be described either in English through the word "chair", but equivalently, for example, in German through the word "Stuhl". For certain purposes, one language is better suited than another. For example, the relatively simple English word "pollution" is translated into German the horribly long word "Umweltverschmutzung". So if you're dealing with an issue surrounding pollution, the English language is better suited.

A bit more precisely a duality is a formal or theoretical equivalence between two theories. Often physicists call the definition of the duality transformation that takes us from one language to another the ‘dictionary’.

Take note that instead of different descriptions of the same thing in different languages, we can have many synonyms for one and the same thing within one language. In the physical context this goes under the name gauge symmetry.

Concrete

This situation is what people call a duality:

However, take note that also if there are three Lagrangians that yield the same QFT, or even n-Lagrangians, people usually don't speak of a triality, or n-ality, but instead call all such situations dualities.

So the thing is that we have two (or more) descriptions - here defined in terms of Lagrangians - that yield the same physical observable results. Hence they describe the same physical situation, but the perspective is changed.

Especially the role of the fundamental degrees of freedom gets switched. The topological excitations of theory 1 become the elementary particles in theory 2 and vice versa.

For example, in Maxwell's theory supplemented with magnetic monopoles in the usual description, our elementary particles are electrons and the magnetic monopoles are topological solitons. However, in the dual description, the magnetic monopoles are the elementary particles and the electrons are topological solitons.

Source

In general, the dual theory will describe the interactions of the topological excitations of the original theory. In the case of U(1), these excitations are monopoles in d=3, and monopole loops in d=4.

page 18 in Gauge Field Theories Theoretical Studies and Computer Simulations by Giarczynski

There should be two "dual equivalent" field formulations of the same theory in which electric (Noether) and magnetic (topological) quantum numbers exchange roles.Magnetic monopoles as gauge particles? by C. Montonen and D. Olive

Examples

Electric-magnetic duality in classical electromagnetism

The Maxwell equations without electrically charged particles, i.e. only with the electromagnetic field do not change their form under the exchange of the electric and magnetic fields. Thus it is said that the electric and the magnetic field are dual to each other.

However, with electrically charged particles this duality does no longer exist. It was realised in 1931 by Dirac that the duality would still exist with electrically charged particles if there exist magnetic monopoles. One way to think about magnetic monopoles is as topological non-trivial configurations of the electromagnetic field for which the electromagnetic field is infinitely large at one point.

While the coupling strength of the electrically charged particles to the electromagnetic field is characterized by the fine structure constant $ \alpha = 1/137$, the duality requires that magnetic charges coupling with $1/\alpha = 137$ to the electromagnetic field. Therefore, they should be easy to be observed, but never were observed.

The hope is that if one can find a similar duality for general Yang-Mills theories such as QCD one could finally understand what goes on in the strong coupling regime by considering the dual theory, which in turn would be very weakly coupled.

(Source: Not Even Wrong by P. Woit page 138)

To quantize the system we must make a choice. It turns out that consistency requires the coupling strengths of the two types of currents to be inversely proportional to each other.Modern Quantum Mechanics by Banks

For a nice discussion of other examples see https://arxiv.org/pdf/1803.09443.pdf

Recommended Resources:

• Comparing Dualities and Gauge Symmetries by Sebastian De Haro, Nicholas Teh, Jeremy N. Butterfield

The crux of many problems in physics is the correct choice of variables with which to label the degrees of freedom. Often the best choice is very different from the obvious choice; a name for this phenomenon is ‘duality’. We will study many examples of it (Kramers-Wannier, Jordan-Wigner, bosonization, Wegner, particle-vortex, perhaps others). This word is dangerous (it is one of the forbidden words on my blackboard) because it is about ambiguities in our (physics) language. Where do quantum field theories come from? by McGreevy

There are many theories, which have more than one Lagrangian. So that's the opposite. Either we have no Lagrangian at all or we have more than one Lagrangian.

Duality and emergent gauge symmetry - Nathan Seiberg

Abstract

While a symmetry is a relation within a single theory (e.g. an automorphism of the space state and-or the set of quantities of the theory), a duality is a relation between different theories. Or between different descriptions of the same theory in case the duality is a self-duality. In such a case, the duality is a symmetry in the usual sense of model theory on the semantic conception of theories. …

Physicists tend to construe duality as an isomorphism of theories: the rough idea is that there is a duality if two theories make the same predictions for all the physical quantities that one can write down in the theory. …

More precisely, we define a duality as an isomorphism between two models of a single theory, where ‘model’ is here understood not in the usual sense, but as a mathematical representation of the theory (i.e. as a homomorphism from the theory to a mathematical structure that does the representing). What we here call the ‘single theory’ is a bare theory, and the originally-given two dual theories are now called ‘models’.

https://arxiv.org/pdf/1803.09443.pdf

Why is it interesting?

Oftentimes in physics, we can only calculate things in terms of a perturbation series. Especially, in quantum field theory, this is almost always the case. The perturbation series in quantum field theory is usually a series in the small coupling constant of the interaction in question. However, when this interaction constant becomes too large, such a perturbative approach no longer yields a good approximation. This is especially problematic for quantum chromodynamics at low energies, where the strong coupling constant becomes too large.

A duality can be an enormously powerful tool in such a situation. A dual theory describes the same physics, but in a different way. Especially the coupling constant of a dual theory is inversely proportional to the original coupling constant. Hence, whenever the coupling constant becomes too large the calculation can be done in the dual theory instead, where we can work with a small coupling constant.

The idea of duality has been at the centre of many important developments in the theoretical physics of the last 50 years. In fundamental physics, the notion of duality has been applied to very different kinds of theories. First, there is the dual resonance model of the late sixties, from which early string theory originated. Successively, one of the most important developments of this idea was the generalization, proposed by Claus Montonen and David Olive in 1977, of electromagnetic duality in the framework of quantum field theory. This was later extended to the context of string theory, where dualities have also spawned recent developments in fundamental physics, offering a window into non-perturbative physics, and motivating both the M theory conjecture and gauge-gravity dualityhttps://arxiv.org/pdf/1803.09443.pdf

Duality symmetries: mapping two theories or two different descriptions of a theory one-to-one to the other. The prime example is self-duality for the free Maxwell theory: you can mutually exchange the electric and the magnetic field, and this replacement leaves Maxwell’s equations invariant. Another example is quantum physics either described by Schrödinger’s wave equation or with Heisenberg’s matrix mechanics. Today, dualities are experiencing a boom, triggered by so called string theory. The five families of string models exhibit astoundingly many dualities, if one introduces branes; for a non-expert this is nicely described in [433]. Another very influential development was the discovery of a duality between anti-de Sitter grav- ity and a conformal field theory, sometimes named after its inventor as Maldacena conjecture. Symmetries in Fundamental Physics by Sundermeyer