What is Montonen-Olive Duality?
It's a little more subtle than just taking the reciprocal of the coupling constant $g\to1/g$. To understand Montonen-Olive Duality, it pays to consider its abelian cousin, the electric-magnetic duality. Note that the term "electric-magnetic duality" is often used as a catch-all for all dualities in supersymmetry gauge theories that resemble it, but here we focus on the motivating idea behind all of them: Maxwell's equations.
The $\mathbf E{-}\mathbf B$ formulation of Maxwell's equations in vacuum make it clear that $(\mathbf E, \mathbf B)\to(\mathbf B, -\mathbf E)$ is a symmetry of the equations. In the Lorentz-covariant formulation, the equations $\mathrm dF=0$ and $\mathrm d\star F=0$ are invariant under $F\to\star F$. When sources for the field strength and dual strength forms are added in, we also require these sources to transform into each other accordingly to preserve the duality.
By analysing the Wick-rotated path integral over the standard abelian gauge action plus a topological term in $\mathbb R^4$, we can conclude that the electric-magnetic duality acts via $$\hat\chi:\tau\to\frac{-1}{\tau}
\\\tau\equiv\frac\theta{2\pi}+\frac{4\pi i}{e^2}
$$
where $\theta$ is the prefactor of the topological term. Finally, given some nice conditions on the manifold, the theory will also be invariant under $\hat\zeta:\tau\to\tau+1$. $\hat\chi$ and $\hat\zeta$ together generate the group $\mathrm{PSL}(2,\mathbb Z)=\mathrm{SL}(2,\mathbb Z)/\mathbb Z_2\subset \mathrm{Aut}(\mathbb C\mathrm P^1)$. It's easy to see that, at least at $\theta = 0$, $\hat\chi$ flips the coupling strength, sending the theory at strong coupling to one at weak coupling and vice versa.
Montonen-Olive duality in its modern usage is the generalisation of this duality from $\mathrm{U}(1)$ theory to non-abelian gauge theory. We perform a similar analysis on the Euclideanised path integral (remembering to sum over the isomorphism classes of the principal bundle defining the theory). However, due to the presence of the connection one-form in the Yang-Mills equations of motion, naïvely copying the abelian case fails. So one must resort to a rather involved analysis of magnetic sources [2] that I will not go into here - but subsequently, Montonen and Olive conjectured that
A Yang-Mills theory with gauge group $G$ and (complex) coupling $\tau$ is dual to a different Yang-Mills theory with gauge group $^LG$ and coupling $-\frac{1}{k\tau}$, where $^LG$ denotes the Langlands dual group of $G$ and $k$ is a constant depending on certain properties of the Lie algebra $\mathfrak g$ of $G$
Additionally, it is invariant under $\tau\to\tau + n$ where $n\in\mathbb Z$ depends on the properties of both the manifold and the gauge group under consideration - so in total there is symmetry under some $\mathfrak N\subset\mathrm{SL}(2,\mathbb R)$. Here I'll take the SQFT viewpoint on "non-perturbatively isomorphic theories" under MO duality, although it also famously descends from S-duality in type II string theory, particularly in D3 background solutions.
So what can you do with it?
Remember that it's still a conjecture, so there are reasons for and against its validity in different regimes (see [1] for a good review) but nothing 100% concrete. However, you can press on and analyse its consequences nonetheless, aided by the force of the deep, far-reaching geometric Langlands program. There is actually an issue involving renormalisation during the derivation of the general MO duality above but we are able to mostly bypass this issue in maximally supersymmetric $\mathcal N = 4$ SYM since it is superconformal (here is a super-cool argument for why the coupling doesn't run). Here are some examples of Langlands dual groups: you'll see that they aren't too exotic and so it seems we do stand a chance of calculating correlators at a well-understood coupling strength and transforming them to the dual coupling.
| $G$ |
$^LG$ |
| $\mathrm{SU}(n)$ |
$\mathrm{SU}(n)/\mathbb Z_n$ |
| $\mathrm{SO}(2n)$ |
$\mathrm{SO}(2n)$ |
| $\mathrm{Sp}(2n)$ |
$\mathrm{SO}(2n+1)$ |
| $G_2$ |
$G_2$ |
| $F_4$ |
$F_4$ |
| $E_8$ |
$E_8$ |
(again, this is not the full story, since the duality can e.g. act non-trivially on the Higgs sector - this is the very surgical-sounding "elliptic endoscopy" [3]). This is a key feature of the Langlands dual: it sends reductive groups to reductive groups, and a reductive Lie algebra is precisely the data that is required for a well-defined Yang-Mills theory.
This means that the partition functions are isomorphic in the dual theories, and a fortiori that correlators of operators should in principle agree with the correlators of some "dual operators" in the dual theory - even though the gauge-invariant observables themselves (for example, Wilson loops) transform non-trivially:
$$
\langle \hat O_{(1)}\hat O_{(2)}...\hat O_{(n)}\rangle\big|_{\{\mathcal M, G, \tau\}}=\langle\tilde O_{(1)}\tilde O_{(2)}...\tilde O_{(n)}\rangle\big|_{\{\mathcal M, ^LG, -1/k\tau\}}
$$
This means that yes, provided the Montonen-Olive conjecture is valid, we can formulate some correlation function and flip to the dual theory where it is easier to evaluate. For the popular case of the $\mathrm{SU}(n)$ theory, it is essentially self-dual on 4-dimensional manifolds with trivial second cohomology group, with the partition function being mapped to itself up to some topological prefactors reflecting additional gravitational couplings.
Unfortunately for non-CP-violating SYM, and I quote [1] here:
[T]he conjecture is untestable unless we get a better handle at strongly
coupled theories—of course, this also means that it cannot be disproved!
Nevertheless, modern evidence in favour of the MO-duality is being procured from techniques in string theory [4], topologically twisted $\mathcal N = 4$ SYM (since this is where the relevance to geometric Langlands becomes evident) and, at a high level, matching OPEs of electric and magnetic sources in SYM (which is of course non-perturbative).
References:
- [1]: J. Figueroa-O'Farrill, Electromagnetic Duality for Children (though, contrary to the name, it is a very comprehensive reference on these matters)
- [2], Goddards, Nuyt, Olive, Gauge Theories and Magnetic Charge
- [3] Argyres, Seiberg, Kapustin, On S-duality for Non-Simply-Laced Gauge Groups
- [4] Vafa, Geometric Origin of Montonen-Olive duality
