# Electro magnetic duality, Strong weak duality and N=4 super Yangmils

How we can interpret this self-dual, or duality in terms of generalized version of electro magneitc duality, or Strong weak duality.

Let me address some basic information. First, electro magnetic duality is a duality of Maxwell's equation in vacuum \begin{align} &\nabla \cdot E =0, \quad \nabla \cdot B=0 \\ & \nabla \times E= - \frac{\partial B}{\partial t}, \quad \nabla \times B= \frac{\partial E}{\partial t} \end{align} above equation is invariant under the transformation of $(E,B)$ to $(B, -E)$. For this relation E,B we call electro-magnetic duality.

From dirac's quantization \begin{align} eg=2\pi n \end{align} and the construction of fine constant $\alpha = \frac{e^2}{4\pi\epsilon \hbar c}$, In QED we know that $\alpha = \frac{1}{137}$.

From above quantization rule, we know that in electric frame, the coupling constant is small, but in magnetic frame its coupling constant is large.

\begin{align} e_{strong} e_{weak} = 2\pi n \end{align}

Thus this can be extended to strong-weak duality.

Now what i want to do is applied this idea to $N=4$ Super Yang Mills theory. How we can analysis this in terms of electro-magnetic duality?

• Electric-Magnetic duality is a large subject. Have you tried reading the original Montonen & Olive paper? Dec 8, 2015 at 16:45
• @user1504, I have read the original paper, Magnetic Monopoles as Gauge Particles, Phys.Lett. B72 (1977) 117. This paper covers the duality between different gauge groups without susy. I Know, N=4 SYM theory is a sort of Mont. Olive duality (Strong-weak duality). What i have question above is how we know N=4 SYM has such kind of duality. Dec 10, 2015 at 12:21
• Actually, i know from many S-duality materials, the complex coupling constant of $N=4$ SYM is given as $\tau = \theta + g^{-2}$ (with some factor), and for $\theta=0$ there is a dualtiy $g\rightarrow g^{-1}$. I want to know how this procedure is allowed and relevant to electro-magnetic duality Dec 10, 2015 at 12:22

The basic idea is that you make a change of variables in the Feynman integral, by mapping the field strength 2-form $F$ to its dual $*F$. Then you observe that, with respect to these new variables, the Feynman integral still describes a gauge theory, except with the coupling constant $g$ interchanged with $\kappa / g$ and the electric and magnetic observables interchanged.
The strongest argument that it's a true conjecture is due to Vafa, who observed that N=4 SYM arises as the low energy limit of both IIA and IIB string theory, compactified on the product $K \times T^2$ of an ALE space $K$ of type ADE and a 2-torus. The strength of the gauge coupling is proportional to the inverse volume of the 2-torus. String theory tells us there is an exact duality which relates IIA and IIB while switch $T^2$ with its dual. In the low energy gauge theory descriptions, this duality flips the electric and magnetic observables. Likewise, it switches the volume of $T^2$ with its inverse, so flips the gauge coupling.