Given a group $G$, there is a so-called Langlands dual group $G^{∨}$.

Given a group $G$, there is also a so-called Goddard-Nyuts-Olive dual group $G^{'}$ that relates to the magnetic charge.

  1. Are the two $G^{∨}$ and $G^{'}$ defined in different settings? Or are their definitions related? What are the intuitions and motivations behind their definitions?

  2. Are the two $G^{∨}$ and $G^{'}$ exactly the same?

  3. What are the constraints on $G$ to give such groups: $G^{∨}$ and $G^{'}$? (eg, compact or not, simple, semi-simple, connected, simply-connected, Lie group or not, topological group, etc.)

A Reference on Goddard-Nyuts-Olive dual group: P. Goddard(Cambridge U.), J. Nuyts(UMH, Mons), David I. Olive(CERN and Bohr Inst.) Dec 1, 1976, Published in: Nucl.Phys.B 125 (1977) 1-28 DOI: 10.1016/0550-3213(77)90221-8 https://www.sciencedirect.com/science/article/pii/0550321377902218?via%3Dihub


The groups $G^{\vee}$ and $G'$ are the same. It is clear from the description in the paper you refer to.

Their definitions are the same as well, just swapping roots and coroots. You can do it in compact Lie groups or reductive algebraic groups.

There is further definition of $G^{\vee}$ from geometric Satake. Its physical significance has been looked into by Kapustin and Witten. They also revisit the Goddard-Nuyts-Olive work, you refer to, from the hills of Geometric Langlands.

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