# Langlands dual group in math vs. Goddard-Nyuts-Olive dual group in physics

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.
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.