Idea

The standard model of particle physics asserts that the fundamental physical fields and particles are modeled as sections of and connections on a vector bundle that is associated to a $G$-principal bundle, where the group $G$ is $\mathrm{SU}(3)\times \mathrm{SU}(2)\times U(1)$ (or perhaps a quotient of this by $Z_6$) and where the representation of $G$ used to form the associated vector bundle looks fairly ad hoc on first sight.

A grand unified theory in this context is an attempt to realize the standard model as sitting inside a conceptually simpler model, in particular one for which the structure group – the gauge group – is a bigger but simpler group $\hat{G}$, preferably a simple group, which contains $G$ as a subgroup.

References

A good introduction to GUTs for mathematicians is

• John Baez and John Huerta, The Algebra of Grand Unified Theories (arXiv)

Blog discussion of this is at

• John Baez, The algebra of grand unified theories I

• John Baez, The algebra of grand unified theories II

• John Baez, The algebra of grand unified theories III