Types of quantum field thories
The standard model of particle physics asserts that the fundamental quantum physical fields and particles are modeled as sections of and connections on a vector bundle that is associated to a -principal bundle, where the group is (or perhaps a quotient of this by ) and where the representation of 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 , preferably a simple group, which contains as a subgroup.
A good introduction to GUTs for mathematicians is
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