Reference no: EM13977488
I have been thinking about various types of compactifications and have been wondering if I have been understanding them, and how they all fit together, correctly.
From my understanding, if we want to compactify spacetime down from D to d dimensions by writing MD= Rd x kD-d. We can do this the following way:
"General" compactification:
Find the universal cover of K, and call it C. G is a group that acts freely on C, and K= C/G.
Then, the D-dimensional Lagrangian only depends on orbits of the group action: LD [Φ(x, y)] = L [Φ (x, Tgy) ].
A necessary and sufficient condition for this is to require that the field transform under a global symmetry: Φ (x, y)=Tg Φ (x, Tgy).
"General" compactifications seem to also be called Scherk-Schwarz compactifications (or dimensional reductions if we only keep the zero modes). An "ordinary" compactification has Tg=Id, and an orbifold compactification has a group action with fixed points.
Assuming this is correct, is this the most general definition of a compactification?
Is it reasonable to introduce gauge fields by demanding that the Tg action be local instead of global? I thought we should generally not expect quantum theories to have global symmetries, but any reference I've seen seems to use only global symmetries in the Lagrangian.
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