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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 $M_D = \mathbb{R}^d \times K_{D-d}$. 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: $\mathcal{L}_D[\phi(x,y)] = \mathcal{L}[\phi(x,\tau_g y)]$, $\forall g \in G$.

A necessary and sufficient condition for this is to require that the field transform under a global symmetry:

$\phi(x,y) = T_g \phi(x,\tau_g y)$.

"General" compactifications seem to also be called Scherk-Schwarz compactifications (or dimensional reductions if we only keep the zero modes). An "ordinary" compactification has $T_g = Id$, 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 $T_g$ 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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Also, most of the literature I see discusses general dimensional reduction, instead of general compactifications. Is this because the dimensional reductions are guaranteed to be consistent? (Is it the case that the general compactifications are not?) Also, are there any general issues in 'upgrading' a dimensional reduction to a compactification? – JMP Nov 10 '11 at 1:44
up vote 5 down vote accepted

The Scherk-Schwarz compactification is just an extremely special kind of compactification of one dimension in which the spacetime fermions are chosen antiperiodic along one circle of the compactification manifold. It's extremely far from a "general compactification" of string/M-theory.

General compactifications of string/M-theory allow many features that can't be discussed by the simple formulae above; in some sense, all of string/M-theory is needed to answer the question what the general compactification can be and cannot be. General compactifications may include a manifold. It doesn't have to be a manifold; it may be an orbifold with orbifols singularities. In fact, orbifold singularities are not the only allowed ones; one may have conifold singularities and probably much more general ones, too. Various fields may have nontrivial monodromies. One may wrap branes, try to incorporate boundaries at the "end of the world", and add many kinds of Wilson lines, fluxes, generalizing the electromagnetic fluxes, with various quantization conditions, and other features, some of which may even be unknown at the present moment.

The compactifications may get strongly coupled at various loci and interpolate between totally different descriptions such as type II string theory and M-theory, too. Moreover, one may have non-geometric compactifications, too. The question as formulated seems to be too broad.

Moreover, I also have to disagree with the first comment written under the question. It's a string/M-compactification, not a dimensional reduction, which is a consistent theory. The simple dimensional reduction is just an approximate effective theory at distances much longer than the size of the compactification manifold and such an effective theory almost always suffers from some kind of an inconsistency. To fix these inconsistencies, one needs to consider the fully consistent theory, namely the string/M-compactification.

There is no "universal algorithm" to "upgrade" a dimensionally reduced theory to a compactification (the term "upgrade" may mean either "dimensional oxidation" if we just try to add dimensions to a theory; or "UV completion" which means finding the precise compactification including all the short-distance physics that may be approximated by a given effective theory). Many compactifications – as many as the notorious number $10^{500}$ – of compactifications may lead to very similar effective theories in the large dimensions. There's no easy way to find the "correct one" among them.

It's also hard to understand in what sense "most of literature" discusses the dimensional reductions rather than compactifications. I don't think it's the case. Of course, if one looks at literature about dimensional reductions only, one may get this conclusion. However, true literature on string theory doesn't agree with the statement. It's mostly about the full physics of compactifications, not just the reductions – otherwise it wouldn't really be a stringy literature. This is true pretty much by definition.

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But by allowing the actions above to be non-free you include conifold and orbifolds, don't you? So that's not too much of a generalization. Or is there more to the most general case than this, aside from including the 'non-geometric' duality twists? I understand that you can 'interpolate' between theories but isn't any particular theory described as a compactification like this? – JMP Nov 13 '11 at 3:44

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