• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

  Scherk-Schwarz and other compactifications?

+ 4 like - 0 dislike

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.

This post has been migrated from (A51.SE)
asked Nov 9, 2011 in Theoretical Physics by JMP (20 points) [ no revision ]
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?

This post has been migrated from (A51.SE)

1 Answer

+ 5 like - 0 dislike

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.

This post has been migrated from (A51.SE)
answered Nov 10, 2011 by Luboš Motl (10,278 points) [ no revision ]
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?

This post has been migrated from (A51.SE)

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights