• 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.


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
5,037 questions , 2,191 unanswered
5,345 answers , 22,706 comments
1,470 users with positive rep
816 active unimported users
More ...

  Does target space have the same structure when we compactify from 3D to 2D?

+ 3 like - 0 dislike

Does target space have the same structure when we compactify from 3D to 2D? I mean when we compactify from 4D to 3D, the moduli space is essentially a torus fibration over the Coulomb branch of $\mathcal{N}=2$ theory and we know that the Coulomb branch has a rigid special Kähler structure. So when we reduce to 2D, why does  the same Hyperkähler target space (moduli space) of 3D theory arise?

asked May 30, 2015 in Mathematics by SI1989 (85 points) [ revision history ]
retagged May 30, 2015 by Dilaton

1 Answer

+ 3 like - 0 dislike

Moduli spaces of supersymmetric vacua of supersymmetric field theories are parametrized by expectation values of massless scalar fields. When we compactify a theory the massless scalars of the original theory are still present in the compactified theory and so the moduli space of the original theory is certainly naturally inside the moduli space of the compactified theory. The moduli space of the compactified theory can be bigger if there are new massless scalar fields in the compactified theory which were not present in the original one. At a generic point of the moduli space, the only way it can happen is by dimensional reduction of the gauge vectors of the original theory along the compact directions: if you have a gauge vector $A_\mu$ in the original theory and if you compactify some of the directions corresponding to some indices $i$ then the corresponding components $A_i$ become massless scalar fields in the compactified theory. This is why when one goes from $6d$ to $5d$, from $5d$ to $4d$, from $4d$ to $3d$ the moduli spaces are in general different. Now the key point in that in spacetime dimension $d$ a gauge vector has only $d-2$ physical components. In particular in $3d$  an abelian (which is the case at a generic point of the moduli space) gauge field can be dualized in a scalar field. If you include this scalar in the description of the moduli space of the $3d$ theory, what is usually done, then there is no longer any non-trivial gauge vector in $3d$. In particular going from $3d$ to $2d$ there are no non-trivial gauge vectors to reduce along the compactified direction to give new massless scalar fields and so the moduli space do not change.

answered May 30, 2015 by 40227 (5,140 points) [ revision history ]

@40227, Thank you very much! I have other questions that I asked here: Coulomb branch of $\mathcal{N}=2$ field theories

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