• 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 ...

  BPS states : Mathematical definition

+ 14 like - 0 dislike

First of all, let me congratulate the theoretical physics community for this site. I am a mathematics student with very little background in phyiscs. The question I want to ask is:

What is the proper mathematical definition of BPS states?

In string theory the BPS states correspond either to coherent sheaves or special Lagrangians of Calabi- Yau manifold depending upon the type of string theory considered. but in SUSY quantum field theories in 4d, there are no CYs as far I know (which is very little) and in gravity theories, these corresponds to some Black Holes. So what is the general mathematical definition of BPS states which is independent of the theory in consideration, say a general SUSY Quantum field theories, be it QFT, string theory, Gravity and in any dimension.

I am very sorry, if the question is not framed properly due to my ignorance of the physics involved. Thanks

This post has been migrated from (A51.SE)
asked Oct 8, 2011 in Theoretical Physics by J Verma (270 points) [ no revision ]
retagged Apr 19, 2014 by dimension10
J Verma, welcome on TP.SE. Please expand BPS acronym at least once.

This post has been migrated from (A51.SE)
BPS = Bogomol'nyi–Prasad–Sommerfield

This post has been migrated from (A51.SE)

2 Answers

+ 13 like - 0 dislike

The BPS bound was discovered independently of supersymmetry, but it was then better understood as general feature of the supersymmetry algebra. Look at the original paper by Witten and Olive. BPS states are states which saturate the BPS bound, forming "short" representations an extended supersymmetry algebra. Such representations have special properties, which often can be thought of as consequence of some fraction of the supersymmetry which remains in their presence. The examples you cite are special cases, in all those cases the special property of the objects you mention is a deduced from the requirement that the states involved form a short representation of the supersymmetry algebra.

This post has been migrated from (A51.SE)
answered Oct 8, 2011 by Moshe (2,405 points) [ no revision ]
+ 12 like - 0 dislike

In any supersymmetric theory, a BPS state is a state which preserves some of the supersymmetry.

If we take as a definition of a supersymmetric theory, some theory (classical or quantum) which admits a Lie superalgebra of symmetries, then a BPS state (or configuration) of such a theory is one which is annihilated by some nonzero odd element in the superalgebra.

Of course, the original meaning comes from the study of magnetic monopoles. Solutions of the Bogomol'nyi equation are precisely those which saturate a bound, the so-called Bogomol'nyi-Prasad-Sommerfield (BPS) bound.

The relation between the two notions is related to the fact that monopole configurations can be thought of as configurations in a four-dimensional $N=2$ supersymmetric gauge theory which preserve half of the supersymmetry.

This post has been migrated from (A51.SE)
answered Oct 8, 2011 by José Figueroa-O'Farrill (2,315 points) [ no revision ]
Here is another question from a mathematician: some physicists have told me that in the context of 4d gauge theory one can think about BPS states as cohomology of the corresponding moduli space of instantons (but some other physicists deny that). In what generality can one claim such thing? For example, what happens if go from pure gauge theory to gauge thery with matter (the instanton moduli space doesn't change...)? What happens if we look at 5d theory compactified on $S^1$? Is there a way to answer such questions in general?

This post has been migrated from (A51.SE)
Alexander: the interpretation of the BPS states depends very strongly on the details of the supersymmetric theory one considers. It is not possible to make such sweeping generalisations. In the context of topologically twisted Yang-Mills theory, the BPS states are indeed annihilated by the BRST operator, which can be interpreted as the de Rham differential in the moduli space of instantons. (More precisely it's the equivariant version of BRST cohomology which can be so interpreted, but this is typical in the gauge theoretic context.)

This post has been migrated from (A51.SE)
Is it important which topological twist you are taking? I suppose you meant the one which leads to Donaldson theory? Also, the context in which I met this statement did not mention topological twist - for example in the last section of Witten's paper "Geometric Langlands from 6 dimensions".

This post has been migrated from (A51.SE)
Alexander: indeed, different twists correspond to different differentials. For 4-d N=2 supersymmetric Yang-Mills, though, all topological twists are equivalent. The story is different in N=4, though, which might be the context of Witten's paper.

This post has been migrated from (A51.SE)
In that part of Witten's paper no topological twist is present. In any case, let me formulate the question in the opposite direction: given particular moduli space (e.g. moduli space of instantons on ${\mathbb R}^4$, or similar space for ${\mathbb R}^3\times S^1$), for which theories can you identify its (equivariant) cohomology with the space of BPS states of that theory? Can you do it for $K$-theory instead of cohomology? What do you do if the moduli space is singular? I think that I know the general of yoga of such things, but I'd like to know how exactly it works in specific examples.

This post has been migrated from (A51.SE)
Alexander: I suggest you ask this question in the site. I'm not sure that I can give you better answers than you might get from others here. (I have not looked at that paper of Witten's in a while, but I would be surprised if this were not in the context of a topologically twisted $N=4$ theory. The geometric Langlands correspondence is basically electromagnetic duality applied to one of the possible $N=4$ twisting, if I recall correctly.)

This post has been migrated from (A51.SE)
Thanks - I'll try to post it as a separate question later. Just one comment: geometric Langlands is related to $N=4$ 4d theory, but the part of Witten's paper that I was asking about is not: it talks about a higher dimensional analog of geometric Langlands and there you must with the (2,0)-theory directly

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