Quantcast
  • 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.

News

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

Attributions

(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 ...

  Why is this form used here?

+ 2 like - 0 dislike
853 views

I was reading this paper, and while I did, I have one question:

Why is that

In the supergravity literature, one often formulates local special geometry in terms of a section of a different bundle. This section is denoted as $V$ and is related to $v$ by

$$V \equiv e^{K/2}v$$, where $K$ is Kaehler potential? That is, why is that it is no longer written as $V$ (aka the symplectic section), but now there is a factor of $e^{K/2}$ hanging around in front of $v$? Why this form?

Please note that

the geometries appearing in $N = 2$ supergravity theories are denoted as local special geometries. In mathemetics literature the local special geometry is called ’projective special geometry’.

Extra notes in order to clarify any vague definitions above:

On p.15 it says

There exists a holomorphic $Sp(2n+ 2;\mathbf{R})$-vector bundle $\mathcal{H}$ over the manifold and a holomorphic section $v(z)$ of $\mathcal{L} \otimes \mathcal{H}$. Note that $\mathcal{L}$ denotes the holomorphic line bundle over the manifold, of which the first Chern class equals the cohomology class of the Kahler form.

This post imported from StackExchange Physics at 2015-12-24 12:30 (UTC), posted by SE-user PhilosophicalPhysics
asked Dec 23, 2015 in Theoretical Physics by PhilosophicalPhysics (20 points) [ no revision ]
Minor comment to the post (v1): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot.

This post imported from StackExchange Physics at 2015-12-24 12:30 (UTC), posted by SE-user Qmechanic

Hi PhilosophicalPhysics, welcome to PhysicsOverflow!

I saw your comments on the original SE question, if you need access to your imported account and content you can follow the instructions here.

1 Answer

+ 4 like - 0 dislike

On the complex line bundle $\mathcal{L}$, we have two natural geometric structures: an holomorphic structure and an hermitian metric. For a section of $\mathcal{L}$ we thus have to natural constraints, in general incompatible, that we can impose on our section: holomorphy or unitarity (i.e. constant norm with respect to the hermitian metric). Up to some i factor, $e^{-K}$ is the norm squared of the holomorphic section $v$. To construct a unitary section $V$, it is enough to rescale $v$ by the inverse of the square root of its norm squared, i.e. by the (non holomorphic) factor $e^{K/2}$. Depending on what one is trying to do it can be more convenient to work with the holomorphic section $v$ or with the unitary section $V$.

answered Dec 24, 2015 by 40227 (5,140 points) [ revision history ]

Thanks @40227 a lot for your answer. I asked this on another forum and maybe someone moved it here , so it was just today that I saw your answer. I have two questions, first you said "Up to some i factor, $e^{−K}$ is the norm squared of the holomorphic section $v$. " How do we know that? Second, concerning this: "it is enough to rescale $v$ by the inverse of the square root of its norm squared, i.e. by the (non holomorphic) factor $e^{K/2}$", I did not understand here the mathematics, why inverse of the square root of its norm squared?

On the first question: I guess it depends on what is your starting point and indeed the paper linked in the question does not exactly use this language. Let me say the following: if $\omega$ is a Kähler form and $\mathcal{L}$ an holomorphic line bundle of first Chern class equals to the class of $\omega$ then a choice of Hermitian metric on $\mathcal{L}$, such that the curvature of the associated Chern connection is $\omega$, is a global version of a choice of Kähler potential for $\omega$.

On the second question: my formulation was maybe a bit strange but I was simply saying that if I have a vector $v$, then to construct a vector of norm 1, I take $v/||v||$ where $||v||=\sqrt{||v||^2}=\sqrt{<v,v>}$.

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:
p$\hbar$ysi$\varnothing$sOverflow
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
...