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 ...

  Bridgeland stability for restricted Kahler moduli?

+ 5 like - 0 dislike
522 views

Let $X$ be a simply-connected, smooth, projective Calabi-Yau threefold. To my understanding, Bridgeland introduced stability conditions on triangulated categories to give a proper mathematical definition of the stringy Kahler moduli space (SKMS) from physics.

Conjecturally, the classical (complexified) Kahler cone $\mathcal{K}_{X}(\mathbb{C})$ of $X$ gives an open chart on the SKMS around the large-volume limit. Coordinates on $\mathcal{K}_{X}(\mathbb{C})$ are called Kahler moduli, and depending on the context, one may prefer to think of them as formal variables tracking degrees along effective curve classes in $X$, i.e. effective classes in $H_{2}(X, \mathbb{Z})$.

Classically, it makes sense to consider only certain Kahler moduli: this would be some sort of sub-cone, or collection of sub-cones, in $\mathcal{K}_{X}(\mathbb{C})$. For example, one setting I'm interested in is when we have a proper surjective map

$$f: X \to \mathbb{P}^{1}$$

whose generic fibers are Calabi-Yau surfaces. You have certain Kahler moduli tracking curve classes in the fibers of $f$, and other Kahler moduli tracking classes "transverse" to the fibers. One might want to focus on just fiber classes, or transverse classes.

So my question is: can one expect submanifolds of the Bridgeland stability manifold/SKMS which correspond to only specific Kahler moduli, as I've described above?

For example, in the case of fiber classes of $f$, one can define the Serre subcategory $Coh(f)_{0}$ of $Coh(X)$ whose objects are coherent sheaves on $X$ supported on the fibers of $f$. You then get a full triangulated subcategory $D^{b}(X)_{f} \subset D^{b}(X)$ consisting of objects whose cohomology sheaves lie in $Coh(f)_{0}$.

By applying the machinery of Bridgeland to $D^{b}(X)_{f}$ or some similar triangulated subcategory, can one find a submanifold of the stability manifold/SKMS corresponding to fiber classes of $f$?

This post imported from StackExchange MathOverflow at 2020-01-22 12:09 (UTC), posted by SE-user Benighted
asked Jul 27, 2019 in Mathematics by Benighted (310 points) [ no revision ]
retagged Jan 22, 2020

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$\hbar$ysicsOverflo$\varnothing$
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
...