• 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
5,075 questions , 2,226 unanswered
5,348 answers , 22,744 comments
1,470 users with positive rep
818 active unimported users
More ...

  Incorporating Divisors (D4-branes) into Donaldson-Thomas Theory?

+ 4 like - 0 dislike

Let $X$ be a Calabi-Yau threefold. Ordinary Donaldson-Thomas theory is formulated as a virtual count of ideal sheaves $\mathcal{I}$ with discrete invariants $\text{ch}(\mathcal{I}) = (1,0, -\beta, -n)$, which is equivalent to counting one-dimensional subschemes $Y \subseteq X$ with $[Y] = \beta \in H_{2}(X, \mathbb{Z})$ and $\chi(Y)=n$. In the derived category, we have the equivalence $\mathcal{I} \cong [\mathcal{O}_{X} \to \mathcal{O}_{Y}]$. For this reason, in the physical literature, they describe the DT theory as enumerating "bound states of D0-D2 branes with a single D6-brane. This is because $\text{ch}(\mathcal{O}_{X})=(1,0,0,0)$ represents a single D6-brane while $\mathcal{O}_{Y}$ represents the D0-D2 branes.

I was thinking it would be nice if divisors (or what a physicist would call a D4-brane) could be placed on the same footing and incorporated into the DT partition function as well. Mathematically, I feel like it would be nice to sort of have all the holomorphic sub-geometry of $X$ in one generating function and physically, it would be preferable to have all the possible D-branes in the Topological B-model (D0,D2,D4,D6) branes sort of on the same footing.

Of course divisors are special as they're given by vanishing of sections of line bundles, so I'm wondering if it's possible to write something like

$$\text{Hilb}_{D, \beta, n}(X) \cong \text{Pic}_{D}(X) \times \text{Hilb}_{\beta' n'}(X)$$

where $\text{Hilb}_{D, \beta, n}(X)$ would be the Hilbert scheme of points, curves, and divisors. Note that $\beta$ and $n$ will be different from $\beta'$ and $n'$.

We have a deformation/obstruction theory for $\text{Hilb}_{\beta' n'}(X)$ with a virtual class. Since $\text{Pic}_{D}(X)$ is smooth, can we compute $[\text{Hilb}_{D, \beta, n}(X)]^{\text{vir}}$ to define enumerative invariants? I was thinking that since the Picard factor is a torus, maybe by some localization argument it wouldn't contribute to an integral over the moduli space?

Perhaps what I'm asking for is nonsense, but even if it's possible, I guess it would probably destroy modularity properties of the partition function? And would almost certainly lose connection to the Gromov-Witten theory via the MNOP conjecture.

This post imported from StackExchange MathOverflow at 2017-05-19 14:25 (UTC), posted by SE-user Stephen Pietromonaco
asked May 11, 2017 in Theoretical Physics by Stephen Pietromonaco (20 points) [ no revision ]
retagged May 19, 2017

1 Answer

+ 2 like - 0 dislike

Indeed your questions are extremely very good ones. 

Daniel Jafferis included (As part of his PhD thesis) D4 branes to the target space description of the topological A-model https://arxiv.org/abs/hep-th/0607032 , the answer is extraordinarily beautiful and many surprises arise (as the relationship with the partition function of the twisted N=4 SYM). I really love the results on the paper, (even when unfortunately look highly underestimated) and I sincerely wish you to enjoy them too.

answered Jun 5, 2019 by Ramiro Hum-Sah (80 points) [ no revision ]

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