• 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

203 submissions , 161 unreviewed
5,006 questions , 2,162 unanswered
5,341 answers , 22,655 comments
1,470 users with positive rep
815 active unimported users
More ...

  Second quantization of SCFT and Geometric Engineering

+ 2 like - 0 dislike

I'm sure I'm not using these terms in their full scope of generality, but when I say "second quantization of a CFT" I will mean in the following instance.  Given a compact Calabi-Yau manifold $X$ of complex dimension $d$, the elliptic genus

$$\text{Ell}_{q,y}(X) = \sum_{n \geq 0,l \in \mathbb{Z}} c(n,l)q^{n} y^{l}$$

is a weak Jacobi form of weight 0 and index $d/2$ which is some partition function in a SCFT with target space $X$.  The second quantization of the elliptic genus is the generating function of the elliptic genera of the symmetric products, which we can write as an infinite product by DMVV

$$\sum_{m=0}^{\infty} Q^{m} \text{Ell}_{q,y}\big(\text{Sym}^{m}(X)\big) = \prod_{m > 0, n \geq 0, l \in \mathbb{Z}}\big( 1- Q^{m}q^{n}y^{l}\big)^{-c(nm, l)}$$

Noting that the elliptic genus is an automorphic object of weight zero, the infinite product form of the DMVV formula is very much aligned with the work of Borcherds on "lifting" weight zero automorphic forms to infinite products.  

By geometric engineering, I am referring to the work of Nekrasov which identifies the instanton partition function in some supersymmetric theory with the topological string partition function on a threefold.  Noting that the infinite product in the DMVV formula looks perhaps like a Donaldson-Thomas, or topological string partition function, basically my question is the following.  Does every instance of geometric engineering involve the second quantization of some weight zero object?  Perhaps some object encoding BPS numbers?  And visa versa, does every second quantization of a SCFT partition function give rise to some topological string partition function via geometric engineering?  

From a purely mathematical point of view, I have an example which involves both and I'm wondering if this is interesting, or totally expected/accidental etc from a physics perspective.  Sparing the details, I have an equivariant elliptic genus whose second quantization (more precisely, the Borcherds lift as it has an extra prefactor) gives rise to Donaldson-Thomas/topological string partition function of a Calabi-Yau threefold.  The equivariant parameter $t$ in the elliptic genus is precisely the DT/GW parameter on the string theory side $t = e^{i \lambda}$.  This seems very much aligned with Nekrasov's geometric engineering picture.  

asked Jul 25, 2018 in Theoretical Physics by Benighted (310 points) [ no revision ]

It would be nice if you provided some references for the claims above (the formulae etc).

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).
To avoid this verification in future, please log in or register.

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights