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 ...

  L-series and Modular Forms in String Theory?

+ 2 like - 0 dislike
2116 views

On the border of algebraic geometry and number theory, there is what is known as the "modularity conjecture" for certain algebraic varieties defined over $\mathbb{Q}$.  Roughly speaking, this is the conjecture that there is a modular object whose Dirichlet series coincides with the L-series of the variety.  

A little more specifically, if $X$ is a Calabi-Yau threefold defined over $\mathbb{Q}$ with $h^{2,1}(X)=0$, then one can form its L-series:

$$L(X, s) = \prod_{p-\text{prime}} \big(1-t_{3}(p)p^{-s} + p^{3-2s}\big)^{-1}.$$

(I'm lying slightly...the product excludes finitely many "bad" primes).  Here, $t_{3}(p)$ are integers closely related to the number of points over the finite field $\mathbb{F}_{p}$ in $X$.  The modularity conjecture is that $t_{3}(p)=a_{p}$ where $a_{p}$ are the Fourier coefficients of a weight 4 modular cusp form for congruence subgroup $\Gamma_{0}(N)$, for some $N$.  For a nice summary, one can see (https://projecteuclid.org/download/pdf_1/euclid.kjm/1250517640).

I know in many areas in enumerative geometry/string theory various partition functions have non-trivial automorphic properties.  Quite simply, I'm curious if there's any well-known or conjectural relationship of these L-series in string theory?  Particularly, perhaps topological string theory on $X$ or something closely related?  

asked Nov 17, 2018 in Theoretical Physics by Benighted (310 points) [ no revision ]

I suggest searching the work of Noriko Yui, she is a guru of modularity in string theory.

@Mitchell Porter Yes!  I've been reading a lot of her work.  But can I ask you where she mentions applications to string theory?  Any sources, slides, or talks that you know of by chance?  Applications of her work on modularity to string theory is exactly what I'm looking for.  

Actually you have a point, she studies Calabi-Yaus more than string theory per se. And from Rolf Schimmrigk e.g. https://arxiv.org/abs/hep-th/0603234 I have a glimpse of the barrier that must be crossed, you need to find a relationship between the properties of the CY, and the worldsheet description. See the paragraph beginning "In general the L-function of a variety..." Schimmrigk has a number of interesting papers on this, up to the start of 2013.

@Mitchell Porter Thanks a lot, indeed his papers look very interesting, and will take me quite a while to digest.  In that paragraph you mention, can I ask if you happen to know what he means by "string theoretic modular form"?  

I don't know how broad a meaning he intends, but https://arxiv.org/abs/hep-th/0211284 describes a specific kind of modular form obtained from the string worldsheet CFT, which is the subject of subsequent papers.

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$\hbar$ysicsOv$\varnothing$rflow
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
...