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

Please welcome our new moderators!

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

122 submissions , 103 unreviewed
3,497 questions , 1,172 unanswered
4,543 answers , 19,337 comments
1,470 users with positive rep
407 active unimported users
More ...

Where do theta functions and canonical Green functions appear in physics

+ 7 like - 0 dislike
31 views

In the beginning of Section 5 in his article, Wentworth mentions a result of Bost and proves it using the spin-1 bosonization formula. This result provides a link between theta functions, canonical Green functions, and Faltings' delta invariant on a compact Riemann surface of positive genus.

Are there any other situations in physics which involve canonical Green functions on compact Riemann surfaces?

This post has been migrated from (A51.SE)
asked Oct 5, 2011 in Theoretical Physics by JaP (60 points) [ no revision ]

5 Answers

+ 7 like - 0 dislike

Theta functions appear when determining various partition functions, e.g. the translational PF of an ideal gas

This post has been migrated from (A51.SE)
answered Oct 5, 2011 by S Huntsman (405 points) [ no revision ]
+ 6 like - 0 dislike

A rich source of applications can be found in string theory. Most straightforwardly, the partition function, or vacuum amplitude, of a conformal field theory on a Riemann surface are essentially given by theta functions (and their generalizations, e.g. Siegel modular forms). Various theta function identites have important physical consequences.

The simplest case to look at would be the torus, discussed at length for example in Polchinski's textbooks - the four spin structures on the torus lead to the four Jacobi theta functions, and the Green's function can also be expressed in terms of theta functions as a consequence. (By the way, bosonization also appears in this context as the worldsheet of a string is 1+1 dimensional (see e.g. Green-Schwarz-Witten's books)).

On the other hand, a very active recent topic would be the occurrence of mock modular forms and the "wall-crossing" phenomenon.

This post has been migrated from (A51.SE)
answered Oct 6, 2011 by Dan (170 points) [ no revision ]
+ 4 like - 0 dislike

Theta functions also appear in the treatment of the quantum Hall effect on the torus, see F.D.M Haldane and E.H.Rezayi, Periodic Laughlin-Jastrow wave functions for the fractional quantized Hall effect, Physical Review B, Vol 31, Number 4, 1985 .

This post has been migrated from (A51.SE)
answered Oct 5, 2011 by jjcale (70 points) [ no revision ]
+ 4 like - 0 dislike

The Greens function for a charged particle moving on a circle which encloses a nonzero magnetic flux is given by the Jacobi theta function. The relation between the expression one obtains using canonical or path integral methods is given by the $\tau \rightarrow -1/\tau$ modular transformation of the theta function.

This post has been migrated from (A51.SE)
answered Oct 7, 2011 by anonymous [ no revision ]
+ 2 like - 0 dislike

I think an example of the kind of result you're looking for might be found in the paper On the Thomae Formula for ZN Curves. Although it proves a result purely about theta functions the proof is derived from work investigating Green's functions arising from conformal field theories on Riemann surfaces with $Z_N$ symmetry.

This post has been migrated from (A51.SE)
answered Oct 6, 2011 by user207442 (20 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:
p$\varnothing$ysicsOverflow
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
...