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

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

176 submissions , 139 unreviewed
4,330 questions , 1,659 unanswered
5,092 answers , 21,632 comments
1,470 users with positive rep
641 active unimported users
More ...

  Gauge theory on schemes

+ 6 like - 0 dislike
51 views

Gauge theories are traditionally formulated in space-time, superspace, manifolds, or supermanifolds.

Are there formulations of gauge theory in the context of algebraic geometry (e.g. defining fields on affine varieties, schemes, or stacks)?

This post imported from StackExchange MathOverflow at 2019-02-02 19:47 (UTC), posted by SE-user Untitled
asked Jan 22 in Theoretical Physics by Untitled (30 points) [ no revision ]
retagged Feb 2
For instance, one particularly algebraic approach for manifolds (which could be promptly applied to schemes) is given in [Section 2.3, DEF$^+$99], where a Lagrangian on a manifold is defined as an element of a particular double chain complex (which is a subcomplex of a complex that is very similar to the de Rham complex of $\mathcal{F}\times M$, with $\mathcal{F}$ the space of fields (see there) on $M$).

This post imported from StackExchange MathOverflow at 2019-02-02 19:47 (UTC), posted by SE-user Untitled
[DEF$^+$99] Pierre Deligne, Pavel Etingof, Daniel S. Freed, Lisa C. Jeffrey, David Kazhdan, John W. Morgan, David R. Morrison, and Edward Witten, editors. Quantum fields and strings: a course for mathematicians. Vol. 1, 2. American Mathematical Society, Providence, RI; Institute for Advanced Study (IAS), Princeton, NJ, 1999. Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997.

This post imported from StackExchange MathOverflow at 2019-02-02 19:47 (UTC), posted by SE-user Untitled
This is a tangent but why gauge theory on schemes?

This post imported from StackExchange MathOverflow at 2019-02-02 19:47 (UTC), posted by SE-user AlexArvanitakis
@AlexArvanitakis I don't have a particular objective in mind, but I think it might lead to interesting algebraic results, as gauge theory on manifolds has lead to (for example) Donaldson invariants. Also, it might be a natural way to study supersymmetry: in an affine scheme, nilpotents are elements of the underlying ring (also seem as functions on the scheme), whereas the sheaf of superspace $\mathbb{R}^{p|q}$ is the freely generated sheaf $\mathscr{C}^\infty[\theta^1,\dots,\theta^q]$ by nilpotent...

This post imported from StackExchange MathOverflow at 2019-02-02 19:47 (UTC), posted by SE-user Untitled
...generators $\theta^1,\dots,\theta^q$ of the sheaf $\mathscr{C}^\infty$ of smooth real-valued functions on ordinary space. That is, schemes include nilpotents by themselves, while superspace adds them a bit artificially.

This post imported from StackExchange MathOverflow at 2019-02-02 19:47 (UTC), posted by SE-user Untitled
fair enough. I was really looking for context that would jog my memory... You could look into papers by Eric Sharpe e.g. the notes arxiv.org/pdf/hep-th/0307245.pdf

This post imported from StackExchange MathOverflow at 2019-02-02 19:47 (UTC), posted by SE-user AlexArvanitakis
@AlexArvanitakis These notes (and his papers) are great! Thank you.

This post imported from StackExchange MathOverflow at 2019-02-02 19:47 (UTC), posted by SE-user Untitled

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$\hbar$ysicsOver$\varnothing$low
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
...