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

146 submissions , 123 unreviewed
3,953 questions , 1,403 unanswered
4,889 answers , 20,762 comments
1,470 users with positive rep
507 active unimported users
More ...

Question on $E_8$ and twistor space

+ 3 like - 0 dislike
55 views

The Kahler $4$ form constructed from two-forms $\{\alpha, \beta\} \in H^2(M,\mathbb Z)$, and $M$ a $4$-manifold, is induced by $\alpha\wedge\beta$ with the map $H^2(M, \mathbb Z)\otimes H^2(M, \mathbb Z) \rightarrow$ $H^4(M, \mathbb Z)$. This defines the topological charge $$ 8\pi k = \langle\omega(\alpha\cup\beta)\rangle = \int\omega. $$ This is geometrically the number of ways these two forms intersect, and the four-form $\omega$ is then an intersection form.

Milnor [On Simply Connected 4-manifolds, Symp. Int. Top. Alg., Mexico (1958) 122-128] demonstrated that for $$ M = \{z_0, z_1, z_2, z_4 \in \mathbb CP^3: z_0^4 + z_1^4 + z_2^4 + z_4^4 = 0\}, $$ the Kummer surface, that this intersection form is given by $$ [E_8]\oplus[E_8]\oplus3\left(\begin{array}{c,c} 0 & 1 \\ 1 & 0\end{array}\right). $$ Here $[E_8]$ is the Cartan center matrix for the exceptional $E_8$ group.

This leads to a couple of questions or observations. One of them is that $\mathbb CP^3$ is projective twistor space $\mathbb P\mathbb T^+$. Twistor projective space is $$ \mathbb P\mathbb T^+ = SU(2,2)/SU(2,1)\times U(1) \simeq SO(4,2)/SO(4,1)\times SO(2). $$ This is a Hermitian symmetric space. The question is then whether the global symmetries given by the Kahler form are a set of global symmetries reduced from the local symmetries of $E_8\times E_8 \sim$ $SO(32)$. This could also be carried to Witten's supertwistor space $\mathbb C\mathbb T^{3|4}$ as well.

String theory requires a background for gravitation. String theory then demands there be some global spacetime symmetry, say a set of global symmetries at the conformal $i^0$. The rest of string theory involves entirely local symmetries. Along the lines of this question, is this relationship between global and local symmetries (assuming my hypothesis here is correct) the reason string theory requires background dependency, such as type II strings on $AdS_5$. The twistor space here has a relationship with the anti-de Sitter spacetime, being quotient groups of $SO(4,2)$ with different divisors.

This post imported from StackExchange Physics at 2016-10-02 10:56 (UTC), posted by SE-user Lawrence B. Crowell
asked May 24, 2016 in Theoretical Physics by Lawrence B. Crowell (590 points) [ no revision ]
retagged Oct 2, 2016
Your introductory paragraph does not make sense to me. A Kähler form is conventially the symplectic form of a Kähler manifold, there is no such thing as a "Kähler form of two 2-forms". The equality $H^4(M,\mathbb{Z}) = H^2(M,\mathbb{Z})\otimes H^2(M,\mathbb{Z})$ is just wrong for a general 4-manifold, just take $S^4$ where the l.h.s. is $\mathbb{Z}$ and the r.h.s. $0$ to see this. What is $\alpha\cup\beta$ and where does $\omega$ suddenly come from? Is it supposed to be a fundamental class? The "intersection form" is the map $(\alpha,\beta)\mapsto\int_M \alpha\wedge\beta$, not some diff-form

This post imported from StackExchange Physics at 2016-10-02 10:56 (UTC), posted by SE-user ACuriousMind
Also, why are there "global symmetries given by the Kähler form"? What does that mean? What is the expression $E_8\times E_8\sim \mathrm{SO}(32)$ supposed to denote? The two groups have the same dimension, but are not the same in any way!

This post imported from StackExchange Physics at 2016-10-02 10:56 (UTC), posted by SE-user ACuriousMind
I changed the wording, that was admittedly awkward. The equality is supposed to be a map induced by the joining or union. The heterotic string theory quite frequently makes use of the similar Dynkin diagrams and the same number of roots between $E_8\times E_8$ and $SO(32)$. This correspondence is to my knowledge not fully understood.

This post imported from StackExchange Physics at 2016-10-02 10:56 (UTC), posted by SE-user Lawrence B. Crowell
Lawrence B. Crowell, John Rennie and @Danu: Actually, moderators can no longer merge accounts. The user should instead go here and follow the instructions. If all that fails, the user should contact the SE team.

This post imported from StackExchange Physics at 2016-10-02 10:56 (UTC), posted by SE-user Qmechanic

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
...