Questions on Yuji Tachikawa's talk Recent advances in Supersymmetry

+ 2 like - 0 dislike
1275 views

I've been watching Tachikawa's talk.

around 38 minutes he talks about objects in $H^3(M,\mathbb{Z}_N)$ with the property that

$\int_C a \in \mathbb{Z}_N$

He says due to self duality in 6D dimensions two intersecting cycles C and C' with $C \cap C' \neq 0$ are mutually non local.

It seems that mutually non-local means $\int_M a \wedge b \neq 0$

But why is this true, what happens when a theory is self dual?

What happens when you add supersymmetry?

He talks about the self dual 3 form what about other fields in the theory?

Is there a fermionic(or supersymmetric) generalization to $H^3(M,\mathbb{Z}_N)$?

What is the recipe to construct the partition vector?

Is there any detailed exposition of the subject ? What books discuss the object $H^3(M,\mathbb{Z}_N)$? I am familiar with the basic ideas of co-homology for U(1) gauge groups, but I have never really studied its generalization to other gauge groups.

Please excuse me if my questions are naive or too simple.

Edit: I have attached the related slides

edited May 19, 2015

What does mutually non-local mean?

@RyanThorngren From what I understood form the talk mutually non local means

$\int a\wedge b \neq 0$

While I did not understand exactly what he meant I will tell you the argument he makes.

He says one wants to fix $\int_C a \in H^3(M,\mathbb{Z})$ so that $\int_C a\in \mathbb{Z}_N$ is the magnetic flux through C.

However he goes on to say one cannot do that because of the self duality of 3 form tensor when you have 2 intersecting cycles C and C' the fluxes will be mutually non local.

Instead He says we have to split $H^2(M,\mathbb{Z}) = A \oplus B$ where A and B are of equal sizes.

Such that the fluxes in A are mutually local to each other and fluxes in b are mutually local to each other.

i.e
$\int a\wedge a' = 0$ and $\int b\wedge b' = 0$

+ 2 like - 0 dislike

This is something that happens for the 2d CFT on the boundary of 3d Chern-Simons theory. The Chern-Simons gauge field is its own canonical conjugate (a self-duality), so you can only specify some of the holonomies (like how the wavefunction depends on position, say, but not momentum). This is why the Hilbert space grows like $K^g$ rather than $K^{2g}$ on a genus $g$ surface.

There is a 7d TQFT that carries the 6d CFT conformal blocks in the same way.

answered May 18, 2015 by (1,925 points)

@RyanThorngren

Interesting I did not there is a 7d TQFT related to 6d theory.

Can you give me a reference that discusses the issue in 3d Chern Simons.

Ed Witten discusses the 7d TQFT here: http://arxiv.org/abs/hep-th/9610234 . There are many references for the 3d story, but Ed Witten also discusses it in quantum field theory and the Jones polynomial".

 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): Email me at this address if my answer is selected or commented on: 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$\varnothing$rflowThen 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