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September 27, 2005

Literature on Exotic R4s

Posted by Urs Schreiber

Today I received the following question by Florent Dieterlen on exotic ℝ 4\mathbb{R}^4 spaces. Since I am no expert on exotic spaces and cannot readily answer this question I reproduce it here in public (with kind permission). If anyone feels like providing help, please do so.

Hello,

I saw your discussion on a forum about exotic ℝ 4\mathbb{R}^4s. I was brought up as a physicist, but did not practice since, except for dynamical systems. I want to study exotic ℝ 4\mathbb{R}^4s, to be able to construct some, following certain conditions from an application. It is not properly a physics application. The problem is that i don’t have the basics. So my question is: starting from level MSc in physics, how do i get the most efficiently to the level i want:

1) do i have to follow Seiberg-Witten instead of Donaldson? The basics are not completely the same.

2) do you counsel Nash and Sen instead of Nakahara for the basics, if i want something very intuitive?

3) If 1) is yes, what book do you counsel for the study of exotic ℝ 4\mathbb{R}^4s following Seiberg-Witten?

Thanks in advance for your answer,
Best regards,
Florent Dieterlen

Posted at 6:03 PM UTC | Permalink | Followups (3)

September 23, 2005

Center for MathPhys Opening Colloquium in Hamburg

Posted by Urs Schreiber

I’ll move to Hamburg next week, where I start a postdoc position in Prof. Schweigert’s group.

There is a fairly new Center for Mathematical Physics which

[…] is a joint venture of DESY, the physics and the mathematics department of Hamburg University. It has been founded in December 2004. Its aim is to foster the activities in mathematical physics in Hamburg. These activities have a long standing tradition both at DESY and at the University. The main focus of its activity are mathematical aspects of string theory and quantum field theory.

The center’s opening colloquium takes place on October 20th-22nd. The registration form is here.

Invited speakers are

* John Cardy
“SLE - a new way of thinking about Conformal Field Theory”

* Philippe DiFrancesco
“Combinatorics and Physics: The Miracles of Integrability”

* Robbert Dijkgraaf

* Nigel Hitchin
“Generalized Geometries”

* Roberto Longo
“The Structure and Classification of local conformal Sets”

* Nikita Nekrasov
“Chiral Algebras, pure Spinors and Superstrings”

* Tudor S. Ratiu

* Nicolai Reshetikhin

* Matthias Staudacher
“Integrability and the AdS/CFT Correspondece”

* Robert M. Wald

* Alexander Zamolodchikov

* Martin Zirnbauer
“From Random Matrices to Supermanifolds”

Should be fun!

Posted at 6:41 PM UTC | Permalink | Trackbacks (3)

A. Gustavsson on Surface Holonomy

Posted by Urs Schreiber

A while ago there appeared a new paper on surface holonomy, which I had missed at that time, being busy with other things, and which I came across by chance yesterday:

Andreas Gustavsson
A reparameterization invariant surface ordering
hep-th/0508243 .

Here is a summary and some discussion.

Update 28 Sept. 2005: As a reaction to the following entry the paper has been replaced by a revised version. For instance the metric factor discussed below no longer appears there.

Continue reading “A. Gustavsson on Surface Holonomy” ...
Posted at 11:27 AM UTC | Permalink | Followups (2)

September 21, 2005

What does the density of vacua predict?

Posted by Robert H.

Imagine a room where the floor has the profile

(1)z(x,y)=sin(π/x). z(x,y) = \sin(\pi/x).

Now, you throw a ball into that room. Where will it end up?

Obviously, the ball will settle into one of the minima of the floor profile, so 1/x1/x will be an odd integer. We can consider the density of these minima (a.k.a. vacua) and this diverges for x→0x\to 0. So it is most likely that the ball will end up at small xx.

You will have noticed that this reasoning is flawed: At the same rate as the vacua get denser, the basin of attraction of each vacuum gets smaller, it is bounded by 1/x1/x being an even integer. So if we do not have any other a priori knowledge of the distribution P(x)P(x), we should conclude that on the average, small xx is as likely as large xx.

I would expect a similar reasoning also to apply to the vacua of string theory.

My understanding of the philosophy of all this anthropic reasoning is that either by quantum probablity (given in terms of some Hartle-Hawking type wave function) the universe pops into being or via some eternal inflation type process there happens to be a region of the universe that expands exponetially. In both cases, the moduli are randomly distributed and from that point on you make statistical predictions.

(Alternatively, we could live in a world described by a Theory of Really Everything that only has one single state. And that state describes all phenomena including the layout of the blog. As explained here I do consider this unlikely.)

So, our fundamental theory is going to have many states. A small fraction of them will be ground states (still, this might be a tricky concept in a theory containing gravity) where the potential energy has a local minimum. So why assume that the random process sends us to one of these? If it doesn’t excess energy will be radiated off and the expansion of the universe also acts like a dissipating system with the Hubble constant giving the amount of friction. So eventually, the universe will settle into (or close to) one of the ground states.

But as the example of the ball above shows, they are not equally likely. In general, a greater density will be counter balanced by smaller regions in parameter space that will settle to that ground state. As long as one does not have a precise understanding of the detailed dynamics of the settling down process, I would assume that by far the most important factor in the determination of the probability distribution of the physical parameters (moduli) is the a priori distribution of states (which is of course unknown) and not the distribution of vacua.

Posted at 2:03 PM UTC | Permalink | Followups (11)

September 2, 2005

Gerbes, Algebroids and Groupoid Bundles, the emerging Picture

Posted by Urs Schreiber

Update (09/14/05): There is a disagreement about the secrecy status of the former content of this entry.

Posted at 8:14 AM UTC | Permalink | Followups (18)