Quantcast
Channel: Conifolds and Exotic Spheres - Mathematics Stack Exchange
Viewing all articles
Browse latest Browse all 2

Conifolds and Exotic Spheres

0
0

First of all, a disclaimer: I'm a physicist trying to understand mathematical aspects of some solutions I've encountered while studying string theory, and am certainly not a mathematician of any sort. In particular I have never taken a class on differential geometry or topology, hence my following question.

I'm reading an older paper in mathematical physics, and I'm trying to understand it. The paper can be found here: http://www.sciencedirect.com/science/article/pii/055032139090577Z

The authors are constructing 5 dimensional Einstein spaces, which they call $\mathcal{N}_{pq}$. I've also seen them denoted $T^{pq}$. I apologize that I don't have a name for them. The spaces are

$ds^2_{pq} = \lambda^2 \left(d\psi+p\cos\theta_1 d\phi_1 + q \cos\theta_2 d\phi_2 \right)^2 + \Lambda_1^{-1} \left(d\theta_1^2 + \sin^2\theta_1 d\phi_1^2 \right) + \Lambda_{2}^{-1} \left(d\theta_2^2 + \sin^2\theta_2 d\phi_2^2 \right). $

Here the lambda constants are fixed to make the space Einstein, $R_{ab} = 4 g_{ab}$:

$4 = \frac{1}{2}\lambda^2 \left[ (p\Lambda_1)^2 + (q \Lambda_2)^2 \right] = \Lambda_1 - \frac{1}{2} \left(\lambda p \Lambda_1\right)^2 = \Lambda_2 - \frac{1}{2} \left(\lambda q \Lambda_2\right)^2. $

For two particular choices of $(p,q)$ these spaces are furthermore fibre bundles over $S^2 \times S^3$. These are $(p,q) = (1,1)$ and $(1,0)$.

I'm confused about the relationship between these two spaces. The authors show that the two spaces are indeed distinct geometries because they have different volumes ($\text{vol}_{11}/\text{vol}_{10} = 2^{13/2} 3^{-5}$). The authors also say that these two spaces are diffeomorphic but that no coordinate transformation exists which brings one space into the other (and they use the unequal volumes to support this conclusion).

I'm confused by this. What does it mean for two spaces to be diffeomorphic and yet "distinct"? I recently learned about exotic spheres (https://en.wikipedia.org/wiki/Exotic_sphere). These are spheres that are homeomorphic but not diffeomorphic to the standard sphere (defined by the embedding $\sum_{i=1}^n x_i^2 = 1$ in $\mathbb{R}^n$). I'm wondering if perhaps the authors used out-dated terminology and the modern and more precise statement would be that $\mathcal{N}_{11}$ and $\mathcal{N}_{10}$ are homeomorphic but not diffeomorphic (just like the connection between exotic spheres and the standard sphere). Is this correct?


Viewing all articles
Browse latest Browse all 2

Latest Images

Trending Articles





Latest Images