3.2 Basics of Space

summary

What can we say about space itself?

We have set the stage for what the basics of particles are. But what about where these membranes or strings live, the space itself? What have we learned in the context of string theory?
One of the basic things about space is that you can take a measuring device like a ruler and measure it. So, is the notion of distance a well-defined concept?
In Einstein’s theory, it is. You can measure the distance from one object to another. For example, if you have a circle and you want to measure its circumference, you can send a light around it and measure the time it takes to make a full circle.
But in string theory, we have learned that length is not a duality invariant concept.
Imagine two corresponding circles, one of radius R and the other of radius 1/R. When the first is big, the second is small, and vice versa. In string theory, there is a duality that maps the the theory of the first circle to that of the second circle. And so sending one kind of light around one circle (a winding state) will result in the same measurement as a different light (momentum state) on the second circle. The notion of distance is not an invariant concept.

Is the topology of space an invariant concept?

Topology of space is a mathematical invariant of space, like how many holes or handles a space has. You can imagine a region of space shaped like a donut. This is called a torus. It has one hole, no matter how the region is distorted.
So, one would assume that the concept of topology should be a duality invariant concept. But it turns out this is not the case. The same physical system in different frames can correspond to strings propagating on spaces with different topologies.
Topology is not a duality frame invariant concept.

What do we know about the dimension of space?

First of all, in the context of string theory, we have learned that space and time cannot just be 3+1 dimensions. There are extra dimensions, very small and curled up.
Imagine looking far away at a telephone wire. From your perspective this wire is one-dimensional. But to an ant crawling on the wire, there are three dimensions in which to travel. So in string theory we have learned that the dimension of spacetime must be greater than 3+1.
Additionally, we have learned from string theory that the dimension itself is not invariant. By changing a parameter known as the coupling constant, an extra dimension in a ten-dimensional space can emerge, and the space appears eleven-dimensional.
Distance, topology, and dimension are all no longer invariant concepts. We have to radically change the way we think about space and time.