8 things you need to know about string theory and higher dimensions with Spiro Karigiannis

Friday, August 27, 2021
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Spiro Karigiannis
Mathematicians and theoretical physicists explore multiple dimensions that go beyond the ones we understand with our common sense.

We can understand the three spatial dimensions as part of our direct lived experience. We can perhaps also understand time as a temporal dimension with some effort, bringing the number of dimensions up to four.

But researchers working in string theory and higher dimensions have shown that our universe might consist of many more dimensions than we can quickly wrap our heads around. Perhaps up to ten or eleven, according to some theories.

Spiro Karigiannis is a professor of pure mathematics at the University of Waterloo and an affiliate member of the Perimeter Institute for Theoretical Physics. He dedicates time and thought to questions of those higher dimensions beyond what we can easily perceive. He’s here to answer a few questions on the eight things you need to know about string theory and higher dimensions in math.

  1. You’re what’s called a geometer in math. What is a geometer, and how does your branch of mathematics relate to higher dimensions?

Pure mathematicians start with ideas and concepts that we can relate to as human beings and then abstract them as much as possible to see how general these concepts really are. The word “geometry” itself is Greek for “measurement of the Earth.” A geometer studies spaces called manifolds that are equipped with a way of measuring distances and angles. These are abstract spaces that can have any “dimension,” which is just the number of independent degrees of freedom needed to describe the space.

  1. Your specific research is on manifolds. So, what are manifolds?

Imagine an old-school thermometer or metre stick. There’s a point which we designate as the origin, and then exactly one real number tells you where you are on the space, positive for one side and negative for the other side, so these are 1-dimensional spaces. A piece of paper is 2-dimensional because there are *two* independent directions you can move from each point: the up/down direction and the left/right direction. Every other direction can be formed by taking a combination of these, so one needs two real numbers to describe where you are on this space.

An N-dimensional manifold is a space that “locally” looks like ordinary N-dimensional space, but “globally” might be quite different. For example, the surface of the Earth is 2-dimensional, because we can describe where we are by two numbers, such as latitude and longitude, but this 2-dimensional manifold is clearly different from a piece of paper. This difference is quantified by geometers as something called curvature.

  1. How many dimensions are there? And how can we understand dimensions we cannot see or experience?

The question of how many physical dimensions there are in the universe in which we exist is important. But one can study manifolds of any dimension independent of any physical interpretation. Whenever you have a set of data that depends in a smooth manner on N independent degrees of freedom, you have an N-dimensional manifold. For example, a meteorologist might want to study certain aspects of the weather. At a given point on Earth (latitude, longitude, height above sea level), they might record the temperature, wind velocity (three numbers), air pressure, humidity, etc. Just with these pieces of data, one already has a 9-dimensional manifold.

  1. What is string theory, and how does it relate to mathematics?

String theory postulates that the fundamental building blocks of the universe are not “point particles” but rather small “loops” or “strings,” which are 1-dimensional manifolds, moving around and interacting with each other in space and time. For technical reasons, for string theory to imply the currently known and experimentally verified laws of physics, including the standard model of quantum mechanics and Einstein’s theory of general relativity, our “spacetime” manifold must include either 6 or 7 additional “curled up” dimensions that we cannot see. For example, a piece of dental floss might look 1-dimensional, but if you zoom in with a microscope, it might look more like a garden hose, which is 2-dimensional, because you can move along the axis of the hose or around its circumference. String theory is entirely formulated in the language of manifolds, so it could not even have been postulated before the development of modern geometry.

  1. Parts of string theory have been called into question by physicists. What does that mean for the direction of research on higher dimensions?

It’s not so much that it’s been called into question, rather than the fact that string theory describes the universe at such large energy scales that, in practice, it is essentially impossible to verify it with our current technology and experimental methods. But as I said before, that has no effect on the validity of the pure mathematics of higher dimensional spaces. Mathematicians use pure logic to prove theorems starting from basic axioms, and the truth of these theorems is eternal and universal. The theorems could then be used to build physical models of various phenomena in science or engineering. For example, higher-dimensional geometry can be used to model robotic motion and many other complicated situations. It’s certainly not just for string theory.

  1. How does the work of mathematicians inform the work of physicists? And how does physics inform mathematics?

Until the 19th century, one did not distinguish between mathematicians and physicists. You’ll hear names like Newton, Laplace, Lagrange, Hamilton, and Euler in both mathematics and physics courses. Advances in mathematics were motivated by physical considerations, and in turn, these advances led to a better understanding of physics. As researchers began to get more specialized, the two disciplines underwent an (amicable) divorce. However, since the second half of the 20th century, there has been a marvelous and extremely fruitful reconciliation, which has led to an explosion of progress in geometry. I’ll just discuss one such result: combining ideas from physicists, in the early 1980’s, Simon Donaldson (Fields medal 1986) proved that while there is only one N-dimensional “flat space” for any N other than 4, there are actually *infinitely many* different 4-dimensional flat spaces! No one saw this coming, and it wouldn’t have been discovered without inspiration from the Yang-Mills theory in physics.

  1. What is something about higher dimensions that is unknown but that you really want to know the answer to? Does anything about the unknown fascinate/frighten you?

The first iteration of string theory (from the 1980s and 1990s) needs six extra dimensions, which form a “Calabi-Yau manifold.” This is a special space whose curvature is related to complex numbers. There is a theorem, due to Shing-Tung Yau (Fields medal 1982, and my thesis supervisor), which completely characterizes which 6-dimensional manifolds are Calabi-Yau spaces. The new iteration of string theory (called “M-theory”) needs seven extra dimensions, which form a “G2-manifold.” Such a space has curvature related to an exotic number system called the “octonions,” in which it both matters what order we multiply *and* where we put our parentheses. (It is “non-commutative” and “non-associative”). We still don’t have a theorem that characterizes which 7-dimensional spaces are G2-spaces, and it’s probably still at least 20 years in the future. It doesn’t frighten me, but it certainly often keeps me up at night thinking about it.

  1. What’s the next big thing in research on higher dimensions?

The physicists studying M-theory actually need G2-manifolds that are not quite smooth, but which have sharp corners, or “singularities” of a very specific type. Apparently, if these singularities weren’t there, then matter would not exist, and that sounds pretty bad. While mathematicians have proved the existence of hundreds of thousands of distinct G2 spaces, these are all smooth. There is ample evidence (both physical and mathematical, some of it due to my work) to suggest that G2-spaces with the right kinds of corners needed for physics do indeed exist, but this has yet to be proved rigorously. There are several groups, including one of which I am a part, that are trying to do this right now. Luckily, these groups are using different ideas, so we could conceivably all succeed, or all fail!

By the Numbers is a weekly series that reflects on the lighter side of student life, research and innovation in the Faculty of Mathematics at the University of Waterloo. Stay tuned to this space for the next installment.