Connections in Gauge Theory

SLMath - November 2023
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A two-dimensional slice of a Calabi–Yau three-manifold. (public domain image)

In the 1980s, physicists who were using mathematical objects known as Calabi–Yau manifolds to create models of string theory noticed that sometimes two very different Calabi–Yau manifolds actually encoded the same physical system. The idea, dubbed mirror symmetry, has guided strands of research in both physics and mathematics in the decades since, and the basic principle of mirror symmetry, the investigation of links between seemingly-unrelated mathematical objects, has proved fruitful in fields far beyond the physics context that originally inspired it.

Jason Lotay, a mathematics professor at the University of Oxford who served as Chancellor’s Professor at the Simons Laufer Mathematical Sciences Institute (SLMath) program on Analytic and Geometric Aspects of Gauge Theory during the fall 2022 semester, is developing new research inspired by mirror symmetry in a field of mathematics called differential geometry. He is working on the relationships between calibrated submanifolds and instantons.

A calibrated submanifold is a type of minimal submanifold. Examples of minimal surfaces, the most famous minimal submanifolds, can be found by minimizing the area among surfaces with a given boundary, and they can be seen in real life by dipping a wire into a soapy solution. The soap film will naturally form a minimal surface that clings to its boundary, the wire. Minimal submanifolds of higher dimensions are similar, and calibrated submanifolds are examples which minimize the higher-dimensional version of volume.

Instantons, on the other hand, are a type of mathematical object called a connection which lives on the larger manifold within which the calibrated submanifold exists. Logically enough, a connection is a way to connect any two points on the manifold; an instanton is an example of a connection that minimizes the energy among similar connections. “Are the instantons, which are connections that minimize the energy, related to the calibrated submanifolds, which are minimal submanifolds that minimize volume?” Lotay asks. “The answer is yes, they are.”

The (pardon the word) connection between calibrated submanifolds and instantons has been known for a while. “I’m trying to build a theory where the submanifolds and the connections are linked from the start,” Lotay says. Rather than simply repackaging information that is already known, this new theory could use pairs of calibrated submanifolds and instantons to create new invariants — quantities associated to manifolds that help researchers determine when two manifolds are the same. “That would be the dream,” Lotay says.

Lotay has been hoping to develop a solid theory connecting calibrated submanifolds and instantons for years, but the SLMath semester program allowed him to make the first concrete progress towards the goal. “It’s something I’ve been thinking about off and on for a while, but this program really kickstarted my enthusiasm to make it work,” he says. Not only was he able to draw from the expertise of the other researchers, the length of the program meant he had time to dig into the new work patiently, methodically and in depth. “You can’t just say, ‘Oh, this week, be smart and come up with a good idea.’ It doesn’t really work like that,” he says. “You have to have some time for the ideas to percolate and bounce around a bit.” Spending the whole semester at SLMath allowed that to happen.

Lotay’s work is still very much in progress. He is currently shoring up the new ideas and testing them out to see whether the invariants they produce will be practical or useful. “What came out of the program was a much better understanding of what is and isn’t known, what people want to know, and how I can contribute something different to the field,” he says.