# New Perspectives from the Intersection of Mathematics and Physics

Yuto Moriwaki

(Special Postdoctoral Researcher, iTHEMS)

When people think of research in quantum field theory, most imagine physicists. However, Yuto Moriwaki is a mathematician. In this interview, we asked Moriwaki about the allure and challenges of working across the fields of mathematics and physics.

Keywords: Functorial Quantum Field Theory, Vertex Operator Algebra, Operad, Moduli Space, Functional Analysis

Affiliation and position are as of the interview date: April 2024

(Written by Naoko Shinozaki ｰ Freelance Announcer / Mathematics Communicator)

## The Intersection of Analysis and Geometry: Mathematical Research in Quantum Field Theory

Quantum field theory is a physical theory that describes a wide range of phenomena, from the world of elementary particles to the scale of the universe. Among these, quantum electrodynamics is an astonishingly precise theory where the agreement between theory and experiment extends beyond ten decimal places. Yet, from a mathematical perspective, many mysteries remain. In quantum field theory, when one attempts to compute physical quantities in a straightforward manner, various values often tend to infinity. However, since the quantities observed in reality are finite, physicists employ techniques to cleverly replace these infinities with finite values in their calculations. Moriwaki’s research focuses on providing a mathematical justification for these procedures and constructing a consistent framework.

In physics, quantities like momentum, pressure, and density are obtained through experiments and observations. These quantities are expressed as functions. One of the objectives of physical theory is to theoretically discover and derive these functions. In quantum field theory, the domains of these functions are the configurations of elementary particles within the universe. Mathematically, this can be interpreted as “functions on geometrical configurations (moduli spaces).”

Physical quantities should maintain consistency with operations like deforming or gluing geometry together, and it is now believed that this consistency provides the mathematical definition of quantum field theory. This approach is called "functorial quantum field theory," but many parts are still incomplete. Moriwaki feels that to complete this theory, it is necessary to re-examine what class of functions should be considered.

As dimensions increase, geometric configurations become more complicated, so Moriwaki is mainly working on theories in two-dimensional spacetime. In two dimensions, special phenomena occur, and certain quantum field theories can be described using holomorphic functions. While these theories are not simple, they are within the scope of modern mathematics.

However, the theories that can be described using holomorphic functions are limited, and two-dimensional quantum field theories are not yet fully understood. Moriwaki is exploring ways to more completely describe two-dimensional quantum field theory using appropriate classes of functions beyond holomorphic functions. Based on this work, he is aiming to understand four-dimensional quantum field theory, the real dimension in which our world exists.

## Viewing Mathematics Through the Lens of Physics: The Appeal of Interdisciplinary Study

Until his master’s program, Moriwaki had studied mathematics with little exposure to physics. He began seriously studying physics in the first year of his Ph.D. program. During a university exchange visit to UC Berkeley, he had the chance to meet Richard Ewen Borcherds, a mathematician who solved the moonshine conjecture and received the Fields Medal. Borcherds, whom Moriwaki had admired since his undergraduate days, told him, "If you understand physics, it will be very useful for mathematics," which motivated him to start studying physics. Around the same time, he also met Professor Masahito Yamazaki, who excels in both mathematics and physics and would later become his advisor. These events at UC Berkeley marked a turning point in Moriwaki's life.

One of the challenges many mathematicians face when learning physics is the difference in motivation. Moriwaki recalls, "If you approach physics with a mathematician’s mindset, it becomes difficult to understand. So, I had to forget all my previous experience in mathematics and study physics as if I were starting from scratch, like a freshman."

Just before his third year in the Ph.D. program, a friend from the mathematics department asked him to teach physics. It was the first time he seriously thought about how to convey physics, which he had been studying separately from mathematics, to mathematicians, and what quantum field theory is as mathematics. Moriwaki reflects, "It was a precious opportunity where my mathematics and physics converged. That’s when I realized that there’s a way to approach mathematics with a different motivation than 'pure mathematics', and my research changed significantly."

Although mathematics and physics are motivated and developed in different ways, Moriwaki says that, just as visiting a different culture or country can give you a new perspective on your own culture, so too can learning physics give you a new perspective on your own mathematics.

"Mathematics is often described as a pursuit of universality, generality, and abstraction, and I think one way modern mathematics progresses is by striving for more universal descriptions. By leveraging the power of physics, I can advance in a different way. In a sense, I aim to directly conceptualize the intuition gained from physics and use it to do mathematics," says Moriwaki.

## iTHEMS: A Place Where Researchers from Various Fields Interact

Studying both mathematics and physics can be challenging. Just as a pianist loses skill after three days without playing, Moriwaki explains that not practicing mathematics or physics for a few days leads to him feel a deterioration in his abilities. That’s why iTHEMS, an environment where researchers from various fields gather and engage in daily conversations, is very helpful to him. “Having people who study physics nearby allows me to understand how physicists typically think and conduct their research. There’s still so much I don’t know in both mathematics and physics, but I believe that understanding the atmosphere and nuances that aren’t written in books, which come through human interactions, is crucial for advancing interdisciplinary research. These conversations have been very helpful for my work,” says Moriwaki.

Mathematics and physics have influenced each other throughout history, as seen in the relationship between calculus and Newtonian mechanics, or Riemannian geometry and relativity. However, in modern times, the division of labor has progressed, and researchers who aim to work in both fields are relatively rare. This is why Moriwaki values the environment at iTHEMS, where interdisciplinary research between mathematics and physics is encouraged.

## The Vast Elephant of Quantum Field Theory

Quantum field theory is vast and has many unknowns. Moriwaki likens it to the parable of the "Six Blind Men and the Elephant," where different people observe various aspects of the same entity in different ways. He believes that to unify quantum field theory as a mathematical theory, it is necessary to connect these seemingly separate aspects.

Just as mathematics has different domains, such as algebra, geometry, and analysis, quantum field theory also has algebraic, geometric, and analytic aspects. These aspects are not necessarily connected, and even where connections exist, much remains to be understood. Moriwaki aims to connect these aspects in two-dimensional quantum field theory to obtain a more complete picture. Initially focused on algebra, he is now also working on analysis and geometry, striving to comprehensively understand quantum field theory.

"I dream of a future where the various aspects of quantum field theory are connected, forming one cohesive picture," says Moriwaki. His research aims to integrate these different aspects to reveal the full structure of quantum field theory.