Waseda Institute for Advanced Study (WIAS)早稲田大学 高等研究所

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Bringing physics notions to mathematics: Mirror symmetry and Stokes structures
SHAMOTO Yota, Assistant Professor

SHAMOTO Yota, Assistant Professor

Clear verbalization of mathematical arguments

My major is mathematics. The widely held image of mathematics is solving equations and other formulas, but since I like to clarify mathematical arguments using formulas and mathematical terms (a process called formulation), that is the kind of research I do most of the time. You might say my research is similar to the proof questions that we see in middle school and high school mathematics. I have always liked to try to understand the essence of things, and for me it’s a thrill to finally organize something that was vague in my mind and describe it beautifully. Experiences like that have led me to where I am today.

We commonly use mathematical concepts to solve physics problems. Recently, however, there has been a reverse trend, where researchers apply notions from physics to solve mathematical problems. One of most famous examples of that phenomenon is the Poincaré conjecture. The Russian mathematician Grigori Perelman used ideas from physics to solve the Poincaré mathematical problem, which until then had gone unsolved for about 100 years.

My research is also deeply related to the physics notion of mirror symmetry, from a theory in physics called superstring theory (the theory that elementary particles are some kind of tiny “strings”). Mirror symmetry refers to the situation where two different theoretical models, Model A and Model B, describe the same physical phenomena with certain parameters. Simply put, the expectation is that the two models, described in two different ways in two different spaces, will yield the same conclusion—and the mathematics describing the two theoretical models should be the same. According to that expectation, the mathematics describing each theoretical model should be the same; however, when you actually describe the two models, the mathematics used to describe them is completely different. Therefore, using a method in which the “structure” of the mathematics describing each theoretical model is extracted from it, we formulate the equivalence of the two theories by assuming that the structures are equivalent.

Extracting the mathematics from different models of the same physical phenomenon

Applying the Landau-Ginzburg model (hereinafter referred to as LG model) of superstring theory, we can extract the differential equations and the category. The category, the world of objects and the arrows that connect them, is a concept often used in mathematics. Although the extracted differential equations and the category seem to represent completely different things, the two can clearly describe the relationship within the geometry (the representation as a diagram) of the LG model.

If we apply the notion of mirror symmetry to this LG model, we have a sigma model, which is a model of a different space that represents the same physical phenomenon. Then we can extract differential equations and the category from the sigma model as well. It is expected that mirror symmetry will be established between the differential equation and the category of the sigma model and those of the LG model. However, the relationship between the category and the differential equation in the sigma model cannot be expressed in the same geometry as in the LG model, and thus the relationship between the differential equation and the category in the sigma model is a mysterious correspondence that is very intriguing to mathematicians (Figure 1).

Figure 1: Diagram showing the relationship between the LG model and the sigma model

Extracting the equivalent Stokes structure from differential equations and the category

This correspondence was formulated in a limited space (part of the Fano manifold) as Dubrovin’s conjecture, but there was a need to broaden the range of application and make the formulation more general.

Therefore, I extracted the structure from the differential equations and the category in the sigma model, and formulated that the two structures are equivalent. Then if I collect all the solutions of the differential equations of the sigma model, I can extract a structure called the Stokes structure.

The Stokes structure is shown in a very simplified example in Fig. 2.

Figure 2: A very simplified example showing the Stokes structure

The singularity (the point excluded as undefined) of this function is x=0. If you look at the graph and focus on the singularity, you will see that as x takes positive values and approaches zero, it diverges to infinity, and as x takes negative values and approaches zero, it converges to zero. At this point, when the singularity 0 is exceeded, the behavior jumps discontinuously. That behavior around the singularity of differential equations is called the Stokes phenomenon, and that phenomenon represents the Stokes structure.

Next, I went on to extract the structure from the category in the sigma model using a method called de-categorification, and I was able to extract the Stokes structure from that category as well. It was possible to extract the same Stokes structures from both the differential equations and the category. In Dubrovin’s conjecture it was formulated in a Stokes matrix, but since I formulated it in terms of Stokes structures, which have a wider range of applicability, the conjecture could be extended to a wider range (expected to be all Fano manifolds).

Currently, I am working on extending this Stokes structure to be able to handle not only differential equations but also the structure of difference equations, with the goal of formulating the relationship between difference equations and category theory.

Interview and composition: AIMONO Keiko
In cooperation with: Waseda University Graduate School of Political Science J-School

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