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Mathematical Physics and Particle Theory Lab
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Research on the mathematics of particle theory and string theory, approached through solitons.
AboutAbout the lab
Unraveling the mathematics of particle theory
Prof. Hamanaka's research themes are particle theory and mathematical physics.
Prof. Hamanaka is interested in the mathematics behind particle theory and string theory in general, and has worked on extending soliton theory and integrable systems to noncommutative spaces.
In the small graduate classes, the plan is to focus on a unified understanding of soliton theory and integrable systems from the framework of string theory.
The particle theory faculty of the graduate school, together with departments such as Physics, hold seminars with invited experts and study sessions on presenting papers, and graduate students also give presentations at the study sessions.
Key terms
- Soliton
- An isolated wave that travels without changing shape and returns to its original form after colliding with other waves
- String theory
- A theory that treats elementary particles not as points but as one-dimensional 'strings'
- Integrable system
- An equation or dynamical system with the special property that its solutions can be written down explicitly. Soliton equations are a representative example
- Noncommutative space
- A space in which the result changes when the order of multiplying coordinates is swapped
ResearchResearch themes
The lab studies the mathematics related to particle theory and string theory. Besides work on extending soliton theory and integrable systems to noncommutative spaces, it is also interested in the relationship between four-dimensional field theories and solvable models.
*Illustrative imageSolitons on Noncommutative Spaces
The lab has worked on extending soliton theory and integrable systems to noncommutative spaces. On noncommutative spaces, singularities are resolved, and in certain situations in string theory, noncommutative solitons correspond directly to D-branes.
*Illustrative imageNoncommutative Ward Conjecture
The lab showed that many soliton equations, such as the KdV equation, can be obtained by dimensional reduction from the four-dimensional noncommutative anti-self-dual Yang–Mills equations. The resulting equations are related to N=2 string theory, and analysis of their soliton solutions can be applied to this theory.
*Illustrative imageBäcklund Transformations and Noncommutative Determinants
In a collaboration with researchers in the UK, a transformation that maps a solution of the noncommutative anti-self-dual Yang–Mills equations to another solution (a Bäcklund transformation) was found, and new solutions were constructed explicitly. These solutions can be written concisely using quasideterminants, a type of noncommutative determinant.
*Illustrative image4D WZW Theory and Solvable Models
It has been shown that four-dimensional Chern–Simons theory and four-dimensional WZW theory can be derived from six-dimensional Chern–Simons theory, and the field has advanced greatly. The two theories are conjectured to be dual to each other, and an international online seminar was launched to clarify the full picture.
MembersFaculty
- LecturerMasashi Hamanaka
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Report an errorAbout the images
The top and research theme images are illustrative and were not taken at the lab. They are cropped from Wikimedia Commons images. Cropped CC BY-SA images are provided under the same CC BY-SA license as the original (version shown on each line).
- Top image: “Collider Detector at Fermilab (CDF) silicon vertex detector cross section” Z22 / CC BY-SA 4.0 (cropped)
- Solitons on Noncommutative Spaces: “Soliton hydro” Christophe.Finot et Kamal HAMMANI / CC BY-SA 2.5 (cropped)
- Noncommutative Ward Conjecture: “The board” Yulia Meshkova / CC BY 4.0 (cropped)
- Bäcklund Transformations and Noncommutative Determinants: “Blackboard bold on a blackboard” Jacob Rus / CC BY-SA 4.0 (cropped)
- 4D WZW Theory and Solvable Models: “Victor Fadin Lecturing QED in BINP 1 June 2014” Andrey Grabovsky, Budker Institute of Nuclear Physics / CC BY-SA 3.0 (cropped)
- Related lab (Izumi Lab): “Lensed Galaxy Arc in Galaxy Cluster RCS” NASA Hubble / CC BY 2.0 (cropped)
- Related lab (Foundations of Mathematics and Mathematical Logic Lab): “Machine de Turing pédagogique” Mdtcreator / CC BY-SA 4.0 (cropped)
- Related lab (Nakashima Lab): “Galton-watson(poisson(1),11gen, 47noeud)” Ipipipourax, S. Roger / CC0 (cropped)
- Related lab (Algebraic Geometry, Representation Theory, Infinite-Dimensional Integrable Systems, and Mathematical Physics Lab (Shintaro Yanagida)): “Truncated icosahedron cherry model by George W. Hart” George W. Hart / CC BY-SA 4.0 (cropped)
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