▶Japanese
Post to SNS
- Home
- > School of Science
- > Mathematics
- > Special courses on advanced topics in Mathematics C
- School of Science
-
School of Engineering
- Group 2
- Group 3
- Group 4
- Group 5
- Group 6
- Metallurgical Engineering
- Organic and Polymeric Materials
- Inorganic Materials
- Chemical Engineering Course
- Chemical Engineering Course(Chemical Engineering)
- Applied Chemistry Course(Chemical Engineering)
- Polymer Chemistry
- Mechanical Engineering and Science
- Mechanical and Intelligent Systems Engineering
- Mechano-Aerospace Engineering
- International Development Engineering
- Control and Systems Engineering
- Industrial and Systems Engineering
- Electrical and Electronic Engineering
- Computer Science
- Civil Engineering
- Architecture and Building Engineering
- Social Engineering
- Common Cource of Engineering
- School of Bioscience and Biotechnology
- Common Course of Undergraduate School
-
Graduate School of Science and Engineering
- Mathematics
- Physics(Particle- Nuclear- and Astro-Physics)
- Physics(Condensed Matter Physics)
- Chemistry
- Earth and Planetary Sciences
- Chemistry and Materials Science
- Metallurgy and Ceramics Science
- Organic and Polymeric Materials
- Applied Chemistry
- Chemical Engineering
- Common Course of Mechanical Engineering
- Mechanical Sciences and Engineering
- Mechanical and Control Engineering
- Mechanical and Aerospace Engineering
- Common Course of Electronic Engineering
- Electrical and Electronic Engineering
- Physical Electronics
- Communications and Integrated Systems
- Communications and Computer Engineering
- Civil Engineering
- Architecture and Building Engineering
- International Development Engineering
- Nuclear Engineering
- Graduate School of Bioscience and Biotechnology
-
Interdisciplinary Graduate School of Science and Engineering
- Built Environment
- Energy Sciences
- Computational Intelligence and Systems Science
- Electronic Chemistry
- Materials Science and Engineering
- Innovative and Engineered Materials
- Environmental Chemistry and Engineering
- Environmental Science and Technology
- Mechano-Micro Engineering
- Information Processing
- Electronics and Applied Physics
- Graduate School of Information Science and Engineering
- Graduate School of Decision Science and Technology
- Graduate School of Innovation Management
- Common Course of Graduate School
- Program for Leading Graduate Schools
- Academic unit or major
- Mathematics
- Instructor(s)
- Nakajima Hiraku
- Class Format
- Lecture (ZOOM)
- Media-enhanced courses
- Day/Period(Room No.)
- Intensive ()
- Group
- -
- Course number
- ZUA.E333
- Credits
- 2
- Academic year
- 2020
- Offered quarter
- 2Q
- Syllabus updated
- 2020/9/18
- Lecture notes updated
- -
- Language used
- English
- Access Index
-
Course description and aims
We give a mathematically rigorous definition of Coulomb branches, which arise supersymmetric gauge theories. They are realized as examples of convolution algebras defined over equivariant homology groups. Spaces for which we take equivariant homology groups are variants of the so-called affine Grassmannian manifolds. They are moduli spaces of principal bundles and sections of associated vector bundles on the 2-dimensional disk. We also explain that this construction can be natually arise in the framework of topological quantum field theories.
In geometric representation theory, we often use the techique constructing algebras and their representations as convolution algebras on equivariant homology groups. One of aims of these lectures is to learn this techique by an example. This technique is often applied to usual manifolds, such as flag manifolds and their cotangent bundles, or quiver varieties. But we graduately learn that it is also applied to infinite dimensional manifolds, such as affine Grassmanian manifolds. It is also natural to treat infinite dimensional manifolds in order to connect them to quantum field theories in theoretical physics. The second aim is to know moduli spaces as examples of infinite dimensional manifolds.
Student learning outcomes
・Learn the definition and basic properties of equivariant homology groups
・Understand the definition and properties of convolution algebras
・Learn the mathematical definition of Coulomb branches of supersymmetric gauge theories
・Understand basics on topological quantum field theories and vaccum.
Keywords
equivariant homology groups, convolution algebras, affine Grassmannian manifolds, Coulomb branches of gauge theories
Competencies that will be developed
| ✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Class flow
This is a standard lecture course. There will be some assignments.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | The following topics will be covered in this order : -- definition and basic properties of equivariant homology groups -- definition and basic properties of convolution algebras -- definitions and basic properties of affine Grassmannian manifolds and varities of triples -- definition of Coulomb branches -- basic properties and examples of Coulomb branches -- summary on topological quantum field theories -- moduli spaces and vacuum | Details will be provided during each class session. |
Textbook(s)
None required
Reference books, course materials, etc.
Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional N=4 gauge theories, arXiv:1706.05154.
Assessment criteria and methods
Assignments (100%).
Related courses
- MTH.B341 ： Topology
- MTH.B301 ： Geometry I
- MTH.B302 ： Geometry II
- MTH.B331 ： Geometry III
- ZUA.A331 ： Advanced courses in Algebra A
Prerequisites (i.e., required knowledge, skills, courses, etc.)
fundamentals of homology and algebra
