2020 Special lectures on advanced topics in Mathematics C

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Academic unit or major
Graduate major in Mathematics
Nakajima Hiraku  Gomi Kiyonori 
Course component(s)
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Day/Period(Room No.)
Intensive ()  
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Syllabus updated
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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.


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.


None required

Reference books, course materials, etc.

Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional N=4 gauge theories, arXiv:1612.09014,1706.05154.

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

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