Perspectives in Mathematical Science I
(2017 Summer, Part 3 -- Hamanaka)
- I am sorry that I seemed to forget to write ``Tr'' in Exercise 1.
(Without Tr, the formula NEVER holds in general!)
This is due to my mistake and hence I have decided
that 15pt. (for the Ex.1) is added to all students in the class.
- The lecture is finished. Thank you very much. (2017/07/25)
- This homepage is open. (2017/05/30)
Syllabus of Part 3 (For total one, see here.)
- Title of Part 3: ADHM construction of Instantons
- Instructor: Masashi Hamanaka
- Date: Tuesday, 14:45--16:15 (during June 27th -- July 25th)
- Room: School of Science Building 1 (Mathematics), Room 109
- Language: English
- Course contents:
I would like to discuss the Atiyah-Drinfeld-Hitchin-Manin (ADHM)
construction of instantons and related topics. Instantons are
finite-action (global) solutions to the Anti-Self-Dual Yang-Mills
equations in four dimension. They play crucial roles in geometry
and physics. The ADHM construction is based on a beautiful duality
(one-to-one correspondence) between a moduli space of the
instantons and a moduli space of the ADHM data. While the former
is specified by a non-linear partial differential equation in four
dimension, the latter is specified by a matrix equation which is
much easier to treat.
The purpose of my part is to give an elementary proof of the
duality in the ADHM construction in four dimensional Euclidean
space, together with a brief introduction to instantons in
geometry and physics. There would be no need to know manifolds,
vector bundles, connections and so on in advance. I would also
like to mention the D-brane interpretations of them,
generalization to noncommutative spaces, and application to
monopoles in three dimension. More detailed syllabus will be
distributed at the beginning of part 3 (on June 27th).
- References (7/18):
- Corrigan and Goddard, ``Construction of Instanton and
Solutions and Reciprocity,''
Annals Phys. 154 (1984) 253 -- 279.
- Atiyah, ``Geometry of Yang-Mills Fields,'' Pisa, Italy:
Sc. Norm. Sup.
(1979, Lezioni Fermiane).
- Donaldson and Kronheimer,
``The Geometry of Four-Manifolds,''
(1990, Oxford UP).
- Nakajima, Monopoles and Nahm's equations,
- Grading: Attendance and Report (Details will be
announced on June 27th.)
Plan --- ADHM construction of Instantons (Update: 7/25)
- 6/27：Overview (Slide), Instantons
- 7/04：Instantons(continued), ADHM data
- 7/11：ADHM construction of instantons
- 7/18：Duality in the ADHM construction(outline),
Nahm construction of monopoles
- 7/25：Noncommutative generalization, D-brane interpretation, Perspective