This is a page of information about the reading group Nahm's equations and infinite dimensional hyperkähler quotient constructions.

The idea is that we will learn about infinite dimensional quotient discussions by studying some examples. The example that we will focus on first is that of Nahm's equations following the lecture notes by Maxence Mayrand found here. These notes and other potentially useful references are linked at the bottom of this page.

We will meet between 13:30 and 15:00 in room 707.

Rough notes of what we have discussed will be periodically updated here.

If anyone wants to see some examples of moment maps being computed there are some given in my notes on moment maps and the Hyperkähler structure of the moduli space of solutions to Hitchin's Self-duality equations. These notes may also be useful if we go on to discuss the quotient construction for the case of Hitchins equations given in [2].

Date | What was read/ will be presented | Presenter | Summary of what we discussed |
---|---|---|---|

1/6/22 | N/A | N/A | We had a brief chat about how we would proceed with the reading group. It was a housekeeping session where we discussed how to proceed. We have decided to aim for 1 hour sessions, 13:30-15:00 UK time , with ~30-45 minutes for a presentation and the rest of the time for a discussion. The plan is for the presenter to change each week though there is no obligation to volunteer. |

8/6/22 | Up to page 9 of [1] and some brief motivation | Calum | We discussed some general motivation of where these quotients show up, e.g. when considering the physical configurations of a theory. Then we met instantons and their dimensional reductions before focussing on the case of Nahm's equations and some examples of solutions for SU(2). |

15/6/22 | Finite dimensional quotients, roughly following Sec.3 of [1] | Jaime | We reviewed the basics of HyperKähler geometry and quotient constructions via moment maps when a compact Lie group acts freely on a Manifold preserving extra structures. e.g. HyperKähler quotients when the group action preserves the HyperKähler structure. We also discussed the example, due to Calabi, of the construction of cotangent bundle of complex projective space as a Hyperkähler quotient the aim is to flesh out this example in the notes. |

22/6/22 | N/A | N/A | No Meeting as ECRM conference is happening this week |

29/6/22 | Enric | The first part of Sec.4 of [1], up to page 27 | Enric covered two simplified cases before we move on to the full Nahm equations next time: The "embryonic" Nahm case e.g. constructing a bi-invariant metric on the Lie group G; and the baby Nahm case e.g. constructing the metric on the cotangent bundle of G. I will flesh out what was discussed in the notes. |

6/7/22 | Jakob | The second part of Sec.4 of [1], from page 27 to 34 TBC. | Jakob recapped some of the discussion from the previous week and clarified the argument for why the gauge orbit only intesects the slice once for a small enough neighbourhood, as well as showing why the group of framed gauge transofmations acts freely. We then discussed the case of the Nahm equations and the various computations that need to be done to show that the hyperkäher quotient gives something finite dimensional. There are a few details that we want to clarify more about uniqueness of the smooth structure and checking that the quotient is Hausdorff Jaime has some ideas about how to do this. |

13/7/22 | Jakob and Jaime | Finishing off sec 4.3.3 and reading upto 4.3.7. | Jakob finished off the discussion of the complex-symplectic description of the "papa" Nahm equations, omitting the sketch proof identifying the complex-symplectic/hyperKahler moduli-spaces. Jaime then spoke about the tri-Hamiltonian G x G-action on the moduli-space, and some generalities about infinite-dimensional Lie groups were discussed with regards to these proofs. |

20/7/22 | N/A | N/A | No meeting as most people are away this week. We decided to wind the reading group down for the summer. Hopefully we will start back up again in September with a similar topic. |

- [1]: Nahmâ€™s equations in hyperkähler geometry, Maxence Mayrand.
- [2]: The Self-Duality Equations on a Riemann Surface, Nigel Hitchin.
- [3]: The page here, is for a course which covered the paper by Atiyah and Bott "The Yang-Mills Equations over Riemann Surfaces", which considers the Yang-Mills equations over Riemann surfaces and discusses some details about their moduli spaces. It may be a useful resource and source of further references.
- [4]: Self-dual Yang-Mills and the Nahm transform, Josh Cork.