Title: Riemannian holonomy groups & Gauge theory and instanton moduli spaces

2018.01.25 |

Date | Mon 06 Aug — Wed 15 Aug |

Time | 09:30 — 12:00 |

Location | Aud D3 (1531-215) / Aud. D2 (1531-119) |

**Dominic Joyce (University of ****Oxford****)**

Title: *Riemannian holonomy groups*

Abstract: Let (X, *g*) be a connected, simply-connected Riemannian n-manifold. The holonomy group Hol(*g*) is a subgroup of SO(*n*) which measures the tensors T on X constant under the Levi-Civita connection ∇ of g. Berger (1955) showed that if *g* is irreducible and nonsymmetric then Hol(*g*) is one of:

(i) SO(*n*) (the boring case, for generic g);

(ii) U(*m*) for *n*=2m (Kähler metrics);

(iii) SU(*m*) for *n*=2m (Calabi-Yau metrics);

(iv) Sp(*m*) for *n*=4m (hyperkähler metrics);

(v) Sp(*m*)Sp(1) for *n*=4*m* (quaternionic Kähler metrics);

(vi) G2 for *n*=7 and (vii) Spin(7) for *n*=8 (exceptional holonomy)

Each of classes (ii)-(vii) have their own interesting geometry. We will review the theory of holonomy groups and Berger’s classification, and discuss Riemannian manifolds in classes (ii)-(vii), especially compact manifolds. If time allows we will also discuss *calibrated submanifolds*, distinguished classes of minimal submanifolds in Riemannian manifolds with special holonomy.

Preliminary lecture notes and reading list (Univ. of Oxford website)

**Yuuji Tanaka (University of ****Oxford****)**

Title: *Gauge theory and instanton moduli spaces*

Abstract: Let (X, *g*) be a compact Riemannian manifold, usually with some additional structure (e.g. *g* may have special holomomy), and *E* → X be a vector bundle or principal bundle, and ∇ be a connection on *E*, with curvature F_{∇}. We call (*E*,∇) an *instanton *if F_{∇} satisfies a linear equation depending on *g* and the additional structure. *Gauge theory* is the study of *moduli space** s M* of instantons, that is, M is the set of isomorphism classes of instantons (E,∇), equipped with a geometric structure which reflects how instantons vary in families.

In good cases, *M* is a smooth manifold, which can be compactified by adding a ‘boundary’ of singular instantons. Instanton moduli spaces may contain interesting information about (X, *g*).

Gauge theory was invented in Physics, but became important in Mathematics because of Donaldson theory of 4-manifolds, which used instanton moduli spaces to define invariants which could distinguish different smooth structures on the same topological 4-manifold. Gauge theory is connected to Riemannian holonomy, as there are interesting instanton-type equations on manifolds (X, *g*) with holonomy SO(3), SO(4), U(*m*), SU(3), SU(4), G_{2 }and Spin(7). Higherdimensional gauge theory (dim > 4) is the focus of much current research.

Topics will include: basics of gauge theory; moduli spaces of instantons on 4- manifolds, the Uhlenbeck compactification and Donaldson theory; holomorphic vector bundles on Kähler manifolds, Hermitian-Einstein connections and the Hitchin-Kobayashi correspondence; G_{2} and Spin(7) instantons; and compactifying instanton moduli spaces in higher dimensions.

**Preliminary time table**

Location: Aud D2 (1531-119)

*Monday 6th – Friday 10th August and Monday 13th – Tuesday 14th August *

09:30-09:45: Coffee and check-in

09.45-10.30: Dominic Joyce

10.45-11.30: Dominic Joyce

11.40-12.00: Q+A's

12.00-13.30: lunch

13.30-14.15: Yuuji Tanaka

14.30-15.15: Yuuji Tanaka

15.30-15.50: Q+A's.

*Wednesday 15th August *

09.30-10.15: Dominic Joyce

10.15-10.30: Yuuji Tanaka

11.30-12.00: Q+A's

12.00 Masterclass finishes and lunch

**List of participants**

Basurto Arzate Efrain (Technische Universität Dortmund)

Gergely Bérczi (QGM, AU)

Francis Bischoff (University of Toronto)

Aleksandra Borowka (Jagiellonian University)

Martin De Borbon (QGM, AU)

Jorge Bravo (Math, AU)

Mondher Chouikhi (Gabes University)

Sungbong Chun (Caltech)

Lennart Döppenschmitt (University of Toronto)

Jan Friedmann (IST Austria)

Jacob Gross (University of Oxford)

Shu-Ting Huang (Academia Sinica)

Roberta Iseppi, (QGM, AU)

Tobias Kildetoft (QGM, AU)

Yuki Koyanagi (QGM, AU)

Fei-Tong Lyu (Université Grenoble Alpes)

Bálint Máté (Central European University)

Yuta Nozaki (University of Tokyo)

Vasileios Ektor Papoulias (University of Oxford)

Corvin Paul (University of Copenhagen)

Du Pei, (QGM/Caltech)

Giovanni Russo (QGM, AU)

Andries Salm (Heriot-Watt University)

Simone Siclari (QGM, AU)

Andreas B. Skovbakke (QGM, AU)

Cristiano Spotti (QGM, AU)

Kelli Francis-Staite (University of Oxford)

Jakob Stein (Heriot-Watt University)

Salvatore Tambasco (University of Pavia)

Chi Nok Enoch Yiu (University of Oxford)

Menelaos Zikidis (University of Heidelberg)

**Organizing committee**Jørgen Ellegaard Andersen (Aarhus University, Denmark)

Dominic Joyce (Oxford University)

Yuuji Tanaka (Oxford University)

Questions? Please send an e-mail to qgm@au.dk

*Video recordings will be available after the masterclass here*