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 D2 (1531-119) |

**Registration**

Not open yet

**Dominic Joyce (Oxford University) **

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.

**Yuuji Tanaka (Oxford University)**

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**Will be updated regularly

**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*