Title of the course: Spin(7) Geometry
Instructor: Dr. Eyüp Yalçınkaya
Institution: TÜBİTAK
Dates: 13-18 September 2021
Prerequisites: Algebra, Linear Algebra
Level: Undergraduate, graduate
Abstract: Spin structures have wide applications to mathematical physics, in particular to quantum field theory. For the special class Spin(7) geometry, there are different approaches. One of them is constructed by holonomy groups. According to the Berger classification (1955), the Spin(7) group is one of these holonomy classes.
Firstly, it is presented its properties. After that, torsion which is another important term in superstring theory will be geometrically introduced and related to Spin(7) geometry.
Let M be an 8-dimensional manifold with the Riemannian metric g and structure group G ⊂ SO(8). The structure group G ⊂ Spin(7), then it is called M admits Spin(7)-structure. M. Fernandez [1] classifies all types of 8-dimensional manifolds admitting Spin(7)-structure. In general, torsion-free Spin(7) manifold are studied considerably.
On the other hand, manifolds admitting Spin(7)-structure with torsion have rich geometry as well. Locally conformal parallel structures have been studied for a long time with Kähler condition being the oldest one. By means of further groups whose holonomy is exceptional, the choices of the G 2 and Spin(7) deserves attention. Ivanov [3], [4], [5] introduces a condition when an 8-dimensional manifold admits locally conformal parallel Spin(7) structure.
Salur and Yalcinkaya [6] studied almost symplectic structure on Spin(7)-manifold with 2-plane field. Then, Fowdar [2] studied Spin(7) metrics from Kähler geometry. In this research, we introduce an 8-manifold equipped with locally conformal Spin(7)-structure with 2-plane field.
Then, almost Hermitian 6-manifold can be classified by the structure of M.
References:
[1] M. Fernandez, A Classification of Riemannian Manifolds with Structure Group Spin (7), Annali di Mat. Pura ed App., vol (143), (1986), 101—122.
[2] U. Fowdar Spin(7) metrics from Kähler Geometry, arXiv:2002.03449, (2020)
[3] S. Ivanov, M. Cabrera, SU(3)-structures on submanifolds of a Spin(7)-manifold, Differential Geometry and its Applications, V 26 (2), (2008) 113–132
[4] S. Ivanov, M. Parton and P. Piccinni, Locally conformal parallel G 2 and Spin(7) manifolds Mathematical Research Letters, V 13, (2006), 167–177
[5] S. Ivanov Connections with torsion, parallel spinors and geometry of Spin(7) manifolds, math/0111216v3.
[6] S. Salur and E. Yalcinkaya Almost Symplectic Structures on Spin(7)-Manifolds, Proceedings of the 2019 ISAAC Congress (Aveiro, Portugal), 2020)
Language: EN