Spin Geometry José Figueroa-O’Farrill* http://empg.maths.ed.ac.uk/Activities/Spin Version of 18th May 2017 These are the notes accompanying the lectures on Spin Geometry, a PG course taught in Edinburgh in the Spring of 2010. The only requirement is a working familiarity with basic differential geometry and basic rep- resentation theory; although scholia on the necessary definitions will be scattered through- out the notes. Any statement which is not proved to your satisfaction is to be thought of as an exercise, even if not explicitly labelled as such! These notes are still in a state of flux and I am happy to receive comments and suggestions either by email or in person. *) j.m.figueroa(at)ed.ac.uk 1 Spin 2010 (jmf) 2 Contents 1 Clifford algebras: basic notions4 1.1 Quadratic vector spaces........................................4 1.2 The Clifford algebra, categorically..................................4 1.2.1 Definition............................................5 1.2.2 Construction..........................................6 1.3 The Clifford algebra as Clifford would have written it.......................7 1.3.1 Clifford algebra in terms of generators and relations...................7 1.3.2 Low-dimensional Clifford algebras..............................8 1.4 The Clifford algebra and the exterior algebra............................8 1.4.1 Filtered and associated graded algebras...........................8 1.4.2 The Z2-grading revisited....................................9 1.4.3 The filtration of the Clifford algebra.............................9 1.4.4 The action of C`(V,Q) on ¤V................................. 10 1.4.5 The Clifford inner product.................................. 11 2 Clifford algebras: the classification 12 2.1 A less-than-useful classification................................... 12 2.2 Complex Clifford algebras....................................... 13 2.3 Filling in the Clifford chessboard................................... 14 2.3.1 The even subalgebra of the Clifford algebra........................ 18 2.4 Classification of complex Clifford algebras............................. 18 3 Spinor representations 20 3.1 The orthogonal group and its Lie algebra.............................. 20 3.2 Pin and Spin............................................... 21 3.3 Pinors and spinors........................................... 23 3.3.1 s t 0 (mod 8)........................................ 25 ¡ Æ 3.3.2 s t 1 (mod 8)........................................ 26 ¡ Æ 3.3.3 s t 2 (mod 8)........................................ 26 ¡ Æ 3.3.4 s t 3 (mod 8)........................................ 26 ¡ Æ 3.3.5 s t 4 (mod 8)........................................ 26 ¡ Æ 3.3.6 s t 5 (mod 8)........................................ 26 ¡ Æ 3.3.7 s t 6 (mod 8)........................................ 26 ¡ Æ 3.3.8 s t 7 (mod 8)........................................ 26 ¡ Æ 3.4 Inner products for pinors and spinors................................ 26 4 Spin manifolds 28 4.1 What is a manifold?........................................... 28 4.2 Fibre bundles.............................................. 28 4.2.1 Basic notions.......................................... 29 4.2.2 Construction from local data................................. 29 4.2.3 Vector and principal bundles................................. 30 4.2.4 Equivalence classes of principal bundles.......................... 31 4.3 Fibre bundles on riemannian manifolds.............................. 31 4.3.1 Orientability and the orthonormal frame bundle..................... 32 4.3.2 The Clifford bundle and the obstruction to defining a pinor bundle.......... 33 4.3.3 Spin structures......................................... 34 Spin 2010 (jmf) 3 5 Connections on principal and vector bundles 36 5.1 Connections on principal bundles.................................. 36 5.1.1 Connections as horizontal distributions.......................... 37 5.1.2 The connection one-form................................... 37 5.1.3 The horizontal projection................................... 38 5.1.4 The curvature 2-form..................................... 38 5.2 Connections on vector bundles.................................... 39 5.2.1 Koszul connections....................................... 40 5.2.2 Basic forms........................................... 40 5.2.3 The covariant derivative.................................... 41 5.2.4 Gauge fields........................................... 42 6 The spin connection 44 6.1 The Levi-Civita connection...................................... 44 6.2 The connection one-forms on O(M), SO(M) and Spin(M).................... 45 6.3 Parallel spinor fields.......................................... 47 7 Holonomy groups 48 7.1 Parallel transport in principal fibre bundles............................ 48 7.2 Parallel transport on vector bundles................................. 49 7.3 The holonomy principle........................................ 50 7.4 Riemannian holonomy groups.................................... 50 7.4.1 Kähler manifolds........................................ 52 7.4.2 Calabi–Yau manifolds..................................... 53 7.4.3 Manifolds of G2 holonomy.................................. 53 7.4.4 Ricci-flatness.......................................... 53 8 Parallel and Killing spinor fields 54 8.1 Manifolds admitting parallel spinor fields.............................. 54 8.1.1 Calabi–Yau 3-folds....................................... 54 8.1.2 Manifolds of G2 holonomy.................................. 55 8.1.3 Some comments about indefinite signature........................ 55 8.2 Manifolds admitting (real) Killing spinor fields........................... 55 8.2.1 The Dirac operator....................................... 56 8.2.2 The Penrose operator and twistor spinor fields...................... 56 8.2.3 Killing spinor fields....................................... 57 8.2.4 The cone construction..................................... 57 8.2.5 The classification........................................ 58 Spin 2010 (jmf) 4 Lecture 1: Clifford algebras: basic notions Consider now a system of n units ¶1,¶2,...,¶n such that the multiplication of any two of them is polar; that is, ¶r ¶s Æ ¶s ¶r . ¡ — William Kingdon Clifford, 1878 In this lecture we define the Clifford algebra of a quadratic vector space and view it from three dif- ferent points of view: the contemporary categorical formulation, Clifford’s original formulation and as a quantisation of the exterior algebra. 1.1 Quadratic vector spaces Throughout K R or C. Let V be a finite-dimensional vector space over K, let B : V V K be a (pos- Æ £ ! sibly degenerate) symmetric bilinear form and let Q : V K denote the corresponding quadratic form, ! defined by Q(x) B(x,x). One can recover B from Q by polarisation, namely Æ (1) B(x, y) 1 ¡Q(x y) Q(x) Q(y)¢. Æ 2 Å ¡ ¡ The pair (V,Q) is called a quadratic vector space (over K). They are the objects of a category QVec with morphisms (V,Q ) (W,Q ) given by linear maps f :V W such that f ¤Q Q , or explicitly that V ! W ! W Æ V Q (f (x)) Q (x) for all x V. The zero vector space with the zero quadratic form is an initial object W Æ V 2 in QVec. The absence of terminal objects and (co)products is due to the fact that projections do not generally preserve norms. We will see that the Clifford algebra C`(V,Q) of a quadratic vector space (V,Q) is an associative, unital K-algebra, with a natural filtration and a Z -grading, and moreover that the assignment (V,Q) 2 7! C`(V,Q) is functorial. There are several ways to understand C`(V,Q): from the very abstract to the very concrete. The latter is good for computations, whereas the former is good to prove theorems which may free us from computations. Therefore we will look at C`(V,Q) in several ways, starting with the categorical definition. j All our associative algebras are unital, unless otherwise stated! 1.2 The Clifford algebra, categorically Let (V,Q) be a quadratic vector space and let A be an associative K-algebra. We say that a K-linear map Á :V A is Clifford if for all x V, ! 2 (2) Á(x)2 Q(x)1 , Æ¡ A where 1A is the unit of A. Clifford maps from a fixed quadratic vector space (V,Q) are the objects of a category Cliff(V,Q), where a morphism from V A to V A0 is given by a commuting triangle ! ! (3) V f A / A0 with f :A A0 a homomorphism of associative algebras. ! Spin 2010 (jmf) 5 1.2.1 Definition Definition 1.1. The Clifford algebra — if it exists — is an initial object in Cliff(V,Q). In other words, it is given by an associative algebra C`(V,Q) together with a Clifford map i :V C`(V,Q) such that for ! every Clifford map Á :V A there is a unique algebra morphism © :C`(V,Q) A making the following ! ! triangle commute: (4) V i Á { © C`(V,Q) / A Remark 1.2. There are several paraphrases of the defining property of the Clifford algebra. One can say that every Clifford map factors uniquely via the Clifford algebra, or that the Clifford algebra is universal for Clifford maps, or that every Clifford maps extends uniquely to a morphism of associative algebras from the Clifford algebra. Remark 1.3. The mathematical literature is replete with such universal definitions. For example, if g is a Lie algebra and A is an associative algebra (over the same ground field) then one can consider linear maps Á : g A such that, for all X,Y g, ! 2 (5) Á(X)Á(Y) Á(Y)Á(X) Á([X,Y]) ¡ Æ Although it is not standard terminology, let us call such maps Lie within the confines of this remark. Then the universal enveloping algebra Ug of g is universal for Lie maps; in