SpinGeometry
                                                     JoséFigueroa-O’Farrill*
                                            http://empg.maths.ed.ac.uk/Activities/Spin
                                                      Versionof18thMay2017
                        These are the notes accompanying the lectures on Spin Geometry, a PG course taught in
                        EdinburghintheSpringof2010.
                        Theonlyrequirementisaworkingfamiliaritywithbasicdifferentialgeometryandbasicrep-
                        resentation theory; although scholia on the necessary definitions will be scattered through-
                        outthenotes.
                        Anystatementwhichisnotprovedtoyoursatisfactionistobethoughtofasanexercise,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
                          Spin2010(jmf)                                                                                                         2
                          Contents
                          1 Cliffordalgebras: basicnotions                                                                                      4
                              1.1 Quadraticvectorspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         4
                              1.2 TheCliffordalgebra,categorically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .          4
                                   1.2.1   Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    5
                                   1.2.2   Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     6
                              1.3 TheCliffordalgebraasCliffordwouldhavewrittenit . . . . . . . . . . . . . . . . . . . . . . .                  7
                                   1.3.1   Clifford algebra in terms of generators and relations . . . . . . . . . . . . . . . . . . .          7
                                   1.3.2   Low-dimensionalCliffordalgebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .           8
                              1.4 TheCliffordalgebraandtheexterioralgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . .               8
                                   1.4.1   Filtered andassociatedgradedalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . .           8
                                   1.4.2   TheZ2-gradingrevisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       9
                                   1.4.3   ThefiltrationoftheCliffordalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . .           9
                                   1.4.4   TheactionofCℓ(V,Q)onΛV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
                                   1.4.5   TheCliffordinnerproduct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
                          2 Cliffordalgebras: theclassification                                                                                12
                              2.1 Aless-than-usefulclassification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
                              2.2 ComplexCliffordalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
                              2.3 FillingintheCliffordchessboard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
                                   2.3.1   TheevensubalgebraoftheCliffordalgebra . . . . . . . . . . . . . . . . . . . . . . . . 18
                              2.4 ClassificationofcomplexCliffordalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
                          3 Spinorrepresentations                                                                                             20
                              3.1 TheorthogonalgroupanditsLiealgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
                              3.2 PinandSpin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
                              3.3 Pinorsandspinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
                                   3.3.1   s −t =0 (mod 8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
                                   3.3.2   s −t =1 (mod 8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
                                   3.3.3   s −t =2 (mod 8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
                                   3.3.4   s −t =3 (mod 8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
                                   3.3.5   s −t =4 (mod 8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
                                   3.3.6   s −t =5 (mod 8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
                                   3.3.7   s −t =6 (mod 8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
                                   3.3.8   s −t =7 (mod 8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
                              3.4 Innerproductsforpinorsandspinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
                          4 Spinmanifolds                                                                                                     28
                              4.1 Whatisamanifold? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
                              4.2 Fibrebundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
                                   4.2.1   Basicnotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
                                   4.2.2   Constructionfromlocaldata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
                                   4.2.3   Vectorandprincipalbundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
                                   4.2.4   Equivalenceclassesofprincipalbundles . . . . . . . . . . . . . . . . . . . . . . . . . . 31
                              4.3 Fibrebundlesonriemannianmanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
                                   4.3.1   Orientability andtheorthonormalframebundle . . . . . . . . . . . . . . . . . . . . . 32
                                   4.3.2   TheCliffordbundleandtheobstructiontodefiningapinorbundle . . . . . . . . . . 33
                                   4.3.3   Spinstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
                        Spin2010(jmf)                                                                                               3
                        5 Connectionsonprincipalandvectorbundles                                                                   36
                           5.1 Connectionsonprincipalbundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
                                5.1.1   Connectionsashorizontaldistributions . . . . . . . . . . . . . . . . . . . . . . . . . . 37
                                5.1.2   Theconnectionone-form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
                                5.1.3   Thehorizontalprojection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
                                5.1.4   Thecurvature2-form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
                           5.2 Connectionsonvectorbundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
                                5.2.1   Koszulconnections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
                                5.2.2   Basicforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
                                5.2.3   Thecovariantderivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
                                5.2.4   Gaugefields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
                        6 Thespinconnection                                                                                        44
                           6.1 TheLevi-Civitaconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
                           6.2 Theconnectionone-formsonO(M),SO(M)andSpin(M) . . . . . . . . . . . . . . . . . . . . 45
                           6.3 Parallelspinorfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
                        7 Holonomygroups                                                                                           48
                           7.1 Paralleltransportinprincipalfibrebundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
                           7.2 Paralleltransportonvectorbundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
                           7.3 Theholonomyprinciple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
                           7.4 Riemannianholonomygroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
                                7.4.1   Kählermanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
                                7.4.2   Calabi–Yaumanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
                                7.4.3   ManifoldsofG holonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
                                                       2
                                7.4.4   Ricci-flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
                        8 ParallelandKillingspinorfields                                                                            54
                           8.1 Manifoldsadmittingparallelspinorfields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
                                8.1.1   Calabi–Yau3-folds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
                                8.1.2   ManifoldsofG holonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
                                                       2
                                8.1.3   Somecommentsaboutindefinitesignature . . . . . . . . . . . . . . . . . . . . . . . . 55
                           8.2 Manifoldsadmitting(real)Killingspinorfields . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
                                8.2.1   TheDiracoperator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
                                8.2.2   ThePenroseoperatorandtwistorspinorfields . . . . . . . . . . . . . . . . . . . . . . 56
                                8.2.3   Killing spinor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
                                8.2.4   Theconeconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
                                8.2.5   Theclassification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
                  Spin2010(jmf)                                                                     4
                  Lecture1: Cliffordalgebras: basicnotions
                                                           Considernowasystemofn unitsι ,ι ,...,ι suchthatthe
                                                                                    1 2   n
                                                           multiplication of any two of them is polar; that is, ι ι =
                                                                                                 r s
                                                           −ι ι .
                                                             s r
                                                                            —WilliamKingdonClifford,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
                  aquantisationoftheexterioralgebra.
                  1.1  Quadraticvectorspaces
                  ThroughoutK=RorC. LetV beafinite-dimensionalvectorspaceoverK,letB:V×V→Kbea(pos-
                  sibly degenerate) symmetric bilinear form and let Q:V →Kdenotethecorrespondingquadraticform,
                  definedbyQ(x)=B(x,x).OnecanrecoverBfromQbypolarisation,namely
                                                     1 ¡                ¢
                  (1)                        B(x,y)= 2 Q(x+y)−Q(x)−Q(y) .
                  Thepair(V,Q)iscalledaquadraticvectorspace(overK). TheyaretheobjectsofacategoryQVecwith
                  morphisms(V,QV)→(W,QW)givenbylinearmaps f :V →W suchthat f∗QW =QV, orexplicitly that
                  QW(f(x))=QV(x)forall x ∈V. The zero vector space with the zero quadratic form is an initial object
                  in QVec. The absence of terminal objects and (co)products is due to the fact that projections do not
                  generallypreservenorms.
                     We will see that the Clifford algebra Cℓ(V,Q) of a quadratic vector space (V,Q) is an associative,
                  unital K-algebra, with a natural filtration and a Z2-grading, and moreover that the assignment (V,Q)7→
                  Cℓ(V,Q)isfunctorial.
                     There are several ways to understand Cℓ(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. ThereforewewilllookatCℓ(V,Q)inseveralways,startingwiththecategoricaldefinition.
                          jAllourassociativealgebrasareunital,unlessotherwisestated!
                  1.2  TheCliffordalgebra,categorically
                  Let(V,Q)beaquadraticvectorspaceandletAbeanassociativeK-algebra. WesaythataK-linearmap
                  φ:V→AisCliffordifforallx∈V,
                                                        2
                  (2)                               φ(x) =−Q(x)1A,
                  where 1A is the unit of A. Clifford maps from a fixed quadratic vector space (V,Q) are the objects of a
                  categoryCliff(V,Q),whereamorphismfromV→AtoV→A′isgivenbyacommutingtriangle
                  (3)                                     V
                                                         f     ′
                                                   A             //A
                  with f :A→A′ ahomomorphismofassociativealgebras.