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Geometric Structures and Special Spinor Fields

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Geometric Structures and Special Spinor Fields Geometric structures and special spinor fields Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.) am Fachbereich der Mathematik und Informatik Philipps-Universit¨atMarburg (Hochschulkennziffer 1180) von Dipl. Math. Jos H¨oll geboren am 13.05.1984 in Herrenberg Erstgutachter: Prof. Dr. habil Ilka Agricola (Universit¨atMarburg) Zweitgutachter: Prof. Dr. Stefan Ivanov (University of Sofia) Eingereicht am 31.07.2014 M¨undliche Pr¨ufungam 17.10.2014 Marburg, 2014 Geometric structures and special spinor fields Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.) am Fachbereich der Mathematik und Informatik Philipps-Universit¨atMarburg (Hochschulkennziffer 1180) von Dipl. Math. Jos H¨oll geboren am 13.05.1984 in Herrenberg Erstgutachter: Prof. Dr. habil Ilka Agricola (Universit¨atMarburg) Zweitgutachter: Prof. Dr. Stefan Ivanov (University of Sofia) Eingereicht am 31.07.2014 M¨undliche Pr¨ufungam 17.10.2014 Marburg, 2014 iii Deutsche Zusammenfassung Gegen Ende der 1920er Jahre legte Paul Dirac den Grundstein f¨urdie Entwicklung von Spinoren und dem nat¨urlicher Weise auf ihnen operierenden Differentialoperator, dem Dirac-Operator ([Di28a, Di28b]). Seit diesem Zeitpunkt spielen Spinoren, die spezielle Differentialgleichungen erf¨ulleneine große Rolle in Physik und Mathematik. Es stellte sich heraus, dass spezielle Spinoren nur auf bestimmten Typen von Mannigfaltigkeiten existieren k¨onnen.Es wurden Korresponden- zen zwischen Differenzialgleichungen f¨urSpinoren und Typen von Mannigfaltigkeiten entdeckt. Einige wichtige Zusammenh¨angesind in der folgenden Liste aufgef¨uhrt. • Parallele Spinoren existieren nur auf Ricci-flachen R¨aumen. • Der Index des Dirac-Operators ist gleich dem (rein topologischen) A^ Geschlecht [AS62]. • Parallele Spinoren erfordern Holonomie Sp(n), SU(n), G2 und Spin(7) [Wa89]. • Korrespondenzen zwischen Killing-Spinoren und geometrischen Strukturen [FK89, FK90, Gr90]. Wir werden die im letzten Punkt genannten Killing-Spinoren verallgemeinern, wie es beispiels- weise schon in [FK01, BGM05] getan wurde, um damit neue Zusammenh¨angemit geometrischen Strukturen herzustellen. So korrespondieren zum Beispiel in Dimension 6 die generalisierten Killing-Spinoren, die f¨urden Levi-Civita Zusammenhang r mit einem symmetrischen Endomor- phismus S durch die Gleichung rX ϕ = S(X) · ϕ beschrieben werden, zu halb-flachen SU(3)-Strukturen. Ebenso ergeben sich Korrespondenzen zum Kern des Dirac-Operators, wie sie in der Physik von großem Interesse sind; in [CCD03] wurden beispielsweise die Restriktionen f¨ureben diese Korrespondenz bereits betrachtet. Im ersten Kapitel werden die geometrischen Strukturen eingef¨uhrt,die in dieser Arbeit behandelt werden und die oben genannten Korrespondenzen zu Spinoren untersucht. Im zweiten Kapitel wenden wir uns der Hyperfl¨achentheorie zu. Ein Spinor ϕ auf einer Mannigfaltigkeit M¯ mit Hyperfl¨ache M ⊂ M¯ erf¨ulltf¨urdie zugeh¨origenLevi-Civita Zusammenh¨ange r und r¯ die Gleichung 1 r¯ ϕ = r ϕ − W (X) · ϕ, (1) X X 2 wobei W die Weingarten-Abbildung ist (z.B. [BGM05]). Dies gibt uns die M¨oglichkeit, Zusam- menh¨angevon Spinoren und durch den ersten Teil der Arbeit damit eben auch zwischen geo- metrischen Strukturen auf M und M¯ herzustellen. Die meisten geometrischen Strukturen tragen einen charakteristischen Zusammenhang rc mit Torsion T , der metrisch ist, die Struktur erh¨altund außerdem die gleichen Geod¨atenwie der Levi-Civita Zusammenhang r besitzt. Wir werden diese Zusammnh¨angenutzen, um Korrespon- denzen von geometrischen Strukturen auf Untermannigfaltigkeiten zu schaffen, die nicht durch Spinoren gegeben sind. Generalisierte Killing-Spinoren, wie sie oben beschrieben wurden, und Killing-Spinoren mit Tor- sion, definiert durch die Gleichung rs · X ϕ = αX ϕ (2) f¨urden Zusammenhang rs = r + 2sT , wobei r der Levi-Civita Zusammenhang und T die charakteristische Torsion ist, sind von immer gr¨oßererBedeutung (siehe z.B. [ABBK13, FK01, BGM05]). Wir werden charakteristische Zusammenh¨angenutzen, um solche Spinoren auf M und M¯ zu untersuchen. In Abschnitt 1 des ersten Kapitels werden metrische fast-Kontakt-Strukturen eingef¨uhrt. Hier wird an die Klassifikation solcher Strukturen erinnert und außerdem ein n¨utzliches Kriterium iv f¨urdie Existenz eines charakteristischen Zusammenhangs gegeben. Wir nutzen dieses Kri- terium, um zu zeigen, dass in der Klasse C13467 der Klassifikation von Chinea und Gonzalez ([CG90]) ein charakteristischer Zusammenhang existiert, nicht aber in C2, C5, C9, C10, C11 oder C12 (Theorem 1.3). Als n¨achstes werden in Abschnitt 2 fast-hermitsche Strukturen definiert und ebenfalls deren Klas- sifikation sowie das bekannte Kriterium ([FI02]) zur Existenz charakteristischer Zusammenh¨ange wiederholt. Im speziellen Fall der Dimension 6 werden in Abschnitt 3 SU(3)-Strukturen eingef¨uhrt.Da die Gruppe SU(3) ⊂ Spin(6) der Stabilisator eines Spinors ist, k¨onnenhier Korrespondenzen zwis- chen dem definierenden Spinor und der resultierenden Struktur geschaffen werden. So wird hier die Klassifikation der SU(3)-Strukturen in spinorielle Gleichungen ¨ubersetzt. Es werden Gr¨oßen wie der Nijenhuis-Tensor und der Dirac-Operator betrachtet. So ist beispielsweise das Kodif- ferential der K¨ahlerformdurch den Dirac-Operator D und die klassifizierende 1-Form η gegeben mittels δ!(X) = 2[(Dϕ, X · j · ϕ) − η(X)]: Zudem wird eine sehr interessante Klasse von SU(3)-Zusammenh¨angenbetrachtet sowie eine spinorielle Bedingung f¨urdie Existenz eines charakteristischen Zusammenhangs entwickelt. Ahnlich¨ wie im vorherigen Abschnitt wird in Abschnitt 4 eine Korrespondenz zwischen Spinoren und G2-Strukturen in Dimension 7 geschaffen. Beispielsweise ist ein Spinor der L¨ange1 genau dann im Kern des Dirac-Operators, wenn die zugeh¨orige G2-Struktur aus der Klasse W23 ist. Wie im vorherigen Abschnitt in Dimension 6, werden auch hier an die G2-Geometrie angepasste Zusammenh¨angen¨ahererl¨autert. In Abschnitt 5 werden kurz Spin(7)-Strukturen in Dimension 8 eingef¨uhrt. Da ein Spinor in dieser Dimension jedoch nicht immer die Gruppe Spin(7) als Stabilisator besitzt, kann hier keine Korrespondenz wie in den obigen F¨allengegeben werden. Im zweiten Kapitel liegt der Fokus auf der Hyperfl¨achentheorie. Tr¨agteine Mannigfaltigkeit M¯ mit Hyperfl¨ache M eine bestimmte G-Struktur so l¨asstsich diese in eine andere geometrische Struktur auf M ¨uberf¨uhren.Mittels der Beziehung aus Gleichung (1) und der jeweilig definieren- den Spinoren wird im Fall dim M = 6 und dim M¯ = 7 in Abschnitt 1 eine Korrespondenz zwischen G2-Strukturen und deren SU(3)-Hyperfl¨achen ausgearbeitet. Es werden außerdem verallgemein- erte Killing-Spinoren mit Torsion (eine Verallgemeinerung der Gleichung (2)) eingef¨uhrtund deren Korrespondenzen auf M und M¯ bestimmt. Wir nutzen die Kegelkonstruktion, um aus einer SU(3)-Mannigfaltigkeit eine G2-Mannigfaltigkeit eines bestimmten Typs zu konstruieren. Außerdem werden Beziehungen zwischen generalisierten Killing-Spinoren mit Torsion auf M und M¯ geschaffen. Aus dem oben im Fall der Spin(7)-Struktur erw¨ahnten Grund lassen sich in anderen Dimen- sionen als dim M = 6 und dim M¯ = 7 mittels Spinoren keine Beziehungen f¨urdie von uns betrachteten geometrischen Strukturen auf Hperfl¨achen herstellen. Daher werden in Abschnitt 2 Zusammenh¨angemit Torsion betrachtet. Diese sind im Folgenden ein wichtiges Werkzeug f¨urdie Untersuchung von Korrespondenzen zwischen geometrischen Strukturen auf M und dem Kegel M¯ , selbst wenn diese Strukturen nicht durch Spinoren gegeben sind. In [B¨a93]wird diese Konstruktion benutzt, um eine Beziehung zwischen Riemannschen Killing-Spinoren und der geometrischen Struktur zu schaffen. Wir benutzen diese Zusammenh¨angeaußerdem, um Killing-Spinoren mit Torsion auf Hyperfl¨achen zu betrachten. So wird in Abschnitt 3 auf dem Kegel einer fast-Kontakt-Struktur eine fast-hermitsche Struktur konstruiert und die jeweiligen Klassifikationen mit einander in Beziehung gesetzt. Wir zeigen beispielsweise, dass eine α-Sasaki Struktur zu einer lokal-konform-K¨ahlerStruktur in Beziehung steht oder dass die beiden Nijenhuis-Tensoren die Gleichen sind. Wir benutzen die in Abschnitt 2 geschaffenen Korrespondenzen zwischen Spinoren, um zu zeigen, dass ein Killing-Spinor mit Torsion auf (M; g) einem Spinor ϕ auf dem Kegel (M;¯ g¯) = (M × R; a2r2g + dr2), der die Bedin- v gung 1 r¯ c ϕ + (Xy(@ y!) ^ !)ϕ = 0 X 2r r erf¨ullt,entspricht. Hier ist ! die K¨ahler-Form und r¯ c der charakteristische Zusammenhang auf M¯ . In Abschnitt 3.4 l¨asstsich anhand der Tatsache, dass die aus den 3 fast-Kontakt-Strukturen kon- struierten Zusammenh¨angenicht die selben sind, nicht wie in den ¨ubrigenAbschnitten verfahren und wir belassen es bei einer kurzen Betrachtung der Situation. Im Falle einer G2-Struktur auf einer 7 dimensionalen Mannigfaltigkeit M tr¨agtder Kegel eine Spin(7)-Struktur. Dieser Fall sowie die Korrespondenz der Klassifikationen und der generali- sierten Killing-Spinoren mit Torsion ist in Abschnitt 4 ausgearbeitet. Hier wird beispielsweise bewiesen, dass eine Spin(7)-Struktur der Klasse U1 (bzw. U2) auf dem Kegel eine G2 Struktur auf M induziert, die niemals aus der Klasse W34 (bzw. W13) sein kann. Einen Teil dieser Ergebnisse (im Wesentlichen sind dies die Ergebnisse des Abschnittes 1 aus Kapitel I sowie die Abschnitte 2, 3 und 4 aus Kapitel II) haben wir bereits in [AH13] publiziert. vii Danksagung Zu aller erst gilt mein Dank meiner Betreuerin Ilka Agricola. Sie zeigte mir den Weg zu einer erfolgreichen Promotion und unterst¨utztemich umfassend
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