The Geometry and Physics of Abelian Gauge Groups in F-Theory

309 Pages·2015·2.2 MB·English

by  Jan Keitel

The Geometry and Physics of
Abelian Gauge Groups in F-Theory
Jan Keitel
Mu¨nchen 2015

The Geometry and Physics of
Abelian Gauge Groups in F-Theory
Jan Keitel
Dissertation
an der Fakult¨at fu¨r Physik
der Ludwig-Maximilians-Universita¨t
Mu¨nchen
vorgelegt von
Jan Keitel
aus Oberursel
Mu¨nchen, 2015
Erstgutachter: Prof. Dr. Dieter Lu¨st
Zweitgutachter: PD Dr. Ralph Blumenhagen
Tag der mu¨ndlichen Pru¨fung: 14.07.2015
Zusammenfassung
Diese Arbeit befasst sich mit der Geometrie und den effektiven physikalischen Theorien Abel-
scher Eichgruppen in F-Theorie-Kompaktifizierungen.
Um passende Calabi-Yau Mannigfaltigkeiten mit Torus-Faserung zu konstruieren, nutzen
wir Methoden der torischen Geometrie. Wir bestimmen Komponenten dieser Calabi-Yau-
Mannigfaltigkeiten, die dazu geeignet sind, unabh¨angig voneinander untersucht zu werden.
Dies erlaubt die Entwicklung von Methoden zur Konstruktion großer Zahlen von Mannigfal-
tigkeiten, die zu gegebenen Eichgruppen fu¨hren. In dem selben Rahmen erreichen wir eine
teilweiseKlassifizierungtorischerEichgruppen.Wirzeigen,dassderFeldinhaltdergew¨ohnlich
betrachtetenF-Theorie-ModellestarkenEinschr¨ankungenunterliegt.UmdieseBegrenzungen
zu umgehen, entwickeln wir einen Algorithmus mittels dessen wir Torus-Faserungen, die als
“complete intersections” definiert sind, untersuchen k¨onnen. Unter Benutzung dieses Algo-
rithmus entdecken wir mehrere neuartige F-Theorie-Kompaktifizierungen. Zuletzt zeigen wir,
wie Torus-Faserungen ohne Schnitt durch ein Netzwerk sukzessiver geometrischer U¨berg¨ange
mit Faserungen mit mehreren Schnitten verbunden werden k¨onnen.
Um die effektive Physik solcher Kompaktifizierungen bei niedrigen Energien zu untersu-
chen, nutzen wir die Dualit¨at zwischen M-Theorie und F-Theorie. Nach der Bestimmung der
effektiven Wirkung von F-Theorie mit Abelschen Eichgruppen in sechs Dimensionen verglei-
chen wir die quantenkorrigierten Chern-Simons-Kopplungen mit topologischen Gr¨oßen der
Kompaktifizierungsmannigfaltigkeit. Dies erlaubt es uns, den Materieinhalt der Theorien zu
bestimmen. Unter bestimmten Bedingungen beweisen wir, dass gravitative und gemischte
Anomalien in F-Theorie automatisch abwesend sind. Weiterhin berechnen wir die effektive
Wirkung von F-Theorie-Kompaktifizierungen ohne Schnitt und schlagen vor, dass die Abwe-
senheiteinessolchenSchnittsdiePr¨asenzeineszus¨atzlichenmassivenEichfeldeszurFolgehat.
ZuletztzeigenwirdurchAusweitungunsererAnalyseaufvierDimensionen,dassU¨berbleibsel
dieses massiven Eichfeldes sich in diskreten Symmetrien und entsprechenden Auswahlregeln
fu¨r die Yukawa-Kopplungen der effektiven Theorie auswirken.

Abstract
In this thesis we study the geometry and the low-energy effective physics associated with
Abelian gauge groups in F-theory compactifications.
To construct suitable torus-fibered Calabi-Yau manifolds, we employ the framework of
toric geometry. By identifying appropriate building blocks of Calabi-Yau manifolds that
can be studied independently, we devise a method to engineer large numbers of manifolds
that give rise to a specified gauge group and achieve a partial classification of toric gauge
groups. Extending our analysis from gauge groups to matter spectra, we prove that the
matter content of the most commonly studied F-theory set-ups is rather constrained. To
circumvent such limitations, we introduce an algorithm to analyze torus-fibrations defined as
complete intersections and present several novel kinds of F-theory compactifications. Finally,
we show how torus-fibrations without section are linked to fibrations with multiple sections
through a network of successive geometric transitions.
In order to investigate the low-energy effective physics resulting from our compactifica-
tions, we apply M- to F-theory duality. After determining the effective action of F-theory
with Abelian gauge groups in six dimensions, we compare the loop-corrected Chern-Simons
terms to topological quantities of the compactification manifold to read off the massless mat-
ter content. Under certain assumptions, we show that all gravitational and mixed anomalies
are automatically canceled in F-theory. Furthermore, we compute the low-energy effective
action of F-theory compactifications without section and suggest that the absence of a sec-
tion signals the presence of an additional massive Abelian gauge field. Adjusting our analysis
to four dimensions, we show that remnants of this massive gauge field survive as discrete
symmetries that impose selection rules on the Yukawa couplings of the effective theory.

Acknowledgments
First and foremost, I would like to express my deep gratitude to Thomas W. Grimm for
taking me on as a member of his group and providing a level of supervision that I have rarely
seen elsewhere. Not only did he immediately suggest research problems tailored towards my
interests to me and constantly provided invaluable support with them, but he also initiated
severalcollaborationsfromwhichIhave hugelybenefitedandthatIhave thoroughlyenjoyed.
Thomashasmadeanenormousefforttoequipmewiththenecessaryskillstoperformresearch
on my own and has never faltered in his support, regardless of my own professional choices.
I am very grateful to Dieter Lu¨st for providing such a productive and friendly environment
to do research at this institute and for kindly offering to be my official supervisor and first
referee at the Ludwig-Maximilians-Universit¨at. Furthermore, I am deeply indebted to Ralph
Blumenhagen for valuable discussions over the course of my PhD and for generously agreeing
to serve as my second referee.
Most of what I know about toric geometry I have learned from Volker Braun and I
would like to sincerely thank him for his expert advise and the many hours he patiently and
cheerfully spent explaining mathematics and Sage to me. A special thanks goes to In˜aki
Garc´ıa-Etxebarria as well as to Lara B. Anderson, Andreas Kapfer, Raffaele Savelli, and
Matthias Weissenbacher for interesting and fruitful collaborations. Finally, I would like to
thank Federico Bonetti, Tom G. Pugh, and Diego Regalado for the numerous discussions and
the many enjoyable times that we have had in our group.
Last, but most certainly not least, I am tremendously grateful to my parents and my
brother for the continuous and unwavering support that I have always received.