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Current Algebra, Generalised Geometry and Integrable Models in String Theory

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Current Algebra, Generalised Geometry and Integrable Models in String Theory Current algebra, generalised geometry and integrable models in string theory David Osten Munchen¨ 2020 Current algebra, generalised geometry and integrable models in string theory David Osten Dissertation an der Fakultat¨ fur¨ Physik der Ludwig-Maximilians-Universitat¨ Munchen¨ vorgelegt von David Osten geb. in Altenburg/Thur.¨ Munchen,¨ den 9. Juni 2020 Treffen sich ein Stein und ein Brett. Datum der mundlichen¨ Prufung:¨ 23.7.2020 Erstgutachter: Prof. Dr. Dieter Lust¨ Zweitgutachter: PD Dr. Ralph Blumenhagen Vorsitzender: Prof. Dr. Thomas Kuhr Weiteres Mitglied: Prof. Dr. Matthias Punk Und darum: Hoch lebe die Physik! Und hoher¨ noch das, was uns zu ihr zwingt, – unsre Redlichkeit! Friedrich Nietzsche, aus: Die frohliche¨ Wissenschaft i ii Zusammenfassung Eine Konsequenz der ausgedehnten Natur des Strings ist, dass in der Stringtheorie allgemeinere Hintergrundgeometrien als (Riemannsche) Mannigfaltigkeiten moglich¨ sind. Insbesondere bei Kartenwechseln sind nicht nur Diffeomorphismen (oder an- dere Eichtransformationen) erlaubt, sondern auch (String-)Dualitatstransformationen.¨ Solche Raume¨ werden auch nicht-geometrische Raume¨ genannt. Ihre mathematische Formulierung basiert auf Hitchins und Gualtieris Verallgemeinerter Geometrie. In dieser Arbeit wird gezeigt, dass entgegen bisheriger Resulte in der Literatur die Poisson-Struktur, genauer die Stromalgebra, eines Strings nicht O(d, d)-invariant ist und deren korrekte Beschreibung so genannte para-Hermitesche Geometrie benotigt.¨ Darauf aufbauend wird eine Hamiltonsche Formulierung von klassischer Stringth- eorie in einem generischen, geometrischen oder nicht-geometrischen Hintergrund vor- geschlagen. Die Essenz dieser Formulierung ist eine Deformation der Stromalgebra, die durch die verallgemeinerten Flusse,¨ die einen solchen Hintergrund beschreiben, charakterisiert wird. Diese Formulierung ist allgemeiner als die durch eine Lagrange/- dichte, da zum Beispiel magnetisch geladene Hintegrunde¨ und solche, die die Sek- tionsbedingung der Verallgemeinerten Geometrie verletzen, hier auch diskutiert wer- den konnen¨ – auf Kosten der Verletzung der Jacobi-Identitat¨ der Stromalgebra. Zwei Anwendungen dieser Formulierung werden diskutiert: Zum einen kann man aus der deformierten Stromalgebra direkt die nicht-kommutative und nicht-assoziative Interpretation der nicht-geometrischen Hintergrunde¨ abgelesen. Zum anderen konnen¨ zweierlei Verallgemeinerungen von nicht-Abelscher T-Dualitat¨ uber¨ Poisson-Lie T-Dua- litat¨ hinaus abgeleitet werden. Es existiert eine nicht-Abelsche T-Dualitatsgruppe,¨ ana- log zu O(d, d) fur¨ Abelsche T-Dualitat.¨ Außerdem existieren Versionen von Poisson- Lie-Dualitat¨ fur¨ Modelle mit generischen konstanten Verallgemeinerten Flussen.¨ Eine Verallgemeinerung dieser Ergebnisse fur¨ M-branen in M-Theorie scheint mog-¨ lich. Fur¨ eine M2-bran in M-Theorie in vier Dimensionen wird gezeigt, dass die Stro- malgebra nicht dualitatsinvariant¨ ist und genauso wie im Falle des Strings eine Lie- Klammer beinhaltet, die in einer para-Hermiteschen Version von exzeptioneller Verall- gemeinerter Geometrie auftaucht. Im Unterschied zur Diskussion des Strings kann Ko- varianz unter der Dualitatsgruppe,¨ hier SL(5), nur durch die Einfuhrung¨ zusatzlicher¨ Objekte, der Membranladungen, wiederhergestellt werden. Mit Hilfe der typischen doppelten dimensionalen Reduktion von M-Theorie zur Typ IIa Superstringtheorie kann man die M2-bran- und Stringstrome¨ miteinander in Beziehung setzen. Ein anderes zentrales Thema sind integrable Modelle im Kontext von Stringtheo- rie. In besonders symmetrischen Hintergrunden,¨ wie der Minkowski-Raumzeit oder bestimmten Anti-de Sitter-Kompaktifizierungen, ist Stringtheorie exakt losbar¨ (inte- grabel). Deformationen dieser Hintergrunde,¨ die die Integrabilitat¨ beibehalten, wur- den in den letzten Jahren ausfuhrlich¨ untersucht. Es stellt sich heraus, dass viele diese Deformationen klare Entsprechungen in der Verallgemeinerten Geometrie haben. In dieser Arbeit wird gezeigt, dass eine große Klasse dieser Deformationen, die homo- genen Yang-Baxter-Deformationen, nichts weiter sind als die b-Transformationen der oben erwahnten¨ nicht-Abelschen T-Dualitatsgruppe.¨ iii iv Abstract One consequence of the extended nature of the string is that more general background geometries than Riemannian manifolds are possible in string theory. In particular, when gluing charts not only diffeomorphisms (or other gauge transformations of the background) but also (string) duality transformations are allowed. These geometries are called non-geometric spaces. Their mathematical formulation is based on Hitchin’s and Gualtieri’s generalised (or O(d, d)-) geometry. In this thesis it is shown that, despite previous results in the literature, the Poisson structure – to be more precise: the current algebra – of a string is not O(d, d)-invariant. Its correct treatment requires the so-called para-Hermitian geometry. Building on that, a Hamiltonian formulation of the classical world-sheet theory in a generic, geometric or non-geometric, background is proposed. The essence of this formulation is that the generalised fluxes, characterising such a background, describe a deformation of the current algebra. This formulation extends to backgrounds for which there is no Lagrangian description of the world-sheet theory – namely magnet- ically charged backgrounds and those that violate the section condition of generalised geometry, at the cost of violating the Jacobi identity of the current algebra. Two applications of this formulation are discussed. On the one hand, one can read off the non-commutative and non-associative interpretation directly from the deformed current algebra. On the other hand, one can derive two generalisations of non-abelian T-duality that go beyond the standard factorised Poisson-Lie T-duality. There is a non- abelian T-duality group, analogous to O(d, d) for abelian T-duality. Moreover, there are generalisations for Poisson-Lie T-duality for models with generic constant generalised fluxes. A generalisation of these results to M-branes in M-theory seems possible. For the membrane in M-theory compactified on a four-dimensional space, it is shown that the current algebra is not U-duality invariant. Exactly as for the string, a Lie bracket ap- pears that is connected to para-Hermitian exceptional generalised geometry. In contrast to the string, even manifest covariance under the U-duality group, here SL(5), is only possible when introducing additional objects, the membrane charges. With the help of the typical double dimensional reduction from M-theory to type IIa superstring theory, one can relate the membrane and string currents. Another central topic of this thesis is integrability in context of string theory. In par- ticularly symmetric backgrounds, like Minkowski spacetime or certain Anti-de Sitter compactifications, string theory is exactly solvable (integrable). Deformations of these backgrounds, that preserve integrability of the world-sheet theory, have been studied extensively in the last years. It turned out that many of these deformations can be de- scribed in terms of generalised geometry. In this thesis it is shown that a big class of these deformations, the homogeneous Yang-Baxter deformations, are nothing else than the b-shifts of the non-abelian T-duality group mentioned above. v vi Acknowledgements The support of my family and my friends is very valuable to me – I am immensely grateful for it. I would like to thank my supervisor Dieter Lust¨ for giving me the opportunity to work in his group, his scientific insights, his support whenever it was needed and his trust in my ideas and my work. I also thank Ralph Blumenhagen for interesting discus- sions and agreeing to referee my thesis. Special thanks goes to my (full- and part-time) office mates Henk Bart, Daniel Klawer,¨ Andriana Makridou, Tobias Stirner and Florian Wolf for the enjoyable and inspiring pastime, and also to all the other members in the Munich string theory group, among others: Saskia Demulder, Veronica Errasti Diez, Michael Fuchs, Michael Haack, Daniela Herschmann, Marvin Luben,¨ Chrysoula Markou, Erik Plauschinn, Felix Rudolph, Dim- itros Skliros and Matthias Traube. The work at the Max-Planck institute and the IMPRS program would not have been as enjoyable if it wouldn’t have been for the work behind the scenes – I thank Monika Goldammer, Thomas Hahn, Frank Steffen, Annette Sturm and all the administrative staff of the MPP. I thank Henk Bart, Franz Herling and Andriana Makridou for proofreading parts of the thesis. vii viii Contents I Introduction1 II Foundations 13 1 Strings 15 1.1 Bosonic strings in flat space.......................... 15 1.1.1 Actions and symmetries........................ 15 1.1.2 Constraints and brackets........................ 17 1.1.3 Quantisation.............................. 19 1.2 Non-linear s-models and the geometric paradigm............. 21 1.2.1 String s-models............................. 22 1.2.2 The geometric paradigm........................ 24 1.3 Superstrings and fluxes............................. 25 1.3.1 Supersymmetric strings........................ 25 1.3.2 Superstring s-models......................... 30 2 Duality 33 2.1 Duality in high energy physics........................ 33 2.1.1 Symmetries, emergence, duality – a disambiguation........ 33 2.1.2 Examples................................ 35 2.2 T-duality..................................... 38 2.2.1 Strings on S1 ............................... 38 2.2.2 Non-linear s-models and abelian
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