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Modularity and Vacua in N = 1* Supersymmetric Gauge Theory Antoine Bourget

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Modularity and Vacua in N = 1* Supersymmetric Gauge Theory Antoine Bourget Modularity and vacua in N = 1* supersymmetric gauge theory Antoine Bourget To cite this version: Antoine Bourget. Modularity and vacua in N = 1* supersymmetric gauge theory. Physics [physics]. Université Paris sciences et lettres, 2016. English. NNT : 2016PSLEE011. tel-01427599 HAL Id: tel-01427599 https://tel.archives-ouvertes.fr/tel-01427599 Submitted on 5 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE DE DOCTORAT DE L’ÉCOLE NORMALE SUPÉRIEURE Spécialité : Physique École doctorale : Physique en Île-de-France réalisée au Laboratoire de Physique Théorique de l’ENS présentée par Antoine BOURGET pour obtenir le grade de : DOCTEUR DE L’ÉCOLE NORMALE SUPÉRIEURE Sujet de la thèse : Vides et Modularité dans les théories de jauge supersymétriques = 1∗ N soutenue le 01/07/2016 devant le jury composé de : M. Ofer Aharony Rapporteur M. Costas Bachas Examinateur M. Amihay Hanany Rapporteur Mme Michela Petrini Examinateur M. Henning Samtleben Examinateur M. S. Prem Kumar Examinateur M. Jan Troost Directeur de thèse Résumé Nous explorons la structure des vides dans une déformation massive de la théorie de Yang-Mills maximalement supersymétrique en quatre dimensions. Sur un espace-temps topologiquement trivial, la théorie des orbites nilpotentes dans les algèbres de Lie rend possible le calcul exact de l’indice de Witten. Nous en donnons les fonctions génératrices pour les algèbres classiques, et recourons à un calcul explicite pour les exceptionnelles. Après compactification sur un cercle, un lien entre les théories de jauge supersymétriques et les systèmes intégrables est exploitable pour réduire la chasse aux vides à une extrémisation du hamiltonien de Calogero-Moser elliptique twisté. Une analyse soigneuse des propriétés globales du groupe de jauge et des opérateurs de ligne est nécessaire pour obtenir un accord parfait. En combinant exploration numérique sur ordinateur et contrôle analytique grâce à la théorie des formes modulaires, nous exhibons la structure des vides massifs pour des algèbres de rang petit, et mettons en évidence de nouvelles propriétés modulaires. Nous montrons que des branches de vides de masse nulle existent, et nous en donnons la structure exacte pour les algèbres de rang deux. Mots clés : Théories de jauge supersymétriques, Systèmes intégrables, Modularité 1 2 Abstract We investigate the vacuum structure of a massive deformation of the maximally supersymmetric Yang-Mills gauge theory in four dimensions. When the topology of spacetime is trivial, the Witten index can be computed exactly for any gauge group using the theory of nilpotent orbits in Lie algebras. We provide generating functions for classical algebras and an explicit calculation for the exceptional ones. Upon compactification on a circle, one can use a bridge between supersymmetric gauge theories and complex integrable systems to reduce the analysis of vacua to the search of extrema of the twisted elliptic Calogero-Moser Hamiltonian. A careful inspection of global properties of the gauge group and line operators are needed to reach total agreement. Using a combination of numerical exploration on a computer and analytical control through the theory of modular forms, we determine the structure of massive vacua for low-rank gauge algebras and exhibit new modular properties. We also show that massless branches of vacua can exist, and provide an analytic description for rank two gauge algebras. Keywords : Supersymmetric gauge theories, Integrable systems, Modularity 3 4 Preface This work is based on three articles [1, 2, 3] by Jan Troost and myself that have been published in the Journal of High Energy Physics in 2015 and 2016. Large portions of these articles have been used with only formatting changes with the agreement of Jan Troost. The table 1 gives a list of sections of this thesis that are reproduced from the three papers. While I have chosen to focus this document on the content of the three aforementioned articles, I have also worked on other topics with Jan during the last three years [4, 5, 6]. Sections in this document [1][2][3] Sections 2.5 and 2.6 × Section 3.3 × Section 3.4 × Chapter 4 × Section 5.3 × Chapter 6 and 7 × Table 1: The sections listed here are in essence taken from the indicated papers. The chapter associated to number 0 is roughly a translation in French of the presentation chapter 1, supplemented with some elements from other chapters and the conclusion. 5 6 Remerciements Dans le travail qui est présenté ici, rien n’aurait vu le jour sans mon directeur de thèse, Jan Troost, qui a rempli son rôle avec une conscience et une présence sans égales. Il a su m’orienter dans l’univers fascinant des théories de jauge supersymétriques, sans toutefois me laisser accroire que l’on peut se passer de la théorie des cordes. Il m’a communiqué son enthousiasme ainsi que son audace inspirée face aux abstractions de la physique théorique. Et surtout, il m’a consacré un temps démesuré tout au long de cette thèse, de la rédaction des articles à la relecture de ce manuscrit, en passant par les longues discussions devant le tableau noir ou le mardi au ministère sur tous les sujets. Je conserverai un souvenir ému de cette collaboration. Je remercie également tous les membres du Laboratoire de Physique Théorique de l’ENS pour m’avoir hébergé, et en particulier son directeur Costas Bachas qui m’a toujours soutenu et qui a accepté de faire partie de mon jury, Jean Iliopoulos qui m’a, le premier, introduit aux mystères de la théorie quantique des champs, Denis Bernard dont le spectre des précieux conseils s’est étalé des systèmes intégrables à l’orientation professionnelle, Marc-Thierry Jaekel qui m’a sauvé plus d’une fois de l’anéantissement numérique, ainsi que Sandrine Patacchini et Viviane Sebille pour leur aide efficace et chaleureuse. Merci également à tous les grands physiciens qui m’ont tant apporté au cours des dernières années, à Paris ou lors des écoles et conférences auxquelles j’ai eu la chance d’assister, et en particulier à Jean-Bernard Zuber, Massimo Bianchi, Amihay Hanany et Sébastien Balibar. Je remercie enfin les autres membres du jury ayant accepté de lire et d’évaluer ce travail, Michela Petrini, Prem Kumar et Henning Samtleben, qui faisant fi du Brexit et des risques de grève, sont pour certains venus de loin pour participer à ma soutenance de thèse. J’ai une pensée particulière pour Ofer Aharony qui, bien qu’il n’ait pas pu se rendre à Paris, s’est plongé en profondeur dans ce manuscrit et a formulé des commentaires inappréciables et des questions pertinentes. J’ai eu le réel privilège de côtoyer des doctorants et post-doctorants exceptionnels au cours des trois dernières années, et je me dois d’être reconnaissant au CPER1 pour m’avoir fourni l’occasion d’avoir autant d’estimables collègues de bureau. L’attribution à chacun ce qu’il m’a 2 apporté dépassant en difficulté la détermination de la structure des vides de E8 , je me contente de laisser cette tâche en exercice au lecteur, et de remercier par ordre d’apparition Jonathan, Sophie, Alice, Dario, Antoine, Thibaud, Bruno, Éric, Thimothée, et Mathieu pour, en vrac, les lundis-piscine, les demi-poulets, les places VIP au debating, les recommandations avisées de blogs, livres, et séries télé, l’eau chaude, Physique pour tous, le sympathique coin sieste du bureau 1La réelle signification de ce sigle peut et doit rester obscure, les intéressés comprendront. 2Non traité dans cette thèse. 7 8 obscur, ainsi que le château d’eau et la poudre du lumineux, les insectes séchés et les moutons, ShareLatex, les nombreux débats parfois enflammés – avec nos sincères excuses à Amir – ; merci de m’avoir éduqué sur les monuments du cyclisme ou sur comment on fait les bébés, en passant par les sous-préfectures ; merci pour l’assertivité savoureuse de certains, mais aussi pour votre tolérance affable envers mes maintes obsessions compulsives, et enfin merci pour l’atmosphère de travail enrichissante et infiniment stimulante que vous avez créée. Et par delà la rue Lhomond, ces trois années de pérégrinations n’auront pas été solitaires car d’autres ont gravi de concert les pentes du doctorat, en particulier Ricardo qu’anime la passion universelle, Jean-Baptiste qui restera toujours délicieux, Siavash qui m’a fait voir Paris autrement, Amaury qui m’a expliqué le monde, Grégoire pour nos inénarrables débats, Ruoting pour nos pots interdisciplinaires, Harold qui m’a prodigué des avis inestimables, mais aussi Shira, Lætitia, Corinne, Jonathan, Jie, Nicolas. Et bien sûr, je remercie la cohorte de mes amis de plus longue date qui m’ont pardonné mes errements théoriques, depuis nos années d’études à Paris, Palaiseau, ou même pour certains depuis l’école primaire. Je ne peux tous les nommer ici, mais s’ils lisent ces lignes, ils sont plus que probablement concernés. Enfin, mes parents et ma famille m’ont apporté tout au long de cette thèse, tout comme au cours des 24 années précédentes, un soutien indéfectible et une affection profonde, et ils sont pour plus qu’il ne le pensent sans doute dans l’achèvement de cette thèse. Au delà de l’aide matérielle de tous les instants pourtant déjà si cruciale, merci surtout à eux d’avoir fait de Sainte-Terre, de Meylan et de la rue Saint-Ferdinand les lieux qui demeurent pour moi les plus chers.
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