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Composite Anomaly in Supergravity and String Amplitude Comparison Soumya Sasmal Composite Anomaly in Supergravity and String Amplitude Comparison Soumya Sasmal To cite this version: Soumya Sasmal. Composite Anomaly in Supergravity and String Amplitude Comparison. High Energy Physics - Theory [hep-th]. Université Paris Saclay (COmUE), 2017. English. NNT : 2017SACLS198. tel-01614382v2 HAL Id: tel-01614382 https://tel.archives-ouvertes.fr/tel-01614382v2 Submitted on 11 Oct 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. NNT : 2017SACLS198 THÈSE DE DOCTORAT DE L’UNIVERSITÉ PARIS-SACLAY PRÉPARÉE À L’UNIVERSITÉ PARIS-SUD AU SEIN DE L’INSTITUT DE PHYSIQUE THÉORIQUE, CEA, SACLAY Ecole doctorale n◦564 Ecole Doctorale Physique en Ile-de-France Spécialité de doctorat: Physique par M. SOUMYA SASMAL Composite Anomaly in Supergravity and String Amplitude Comparison Thèse présentée et soutenue à l’Institut de Physique Théorique, CEA, Saclay, le 06 Septembre 2017. Composition du Jury : M. ULRICH ELLWANGER Professeur (Président du jury) LPT Orsay M. CARLO ANGELANTONJ Chargé de recherche (Rapporteur) Turin University M. GUILLAUME BOSSARD Chargé de recherche (Examinateur) CPHT Ecole Polytechnique M. STEFAN HOHENEGGER Maître de conférence (Rapporteur) IPNL, Université de Lyon M. RUBEN MINASIAN Directeur de recherche (Directeur de thèse) IPHT, CEA, Saclay M. DANIEL WALDRAM Professeur (Examinateur) Imperial College London Titre : Autour de Supergravité par l’Anomalie Composée et l’Amplitude en Théorie de Cordes Keywords : QFT Anomalie, Supergravité, Amplitude en théorie de cordes Résumé : Dans ce projet de thèse, nous étudions le rôle joué par l’anomalie dû à la connexion com- posée dans les théories de supergravités dans l’espace-temps à huit dimensions. Ce genre d’anomalie est en effet engendrée par la structure quotient d’espace des champs moduli de la supergravité là où le nombre des super-charges posent des contraintes rigoureux. Notre accomplissement principal est de proposer des termes supplémentaires pour annuler cette anomalie dans la théorie de supergravité en huit dimensions avec seize supercharges. Ces termes, en outre, peuvent être considérés comme une manifestation des corrections provenant de la théorie de super-cordes et nous montrons par des calculs explicites qu’une amplitude sur une boucle dans la théorie de cordes correspondante reproduit ces ter- mes. Motivés par cette démonstration de la cohérence entre la supergravité et la théorie de cordes, nous proposons un seuil mathématique pour la compactification de ces théories dans huit dimensions vers six dimensions sur une sphère en présence des branes de co-dimension 2. Ceci est une simulation de compactification sur une surface K3 à l’aide des branes. Nous montrons que la présence d’anomalie composée ne peut être justifiée que par des branes de co-dimensions deux. Nous discutons la dualité entre la théorie Heterotic et la théorie-F sous la lumière de 7-branes et puis la compactification des su- pergravités de dix dimensions sur K3 en présence des 5-branes. Tous cela nous ouvrent nouvelles voies pour étudier des aspects non-perturbatives de la théorie de cordes. Nous concluons avec un calcul sur deux boucles dans la théorie de cordes Heterotic de dix dimensions qui n’était pas beaucoup exploré dans la littérature. Université Paris-Saclay Espace Technologique / Immeuble Discovery Route de l’Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France iii Title : Composite Anomaly in Supergravity and String Amplitude Comparison Keywords : QFT Anomaly, Supergravity, String Amplitude Abstract : We examine the structure of composite anomaly in maximal and half-maximal super- gravity theories especially in eight space-time dimensions. The number of super-charges dictates the structure of the coset space of the moduli fields of the theory which in turn engenders the composite anomaly in such theories. Our main achievement lies in proposing counter-terms for such anomalies. These terms are of stringy nature and we show by explicit 1-loop amplitude calculations in correspond- ing string theories that those counter-terms are consistently provided by string amplitude. In the light of non-perturbative higher dimensional theories like F-theory, the anomaly cancelling counter-terms are seen to be related to co-dimension two branes e.g. 7-branes. We then use these results of 8-dimensional theories to provide for supergravity theories in six-dimensions by compactifying on a sphere in the pres- ence of 5-branes. This is in fact a simulation of K3 compactification and our knowledge of composite connection provide us with threshold conditions to achieve such compactifications. All these analysis provide for greater insight into the non-perturbative regime of string theory. We then conclude with a calculation of 2-loop Heterotic string amplitude which has been very less explored in the literature. Université Paris-Saclay Espace Technologique / Immeuble Discovery Route de l’Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France v Hello World Acknowledgements The obvious premier acknowledgement goes to Ruben Minasian for his support, sci- entific insights and his patience. Next comes Raffaelle Savelli for his collaboration and immense input in course of my thesis. I then like to thank the whole group of string theory of IPHT, especially Mariana Grana, Iosif Bena, Pierre Vanhove and all the post-docs who came during my office at Saclay and enriched my experience. I learned a lot from Pierre from our discussions and his insights are reflected in many places of this thesis. In the same line I would like to thank Guillaume Bossard, Jonathan Heckman, Boris Pioline, Piotr Tourquin for helpful discussions. My grati- tude to the members of the jury for agreeing to assist my defence and for their flexi- bility of precious time for the changed schedule of defence. Lastly, I cannot but thank the interns and other graduate students for the cordiality which held us together in all thin and thick during the course of this long work. vii Contents Acknowledgements vii Résumé en Français1 1 Introduction5 1.1 Quantization of gravity and quest for unification.............5 1.2 What this thesis stands for?.........................7 1.3 Outline of the thesis..............................8 2 Supersymmetry and Supergravity9 2.1 Supersymmetry................................9 2.2 Supersymetry algebra in various dimensions............... 10 2.3 Supergravity.................................. 12 2.4 Composite connection in supergravity................... 16 2.5 Anomaly in supergravity........................... 18 3 Perturbative and non-perturbative aspects of string theory 23 3.1 Worldsheet perspective of (super)string.................. 24 3.2 String theories in 10D............................. 27 3.3 String interactions and effective actions.................. 29 3.3.1 Scattering amplitudes in string theory............... 30 3.3.2 String coupling to background fields and low energy effective action.................................. 34 0 3.3.3 String loop expansion in gs and α ................. 35 3.4 Branes in string theory............................ 37 3.5 Compactification in string theory...................... 40 3.5.1 Circle compactification........................ 41 3.5.2 Toroidal compactification...................... 42 3.5.3 Calabi-Yau compactification..................... 42 3.6 Duality in string theory............................ 45 3.6.1 T-duality................................ 45 3.6.2 S-duality................................ 47 3.6.3 S duality of type IIB string theory................. 47 3.6.4 U-duality............................... 49 3.7 M-theory: relation with type IIA and Heterotic theory.......... 49 3.7.1 Heterotic theory from M-theory: Horava-Witten mechanism.. 50 3.8 F-theory: relation with type IIB and Heterotic theory........... 53 4 Discrete anomaly in maximal supergravity 59 4.1 The Green-Gaberdiel counterterm..................... 60 4.1.1 F theory on K3 and 7-brane coupling................ 62 4.2 Discrete anomaly in D=8, N=2 supergravity................ 65 4.2.1 Comparison with 5-point String amplitude and M-theory... 67 ix 4.2.2 Five point string amplitude for type II strings on T 2 : a com- plemntary review........................... 68 5 Discrete anomalies in half-maximal Supergravity and string amplitude 73 5.1 The Green-Schwarz term in D=10, N=1 supergravity effective action. 73 5.2 Green-Schwarz term from string 1-loop amplitude: Elliptic genus... 76 5.3 Discrete anomaly in D=8, N=1 supergravity and string amplitude com- parison..................................... 86 5.3.1 Case 1: G= SO(32) and E8 E8 .................. 93 × The anomaly counterterm...................... 93 Comparison with 5-point String amplitude............ 94 5.3.2 Case 2: G= SO(16) SO(16) .................... 99 × The anomaly counter-term and group traces........... 99 Comparison with 5-point String amplitude............ 100 5.3.3 Case 3: G= SO(8)4 .......................... 105 Anomaly cancelling term and group trace............. 105 String amplitude with G= SO(8)4 ................. 106 Calculating string amplitude with Hecke
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