10335_9789813202832_tp.indd 1 30/11/17 3:44 PM b2530 International Strategic Relations and China’s National Security: World at the Crossroads TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk b2530_FM.indd 6 01-Sep-16 11:03:06 AM 10335_9789813202832_tp.indd 2 30/11/17 3:44 PM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Marcolli, Matilde, author. Title: Noncommutative cosmology / by Matilde Marcolli (California Institute of Technology, USA). Description: New Jersey : World Scientific, 2018. | Includes bibliographical references. Identifiers: LCCN 2017042860| ISBN 9789813202832 (hardcover : alk. paper) | ISBN 9789813202849 (pbk : alk. paper) Subjects: LCSH: Noncommutative differential geometry. | Mathematical physics. | Cosmology. Classification: LCC QC20.7.G44 M37 2018 | DDC 523.101/51636--dc23 LC record available at https://lccn.loc.gov/2017042860 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2018 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. For any available supplementary material, please visit http://www.worldscientific.com/worldscibooks/10.1142/10335#t=suppl Printed in Singapore RokTing - 10335 - Noncommutative Cosmology.indd 1 30-11-17 1:34:43 PM November22,2017 10:31 BC:10335–NoncommutativeCosmology NCGCosmoBookIndex pagev Preface This book explores the use of mathematical methods from Noncommuta- tive Geometry to construct new geometric models for classical and quan- tum cosmology and for (Euclidean) quantum gravity. A theme we will be discussing at length is a class of modified gravity models based on Dirac operators and the spectral action functional. We will also discuss the use of Algebraic Geometry and Arithmetic in cosmological models. The book is based on a series of lectures given by the author at the summerschool“Geometric,algebraicandtopologicalmethodsforQuantum Field Theory”, organized by Alexander Cardona, Hern´an Ocampo, Sylvie Paycha and Andr´es Reyes at Villa de Leyva, Colombia, July 4–22, 2011; additionalmaterialwasintroducedbasedonthegraduatecourses“Topicsin Geometry and Physics” (Ma193b) and “Mathematical Physics” (Ma148b) taught by the author at Caltech in 2012 and 2016, and on several talks given by the author in the Cosmology Seminar at the Perimeter Institute for Theoretical Physics. The research work surveyed in this book is based on the results of sev- eral extensive collaborations the author has had over the past few years. Many of the topics discussed in the book were developed in collaborations with Yuri Manin and with Elena Pierpaoli, on a range of different aspects of the relation between geometry and cosmology. Other parts of the book reflect past and ongoing work with Wentao Fan and Farzad Fathizadeh and with Walter van Suijlekom. Other results reported in the volume arise from work carried out by the author with several of her students: Adam Ball,DomenicDenicola,ChristopherDuston,ChristopherEstrada,Yeorgia Kafkoulis, Daniel Kolodrubetz, Nicolas Tedeschi, Ahmed Zainy al-Yasry, and Nick Zolman who worked on various parts of this project, and espe- ciallyBranimirC´a´ci´candKevinTeh,whocontributedessentiallytoseveral v November22,2017 10:31 BC:10335–NoncommutativeCosmology NCGCosmoBookIndex pagevi vi Noncommutative Cosmology crucial aspects of this work on noncommutative cosmology. The volume is structured as follows: the first chapter recalls general backgroundonNoncommutativeGeometryandonthespectralactionfunc- tional. Part of this chapter also discusses particle physics models based on the spectral action and summarizes previous work of the author with Ali Chamseddine and Alain Connes. The chapter also briefly reviews results of Beenakker, van den Broek, and van Suijlekom on particle physics mod- els with supersymmetry and the spectral action, and work of the author and Zolman on the spectral action and supersymmetry algebras, as well as the work of Chamseddine, Connes and van Suijlekom on spectral ac- tion models of grand unified theories. It also briefly surveys work of Boyle and Farnsworth on the fused algebra approach to the noncommutative ge- ometry of the Standard Model of particle physics. The second chapter is based on joint work with Elena Pierpaoli on Eary Universe models based on the spectral action functional of gravity coupled to matter and its large energiesasymptoticexpansion. ItalsodiscussesrecentresultsontheHiggs mass problem in spectral action models, based on results of Chamseddine– Connes and of the author and Chris Estrada. The third chapter is based on results of the author with Elena Pierpaoli, Branimir C´a´ci´c, and Kevin Teh on the problem of Cosmic Topology in the context of models of grav- ity based on the spectral action and associated slow-roll inflation mod- els. The fourth chapter is based on joint work with Yuri Manin on the use of algebro-geometric techniques in cosmology and associated models of conformally cyclic cosmologies and of eternal inflation scenarios based on algebro-geometric blow-ups. The fifth chapter is based on work with YuriManinandonworkwithChrisEstradaonmixmasteruniversemodels andtheirarithmeticproperties,andontheirnoncommutativedeformations and possible Early Universe scenario. The case of the SU(2) Bianchi IX gravitational instantons is discussed based on joint work with Yuri Manin. In the sixth chapter the discussion of the SU(2) Bianchi IX gravitational instantons is further expanded, based on joint work with Wentao Fan and Farzad Fathizadeh, with explicit computations of the coefficients of the spectral action and a result on the arithmetic structure of the spectral ac- tion, based on modular forms. The seventh chapter summarizes work in collaborationwithFarzadFathizadehontheoccurrenceofmixedTatemo- tives and periods in the asymptotic expansion of the spectral action for Robertson–Walker spacetimes. The eighth chapter is based on joint work of the author with Adam Ball and with Nicolas Tedeschi, on multifractal models of cosmology treated with the methods of noncommutative geome- November22,2017 10:31 BC:10335–NoncommutativeCosmology NCGCosmoBookIndex pagevii Preface vii try. Inparticularitcoversthespectralactionmodelofgravityandslow-roll inflation scenarios for Packed Swiss Cheese Cosmologies, and a version of the“EternalSymmetree”p-adicmodelofeternalinflation. Theninthchap- ter deals with an approach to Euclidean quantum cosmology, in the style of Hartle–Hawking, based on the spectral action functional. The chapter includes a survey of results of Chris Duston on exotic smoothness in Eu- clidean quantum cosmology, results of the author and Domenic Denicola and Ahmed Zainy al-Yasry, and of the author and Walter van Suijlekom, on topological spin foam models and gauge spin networks models. It also surveys recent work of John Barrett and Lisa Glaser on random finite non- commutative geometries. Matilde Marcolli California Institute of Technology Perimeter Institute for Theoretical Physics University of Toronto b2530 International Strategic Relations and China’s National Security: World at the Crossroads TTTThhhhiiiissss ppppaaaaggggeeee iiiinnnntttteeeennnnttttiiiioooonnnnaaaallllllllyyyy lllleeeefffftttt bbbbllllaaaannnnkkkk b2530_FM.indd 6 01-Sep-16 11:03:06 AM November22,2017 10:31 BC:10335–NoncommutativeCosmology NCGCosmoBookIndex pageix Acknowledgments This work was supported in part by NSF grants DMS-0901221, DMS- 1007207, DMS-1201512, DMS-1707882, PHY-1205440. The author also thankstheMathematicalSciencesResearchInstituteinBerkeleyforhospi- tality and support during some of the writing of this book in 2013. Part of this work was done at the Perimeter Institute for Theoretical Physics, over the course of several visits between 2014 and 2016. The Perimeter Insti- tute is supported by the Government of Canada through Industry Canada andbytheProvinceofOntariothroughtheMinistryofEconomicDevelop- mentandInnovation. Theauthorthanksallhercollaboratorsandstudents listed in the Preface above, for making this book possible and for the very rewarding joint work, especially among them Yuri Manin, Elena Pierpaoli, FarzadFathizadeh,BranimirC´a´ci´candKevinTeh. Theauthoralsothanks LathamBoyleandShaneFarnsworthformanyusefuldiscussions,andespe- ciallythelatterforreadinganearlierdraftofthemanuscriptandproviding extensive comments. She also thanks Leon Kot for his involvement and in- valuable help with the writing of this volume. ix