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Universidad Complutense de Madrid Topological Quantum Field Theories for Character Varieties Teor´ıasTopol´ogicasde Campos Cu´anticos para Variedades de Caracteres Angel´ Gonzalez´ Prieto Directores Prof. Marina Logares Jim´enez Prof. Vicente Mu~nozVel´azquez Tesis presentada para optar al grado de Doctor en Investigaci´onMatem´atica Departamento de Algebra,´ Geometr´ıay Topolog´ıa Facultad de Ciencias Matem´aticas Madrid, 2018 Universidad Complutense de Madrid Topological Quantum Field Theories for Character Varieties Angel´ Gonzalez´ Prieto Advisors: Prof. Marina Logares and Prof. Vicente Mu~noz Thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematical Research Departamento de Algebra,´ Geometr´ıay Topolog´ıa Facultad de Ciencias Matem´aticas Madrid, 2018 \It's not who I am underneath, but what I do that defines me." Batman ABSTRACT Topological Quantum Field Theory for Character Varieties by Angel´ Gonzalez´ Prieto Abstract en Espa~nol La presente tesis doctoral est´adedicada al estudio de las estructuras de Hodge de un tipo especial de variedades algebraicas complejas que reciben el nombre de variedades de caracteres. Para este fin, proponemos utilizar una poderosa herramienta de naturaleza ´algebro-geom´etrica,proveniente de la f´ısicate´orica,conocida como Teor´ıaTopol´ogicade Campos Cu´anticos (TQFT por sus siglas en ingl´es). Con esta idea, en la presente tesis desarrollamos un formalismo que nos permite construir TQFTs a partir de dos piezas m´assencillas de informaci´on: una teor´ıade campos (informaci´ongeom´etrica)y una cuantizaci´on(informaci´onalgebraica). Como aplicaci´on,construimos una TQFT que calcula las estructuras de Hodge de variedades de representaciones y la usamos para calcular expl´ıcitamente los polinomios de Deligne-Hodge de SL2(C)-variedades de caracteres parab´olicas. Palabras clave: TQFT, variedades de caracteres, estructuras de Hodge, espacios de m´oduli, GIT. Abstract in English This PhD Thesis is devoted to the study of Hodge structures on a special type of complex algebraic varieties, the so-called character varieties. For this purpose, we propose to use a powerful algebro- geometric tool coming from theoretical physics, known as Topological Quantum Field Theory (TQFT). With this idea in mind, in the present Thesis we develop a formalism that allows us to construct TQFTs from two simpler pieces of data: a field theory (geometric data) and a quantisation (algebraic data). As an application, we construct a TQFT computing Hodge structures on representation varieties and we use it for computing explicity the Deligne-Hodge polynomials of parabolic SL2(C)-character varieties. Key words: TQFT, character varieties, Hodge structures, moduli spaces, GIT. 2010 Mathematics Subject Classification. Primary: 57R56. Secondary: 14C30, 14D21, 14L24. Acknowledgements El camino hacia la consecuci´ondel Doctorado en Investigaci´onMatem´aticaque culmina con esta tesis ha sido largo, complejo y lleno de curvas. Sin embargo, de ninguna manera puedo decir que haya sido una senda dura o desagradable. Durante los tres a~nosen los que he realizado este doctorado he sido, simple y llanamente, feliz. Me he podido dedicar en cuerpo y alma a lo que siempre hab´ıa deseado y, lejos de defraudarme, ha sido una experiencia enormemente enriquecedora y, por qu´eno decirlo, divertida. Sin embargo, la causa ´ultimade toda esta felicidad no es sino la gente que me ha rodeado a lo largo de todo este proceso y, sin cuya ayuda y apoyo, esta tesis no habr´ıasido posible. Normalmente en este tipo de escritos se suele argumentar el manido \no mencionar´enombres para evitar omisiones". Yo creo que me arriesgar´e. En primer lugar, quiero dedicar esta tesis a mis padres. Durante toda mi vida, ellos han sido un gran apoyo, sustento y acicate para crecer y progresar. Si decid´ıhacerme cient´ıficofue gracias a ellos. De mis padres aprend´ıque la esencia del ser humano es cuestionarse cosas y que la cultura nos hace libres. Ellos me ense~naronque la oportunidad de aprender es la mayor fortuna que se puede tener. No pod´ıafaltar aqu´ıuna menci´ona todos mis amigos que, de una u otra manera, me han acompa~nado y empujado a lo largo de este viaje. Gracias a Adri, Alberto, Nico y Nuria por esas fant´asticascervezas en el Ferm´ınenvueltas en risas y seguidas de una (mala) pel´ıcula de terror. Tambi´enquiero dar las gracias a mis amigos murcianos (de coraz´on,solo dos de ellos de nacimiento) Germ´an, Jaime, Luis, Miguel y Sara. Y a mi cu~nadaCristina y a mis suegros, Juanjo y Mar´ıa,porque siempre me han apoyado y me han hecho sentir uno m´asde la familia. Y, por supuesto, no me puedo olvidar de Elena, Kike y Tania por todas esas tardes en Principe P´ıo,probablemente en el Burger King, en las que las horas se nos van volarnos sin darnos ni cuenta. Pero no solo he contado con un gran apoyo fuera de la universidad, sino tambi´endentro de ella. Por eso, tambi´enquiero dedicar esta tesis a Angel´ Ramos, Antonio Br´u,Benjamin Ivorra, David G´omez- Castro y Susana G´omezpor todas esas interesantes charlas, buenos consejos y risas en tantas y tantas comidas. Muchas gracias de todo coraz´on. Gracias a vosotros, ir a la universidad cada d´ıaha sido fant´astico. Tambi´enquiero dedicar unas l´ıneas de estos agradecimientos a mis compa~nerosde departamento Giovanni, Juan Angel´ y Luc´ıa,por tantas largas conversaciones matem´aticasy por siempre ayudarme a responder todas mis preguntas sobre spinores, ´algebras de Clifford y operadores de Dirac. Vosotros me hab´eisense~nadoque la matem´aticano debe ser una actividad solitaria y que la verdadera comprensi´on solo se alcanza cuando se discute sobre ella. Y, por supuesto, tambi´enquer´ıadedicar esta tesis a Antonio Vald´espor todos estos a~nosque hemos compartido mientras le ayudaba en la docencia. Quiero darte las gracias sinceramente porque siempre me has escuchado, valorado y tenido muy en cuenta mi opini´on. A tu lado nunca me he sentido un mero ayudante, sino un profesor m´asy eso me ha permitido encontrar en la docencia una verdadera vocaci´on.Nuestros tes con porra y nuestras vi comidas de fin de cuatrimestre son siempre un aut´entico placer y en ti he encontrado no solo un colega sino tambi´en un amigo. I would like to thank the reviews of this thesis, David Ben-Zvi and Peter Newstead, for their careful reading of the manuscript. Their very helpful comments and constructive suggestions have improved the exposition of this thesis. I would like to specially thank Thomas Wasserman. I am greatly indebted for your invaluable help through the development of this thesis. It was you who taught me all the secrets of TQFTs and who gave me new tools and perspectives for addressing the problems I had found. I really enjoy all our talks about maths, physics and life in general. This thesis would have been impossible without your help and support. Por descontado, tambi´enme gustar´ıaagradecer a mis directores, Marina Logares y Vicente Mu~noz, todo el apoyo, cari~noy confianza que han depositado en m´ıdurante todos estos a~nos. Sinceramente, creo que no podr´ıahaber tenido mejores gu´ıaspara esta ruta. Gracias por haber escuchado con gran atenci´ontodas y cada una de mis ideas, por alocadas que fuesen, y por siempre haber sido capaces de extraer algo bueno de ellas. Nunca me hab´eisdesalentado y siempre me hab´eisanimado a no desistir de mis intentos por atacar un problema. Creo sinceramente que eso es lo que distingue a un mero supervisor de un buen director. Ojal´apueda estar a la altura cuando me llegue el momento de ser yo el director. Cuando os conoc´ı,podr´ıasaber m´aso menos matem´aticas(m´asbien menos), pero vosotros me hab´eisense~nadoa ser Matem´atico,con may´usculas.Sois mucho m´asque mis directores, sois mis maestros. Finalmente, mis ´ultimasl´ıneasde esta secci´onson, como no podr´ıaser de otro modo, para Laura. Gracias. Gracias por todo tu amor, tu apoyo, tu cari~noy tu comprensi´ondurante toda la realizaci´onde esta tesis (y antes). T´uhas sido mi energ´ıa,mi motor y mi fuerza para seguir estudiando, trabajando y avanzando durante todo el doctorado. T´usiempre confiaste en m´ı,incluso cuando ni yo mismo lo hac´ıay los teoremas no sal´ıan. Siempre cre´ısteen m´ı. Simplemente, gracias por existir y por estar a mi lado. Contents Abstract iv Acknowledgements vi Table of Contents viii Introduction xii Introducci´on xxii 1 Topological Quantum Field Theories1 1.1 Review of category theory and TQFTs...........................1 1.1.1 Monoidal categories.................................4 1.1.2 Duality properties and low dimensional classification of TQFTs.........6 1.1.3 Lax monoidality and 2-categories.......................... 10 1.2 Lax monoidal Topological Quantum Field Theories.................... 14 1.2.1 C-algebras....................................... 14 1.2.2 Quantisation..................................... 19 1.2.3 Field theory...................................... 21 1.3 Other versions of Topological Quantum Field Theories.................. 26 1.3.1 TQFTs over a sheaf................................. 26 1.3.2 Almost-TQFTs and the category of tubes..................... 32 1.3.3 Reduction of a TQFT................................ 34 1.3.4 Soft TQFTs...................................... 37 2 Hodge Theory 41 2.1 A panoramic view of K-theory............................... 41 2.1.1 Review of derived categories............................. 43 2.1.2 Grothendieck group................................. 46 2.2 Hodge structures....................................... 50 2.2.1 Pure Hodge structures................................ 50 2.2.2 Mixed Hodge structures............................... 53 2.3 Mixed Hodge modules.................................... 57 2.3.1 Riemann-Hilbert correspondence.......................... 57 2.3.2 Saito's theory of mixed Hodge modules...................... 64 2.4 Monodromy as mixed Hodge module...........................
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  • D-Modules, Perverse Sheaves, and Representation Theory

    D-Modules, Perverse Sheaves, and Representation Theory

    Progress in Mathematics Volume 236 Series Editors Hyman Bass Joseph Oesterle´ Alan Weinstein Ryoshi Hotta Kiyoshi Takeuchi Toshiyuki Tanisaki D-Modules, Perverse Sheaves, and Representation Theory Translated by Kiyoshi Takeuchi Birkhauser¨ Boston • Basel • Berlin Ryoshi Hotta Kiyoshi Takeuchi Professor Emeritus of Tohoku University School of Mathematics Shirako 2-25-1-1106 Tsukuba University Wako 351-0101 Tenoudai 1-1-1 Japan Tsukuba 305-8571 [email protected] Japan [email protected] Toshiyuki Tanisaki Department of Mathematics Graduate School of Science Osaka City University 3-3-138 Sugimoto Sumiyoshi-ku Osaka 558-8585 Japan [email protected] Mathematics Subject Classification (2000): Primary: 32C38, 20G05; Secondary: 32S35, 32S60, 17B10 Library of Congress Control Number: 2004059581 ISBN-13: 978-0-8176-4363-8 e-ISBN-13: 978-0-8176-4523-6 Printed on acid-free paper. c 2008, English Edition Birkhauser¨ Boston c 1995, Japanese Edition, Springer-Verlag Tokyo, D Kagun to Daisugun (D-Modules and Algebraic Groups) by R. Hotta and T. Tanisaki. All rights reserved. This work may not be translated or copied in whole or in part without the writ- ten permission of the publisher (Birkhauser¨ Boston, c/o Springer Science Business Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter de- veloped is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.