University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2019 Deformations Of G2-Structures, String Dualities And Flat Higgs Bundles Rodrigo De Menezes Barbosa University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Mathematics Commons, and the Quantum Physics Commons Recommended Citation Barbosa, Rodrigo De Menezes, "Deformations Of G2-Structures, String Dualities And Flat Higgs Bundles" (2019). Publicly Accessible Penn Dissertations. 3279. https://repository.upenn.edu/edissertations/3279 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/3279 For more information, please contact [email protected]. Deformations Of G2-Structures, String Dualities And Flat Higgs Bundles Abstract We study M-theory compactifications on G2-orbifolds and their resolutions given by total spaces of coassociative ALE-fibrations over a compact flat Riemannian 3-manifold Q. The afl tness condition allows an explicit description of the deformation space of closed G2-structures, and hence also the moduli space of supersymmetric vacua: it is modeled by flat sections of a bundle of Brieskorn-Grothendieck resolutions over Q. Moreover, when instanton corrections are neglected, we obtain an explicit description of the moduli space for the dual type IIA string compactification. The two moduli spaces are shown to be isomorphic for an important example involving A1-singularities, and the result is conjectured to hold in generality. We also discuss an interpretation of the IIA moduli space in terms of "flat Higgs bundles" on Q and explain how it suggests a new approach to SYZ mirror symmetry, while also providing a description of G2-structures in terms of B-branes. The net result is two algebro-geometric descriptions of the moduli space of complexified G2-structures: one as a character variety and a mirror description in terms of a Hilbert scheme of points. Usual G2-deformations are parametrized by spectral covers of flat Higgs bundles. We also discuss a few ongoing developments: (1) we propose a heterotic dual to our main example, (2) we explain how the moduli space of flat Higgs bundles fits into a family of moduli spaces of extended Bogomolnyi monopoles, and (3) we introduce a natural variation of Hodge structures over the complexified G2-moduli space, and conjecture this space admits the structure of a complex integrable system. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) Graduate Group Mathematics First Advisor Tony G. Pantev Keywords Deformation Theory, G2-Manifolds, Higgs Bundles, Mirror Symmetry, Moduli Spaces, M-Theory Subject Categories Mathematics | Quantum Physics This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/3279 DEFORMATIONS OF G2-STRUCTURES, STRING DUALITIES AND FLAT HIGGS BUNDLES Rodrigo de Menezes Barbosa A DISSERTATION in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2019 Supervisor of Dissertation Tony Pantev, Professor of Mathematics Graduate Group Chairperson Julia Hartmann, Professor of Mathematics Dissertation Committee: Ron Donagi, Professor of Mathematics Wolfgang Ziller, Professor of Mathematics Acknowledgments It is a pleasure to thank my wonderful Ph.D. advisor Tony Pantev for his mentorship during my time at Penn. Tony's profound insight into Mathematics has been a constant source of inspiration for me and also a major driving force behind this project. It still amazes me that I have learned something in every single one of our conversations - and that sometimes a side remark of his could lead me to hours sitting at the library, trying to crack down an intricate construction. I will always be grateful to him for his fundamental guidance in my journey to become a researcher in Mathematics. A special thanks goes to Ron Donagi and Wolfgang Ziller for their crucial role during my mathematical formation at Penn. I am thankful also to Bobby Acharya for important comments on this thesis and for drawing my attention to his paper [Ach98], which clarified a few issues at the final stage of this work; and to Dave Morrison, for inviting me to a crucial conference on Physics and Holonomy and for the opportunity to present this work there. I am grateful to quite a few Penn Math and Physics professors, including ii Jonathan Block, Mirjam Cvetic,ˇ Antonella Grassi, David Harbater, Jonathan Heck- man and Julius Shaneson. Many thanks also to the Math Office staff: Monica, Paula, Reshma and Robin, for making sure the department is always running smoothly; and to the many workers of DRL for keeping it an amazing work en- vironment. Thanks to the University of Pennsylvania for funding me throughout my PhD and also for providing wonderful facilities and resources. I am thankful to my fellow Penn Math graduate students, especially those in the Algebraic Geometry and Differential Geometry groups. A big thanks in particular to Benedict, Jia-Choon and Seokjoo for the friendship and for proof-reading and sharing their thoughts on this thesis. A special thanks goes to my good friend Matei, for all of that and also for bearing my years-long never-ending morning rambles on variations of Hodge structures in the most perfectoid room of Van Pelt. Thanks also to my office mates over the years: Aline, Ben, Josh, Kostis and Martin; and to Artur, Justin, Michael and Roberta for many enjoyable conversations, about Math or anything else. Among non-Penn Math people, thanks to my good friends Isabel Leal and Nuno Cardoso for our now decade-old friendship, and Alex Kinsella for many insightful conversations on Heterotic/M-theory duality and for hosting me in Santa Barbara on the occasion of my first major conference talk. I am thankful also to my Mas- ter's advisor Marcos Jardim for guiding me in my early steps as a mathematician, to my undergraduate advisor Alberto Saa for almost making me become a general iii relativist, to my UNICAMP professors Luiz San Martin and Adriano Moura, and to Elizabeth Gasparim for essentially bridging my way to Penn. A big thanks to everyone involved with the Simons Collaboration on Homological Mirror Symme- try and the Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics: participating in these collaborations has essentially shaped me as a math- ematician, and I am sure I will carry these impressions on all my future work. In particular, I am grateful to James Simons and the Simons Foundation for making all of this possible. Finally, thanks to Szil´ardSzab´oand Dennis DeTurck for their help during the job market months. I was lucky to have a consistent group of close friends during my time in Philadel- phia. This provided the stability necessary to bear the emotional burdens that a PhD inevitably entails. Among these, I am mostly grateful to my \Philly family", who I can safely say are among my best friends in life: Andr´e,Charu, David, Sara and Stefano. I am also thankful to my amazing friends from the Brazilian and Ital- ian groups, and all other friends I made during these five years: there are too many of you to name here, but I am sure you know who you are. Thanks to my lifetime friends Diego and Rafael for having my back since the Wise times (and even before), to all my good friends in Bras´ıliafor making my time back home so enjoyable, and also to those who have spent time with me in Europe or Japan during quite a few memorable work trips. I am lucky to have an amazing family that even from a distance keeps me on iv track with their unwavering support. Thanks to my grandmothers Eneida and Leide, and to all my uncles, aunts and cousins in Brazil for always believing in me. Finally, and above all, I am deeply thankful to my parents Wagner and S^onia,my sister Nat´aliaand my sweetheart Laura for their unconditional love and patience. Math is fun, but the true purpose of life is to treasure your loved ones every single day. You are the foundation of my life. I love you. v \L'hydre-Univers tordant son corps ´ecaill´ed'astres." Victor Hugo, Les Contemplations, VI.26 \Meu inc^endio´euma metamorfose e a minha metamorfose ´ea madeira de um inc^endio." vi ABSTRACT DEFORMATIONS OF G2-STRUCTURES, STRING DUALITIES AND FLAT HIGGS BUNDLES Rodrigo de Menezes Barbosa Tony Pantev, Advisor We study M-theory compactifications on G2-orbifolds and their resolutions given by total spaces of coassociative ALE-fibrations over a compact flat Riemannian 3- manifold Q. The flatness condition allows an explicit description of the deformation space of closed G2-structures, and hence also the moduli space of supersymmetric vacua: it is modeled by flat sections of a bundle of Brieskorn-Grothendieck resolu- tions over Q. Moreover, when instanton corrections are neglected, we also have an explicit description of the moduli space for the dual type IIA string compactifica- tion. The two moduli spaces are shown to be isomorphic for an important example involving A1-singularities, and the result is conjectured to hold in generality. We also discuss an interpretation of the IIA moduli space in terms of “flat Higgs bun- dles" on Q and explain how it suggests a new approach to SYZ mirror symmetry, while also providing a description of G2-structures in terms of B-branes. The net result is two algebro-geometric descriptions of the moduli space of complexified G2- C structures MG2 : one as a character variety and a mirror description in terms of a Hilbert scheme of points. Usual G2-deformations are parametrized by spectral vii covers of flat Higgs bundles. We also discuss a few ongoing developments: p1q we propose a heterotic dual to our main example, p2q we explain how the moduli space of flat Higgs bundles fits into a family of moduli spaces of extended Bogomolnyi monopoles, and p3q we C introduce a natural variation of Hodge structures over MG2 , and conjecture this space admits the structure of a complex integrable system.