University of London Imperial College of Science, Technology and Medicine Department of Physics Iter Geometria { Vacuum Moduli Spaces of Supersymmetric Quiver Gauge Theories with 8 Supercharges Anton Zajac Supervised by professor Amihay Hanany Submitted in part fulfilment of the requirements for the degree of Doctor of Philosophy in Physics of Imperial College London and the Diploma of Imperial College, October 2020 Abstract Supersymmetric gauge theories have been at the heart of research in theoretical physics for the past five decades. The study of these theories holds not only the promise for the ulti- mate understanding of Nature, but also for discoveries of new mathematical constructions and phenomena. The latter results from a highly geometrical nature of these theories, which is prominently carried by the moduli space of vacua. This thesis is dedicated to the study of moduli spaces of supersymmetric quiver gauge the- ories with 8 supercharges. Typically, such moduli spaces consist of two branches, known as the Coulomb branch and the Higgs branch. Following a review of the Higgs and the Coulomb branch computational techniques in the first part of this thesis, it is shown how the study of Coulomb branches of 3d N = 4 minimally unbalanced theories is used for developing a classification of singular hyperK¨ahlercones with a single Lie group isometry. As another appli- cation, three-dimensional Coulomb branches are used to study Higgs branches of a stack of n M5 branes on an A-type orbifold singularity. Analysis of such systems gives rise to a discrete gauging phenomenon with importance for both physics and mathematics. From the physics per- spective, discrete gauging solves the problem of understanding the non-classical Higgs branch phases of the corresponding 6d N = (1; 0) world-volume theory, even when coincident subsets of M5 branes introduce tensionless BPS strings into the spectrum. From the mathematical perspective, discrete gauging provides a new method for constructing non-Abelian orbifolds with certain global symmetry. The thesis also includes an investigation of theories associated with non-simply laced quivers. Remarkably, the formalized analysis of the so-called ungauging schemes and the corresponding Coulomb branches reproduce orbifold relations amid closures of nilpotent orbits of Lie algebras studied by Kostant and Brylinski. Finally, three-dimensional Coulomb branches are employed to understand the Higgs mechanism in supersymmetric gauge theories with 8 supercharges in 3; 4; 5, and 6 dimensions. It is illustrated that the physical phenomenon of partial Higgsing is directly related to the mathematical structure of the moduli space, and in particular, to the geometry of its singular points. The developed techniques pro- vide a new set of algorithmic methods for computing the geometrical structure of symplectic singularities in terms of Hasse diagrams. i ii Declaration The results presented in chapters 3 and 4 are based on the research collaborations with Amihay Hanany and Santiago Cabrera. This work has been published in: • S. Cabrera, A. Hanany, and A. Zajac, "Minimally Unbalanced Quivers," JHEP 02 (2019) 180, arXiv: 1810.01495 [hep-th] • A. Hanany, A. Zajac, "Discrete Gauging in Coulomb branches of Three Dimensional N=4 Supersymmetric Gauge Theories," JHEP 08 (2018) 158, arXiv: 1807.03221 [hep-th] Material in chapter 5 is based on the publication: • A. Hanany, and A. Zajac, "Ungauging Schemes and Coulomb Branches of Non-simply Laced Quiver Theories," JHEP 09 (2020) 193, arXiv:2002.0571 [hep-th] Results in chapter 6 are based on a collaboration with Antoine Bourget, Santiago Cabrera, Julius. F. Grimminger, Amihay Hanany, Marcus Sperling and Zhenghao Zhong: • A. Bourget, S. Cabrera, J.F. Grimminger, A.Hanany, M. Sperling, A. Zajac and Z. Zhong, "The Higgs Mechanism - Hasse Diagrams for Symplectic Singularities", JHEP 01 (2020) 157, arXiv:1908.04245 [hep-th] To author's best knowledge the results and work of other researchers appearing in this thesis are properly cited and acknowledged. `The copyright of this thesis rests with the author. Unless otherwise indicated, its contents are licensed under a Creative Commons Attribution-Non Commercial 4.0 International Licence (CC BY-NC). Under this licence, you may copy and redistribute the material in any medium or format. You may also create and distribute modified versions of the work. This is on the condition that: you credit the author and do not use it, or any derivative works, for a commercial purpose. When reusing or sharing this work, ensure you make the licence terms clear to others by naming the licence and linking to the licence text. Where a work has been adapted, you should indicate that the work has been changed and describe those changes. Please seek permission from the copyright holder for uses of this work that are not included in this licence or permitted under UK Copyright Law.' iii iv Acknowledgements Since the very moment when I first heard of String theory as a fresh undergraduate I have been drawn towards pursuing this field of physics. At that time, I had utterly no idea about what kind of mathematics and physics had been involved in the subject. After my undergraduate studies, I was given the opportunity to scratch the surface of string theory in the wonderful Quantum Fields and Fundamental Forces programme at ICL. I could not have been more fortunate as to have professor Amihay Hanany as my String theory lecturer and my supervisor. I remain extremely grateful to professor Amihay Hanany for guiding me into the beautiful world of String theories and quivers. Without him paving my path, I would have never had a chance to penetrate into the exciting realm of branes and quiver gauge theories. His playful approach to String Theory made it at times seem like a game - making one depressed when loosing, elevated when winning - but overall bringing a lot of joy from merely playing it. All humor aside, doing fundamental theoretical physics really is like playing a game of hide-and-seek with Nature. In this aspect, I consider the opportunity to join the game and the opportunity to be introduced to the game by professor Amihay Hanany the best I could have possibly wished for. The joy from the game was without the puniest doubt amplified by having a raft of outstanding co-players to whom I also wish to extend my deep gratitude. I would like to thank Santiago Cabrera whom I first met during QFFF programme and who since then kept on surprising me with his deep and clear insight into the world of branes and String theory. I am very thankful to Julius Grimminger for being there for me, always ready to help both as a friend and as a research partner. My gratitude also goes to Rudolph Kalveks for his immense help with computations, fruitful discussions and witty english humor. I also thank Antoine Bourget, who has been a big inspiration and help to me. I am also thankful to Marcus Sperling and Zhenghao Zhong and I value each discussion we had in various places all around the world. The entire Theory Group of Imperial College deserves to be praised for the exceptional research environment it creates and for being tremendously welcoming to every newcomer. A special thanks belongs to my parents, who were there with me the entire time, always supportive, always bestowing hope. I would like to say thank you for all that you have done and for believing in me. Without you I would never have found myself here - figuratively as well as literally. I would also like to thank my beloved Jarka for her support during the last year of my PhD and during the time this writing was taking place. v vi Dedication This thesis is dedicated to my parents and Jarka. vii \There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There is another theory which states that this has already happened." Douglas Adams, The Restaurant at the End of the Universe viii Contents Abstract i Acknowledgements v 1 Introduction 1 1.1 Historical Aper¸cu. 1 1.2 Gauge Theory and the Moduli Space of Vacua . 4 1.3 Outline of the Thesis . 8 2 Preliminary Background 11 2.1 Supersymmetry . 11 2.2 Moduli Space of Vacua . 13 2.2.1 Two Branches of the Moduli Space . 15 2.3 The Hilbert Series . 18 2.4 Computing Higgs branch via HyperK¨ahlerQuotient . 23 2.4.1 Hilbert series for the Higgs branch . 25 2.5 Computing Coulomb Branch using Monopole Formula . 29 2.5.1 Hilbert series for the Coulomb branch . 32 ix x CONTENTS 2.6 Quiver . 34 2.7 Branes . 36 2.8 Branes and the Moduli Space of 3d SQED with 8 Supercharges . 39 2.8.1 3d Mirror Symmetry . 42 3 Minimally Unbalanced Quivers 47 3.1 Introduction to MUQ . 47 3.2 Coulomb Branch with SU(N) isometry and a minimal set of generators . 50 3.2.1 Example: GF = SU(3) .............................. 51 3.2.2 Example: GF = SU(10) .............................. 53 3.3 Classification of Minimally Unbalanced Quivers . 54 3.4 Simply Laced Minimally Unbalanced Quivers . 57 3.4.1 G of Type AN ................................... 57 3.4.2 G of Type DN ................................... 64 3.4.3 G of Type EN ................................... 66 3.5 Concluding Comments for Minimally Unbalanced Quivers . 71 4 Discrete Gauging 73 4.1 Introduction to Discrete Gauging . 73 4.2 First Family: Quivers with central 2 node and a bouquet of 1 nodes . 81 4.2.1 Case: k=2, n=1 . 81 4.2.2 Case: k=2, n=2 . 95 n1 4.3 Second Family: Bouquet quivers with A1 × Dn2+1 global symmetry .