On N = 6 Superconformal Field Theories Damon John Binder A Dissertation Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy Recommended for Acceptance by the Department of Physics Adviser: Silviu Pufu September 2021 © Copyright by Damon John Binder, 2021. All Rights Reserved Abstract In this thesis we study N = 6 superconformal field theories (SCFTs) in three dimensions. Such theories are highly constrained by supersymmetry, allowing many quantities to be computed exactly. Yet though constrained, N = 6 SCFTs still exhibit a rich array of behaviors, and in various regimes 7 3 can be dual holographically to M-theory on AdS4 ×S , IIA string theory on AdS4 ×CP , and higher- spin gravity on AdS4. We will use tools from conformal bootstrap and supersymmetric localization to study N = 6 theories, both in general and in holographic regimes. We begin in Chapter 2 by deriving the supersymmetric Ward identities and the superconformal block expansion for the four-point correlator hSSSSi of stress tensor multiplet scalars S. Chapter 3 then studies the mass-deformed sphere partition function, which can be computed exactly using supersymmetric localization, and relates derivatives of this quantity to specific integrals of hSSSSi. In Chapter 4 we study the IIA string and M-theory limits of the ABJ family of N = 6 SCFTs. Using the supersymmetric Ward identities and localization results, we are able to fully determine the R4 corrections to the hSSSSi correlator in both limits. By taking the flat space limit, we can compare to the known R4 contribution to the IIA and M-theory S-matrix, allowing us to perform a check of AdS/CFT at finite string coupling. In Chapter 5 we study the higher-spin limit of N = 6 theories. Using the weakly broken higher- spin Ward identity, we completely determine the leading correction to hSSSSi in this limit up to two free parameters, which for ABJ theory we then fix using localization. Finally, in Chapter 6we perform the first numerical bootstrap study of N = 6 superconformal field theories, allowing us to derive non-perturbative bounds on the CFT data contributing to hSSSSi. iii Acknowledgements First and foremost I would like to thank Silviu Pufu for advising me over the course of my PhD. He has always found time to talk, to listen and to explain things to me. I would also like to thank my collaborators over the years: Shai Chester, Yifan Wang, Slava Rychkov, Dan Freedman, and Max Jerdee. I have enjoyed learning from, and arguing with, you all. Thanks go to Simone Giombi for agreeing to read this thesis, and Igor Klebanov and Lyman Page for serving on my FPO committee. Thanks also to Lyman for supervising my experimental project, and to Himanshu Khanchandani, Roman Kolevatov, and Nikolay Sukhov, for making the search for axions a joy — even if the only thing we detected were the lights. I would like to thank the many denizens of Jadwin for countless lunches, Friday afternoons, and trivia nights at D-Bar. In particular I would like to thank Luca Iliesiu and Christian Jepsen, elderly grad students I could always look up to. Thanks are due to my housemates Tom Postma, Albi Padovan, and Ian McKeachie, for reminding me there is life outside physics, and to Buck Shlegeris for the same, but in his own unique way. I am grateful to Nick Haubrich and the Sci-Fi Sunday crew for many evenings of pie, The Prisoner and Blake’s 7; to Andrew Ferdowsian, Dan McGee, and all the rest for many Saturday nights playing board games and Blood on the Clocktower; and to everyone at Princeton Effective Altruism. These activities helped keep me sane, especially after I fled the US due to coronavirus. It saddens me that I will finish my PhD without havingmetwith any of you in person for over a year. Be seeing you. I would like to thank Angus Gruen for a great two weeks of quarantining, and his family for putting up with the both of us. Thanks also to the rest of my friends in Canberra, to the Australian Government for not completely dropping the ball on COVID, and to Cedric Simenel for providing me a place to work. I would like to thank the General Sir John Monash Foundation for supporting my studies, and for organizing an amazing community of scholars which I am glad to be part of. I am grateful for the many teachers, professors, and advisors who have taught me over the years, at Princeton, at the Australian National University, and in Townsville. Special thanks go to Brad Cooper and Shane Gallagher, who helped stoke my early interests in mathematics and in science, and who both sadly passed away while I was studying at Princeton. And of course, thank you to my family, for their love and support throughout the years. Finally, I would like to thank my fiancée Abigail Thomas, for sharing in the highs and lows, and for constantly supporting me, even when it meant years living apart. iv For Abigail May Oxford prove closer to Hong Kong Then Princeton is to Canberra v Contents Abstract . iii Acknowledgements . iv 1 Introduction 1 1.1 Conformal Fields Theories with N = 6 Supersymmetry . 6 1.1.1 Conformal Symmetry . 6 1.1.2 N = 6 Superconformal Symmetry . 10 1.1.3 The N = 6 Stress Tensor Multiplet . 13 1.2 Known N = 6 Superconformal Field Theories . 15 1.3 ABJ Triality . 19 1.3.1 The AdS/CFT Duality . 19 1.3.2 Holographic Regimes of ABJ Theory . 21 2 The hSSSSi Superconformal Block Expansion 25 2.1 Superconformal Ward Identities . 25 2.1.1 Conformal and R-symmetry Invariance . 25 2.1.2 Discrete Symmetries . 27 2.1.3 The Q Variations . 29 2.1.4 Parity Odd Superconformal Ward Identity . 31 2.2 The OPE Expansion . 31 2.2.1 Operators in the S × S OPE . 33 2.2.2 Constraining so(6) Singlets and Adjoints . 39 2.3 Superconformal Casimir Equation . 41 2.4 Examples: GFFT and Free Field Theory . 50 vi 3 Exact Results from Supersymmetric Localization 53 3.1 Integrated Correlators on S3 ............................... 55 3.2 U(N)k × U(N + M)−k Theory . 62 3.2.1 Simplifying the Partition Function . 62 3.2.2 Finite M; N; k Calculations . 64 3.2.3 Supergravity Limit . 65 3.2.4 Higher-Spin Limit . 69 3.3 SO(2)2k × USp(2 + 2M)−k Theory . 73 3.3.1 Simplifying the Partition Function . 73 3.3.2 Finite M; k Calculations . 75 3.3.3 Higher-Spin Limit . 76 3.4 Additional U(1) Factors . 78 4 String and M-Theory Limits 80 4.1 Mellin Space . 82 4.1.1 The Flat-Space Limit . 83 4.2 Scattering Amplitudes with N = 6 Supersymmetry . 84 4.2.1 Spinor-Helicity Review . 85 4.2.2 N = 6 On-Shell Formalism . 87 4.2.3 Counterterms in N = 6 Supergravity . 91 4.2.4 Implications of Flat-Space Amplitudes for N = 6 SCFTs . 92 4.2.5 Exchange Amplitudes . 96 4.3 Holographic Correlators with N = 6 Supersymmetry . 97 4.3.1 Computing M3(s; t) ................................ 97 4.3.2 Computing M4(s; t) and Supergravity Exchange . 100 4.4 ABJ Correlators at Large cT ............................... 102 4.4.1 Large cT , finite k .................................. 104 4.4.2 ’t Hooft strong coupling limit . 105 4.4.3 Large cT , finite µ .................................. 107 4.5 Constraints from CFT Data . 108 4.5.1 Fixing the Supergravity Terms . 108 4.5.2 Integrating Holographic Correlators . 109 4.5.3 Fixing hSSSSi with Localization . 113 vii 5 The Higher-Spin Limit 115 5.1 Weakly Broken Higher-Spin Symmetry . 116 5.1.1 N = 6 Conserved Currents . 116 5.1.2 The so(6) Pseudocharge . 119 5.1.3 Pseudocharge Action on Scalars . 122 5.2 hSSSSi in the Higher-Spin Limit . 126 5.2.1 Three-Point Functions . 126 5.2.2 Ansatz for hSSSSi ................................. 130 5.2.3 hSSP P i and hPPPP i ............................... 132 5.3 hSSSP i in the Higher-Spin Limit . 133 5.4 Constraints from Localization . 137 5.4.1 Integrating Higher-Spin Correlators . 138 5.4.2 Applying the Constraints . 140 5.4.3 Extracting CFT Data . 142 5.5 Discussion . 147 6 Numeric Conformal Bootstrap 151 6.1 Crossing Equations . 152 6.2 Numeric Bootstrap Bounds . 154 6.2.1 Short OPE Coefficients . 154 6.2.2 Semishort OPE Coefficients . 156 6.2.3 Bounds on Long Scaling Dimensions . 159 6.3 Islands for Semishort OPE Coefficients . 162 6.4 Discussion . 166 7 Conclusion 168 A Supersymmetric Ward Identities 172 B Characters of osp(6j4) 179 C Decomposing N = 8 Superblocks to N = 6 181 D D¯ Functions 184 viii Chapter 1 Introduction In this thesis we study a class of highly symmetric quantum field theories, N = 6 superconformal field theories (SCFTs) in three spacetime dimensions. Much like the quantum field theories we use in particle physics — the Standard Model, QCD, QED, and so on — N = 6 SCFTs have Poincaré symmetry: they are invariant under translations, rotations, and boosts. But N = 6 SCFTs are also invariant under two additional kinds of spacetime symmetries: conformal symmetries and supersymmetries. We will now motivate each of these in turn. Conformal Symmetry Physical theories come with characteristic length scales. At distances much greater than this length scale, the precise details of the short-distance physics become unimportant and so we expect the theory to become scale-invariant. Under broad but still not entirely understood circumstances,1 scale symmetry is enhanced to the larger group of conformal symmetries, which are the spacetime transformations which locally look like rotations and rescalings.