Algebraic aspects of conformal field theory Christopher Raymond B.Sc. (Hons I), M.Phil 0000-0001-5535-0823 A thesis submitted for the degree of Doctor of Philosophy at The University of Queensland in 2020 School of Mathematics and Physics Abstract This thesis presents an algebraic study of two areas of conformal field theory. The first part extends the theory of Galilean conformal algebras. These algebras are extended conformal symmetry algebras for two-dimensional quantum field theories, formed through a process known as Galilean contraction. Analogous to an Inonu-Wigner contraction, the Galilean contraction procedure takes two conformal symmetry algebras, equivalent up to central charge, as input and produces a new conformal symmetry algebra. We extend the theory of Galilean contractions in several directions. First, we develop the theory to allow input of any number of symmetry algebras. These generalised algebras have a truncated graded structure. We develop a theory for multi-graded Galilean algebras, whereby we can extend structures which are graded by sequences. Finally, we present a comprehensive analysis of the possible algebras which arise when the input algebras are no longer required to be equivalent. We refer to this as the asymmetric Galilean contraction. For each stage of generalisation, we present several pertinent examples, discuss the Sugawara construction of a Galilean Virasoro algebra given an affine Lie algebra, and apply our results to the W-algebra W3. The second part of this thesis presents an exploratory study of reducible but indecomposable modules of the N = 2 Superconformal algebras, known as staggered modules. These modules are characterised by a non-diagonalisable action of L0, the Virasoro zero-mode. Using recent results on the coset construction of N = 2 minimal models, we are able to construct the first examples of staggered modules for the N = 2 algebras. We determine the structure of a family of such modules for all admissible values of the central charge. Furthermore, we investigate the action of symmetries such as spectral flow on these modules. We present the results along with a range of examples, and discuss possible paths towards a classification of such modules for the Neveu-Schwarz and Ramond N = 2 superconformal algebras. Declaration by author This thesis is composed of my original work, and contains no material previously published or written by another person except where due reference has been made in the text. I have clearly stated the contribution by others to jointly-authored works that I have included in my thesis. I have clearly stated the contribution of others to my thesis as a whole, including statistical assistance, survey design, data analysis, significant technical procedures, professional editorial advice, financial support and any other original research work used or reported in my thesis. The content of my thesis is the result of work I have carried out since the commencement of my higher degree by research candidature and does not include a substantial part of work that has been submitted to qualify for the award of any other degree or diploma in any university or other tertiary institution. I have clearly stated which parts of my thesis, if any, have been submitted to qualify for another award. I acknowledge that an electronic copy of my thesis must be lodged with the University Library and, subject to the policy and procedures of The University of Queensland, the thesis be made available for research and study in accordance with the Copyright Act 1968 unless a period of embargo has been approved by the Dean of the Graduate School. I acknowledge that copyright of all material contained in my thesis resides with the copyright holder(s) of that material. Where appropriate I have obtained copyright permission from the copyright holder to reproduce material in this thesis and have sought permission from co-authors for any jointly authored works included in the thesis. Publications included in this thesis Publications included in this thesis are the following 1. [1] J. Rasmussen, C. Raymond, Galilean contractions of W-algebras, Nucl. Phys. B922 (2017), 435–479, arXiv:1701.04437 [hep-th]. 2. [2] J. Rasmussen, C. Raymond, Higher-order Galilean contractions, Nucl. Phys. B945 (2019), 114680, arXiv:1901.06069 [hep-th]. 3. [3] E. Ragoucy, J. Rasmussen, C. Raymond, Multi-graded Galilean conformal algebras, Nucl. Phys. B957, 115092, arXiv:2002.08637 [hep-th]. Submitted manuscripts included in this thesis No manuscripts submitted for publication. Other publications during candidature No other published articles are included in this thesis. Contributions by others to the thesis The work on Galilean algebras was conceptualised, and guided by Jørgen Rasmussen. The initial work on multi-graded, and asymmetric, Galilean algebras came from correspondence between Jørgen Rasmussen and Eric Ragoucy. Jørgen Rasmussen and Eric Ragoucy contributed to technical results on the product structure of multi-graded and asymmetric Galilean algebras. The project on N = 2 superconformal staggered modules was conceived and designed in conversations with David Ridout and Jørgen Rasmussen, designed to extend previous work of David Ridout and collaborators. David Ridout contributed to early module calculations, and interpretation of the resulting modules that were found. Jørgen Rasmussen, David Ridout, and Eric Ragoucy have provided feedback on the manuscript for clarity and accuracy. Statement of parts of the thesis submitted to qualify for the award of another degree The beginning of Chapter 2 contains background theory and instructional examples which were previously submitted in [4] for the degree of M.Phil at The University of Queensland. We do not report these results as new in this thesis, and include them only as background necessary to understand further work based upon them. Research involving human or animal subjects No animal or human subjects were involved in this research. Acknowledgements First, I would like to thank my supervisors Jørgen and David for their time and patience. I have tried to learn as much as I can from both of you, not just about mathematics, but also about life. Thank you for making this experience truly enjoyable. Throughout my PhD, there have been so many valuable interactions with academics who were always willing to take the time to talk and encourage my progression. In particular, I’d like to thank Eric Ragoucy, Mark Gould, Jon Links, Thomas Quella, Phil Isaac, Yao-Zhong Zhang, Ole Warnaar, and Masoud Kamgarpour. I’d also like to thank everyone involved with the UQ Maths QFT seminar series over the years. It has always been a welcoming environment for new students. Last but not least, I’d like to thank my family and friends for their unwavering support. My parents, my siblings, my partner, and those that have been there from the beginning and every day for lunch. Financial support This project was funded by a University of Queensland Research Higher Degree Award scholarship. Some travel undertaken during the PhD to present or discuss results presented in this thesis was funded by the Australian Research Council under the Discovery Project scheme, project number DP16010137. Keywords conformal field theory, representation theory, infinite dimensional lie algebras, Galilean contractions Australian and New Zealand Standard Research Classifications (ANZSRC) ANZSRC code: 010505 Mathematical Aspects of Quantum and Conformal Field Theory, Quantum Gravity and String Theory, 100% Fields of Research (FoR) Classification FoR code: 0105 Mathematical Physics, 100% Contents Abstract . ii Contents viii 1 Introduction 1 1.1 Conformal symmetry in two dimensions . 1 1.2 The Virasoro algebra and its representation theory . 2 1.3 The state-field correspondence and the operator product expansion . 3 1.4 Unitarity and the Kac determinant . 10 1.5 The unitary Virasoro minimal models and fusion . 12 1.6 Conformal field theories with affine Lie algebra symmetries . 15 1.7 The coset construction . 17 1.8 Extended symmetry conformal field theories and W-algebras . 20 2 Higher-order Galilean algebras 25 2.1 Introduction to the Galilean contraction procedure . 25 2.2 Operator product algebras and the Galilean contraction . 27 2.2.1 Galilean algebras and the contraction procedure . 29 2.3 Higher-order Galilean contractions . 32 2.3.1 Contraction prescription . 32 2.3.2 Examples: Galilean Virasoro and affine algebras . 33 2.4 General properties of higher-order Galilean contractions . 34 N 2.4.1 Truncated graded structure of AG ....................... 34 2.4.2 Relation to Takiff algebras . 35 2.5 Higher-order Galilean Sugawara constructions . 36 2.5.1 Galilean Sugawara construction . 37 2.5.2 Sugawara before Galilean contraction . 39 2.6 Higher-order Galilean W3 algebras . 41 2.6.1 The W3 algebra . 41 2.6.2 Higher-order W3 algebras . 42 2.6.3 Renormalisation . 44 viii CONTENTS ix 3 Multi-graded Galilean algebras 47 3.1 Introduction . 47 3.2 Multi-graded Galilean algebras . 48 3.2.1 Preliminary theory . 48 3.2.2 Introduction of grading sequences . 49 3.2.3 Example: Multi-graded Galilean Virasoro algebras . 51 3.2.4 Example: Multi-graded Galilean affine Lie algebras . 51 3.3 General properties of multi-graded Galilean algebras . 51 3.3.1 The grading on multi-graded Galilean algebras . 51 3.3.2 Permutation invariance . 52 3.3.3 Relation to multi-variable Takiff algebras . 53 3.4 Multi-graded Galilean Sugawara construction . 53 3.5 Multi-graded W3 algebras . 60 4 Asymmetric Galilean algebras 63 4.1 Introduction . 63 4.2 Asymmetric contraction theory . 63 4.3 Examples . 66 4.3.1 The Galilean Lie algebra slb(2);Hb G ..................... 67 4.3.2 The Galilean Lie algebra slb(2);b G ...................... 67 2 4.3.3 The asymmetric Galilean Virasoro algebra (Vir)G;Vir G . 68 4.3.4 The asymmetric N = 2 superconformal algebra SCA2;Vir1 G . 69 4.3.5 The asymmetric W3;Vir G algebra .