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UC Santa Barbara UC Santa Barbara Electronic Theses and Dissertations UC Santa Barbara UC Santa Barbara Electronic Theses and Dissertations Title Conformal Perturbation Theory and LLM Geometries Permalink https://escholarship.org/uc/item/23d7z11w Author Miller, Alexandra Publication Date 2018 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California University of California Santa Barbara Conformal Perturbation Theory and LLM Geometries A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics by Alexandra Patrea Mathisen Miller Committee in charge: Professor David Berenstein, Chair Professor Gary Horowitz Professor Elisabeth Gwinn September 2018 The Dissertation of Alexandra Patrea Mathisen Miller is approved. Professor Gary Horowitz Professor Elisabeth Gwinn Professor David Berenstein, Committee Chair June 2018 Conformal Perturbation Theory and LLM Geometries Copyright ⃝c 2018 by Alexandra Patrea Mathisen Miller iii This dissertation is dedicated to the memory of Joe Polchinski. iv Acknowledgements I am grateful for so many people who have helped me during my time at UCSB. My graduate school experience has been filled with many ups and downs and I am certain that I would not have made it to where I am today without the support of so many wonderful people. First and foremost, I must thank my advisor, David Berenstein. He went above and beyond in his duties by not only advising me in my research, but also by serving as a mentor more generally. During all of my times of self doubt, he was always there to support and encourage me. Thank you, David, for all you have taught me! To my defense, advancement, and supervisory committee members: Gary Horowitz, Joe Polchinski, Beth Gwinn, and Mark Srednicki. Thank you for your words of wisdom over the years. To my undergraduate research advisor, Zhigang Chen, and my postdoc mentor, Peng Zhang. Their mentoring and teachings were crucial in both helping me to get into graduate school and to be successful during my time here. Though I ended up completely switching research fields, my experience in the Chen lab has always helped me to be a more well rounded physicist. It's always a plus when a theorist has some idea of what actually goes on in a lab! To the many professors and teachers throughout my academic career, from my ele- mentary school math and science teachers (Mr. Adamick and Mrs. Mariano), to my high school teachers (Ms. Centeno, Mr. Green, Mr. Kenyon, Ms. Shields, Mr. Souza, and Ms. Zwicker), the wonderful professors at SFSU (Joe Barranco, Roger Bland, Adrienne Cool, Jeff Greensite, Susan Lea, Weining Man, and Barbara Neuhauser), and finally the UCSB faculty (David Berenstein, Nathaniel Craig, Matthew Fisher, Steve Giddings, Gary Horowitz, Don Marolf, Joe Polchinski, and Mark Srednicki). Thank you for all you v have taught me, both in and out of the classroom, and for inspiring me to love to learn! To my teaching mentors: Tengiz Bibilashvili, Lisa Berry, and Mindy Collin. Being a teacher takes a lot of work. Thank you for giving me the tools necessary to one day achieve my goal of being a great physics teacher! To Jennifer Farrar, who is the glue that holds the UCSB physics department together and has helped me in so many ways throughout my graduate school career. To all the wonderful friends I've had throughout my life who are most certainly too numerous to name individually, but categorically can be listed as: UCSB Physics Grads, HEP JC, Broida 6234, Physics Force Soccer, Sting Soccer, Track Tuesday, Pali and other Funk zone homies, UCSB Geologists, SFSU PAC, SFSU Pizza and Milkshake Enthusiasts, and Vanden Drama Club. You know who you are! And special shout outs to my partner, Alex Schrader. To Netta Englehardt, my hero. And to Sebastian Fischetti and Kurt Fujiwara, two of the best friends I could ever ask for!!! To all of the doctors, nurses, and staff at the UCSB Health Center, the Ridley Tree Cancer Center, Sansum Clinic, SB Fertility Center, Cottage Health, and UCSB CAPS. Especially thanks to Dr. Newman, Dr. Hughes, Dr. Bagalio, and Dr. Lantrip! Finally, to my wonderful family who have helped shape me into the person I am today. I am eternally grateful for the love and support throughout my whole life. Thank you, Mom and Dad, for encouraging me to be whoever I wanted to be. I love you to the moon and back! This dissertation was supported by the National Science Foundation Graduate Re- search Fellowship Program, the Broida-Hirschfelder Fellowship, the Graduate Division Dissertation year Fellowship, funds by the University of California, and the Department of Energy (grants DE-FG02-91ER40618 and DE-SC 0011702). vi Curriculum Vitæ Alexandra Patrea Mathisen Miller Education 2018 Ph.D. in Physics (Expected), University of California, Santa Bar- bara. 2015 M.A. in Physics, University of California, Santa Barbara. 2011 B.S. in Physics, San Francisco State University Publications 1. D. Berenstein and A. Miller, \Code Subspaces for LLM Geometries," accepted for publication in Class. Quant. Grav. [arXiv:1708.00035 [hep-th]]. 2. D. Berenstein and A. Miller, \Superposition Induced Topology Changes in Quantum Gravity," JHEP 1711, 121 (2017) [arXiv:1702.03011 [hep-th]]. 3. D. Berenstein and A. Miller, \Logarithmic Enhancements in Conformal Perturba- tion Theory and Their Real Time Interpretation," [arXiv:1607.01922 [hep-th]]. 4. D. Berenstein and A. Miller, \Can Topology and Geometry be Measured by an Operator Measurement in Quantum Gravity?," Phys. Rev. Lett. 118, 261601 (2017) [arXiv:1605.06166 [hep-th]]. 5. D. Berenstein and A. Miller, \Reconstructing Spacetime from the Hologram, Even in the Classical Limit, Requires Physics Beyond the Planck Scale," Int. J. Mod. Phys. D 25, 1644012 (2016) [arXiv:1605.05288 [hep-th]]. Received honorable mention in 2016 Gravity Research Foundation essay contest. 6. D. Berenstein and A. Miller, \Conformal Perturbation Theory, Dimensional Reg- ularization, and AdS/CFT," Phys. Rev. D 90, 086011 (2014) [arXiv:1406.4142 [hep-th]]. 7. J. Yang, D. Gallardo, A. Miller, and Z. Chen, \Elimination of Transverse Instability in Stripe Solitions by One-Dimensional Lattices," Opt. Lett. 37, 1571-1573 (2012) [arXiv:1205.0767 [physics]]. 8. J. Wang, N.K. Efremidis, A. Miller, C. Lu, P. Zhang, and Z. Chen, \Nonlinear Beam Deflection in Photonic Lattices with Negative Defects" Phys. Rev. A 83, 033836 (2011). 9. P. Zang, N.K. Efremidis, A. Miller, P. Ni, and Z. Chen, “Reconfigurable 3D Pho- tonic Lattices by Optical Induction for Optical Control of Beam Propagation" Appl. Phys. B 104, 553 (2011). 10. P. Zang, N.K. Efremidis, A. Miller, Y. Hu, and Z. Chen, \Observation of Coherent Destruction of Tunneling and Unusual Beam Dynamics due to Negative Coupling in Three-Dimensional Photonic Lattices" Opt. Lett. 35, 3252 (2010). vii Abstract Conformal Perturbation Theory and LLM Geometries by Alexandra Patrea Mathisen Miller This dissertation will focus on various aspects of the AdS/CFT correspondence. Each new result can be thought of as doing at least one of three things: 1) providing support of the duality, 2) using the duality to learn about quantum gravity, and 3) helping to further develop our understanding of the duality. The dissertation is divided into two parts, each dealing with a different physical system. In the first part, we derive universal results for near conformal systems, which we have perturbed. In order to do this, we start by looking at the conformal correlation functions and compute the corrections that arise when he hit the system with a new operator. We were able to analyze what happens to the dual gravitational system under such circumstances and see that our answers agree, providing support for the AdS/CFT conjecture. These universal results also provided a previously lacking interpretation of the universality of energy found in a quenching your system between the perturbed and unperturbed set-ups. In order to perform these computations, we put our CFT on a cylinder, which happens to be the boundary of global AdS. This provided an IR regulator and we found that the remaining divergences were of the same form as one expects in dimensional regularization. Following along these same lines, we further analyzed the divergence structure of correlators in conformal perturbation theory. We found that on the plane, the logarithmic divergences that show up can be understood in terms of resonant behavior in time dependent perturbation theory, for a transition between states viii that is induced by an oscillatory perturbation on the cylinder. In part two, we restrict to the set of LLM geometries, which are the set of 1/2 BPS solutions to IIB supergravity. In our first work, we analyzed limitations of the duality, showing that boundary expectation values are not enough to determine the classical bulk geometry. Next, we used this system in order to learn about quantum gravity. We first were able to show that a quantum superposition of states with a well defined spacetime topology leads to a new state with a different topology. From this, we were able to prove that for this set of states there cannot exist a quantum topology measuring operator, bringing to doubt whether such an operator can exist in quantum gravity more generally. Finally, we were able to advance our understanding of the dictionary itself by reinterpreting these results in terms of the language of quantum error correction, showing that questions about topology perhaps only make sense within a particular (code) subspace of states. ix Contents Curriculum Vitae vii Abstract viii 0 Introduction 1 0.1 Scales in Physical Theories and the Road to Quantum Gravity ...... 1 0.2 The Search for a Theory of Quantum Gravity ............... 4 0.3 AdS/CFT ................................... 6 0.4 Conformal Perturbation Theory ....................... 13 0.5 LLM Geometries ..............................
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