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Aspects of Local Conformal Symmetry in 1+1 Dimensions Aspects of local conformal symmetry in 1+1 dimensions The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Collier, Scott Andrew. 2020. Aspects of local conformal symmetry in 1+1 dimensions. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. Citable link https://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37365728 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA ©2020 – Scott Andrew Collier all rights reserved. Thesis advisor: Professor Xi Yin Scott Andrew Collier Aspects of local conformal symmetry in 1 + 1 dimensions Abstract In this dissertation we describe progress towards carving out the space of non-integrable conformal field theories (CFTs) in 1 + 1 dimensions using conformal bootstrap techniques. We develop and apply numerical and analytical techniques to explore the non-perturbative consequences of conformal symmetry, unitarity and locality on the spectrum and dynamics of 2D CFTs. Throughout we focus on compact irrational CFTs, which have a discrete spectrum consisting of an infinite number of primary operators and exhibit chaotic dynamics. In the first part of the dissertation we report on numerical advances in charting the space of2DCFTs. We use semidefinite programming to compute bounds on the gap in the spectrum of primary operators asa function of the central charge and on spectral functions that capture the spectral density of primary operators. We then develop recursion relations for arbitrary Virasoro conformal blocks, and explore the constraints of modular invariance of the genus-two partition function, leading to “critical surfaces” that bound the heavy structure constants of a CFT. In the second part of the dissertation we develop the analytic bootstrap program for two dimensional con- formal field theories, focusing on universal aspects of irrational unitary CFTs and holography. We showthat the large-spin sector of irrational CFTs is universally governed by a “Virasoro mean field theory,” incorporat- ing exact stress tensor dynamics into the analytic bootstrap and leading to non-perturbative insights about the bound-state dynamics of three-dimensional quantum gravity. We then leverage a generalized notion of modular invariance to derive a universal asymptotic formula for the average value of the structure constants whenever one or more of the operators has large conformal dimension. The large central charge limit makes contact with quantum gravity in three dimensions, where the averaging over heavy states corresponds to iii Thesis advisor: Professor Xi Yin Scott Andrew Collier coarse-graining over black hole microstates. Finally, we study the Euclidean gravitational path integral of pure AdS3 quantum gravity, and use a modular bootstrap analysis to suggest new saddle point configura- tions that can be consistently included in the gravitational path integral and whose inclusion renders the dual torus partition function unitary. iv Contents Title page i Abstract iii Contents v Citations to previously published research x Acknowledgements xii 1 Introduction and summary of results 1 1.1 Abstract conformal field theory, CFT data and conformal bootstrap .............. 2 1.2 Conformal field theories in 1 + 1 dimensions ........................... 5 1.3 Organization of this dissertation .................................. 8 2 Modular bootstrap revisited 12 2.1 Introduction ............................................. 12 2.2 Constraining gaps in the operator spectrum ........................... 15 2.2.1 Basic setup ......................................... 15 2.2.2 The twist gap ........................................ 19 2.2.3 Refinement of Hellerman-Friedan-Keller bounds ..................... 21 2.2.4 Bounds from full modular invariance ........................... 24 2.2.5 The optimal linear functional ............................... 26 2.3 Spin-dependent bounds ....................................... 28 2.3.1 Gap in the spectrum of scalar primaries ......................... 28 2.3.2 Kinks and CFTs with generic type partition function . 31 2.3.3 Allowing for primary conserved currents ......................... 32 2.3.4 Turning on a twist gap ................................... 34 2.4 Operator degeneracies and extremal spectra ........................... 37 2.4.1 Bounds on the degeneracy at the gap ........................... 37 2.4.2 Extremal spectrum from the optimal linear functional ................. 40 2.4.3 Extremal spectra with maximal gap ........................... 43 2.5 Discussion and open questions ................................... 47 3 Bootstrapping the spectral function 49 3.1 Introduction ............................................. 49 3.2 Spectral function bounds from semidefinite programming .................... 54 3.2.1 A sphere four-point spectral function ........................... 54 3.2.2 Bounding the spectral function in a CFT with only scalar primaries . 59 3.3 The linear method and the uniqueness of Liouville theory .................... 63 3.3.1 Solution of the linear constraints on the spectral function . 63 v 3.3.2 Constraints from modular invariance ........................... 69 3.3.3 Degenerate spectrum and TQFT ............................. 72 3.4 The modular spectral function ................................... 76 3.4.1 The minimization problem ................................. 76 3.4.2 Some consistency checks .................................. 78 3.4.3 CFTs at large c with large gap .............................. 83 3.5 On the universality of the BTZ spectral density ......................... 87 4 Recursive representations of arbitrary Virasoro conformal blocks 90 4.1 Introduction ............................................. 90 4.2 Review of the sphere 4-point Virasoro block ........................... 95 4.2.1 Definition of Virasoro conformal block .......................... 95 4.2.2 Simple pole structure and its residue ........................... 97 4.2.3 Determining the regular part ...............................100 4.3 h-recursion for torus N-point Virasoro conformal blocks in the necklace channel (and sphere N-point blocks in the linear channel) ...............................102 4.3.1 Definition of the Virasoro block in the necklace channel . 103 4.3.2 Polar part ..........................................104 4.3.3 Regular part ........................................105 4.3.4 h-recursion representation .................................107 4.3.5 Sphere N-point block in the linear channel . 108 4.4 c-recursion for all sphere and torus Virasoro conformal blocks . 111 4.4.1 Factorization of 3-point functions with degenerate representations and the poles of conformal blocks ......................................112 4.4.2 Large c, fixed hi limit of Virasoro conformal blocks . 115 4.4.3 Global SL(2) blocks ....................................119 4.4.4 Examples of c-recursive representations . 121 4.5 Generalization to higher genus ...................................125 4.6 Discussion ..............................................130 5 Genus two modular bootstrap 132 5.1 Introduction .............................................132 5.2 The genus two conformal block ...................................136 3 5.2.1 OPE of Z3-twist fields in Sym (CFT) . 137 5.2.2 The conformal block decomposition of hσ3(0)¯σ3(z)σ3(1)¯σ3(1)i . 140 5.2.3 Recursive representation ..................................141 5.2.4 Mapping to the 3-fold-pillow ................................144 5.3 The genus two modular crossing equation .............................145 5.3.1 Some preliminary analysis .................................145 5.3.2 Critical surfaces .......................................147 5.4 Beyond the Z3-invariant surface ..................................152 5.5 Discussion ..............................................154 6 Quantum Regge trajectories and the Virasoro analytic bootstrap 157 6.1 Introduction and summary .....................................157 6.1.1 Motivation by inversion ..................................158 6.1.2 The Virasoro fusion kernel .................................160 vi 6.1.3 Summary of physical results ................................162 6.2 Analyzing the fusion kernel .....................................170 6.2.1 Integral form of the kernel .................................173 6.2.2 Computing properties of the kernel ............................177 6.2.3 Cross-channel limit of Virasoro blocks . 181 6.3 Extracting CFT data ........................................184 6.3.1 Quantum Regge trajectories and a “Virasoro Mean Field Theory” . 184 6.3.2 Large spin ..........................................188 6.3.3 Large spin and large c ...................................193 6.3.4 Large conformal dimension ................................195 6.4 Global limit .............................................196 6.4.1 Vacuum kernel .......................................198 6.4.2 Non-vacuum kernel .....................................201 6.5 Gravitational interpretation of CFT results ............................203 6.5.1 Generic c ..........................................203 6.5.2 Gravitational interpretation of anomalous twists . 205 6.6 Semiclassical limits and late-time physics .............................209 6.6.1 Heavy-light limit ......................................209
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