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Register a new userA tauberian theorem for the conformal bootstrap
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For expansions in one-dimensional conformal blocks, we provide a rigorous link between the asymptotics of the spectral density of exchanged primaries and the leading singularity in the crossed channel. Our result has a direct application to systems of SL(2,R)-invariant correlators (also known as 1d CFTs). It also puts on solid ground a part of the lightcone bootstrap analysis of the spectrum of operators of high spin and bounded twist in CFTs in d>2. In addition, a similar argument controls the spectral density asymptotics in large N gauge theories.
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Current numerical conformal bootstrap techniques carve out islands in theory space by repeatedly checking whether points are allowed or excluded. We propose a new method for searching theory space that replaces the binary information allowed/excluded with a continuous navigator function that is negative in the allowed region and positive in the excluded region. Such a navigator function allows one to efficiently explore high-dimensional parameter spaces and smoothly sail towards any islands they may contain. The specific functions we introduce have several attractive features: they are everywhere well-defined, can be computed with standard methods, and evaluation of their gradient is immediate due to an SDP gradient formula that we provide. The latter property allows for the use of efficient quasi-Newton optimization methods, which we illustrate by navigating towards the 3d Ising island.
Extending previous work on 2 -- and 3 -- point functions, we study the 4 -- point function and its conformal block structure in conformal quantum mechanics CFT$_1$, which realizes the SO(2,1) symmetry group. Conformal covariance is preserved even though the operators with which we work need not be primary and the states are not conformally invariant. We find that only one conformal block contributes to the four-point function. We describe some further properties of the states that we use and we construct dynamical evolution generated by the compact generator of SO(2.1).
Infrared fixed points of gauge theories provide intriguing targets for the modern conformal bootstrap program. In this work we provide some preliminary evidence that a family of gauged fermionic CFTs saturate bootstrap bounds and can potentially be solved with the conformal bootstrap. We start by considering the bootstrap for $SO(N)$ vector 4-point functions in general dimension $D$. In the large $N$ limit, upper bounds on the scaling dimensions of the lowest $SO(N)$ singlet and traceless symmetric scalars interpolate between two solutions at $Delta =D/2-1$ and $Delta =D-1$ via generalized free field theory. In 3D the critical $O(N)$ vector models are known to saturate the bootstrap bounds and correspond to the kinks approaching $Delta =1/2$ at large $N$. We show that the bootstrap bounds also admit another infinite family of kinks ${cal T}_D$, which at large $N$ approach solutions containing free fermion bilinears at $Delta=D-1$ from below. The kinks ${cal T}_D$ appear in general dimensions with a $D$-dependent critical $N^*$ below which the kink disappears. We also study relations between the bounds obtained from the bootstrap with $SO(N)$ vectors, $SU(N)$ fundamentals, and $SU(N)times SU(N)$ bi-fundamentals. We provide a proof for the coincidence between bootstrap bounds with different global symmetries. We show evidence that the proper symmetries of the underlying theories of ${cal T}_D$ are subgroups of $SO(N)$, and we speculate that the kinks ${cal T}_D$ relate to the fixed points of gauge theories coupled to fermions.
Conformal field theories play a central role in theoretical physics with many applications ranging from condensed matter to string theory. The conformal bootstrap studies conformal field theories using mathematical consistency conditions and has seen great progress over the last decade. In this thesis we present an implementation of analytic bootstrap methods for perturbative conformal field theories in dimensions greater than two, which we achieve by combining large spin perturbation theory with the Lorentzian inversion formula. In the presence of a small expansion parameter, not necessarily the coupling constant, we develop this into a systematic framework, applicable to a wide range of theories. The first two chapters provide the necessary background and a review of the analytic bootstrap. This is followed by a chapter which describes the method in detail, taking the form of a practical guide to large spin perturbation theory by means of a step-by-step implementation. The second part of the thesis presents several explicit implementations of the framework, taking examples from a number of well-studied conformal field theories. We show how many literature results can be reproduced from a purely bootstrap perspective and how a variety of new results can be derived.
We consider 5-point functions in conformal field theories in d > 2 dimensions. Using weight-shifting operators, we derive recursion relations which allow for the computation of arbitrary conformal blocks appearing in 5-point functions of scalar operators, reducing them to a linear combination of blocks with scalars exchanged. We additionally derive recursion relations for the conformal blocks which appear when one of the external operators in the 5-point function has spin 1 or 2. Our results allow us to formulate positivity constraints using 5-point functions which describe the expectation value of the energy operator in bilocal states created by two scalars.
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