JuliBootS: a hands-on guide to the conformal bootstrap

We introduce {\tt JuliBootS}, a package for numerical conformal bootstrap computations coded in {\tt Julia}. The centre-piece of {\tt JuliBootS} is an implementation of Dantzig's simplex method capable of handling arbitrary precision linear programming problems with continuous search spaces. Current supported features include conformal dimension bounds, OPE bounds, and bootstrap with or without global symmetries. The code is trivially parallelizable on one or multiple machines. We exemplify usage extensively with several real-world applications. In passing we give a pedagogical introduction to the numerical bootstrap methods.Comment: 29 page

Bootstrapping the 3d Ising twist defect

Recent numerical results point to the existence of a conformally invariant twist defect in the critical 3d Ising model. In this note we show that this fact is supported by both epsilon expansion and conformal bootstrap calculations. We find that our results are in good agreement with the numerical data. We also make new predictions for operator dimensions and OPE coefficients from the bootstrap approach. In the process we derive universal bounds on one-dimensional conformal field theories and conformal line defects.Comment: 24+8 pages, 12 figures, references adde

Holographic phase space: cc-functions and black holes as renormalization group flows

We construct a $\mathcal N$-function for Lovelock theories of gravity, which yields a holographic $c$-function in domain-wall backgrounds, and seemingly generalizes the concept for black hole geometries. A flow equation equates the monotonicity properties of $\mathcal N$ with the gravitational field, which has opposite signs in the domain-wall and black hole backgrounds, due to the presence of negative/positive energy in the former/latter, and accordingly $\mathcal N$ monotonically decreases/increases from the UV to the IR. On $AdS$ spaces the $\mathcal N$-function is related to the Euler anomaly, and at a black hole horizon it is generically proportional to the entropy. For planar black holes, $\mathcal N$ diverges at the horizon, which we interpret as an order $N^2$ increase in the number of effective degrees of freedom. We show how $\mathcal N$ can be written as the ratio of the Wald entropy to an effective phase space volume, and using the flow equation relate this to Verlinde's notion of gravity as an entropic force. From the effective phase space we can obtain an expression for the dual field theory momentum cut-off, matching a previous proposal in the literature by Polchinski and Heemskerk. Finally, we propose that the area in Planck units counts states, not degrees of freedom, and identify it also as a phase space volume. Written in terms of the proper radial distance $\beta$, it takes the suggestive form of a canonical partition function at inverse temperature $\beta$, leading to a "mean energy" which is simply the extrinsic curvature of the surface. Using this we relate this definition of holographic phase space with the effective phase space appearing in the $\mathcal N$-function.Comment: 31 pages, v2: typos fixed, references adde

Star Integrals, Convolutions and Simplices

We explore single and multi-loop conformal integrals, such as the ones appearing in dual conformal theories in flat space. Using Mellin amplitudes, a large class of higher loop integrals can be written as simple integro-differential operators on star integrals: one-loop $n$-gon integrals in $n$ dimensions. These are known to be given by volumes of hyperbolic simplices. We explicitly compute the five-dimensional pentagon integral in full generality using Schl\"afli's formula. Then, as a first step to understanding higher loops, we use spline technology to construct explicitly the $6d$ hexagon and $8d$ octagon integrals in two-dimensional kinematics. The fully massive hexagon and octagon integrals are then related to the double box and triple box integrals respectively. We comment on the classes of functions needed to express these integrals in general kinematics, involving elliptic functions and beyond.Comment: 23 page