Abstract
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 well-defined in large regions of parameter space, 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.
Highlights
We have presented in this work a powerful alternative to the scanning-based approach employed so far in the numerical conformal bootstrap program
This came from the realization that there exist functions, for which we have coined the term “navigator functions,” which measure how far a given point is from the boundary between allowed and disallowed regions and can be used to efficiently find an allowed point as well as the boundary of an allowed region
Adding the generalized free field solution to the crossing equation has led us in Section 2.1.1 to the definition of the Generalized Free Field (GFF) navigator
Summary
The numerical conformal bootstrap program has relied on the idea [4] that for any point in CFT parameter space it is possible to check if the point is allowed or excluded by constructing positive linear functionals. One knows that one is close to ∂ R if one can find two nearby trial points xi and xi such that they are on two different sides of the boundary We will modify this setup so that a single SDPB run computes a continuous function N (x), called a navigator, which will give a more nuanced measure of success than “allowed/ excluded.”. Our third important result is to demonstrate how a quasi-Newton method—the BFGS algorithm [11]—successfully overcomes these difficulties (Section 5) This algorithm finds first the allowed region, and the navigator minimum, in a relatively small number of steps. This represents an attractive alternative to the tiptop algorithm recently introduced for this purpose in the feasibility setup [12]. Appendix C shows how one can evaluate the navigator Hessian, in addition to the gradient, and provides numerical tests of these procedures
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