Abstract
We study 2d N=4 superconformal field theories, focusing on its application on numerical bootstrap study. We derive the superconformal block by utilizing the global part of the super Virasoro algebra and set up the crossing equations for the non-BPS long-multiplet 4-point function. Along the way, we build global N=4 superconformal short and long multiplets and compute all possible 2,3-point functions of long-multiplets that are needed to construct the superconformal blocks and the crossing equations. Since we consider a long-multiplet 4-point function, the number of crossing equations is huge, and we expect it to give a strong constraint than the usual superconformal bootstrap analysis, which relies on BPS 4-point functions. In addition, we present an alternative way to derive crossing equations using N=4 superspace and comment on a puzzle.
Highlights
For a decade, there has been extensive work on solving various conformal field theories using only first principles — unitarity, associativity of the operator product algebra, and the so called conformal bootstrap program
We study 2d N=4 superconformal field theories, focusing on its application on numerical bootstrap study
Since we consider a long-multiplet 4-point function, the number of crossing equations is huge, and we expect it to give a strong constraint than the usual superconformal bootstrap analysis, which relies on BPS 4-point functions
Summary
There has been extensive work on solving various conformal field theories using only first principles — unitarity, associativity of the operator product algebra, and the so called conformal bootstrap program. Since the coefficients placed in front of each decomposed Virasoro blocks are independent, the set of crossing equations is distinguished from non-supersymmetric 2d CFT 4-point functions and at the same time captures structure of N = 4 This fact was used in [9] to do the N = 2 long-multiplet bootstrap analysis. Our goal is to use Casimir differential equation to solve the superconformal block, but N = 4 superspace does not seem to fully represent small N = 4 superconformal algebra As it is not a complete treatment, we point out some limitations that we encountered. It would be far more constraining to use the infinite dimensional super-Virasoro algebra when one tries to bootstrap two dimensional conformal field theories, but the full recursion relation that leads to the approximate expression for conformal block for extended supersymmetry has not been worked out in the literature.
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