[Paper Review] autoboot: A generator of bootstrap equations with global symmetry

Mocho Go, Yuji Tachikawa|arXiv (Cornell University)|Mar 25, 2019

Theoretical and Computational Physics61 references4 citations

TL;DR

This paper introduces *autoboot*, a Mathematica-based tool that automatically generates mixed-correlator bootstrap equations for conformal field theories with global symmetries, using group representation data. It compiles these into optimized Python code for use with the *sdpb* semi-definite programming solver, significantly accelerating numerical bootstrap computations via a novel 'hot-starting' technique that reuses solver states between nearby parameter points.

ABSTRACT

We introduce autoboot, a Mathematica program which automatically generates mixed-correlator bootstrap equations of an arbitrary number of scalar external operators, given the global symmetry group and the representations of the operators. The output is a Python program which uses Ohtsuki's cboot which in turn uses Simmons-Duffin's sdpb. In an appendix we also discuss a simple technique to significantly reduce the time to run sdpb, which we call hot-starting.

Motivation & Objective

• To automate the derivation of mixed-correlator bootstrap equations in conformal field theories with arbitrary global symmetries, reducing manual error and effort.
• To enable efficient numerical bootstrap studies of CFTs with non-Abelian global symmetries, such as $D_8$, $O(N)$, and $O(2)$, which have been underexplored due to technical complexity.
• To streamline the pipeline from group theory data to executable semi-definite programming input for *sdpb*, integrating with existing tools like *cboot* and *PyCFTBoot*.
• To introduce and validate a 'hot-starting' technique that reuses dual and primal solver states between consecutive parameter evaluations, drastically reducing runtime in exclusion plot generation.

Proposed method

• The tool uses the SmallGrp library via GAP to retrieve group theory data, including character tables and Clebsch-Gordan coefficients, for finite groups up to order 2000.
• It constructs operator product expansion (OPE) decompositions for scalar operators transforming in specified group representations, enforcing symmetry constraints on four-point functions.
• Bootstrap equations are symbolically derived from crossing symmetry and conformal block decomposition, then reformulated as semi-definite programming (SDP) constraints.
• The SDP is translated into a Python script using the *cboot* interface, which generates XML input for the *sdpb* solver.
• A novel 'hot-starting' technique is implemented by reusing the final iterate $(x,X,y,Y)$ of the *sdpb* solver from a previous run as the initial state for a nearby parameter point.
• The method includes heuristics for robustness: restarting from earlier states if the solver enters a pathological state with decreasing step sizes and increasing barrier parameter $\mu$.

Experimental results

Research questions

• RQ1 Can the derivation of mixed-correlator bootstrap equations for CFTs with global symmetries be fully automated to reduce human error and effort?
• RQ2 How can the computational cost of generating exclusion plots for scaling dimensions be reduced in the numerical bootstrap framework?
• RQ3 To what extent does reusing the interior point iterates of the *sdpb* solver between nearby parameter points accelerate convergence?
• RQ4 What heuristics are necessary to maintain stability when reusing solver states across multiple runs?

Key findings

• The *autoboot* tool successfully generates bootstrap equations for CFTs with global symmetries, demonstrated on $D_8$, 3d Ising, and $O(2)$ models with scalar operators in singlet and fundamental representations.
• The hot-starting technique reduces *sdpb* runtime by approximately 10 to 20 times compared to default initialization, based on empirical benchmarks.
• Reusing solver states from one run as initial values for the next significantly accelerates convergence, especially when scanning over nearby scaling dimensions.
• The method remains robust when combined with dual feasibility detection and primal jump detection, even though the original *sdpb* assumption about primal feasibility precluding dual feasibility no longer holds under hot-starting.
• Restarting from a previously stable state is necessary when the barrier parameter $\mu$ begins to increase and step lengths $\alpha_P$, $\alpha_D$ shrink to near-zero, indicating stagnation.