• SDPB
• Installation and Usage
• Attribution
• Acknowledgements
• Works Using SDPB

SDPB is an open-source, arbitrary-precision, parallelized semidefinite program solver, designed for the conformal bootstrap. It solves the following problem:

Let $S^{m\times m}[x]$ be the space of symmetric $m\times m$ matrices whose entries are polynomials in $x$.

• Given:
  • A collection of $J\times (N+1)$ polynomial matrices $M_{nj}(x) \in S^{m_j\times m_j}[x]$, ($j=1,\dots,J$ and $n=0,\dots,N$),
  • A constant $b_0 \in \mathbb{R}$,
  • A vector $b\in \mathbb{R}^N$,
• Maximize $b_0 + b\cdot y$ over decision variables $y\in \mathbb{R}^N$
• Such that: $M_{0j}(x) + \Sigma_{n=1}^N y_n M_{nj}(x) \succeq 0$ for all $x\geq 0$ and $j=1,\dots,J$.

Here, $M\succeq 0$ means "M is positive semidefinite."

For more information, see A Semidefinite Program Solver for the Conformal Bootstrap and the manual.

Authors: David Simmons-Duffin ([email protected]), Walter Landry ([email protected]), Vasiliy Dommes ([email protected])

The easiest way to run SDPB on a Windows or Mac machine is to follow the Docker instructions. For Linux and HPC centers, the Singularity instructions will probably work better. If you want to build it yourself, there are detailed instructions in Install.md.

Usage instructions are detailed in Usage.md.

Two python wrappers for SDPB are available:

• PyCFTBoot by Connor Behan (arXiv:1602.02810)
• cboot by Tomoki Ohtsuki (arXiv:1602.07295).

An unofficial Haskell wrapper is available:

• sdpb-haskell by David Simmons-Duffin

If you use SDPB in work that results in publication, consider citing

• D. Simmons-Duffin, A Semidefinite Program Solver for the Conformal Bootstrap, JHEP 1506, 174 (2015) arXiv:1502.02033.
• W. Landry and D. Simmons-Duffin, Scaling the semidefinite program solver SDPB arXiv:1909.09745.

Depending on how SDPB is used, the following other sources might be relevant:

The first use of semidefinite programming in the bootstrap:

• D. Poland, D. Simmons-Duffin and A. Vichi, Carving Out the Space of 4D CFTs, JHEP 1205, 110 (2012) arXiv:1109.5176.

The generalization of semidefinite programming methods to arbitrary spacetime dimension:

• F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N) Vector Models, JHEP 1406, 091 (2014) arXiv:1307.6856.

The generalization of semidefinite programming methods to arbitrary systems of correlation functions:

• F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping Mixed Correlators in the 3D Ising Model, JHEP 1411, 109 (2014) arXiv:1406.4858.

Derivation of linear and quadratic variations of the objective function, used in approx_objective:

• M. Reehorst, S. Rychkov, D. Simmons-Duffin, B. Sirois, N. Su, B. van Rees, Navigator Function for the Conformal Bootstrap, arXiv:2104.09518.

Spectrum extraction was originally written for use in:

• Z. Komargodski and D. Simmons-Duffin, The Random Bond Ising Model in 2.01 and 3 Dimensions, arXiv:1603.04444

An explanation of how it works appears in:

• D. Simmons-Duffin, The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT, arXiv:1612.08471

• Versions 2 and 3 of SDPB were made possible by support from the Simons Collaboration on the Nonperturbative Bootstrap.

• The design of SDPB was partially based on the solvers SDPA and SDPA-GMP, which were essential sources of inspiration and examples.

• Thanks to Filip Kos, David Poland, and Alessandro Vichi for collaboration in developing semidefinite programming methods for the conformal bootstrap and assistance testing SDPB.

• Thanks to Amir Ali Ahmadi, Hande Benson, Pablo Parrilo, and Robert Vanderbei for advice and discussions about semidefinite programming.

• Thanks also to Noah Stein, who first suggested the idea of semidefinite programming to me in this Math Overflow question.

Here is a list of papers citing SDPB.