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virasoro_blocks: Software for the 2d bootstrap

This paper introduces *virasoro_blocks*, a high-precision software tool that optimizes the computation of Virasoro conformal blocks by reducing pole complexity via optimal interpolation, thereby enabling robust numerical bootstrap bounds on 2D CFT spectra that align closely with known solutions like Liouville theory.

Original authors: David Poland

Published 2026-10-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: David Poland

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of theoretical physics, there exists a powerful method for understanding the universe without needing to know the specific particles or forces at play. This approach, known as the conformal bootstrap, treats the laws of nature as a set of logical puzzles. It relies on two fundamental rules: that the laws of physics look the same regardless of how you rotate or scale your view, and that probabilities must always add up to a positive number. By applying these strict rules to the way particles interact, physicists can map out the possible "spectrum" of a theory—the list of all the different types of particles that can exist and how heavy they are. In two-dimensional universes, a simplified version of our own, these interactions are governed by a particularly rich symmetry that organizes particles into families. For decades, calculating the behavior of these families has been a monumental task, requiring complex mathematics that often breaks down when pushed to the limits of precision.

A new software tool called virasoro blocks, developed by David Poland at Yale University, has finally cracked this computational bottleneck. The program is designed to calculate the behavior of these particle families in two-dimensional theories with unprecedented speed and accuracy. Before this tool existed, researchers were often forced to stop their calculations early because the mathematical expressions became too messy, filled with too many "poles"—points where the equations blow up and become undefined. These poles are like sudden cliffs in a mathematical landscape; if you try to drive a car (or run a computer program) over them, the journey ends abruptly. The new software uses a clever technique called optimal interpolation to smooth over these cliffs. Instead of trying to navigate every single jagged edge, the program selects the most important ones and fills in the gaps with a highly accurate approximation. This allows the computer to handle the complex math of two-dimensional physics without getting stuck, opening the door to rigorous tests of theories that were previously out of reach.

The researchers used this new software to explore the boundaries of what is possible in these two-dimensional worlds. They asked a simple question: if a universe contains only the simplest type of particles, what are the limits on how light or heavy those particles can be? By running thousands of simulations, they found that the allowed weights of these particles fall into a specific window. The lower limit of this window sits at a precise fraction of the universe's central charge, a number that measures the complexity of the theory, while the upper limit hovers just above a famous threshold known from a theory called Liouville theory. This result is significant because it provides strong numerical evidence that if you strip a two-dimensional universe down to its bare essentials, the only way to satisfy the laws of physics is to follow the rules of Liouville theory. The software calculated bounds that approach the window where this specific theory is expected to be the unique solution, suggesting that no other arrangement of particles could exist within these strict constraints.

Beyond this specific case, the team used the software to map out the average weight of particles in various scenarios, looking for patterns across different types of universes. They discovered a landscape rich with structure, filled with sharp peaks and deep valleys that correspond to known, real-world theories. Many of the theories that physicists have studied for decades, including those describing phase transitions in materials, sit right on the edge of these calculated limits. This alignment suggests that nature, even in its most abstract forms, tends to push against the boundaries of what is mathematically possible. As the researchers pushed the simulations closer to a specific limit where the universe becomes slightly more complex, they noticed that the rules governing the particles began to change in unexpected ways. The results indicate that the transition between different types of universes is not smooth but rather filled with narrow, intricate features that only appear when the calculations are precise enough to see them.

The power of this work lies not just in the answers it provides, but in the clarity it brings to a field that has long been shrouded in mathematical fog. By turning a chaotic set of equations into a manageable, high-precision tool, the software allows physicists to test their ideas against the hard constraints of logic and symmetry. It reveals that the space of possible two-dimensional universes is far more structured than previously thought, with known theories acting as signposts along the edges of allowed possibilities. The findings do not just confirm old ideas; they provide a detailed map of where new physics might hide, waiting to be discovered by those who can look closely enough at the mathematical boundaries. As the software continues to be refined, it promises to turn the abstract art of theoretical physics into a precise science, where the limits of the possible are drawn with the same certainty as the laws of gravity.

Title: virasoro blocks: Software for the 2d bootstrap
Author: David Poland (Yale University)

Problem Statement

The numerical conformal bootstrap relies on formulating crossing symmetry and unitarity constraints as semidefinite programs (SDPs). In dimensions $d > 2$ , general-purpose software exists to compute conformal block derivatives required for these programs. However, in two dimensions, the enhanced local conformal symmetry organizes four-point functions into Virasoro blocks. While Zamolodchikov's recursion relation allows for the efficient computation of these blocks as power series in the elliptic nome $q$ , the resulting expressions are rational functions of the exchanged weight $h$ with a large number of Kac poles.

The presence of these poles presents a significant obstacle for SDP solvers. At finite order $N$ in the nome expansion, the number of poles grows as $N \log N$ . For the derivative orders required in modern bootstrap studies ( $N \gtrsim 60$ ), the resulting polynomials have degrees in the hundreds. Furthermore, for central charges $1 < c < 25$ , most Kac poles are complex and lack a natural ordering, leading to ill-conditioned bases if all poles are retained. Previous attempts to use Virasoro blocks in rigorous convex optimization were limited to low derivative orders due to these computational and numerical stability issues.

Methodology

The paper introduces virasoro blocks, a software package designed to compute Virasoro conformal blocks suitable for numerical bootstrap applications. The core methodology involves optimal interpolation to reduce the complexity of the block representation while maintaining high accuracy.

Optimal Interpolation: Instead of using the exact rational function with hundreds of poles, the software approximates the block derivatives using a "damped rational" form:
$T^{m,n}_\ell(\Delta) \approx \chi_\ell(\Delta) P^{m,n}_\ell(\Delta)$
where $\chi_\ell(\Delta)$ contains a positive prefactor and a product of a small number of retained Kac poles, and $P^{m,n}_\ell$ is a low-degree polynomial.
- Node Selection: The interpolation nodes are chosen based on the equilibrium density of unit charges with logarithmic repulsion in an external potential derived from the prefactor $\chi_\ell$ . This construction minimizes the weighted Lebesgue constant.
- Pole Selection: The software employs a "contribution optimal" rule to select which poles to retain. This algorithm iteratively estimates the error introduced by dropping a pole, considering the residue and the pole's distance from the physical region. This approach retains poles that are difficult to approximate with polynomials while discarding those whose effects can be absorbed by the polynomial term.
- Handling Singularities: The software handles singular central charges (minimal models, $c=1$ , $c=25$ , and rational $b^2$ ) by performing exact Laurent expansions around the singular values, merging colliding Kac poles, and computing finite limits.
Arbitrary Precision Arithmetic: The implementation uses Python with mpmath accelerated by FLINT/Arb. It operates in arbitrary precision to manage the catastrophic cancellations that occur when evaluating high-order rational functions.
Scope of Computation: Beyond standard sphere Virasoro blocks, the software computes:
- Global $SL(2) \times SL(2)$ blocks.
- $N=1$ superconformal blocks (Neveu–Schwarz and Ramond sectors).
- Torus characters and torus one-point blocks.
- Large-spin towers for fixed twist.

Key Contributions

Software Implementation: A robust, open-source package (virasoro blocks) that generates derivative tables for semidefinite programming solvers (specifically SDPB).
Algorithmic Improvement: The application of optimal interpolation to the Virasoro bootstrap, reducing the polynomial degree from hundreds to manageable levels (comparable to 3D bootstrap) while achieving accuracies of $10^{-40}$ or better.
Handling of Complex Poles: A systematic method for selecting and retaining complex-conjugate pole pairs for $1 < c < 25$ , ensuring the resulting basis remains well-conditioned.
Singular Charge Regularization: Exact handling of recursion singularities at minimal models and $c=1$ via Laurent expansions, allowing the bootstrap to be applied to these specific theories without numerical instability.

Results

The author illustrates the software's capabilities with two specific applications:

Scalar Gap Bounds: Assuming a theory where all Virasoro primaries are scalars, the author computed bounds on the scalar gap $\Delta_{\text{gap}}$ .
- The results show an allowed window for the gap bounded below by $\Delta = (c-1)/16$ and above by a value approaching the Liouville threshold $(c-1)/12$ .
- As the derivative cutoff $\Lambda$ increases, the upper bound approaches the Liouville threshold, supporting the conjecture that Liouville theory is the unique solution for scalar-only spectra.
- The lower bound coincides with $(c-1)/16$ , consistent with analytic continuation arguments regarding the unitarity of the Liouville four-point function.
Moment Bounds: The author bounded the average exchanged dimension $\langle \Delta \rangle$ in four-point functions of identical scalars at various central charges ( $c = 8, 4, 2$ and rational values like $3/2, 6/5$ ).
- The bounds exhibit a rich structure of kinks, spikes, and dips.
- Many known 2D CFTs (including rational theories and $c=1$ theories) lie on or near the boundary of these allowed regions.
- Near $c=1$ , narrow features develop at degenerate external dimensions, suggesting that the limits $c \to 1$ and $\Lambda \to \infty$ do not commute, implying stronger constraints for theories just above $c=1$ .

Significance

The paper claims that virasoro blocks makes rigorous numerical bootstrap studies of 2D CFTs with Virasoro symmetry widely accessible, similar to how existing tools have democratized $d>2$ studies. By overcoming the obstacle of Kac poles through optimal interpolation, the software enables high-precision constraints on the spectrum and OPE coefficients of 2D theories. The results suggest that the bootstrap can effectively distinguish between known theories and potential new solutions, particularly in the context of Liouville theory and theories with extended chiral algebras. The author notes that the software is a companion to virasoro bootstrap, which assembles the crossing equations and SDPs, providing a complete pipeline for 2D bootstrap investigations.

Problem Statement

Methodology

Key Contributions

Results

Significance

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