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arXiv:2610.39106

virasoro_blocks: Software for the 2d bootstrap

David Poland

Published Oct 29, 2026• 0 citations
arXivHTMLPDF

Abstract

We introduce virasoro_blocks, software that computes Virasoro conformal blocks at any central charge in a form suitable for the numerical bootstrap. At each finite order in the elliptic nome, Zamolodchikov's recursion gives the block as a positive prefactor times a rational function of the exchanged weight. However, its many Kac poles make it difficult to use in semidefinite programming approaches to the bootstrap. In this software we use optimal interpolation to reduce the number of poles while keeping high accuracy. The software works in arbitrary precision and handles singular central charges, including the minimal models and $c=1$. It also computes global blocks, $N=1$ superconformal blocks, torus characters, and torus one-point blocks. We illustrate it with two applications. Assuming only scalar Virasoro primaries, we compute bounds on the scalar gap that approach the window of Liouville solutions between $(c-1)/16$ and $(c-1)/12$. We also bound the average exchanged dimension in four-point functions of identical scalars at several central charges. These bounds have a rich structure, and many known 2d CFTs lie on or near their boundary.

DOI
10.48550/arxiv.2610.39106

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We introduce virasoro_blocks, software that computes Virasoro conformal blocks at any central charge in a form suitable for the numerical bootstrap. At each finite order in the elliptic nome, Zamolodchikov's recursion gives the block as a positive prefactor times a rational function of the exchanged weight. However, its many Kac poles make it difficult to use in semidefinite programming approaches to the bootstrap. In this software we use optimal interpolation to reduce the number of poles while keeping high accuracy. The software works in arbitrary precision and handles singular central charges, including the minimal models and c=1c=1c=1. It also computes global blocks, N=1N=1N=1 superconformal blocks, torus characters, and torus one-point blocks. We illustrate it with two applications. Assuming only scalar Virasoro primaries, we compute bounds on the scalar gap that approach the window of Liouville solutions between (c−1)/16(c-1)/16(c−1)/16 and (c−1)/12(c-1)/12(c−1)/12. We also bound the average exchanged dimension in four-point functions of identical scalars at several central charges. These bounds have a rich structure, and many known 2d CFTs lie on or near their boundary.
We derive expressions for the Virasoro OPE and four-point conformal blocks on the sphere via the resolution of identity recently determined in [J.-F. Fortin et al., Phys. Rev. D 111, 085010 (2025)]. Even though the resolution of the identity depends on Virasoro singular vectors, our expression for the blocks does not depend on their precise form, but just on their well-known conformal weights. We verify that our expression is compatible with—but differs from—Zamolodchikov’s h h -recursion relation and we also examine the impact of various large central charge limits in our formula. A Mathematica notebook with a simple implementation of our expression for the Virasoro conformal blocks is provided as an ancillary file.
Abstract Virasoro conformal blocks are universal ingredients of correlation functions of two-dimensional conformal field theories (2d CFTs) with Virasoro symmetry. It is acknowledged that in the (classical) limit of large central charge of the Virasoro algebra and large external, and intermediate conformal weights with fixed ratios of these parameters Virasoro blocks exponentiate to functions known as Zamolodchikovs’ classical blocks. The latter are special functions which have awesome mathematical and physical applications. Uniformization, monodromy problems, black holes physics, quantum gravity, entanglement, quantum chaos, holography, N \mathcal{N} N N = 2 gauge theory and quantum integrable systems (QIS) are just some of contexts, where classical Virasoro blocks are in use. In this paper, exploiting known connections between power series and integral representations of (quantum) Virasoro blocks, we propose new finite closed formulae for certain multi-point classical Virasoro blocks on the sphere. Indeed, combining classical limit of Virasoro blocks expansions with a saddle point asymptotics of Dotsenko-Fateev (DF) integrals one can relate classical Virasoro blocks with a critical value of the “Dotsenko-Fateev matrix model action”. The latter is the “DF action” evaluated on a solution of saddle point equations which take the form of Bethe equations for certain QIS (Gaudin spin models). A link with integrable models is our main motivation for this research line. For instance, analogous quantities as the “DF on-shell action” appear in 2d CFT realization of the so-called Richardson’s solution of the reduced BCS model describing physics of ultra-small superconducting grains. Precisely, the Richardson solution and its particular limit characterizing the rational Gaudin model have known implementation in 2d CFT in terms of a free field representation of certain (perturbed and unperturbed) WZW blocks. The WZW analogue of the “DF on-shell action” appears here and plays a crucial role, it is a generating function for eigenvalues of quantum integrals of motion. An exploration of relationships between power expansions and Coulomb gas representations of the aforementioned WZW blocks could pave a way for new analytical tools useful in the study of models of the Richardson-Gaudin type.
We constrain the spectrum of two-dimensional unitary, compact conformal field theories with central charge c > 1 using modular bootstrap. Upper bounds on the gap in the dimension of primary operators of any spin, as well as in the dimension of scalar primaries, are computed numerically as functions of the central charge using semi-definite programming. Our bounds refine those of Hellerman and Friedan-Keller, and are in some cases saturated by known CFTs. In particular, we show that unitary CFTs with c < 8 must admit relevant deformations, and that a nontrivial bound on the gap of scalar primaries exists for c < 25. We also study bounds on the dimension gap in the presence of twist gaps, bounds on the degeneracy of operators, and demonstrate how "extremal spectra" which maximize the degeneracy at the gap can be determined numerically.
A bstract In this paper, we attempt to explore the landscape of two-dimensional conformal field theories (2d CFTs) by efficiently searching for numerical solutions to the modular bootstrap equation using machine-learning-style optimization. The torus partition function of a 2d CFT is fixed by the spectrum of its primary operators and its chiral algebra, which we take to be the Virasoro algebra with c > 1. We translate the requirement that this partition function is modular invariant into a loss function, which we then minimize to identify possible primary spectra. Our approach involves two technical innovations that facilitate finding reliable candidate CFTs. The first is a strategy to estimate the uncertainty associated with truncating the spectrum to the lowest dimension operators. The second is the use of a new singular-value-based optimizer (Sven) that is more effective than gradient descent at navigating the hierarchical structure of the loss landscape. We numerically construct candidate truncated CFT partition functions with central charges between 1 and 87\frac{8}{7}78​ , a range devoid of known examples, and argue that these candidates likely come from a continuous space of modular bootstrap solutions. We also provide evidence for a more stringent constraint on the spectral gap near c = 1 than the existing bound of Δgap≤c6+13{\Delta}_{\text {gap}}\le \frac{c}{6}+\frac{1}{3}Δgap​≤6c​+31​ .
Abstract We constrain the spectrum of two-dimensional unitary, compact conformal field theories with central chargec >1 using modular bootstrap. Upper bounds on the gap in the dimension of primary operators of any spin, as well as in the dimension of scalar primaries, are computed numerically as functions of the central charge using semi-definite programming. Our bounds refine those of Hellerman and Friedan-Keller, and are in some cases saturated by known CFTs. In particular, we show that unitary CFTs withc <8 must admit relevant deformations, and that a nontrivial bound on the gap of scalar primaries exists forc <25. We also study bounds on the dimension gap in the presence of twist gaps, bounds on the degeneracy of operators, and demonstrate how “extremal spectra” which maximize the degeneracy at the gap can be determined numerically.
Virasoro conformal blocks are fixed in principle by symmetry, but a closed-form expression is unknown in the general case. In this work, we provide three closed-form expansions for the four-point Virasoro blocks on the sphere, for arbitrary operator dimensions and central charge c. We do so by solving known recursion relations. One representation is a sum over hypergeometric global blocks, whose coefficients we provide at arbitrary level. Another is a sum over semiclassical Virasoro blocks obtained in the limit in which two external operator dimensions scale linearly with large c. In both cases, the 1/c expansion of the Virasoro blocks is easily extracted. We discuss applications of these expansions to entanglement and thermality in conformal field theories and particle scattering in three-dimensional quantum gravity.