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 $$\frac{8}{7}$$ , 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 $${\Delta}_{\text {gap}}\le \frac{c}{6}+\frac{1}{3}$$ .
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 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≤6c+31 .
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.
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.
Two-dimensional Conformal Field Theories (CFTs) allow us to describe quantum field theories at critical points and in general systems without a characteristic scale. It is possible to implement such theories on topologically non-trivial surfaces. If we formulate a two-dimensional CFT on a toroidal surface, a new symmetry emerges known as modular invariance. This symmetry corresponds to the set of transformations that act without changing the toroidal surface, consequently, the partition function of the theory must remain invariant under these transformations. This invariance is used as a constraint by the modular bootstrap method to obtain information about the spectrum without using the Lagrangian and operator content of the theory. By imposing that the theory is consistent with modular symmetry, it is possible to determine the limits of the spectrum beyond which modular invariance can no longer be satisfied, thus constraining the space of admissible two-dimensional conformal field theories. To study CFTs that present discrete global symmetries, it is essential to introduce Topological Defect Lines (TDLs). These are one-dimensional, topologically invariant operators that implement the action of the symmetry group on the Hilbert space. The presence of TDLs modifies the topology of the system and introduces new partition functions associated with different symmetry sectors. Under modular transformations, the TDLs mix among themselves, and consequently, the corresponding partition functions also mix, thus promoting the modular constraint to a vector-valued condition. This formalism makes it possible to implement discrete symmetries also considering the possible presence of anomalies. The objective of this thesis is therefore to understand how discrete symmetries and anomalies affect the spectrum of a CFT. Based on the modular bootstrap framework developed by Lin and Shao for the Z2 symmetry, the analysis was first analytically extended to the Abelian case Z3. This was useful in extending the discussion to a more complex case, namely the non-Abelian symmetry S3. This extension constitutes the original theoretical contribution of the thesis, providing the explicit formulation of modular bootstrap equations for CFTs with non-Abelian discrete symmetry. The analytical modular constraints were reformulated numerically, bridging the modular bootstrap framework with semidefinite programming in SDPB. This analysis paves the way for a numerical study based on convex optimization techniques to explore the constraints arising from modular bootstrap equations.
In this dissertation we describe progress towards carving out the space of non-integrable conformal field theories (CFTs) in 1+1 dimensions using conformal bootstrap techniques. We develop and apply numerical and analytical techniques to explore the non-perturbative consequences of conformal symmetry, unitarity and locality on the spectrum and dynamics of 2D CFTs. Throughout we focus on compact irrational CFTs, which have a discrete spectrum consisting of an infinite number of primary operators and exhibit chaotic dynamics. In the first part of the dissertation we report on numerical advances in charting the space of 2D CFTs. We use semidefinite programming to compute bounds on the gap in the spectrum of primary operators as a function of the central charge and on spectral functions that capture the spectral density of primary operators. We then develop recursion relations for arbitrary Virasoro conformal blocks, and explore the constraints of modular invariance of the genus-two partition function, leading to ``critical surfaces'' that bound the heavy structure constants of a CFT. In the second part of the dissertation we develop the analytic bootstrap program for two dimensional conformal field theories, focusing on universal aspects of irrational unitary CFTs and holography. We show that the large-spin sector of irrational CFTs is universally governed by a ``Virasoro mean field theory,'' incorporating exact stress tensor dynamics into the analytic bootstrap and leading to non-perturbative insights about the bound-state dynamics of three-dimensional quantum gravity. We then leverage a generalized notion of modular invariance to derive a universal asymptotic formula for the average value of the structure constants whenever one or more of the operators has large conformal dimension. The large central charge limit makes contact with quantum gravity in three dimensions, where the averaging over heavy states corresponds to coarse-graining over black hole microstates. Finally, we study the Euclidean gravitational path integral of pure AdS3 quantum gravity, and use a modular bootstrap analysis to suggest new saddle point configurations that can be consistently included in the gravitational path integral and whose inclusion renders the dual torus partition function unitary.
Modular invariance imposes rigid constraints on the partition functions of two-dimensional conformal field theories (CFTs). Many fundamental results follow strictly from modular invariance and unitarity, giving rise to the numerical modular bootstrap program. Here we report on a way to relate a particular family of quantum error correcting codes to a family of "code CFTs," which forms a subset of the space of Narain CFTs. This correspondence reduces modular invariance of the 2D CFT partition function to a few simple algebraic relations obeyed by a multivariate polynomial characterizing the corresponding code. Using this correspondence, we construct many explicit examples of physically distinct isospectral theories, as well as many examples of nonholomorphic functions, which satisfy all the basic properties of a 2D CFT partition function, yet are not associated with any known CFT.
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.