Fixed-point tensor network for compactified boson conformal field theory

Abstract: Fixed-point (FP) tensor networks provide a discrete spacetime representation of conformal field theories (CFTs), offering a new route toward understanding holographic duality, generalized symmetries, and even quantum gravity. In this work, we construct FP tensors for the 2D compactified boson theory at a generic compactification radius, an archetypal irrational CFT, using boundary (open-string) data with conformal boundary conditions. We show that the resulting tensors reproduce the closed-string spectrum with high accuracy and generate stable renormalization-group (RG) flows under the tensor complex renormalization algorithm. Moreover, we identify a controllable exactly marginal deformation at the level of a single tensor, enabling flows that move continuously along the $c=1$ moduli space. This framework establishes a concrete lattice-level route toward describing a broad class of 2D irrational CFTs.

• The paper demonstrates an FP tensor construction for the compactified boson CFT, enabling the discretization of continuous operator spectra.
• It employs three- and four-point boundary correlators to encode U(1) charge structures and implement exact marginal current-current deformations.
• Numerical validations show convergence of the c=1 spectrum and RG stability across varying radii by suppressing marginal perturbations.

Fixed-Point Tensor Networks for Compactified Boson CFT

Introduction and Motivation

The compactified boson conformal field theory (CFT) serves as an archetypal irrational CFT, pivotal in string theory, condensed matter, and statistical mechanics. Unlike rational CFTs (RCFTs), where the structure of local operator content and fusion categories is finite and algebraically tractable, the compactified boson CFT exhibits a continuous moduli space parameterized by the compactification radius $R$ and generically possesses an infinite set of representations and generalized symmetries. Previous fixed-point (FP) tensor constructions for RCFTs established a method to discretize CFT path integrals with tensor networks built from boundary changing operators (BCOs) and their correlators. Extending this framework to irrational CFTs, specifically to the compactified boson at arbitrary radius, is nontrivial due to the lack of finite fusion rules and the necessity for continuum or infinite sums in the tensor structure.

Construction of FP Tensors for the Compactified Boson

The paper demonstrates the construction of FP tensors for the compactified boson CFT at generic $R$, both for boundary Dirichlet and Neumann conformal boundary conditions (CBCs). The local tensor ingredients are three-point and four-point boundary correlators of open-string vertex operators that encode the $U(1)$ charge structures, with precise geometric and topological components reflecting the conformal and generalized symmetries.

For a triangulation of the underlying two-dimensional manifold, each tensor leg hosts BCOs, labeled by conformal families and their descendants; tensor entries are determined by the corresponding three- or four-point functions, incorporating the fusion rules implicit in the charge conservation of the boson. Importantly, the bond dimension of the tensor network is not fixed but may be made arbitrarily large or even continuous, reflecting the irrationality of the CFT.

To make the construction numerically feasible and validate the continuous (irrational) structure using finite computational resources, the continuous CBCs are discretized onto grids—with grid size determined by irrelevance conditions on induced marginal and higher-charge deformations. Analytical conditions show that in the large-bond-dimension limit (and in the continuum grid limit), the construction converges to the exact CFT.

Marginal Deformations, T-Duality, and the MSBC

An intrinsic property of the $c=1$ compactified boson is the exactly marginal current-current deformation that tunes $R$ and generates the $c=1$ moduli space. The tensor network construction explicitly realizes this deformation: the FP tensors possess tunable geometric parameters, either through their structure constants or through the continuous families of CBCs, that exactly generate the marginal operators at the tensor level. The formal framework supports marginal flows along the $c=1$ line, with associated changes in the physical spectrum captured nonperturbatively.

A key technical challenge is the suppression of residual marginal deformations that arise due to truncation (finite bond dimension) and the use of only a single type of CBC. The authors introduce a mixed shrinkable boundary condition (MSBC)—a symmetric linear combination of Dirichlet and Neumann Cardy states with specific weights. Implementation of the MSBC at the tensor level cancels the leading marginal $(1,1)$ descendant, thereby stabilizing the radius under tensor network renormalization (TNR) and complex RG flow.

The construction is shown to be compatible with T-duality (interchange of Dirichlet and Neumann boundary grid descriptions under $R \to 1/(2R)$). The continuous and discrete lattice representations yield physically equivalent path integrals in the limit of infinite bond dimension.

Numerical Verification: RG Flow and Spectrum Extraction

The validity of the FP tensor construction is benchmarked by computation of the closed-string spectrum using transfer matrices built from the constructed tensors. The spectrum, as extracted numerically from both rank-3 and rank-4 tensors (including those with MSBC), converges to the exact compactified boson spectrum at all tested radii as the number of descendants and bond dimension increases. The RG flow under tensor complex renormalization (TCR) remains stable, reflecting the underlying CFT fixed point, and the extracted central charge is universally $c=1$ within numerical uncertainty. At selected radii, the tensor network reproduces not only the correct low-lying conformal dimensions but also the necessary degeneracy structure.

The rank-4 FP tensor structure further suppresses marginal perturbations, even with primary-only truncations, making it highly suited for numerical computation and for controlling marginal flows. The paper documents high-precision numerical data at the self-dual ($R$0), fermionic ($R$1), and KT ($R$2) points.

Marginal Flows and Tensor Geometry

The tensor network explicitly supports tuning along the CFT marginal line. By deforming geometric parameters (e.g., the size of "holes" in the tensor patch), the network realizes an explicit marginal current-current perturbation, with the effect of continuously varying the compactification radius. This is directly confirmed by extracting scaling dimensions from transfer matrices for different geometric deformations and matching the change in conformal data along the $R$3 moduli space. The procedure provides a robust, nonperturbative tool to study exactly marginal deformations, circle and orbifold branches, and possibly emergent dualities in tensor network constructions.

Theoretical and Practical Implications

This work provides a concrete, algebraic framework for constructing discrete spacetime tensor networks for irrational CFTs, explicitly the compactified boson. The ability to work with continuous families of CBCs and to encode exactly marginal deformations at the tensor level opens the possibility of directly simulating a range of $R$4 theories, including Luttinger liquids and multi-component critical field theories, in nonperturbative lattice or quantum computational settings.

The construction paves the way for future exploration of continuous CFT moduli spaces, topological holography in irrational settings, and connections to generalized global symmetries and higher-form structures in quantum field theory. The framework is amenable to extension to other irrational theories with continuous operator spectra and could inform the design of tensor network states for higher-dimensional or supersymmetric theories. Potential future developments include direct construction of ground state wave functions for CFTs and incorporation of more general boundary and defect data for holographic or gravitational models.

Conclusion

The work establishes a practical and theoretically robust method for encoding the full content of compactified boson CFTs at arbitrary radius within fixed-point tensor network constructions. This achievement extends algebraic tensor network methods beyond rational models and provides explicit tools for controlling marginal deformations, extracting spectra, and exploring the RG structure of irrational CFTs. The framework is expected to influence both practical tensor network simulations in condensed matter and the study of algebraic and topological features of quantum field theories.