Exactly Solvable 1+1d Chiral Lattice Gauge Theories

Abstract: Using the modified Villain lattice Hamiltonian formulation of the 1+1d compact boson theory, we construct exactly solvable abelian chiral lattice gauge theories in two spacetime dimensions. As a concrete example, we derive an explicit quadratic lattice Hamiltonian for the "34-50" chiral gauge theory. We further show that $N$ copies of the modified Villain theory realize the $O(N,N;\mathbb{Z})$ T-duality transformations, which we then use to solve and analyze these lattice gauge theories.

• The paper introduces a modified Villain Hamiltonian enabling exact diagonalization of chiral lattice symmetries and reproducing continuum anomaly structures.
• It demonstrates anomaly cancellation via charge assignments and explicit constructions like the 34-50 U(1) theory using bosonization and T-duality.
• The framework provides a robust basis for non-perturbative studies and quantum simulations of chiral gauge theories in 1+1 dimensions.

Exactly Solvable 1+1d Chiral Lattice Gauge Theories

Introduction and Motivation

The formulation of anomaly-free chiral gauge theories on the lattice remains a central challenge for the non-perturbative definition of the Standard Model, particularly due to constraints imposed by the Nielsen-Ninomiya theorem in even-dimensional spacetimes. The Standard Model evades this obstacle via nontrivial anomaly cancellation, motivating the search for lattice regularizations of chiral gauge theories in lower dimensions. The paper "Exactly Solvable 1+1d Chiral Lattice Gauge Theories" (2601.14359) presents a rigorous construction of exactly solvable 1+1d abelian chiral lattice gauge theories, leveraging the modified Villain formulation of the compact boson theory, with explicit focus on quadratic Hamiltonians amenable to exact solutions.

Modified Villain Hamiltonian and Chiral Symmetries on the Lattice

The modified Villain Hamiltonian provides a lattice regularization for the compact boson CFT, enabling the exact realization of chiral U(1) symmetries in 1+1 dimensions. These symmetries are distinguished into momentum and winding sectors, with corresponding charges $Q_m$ and $Q_w$ that are both quantized and gauge invariant after imposing Gauss law constraints. Importantly, the algebra of local charges reproduces the characteristic Schwinger terms of the continuum, manifesting mixed 't Hooft anomalies between the momentum and winding symmetries, and ensuring the correct anomaly structure for subsequent gauging.

The Hamiltonian's quadratic form allows for explicit diagonalization. By reexpressing the degrees of freedom as variables $x_\ell$ and analyzing their Fourier modes, both zero-modes and non-zero oscillatory modes can be solved, confirming agreement with the continuum boson CFT spectrum at arbitrary radius $R$ in the $L \to \infty$ limit.

Gauging Chiral Symmetries: Construction of Lattice Gauge Theory

The construction of anomaly-free chiral gauge theories on the lattice proceeds from $N$ flavors of compact bosons, each formulated via the modified Villain model. Carefully chosen charge assignments $(n_m^{(I)}, n_w^{(I)})$ allow the definition of anomaly-free chiral U(1) symmetries, which can then be gauged following a prescription that modifies the lattice constraints or couples gauge fields directly to the Hamiltonian.

Two complementary presentations are given:

• Constraint-driven coupling: Background U(1) gauge fields are introduced via modified Gauss law constraints. Making these fields dynamical, subject to the anomaly cancellation condition $\sum_{I=1}^N n_m^{(I)} n_w^{(I)} = 0$, the construction guarantees commutativity of Gauss's law at distinct lattice sites, preserving locality and solvability.
• Direct Hamiltonian coupling: Through a similarity transformation, gauge fields are coupled directly to the bosonic degrees of freedom in the Hamiltonian, yielding a more canonical form for analysis and numerics.

The explicit formulation encompasses both the kinetic term for electric fields and the coupling to compact bosons (or, after fermionization, to chiral lattice fermions).

Illustration: The 34-50 Theory

A concrete example of this construction is the so-called "34-50" U(1) chiral gauge theory, where two Dirac fermions have left-moving charges $(3,4)$ and right-moving charges $(5,0)$. Both gravitational and gauge anomalies vanish, as $3^2+4^2=5^2+0^2$, enabling consistent lattice regularization.

Via bosonization at $R=1/\sqrt{2}$, the corresponding compact boson model admits charge assignments $(8,4,-1,2)$ for $(n_m^{(1)}, n_m^{(2)}, n_w^{(1)}, n_w^{(2)})$. The model is shown to be reducible via an $O(2,2;\mathbb{Z})$ T-duality to a quadratic form involving a massive boson coupled to exactly solvable massless compact boson modes. Majorana fermion variables are introduced to achieve the fermionization required for the implementation of lattice chiral fermion dynamics.

T-duality and Symmetry Realization

The work further exploits T-duality transformations, showing that $N$ copies of the modified Villain model possess the full $O(N,N;\mathbb{Z})$ duality group, with explicit generators constructed for $N=2$. This enables alternative lattice realizations of the modular space of compact boson theories and, via duality, facilitates solution and analysis of arbitrarily assigned anomaly-free chiral lattice gauge theories.

Solution of the Schwinger Model and Benchmarking

As a consistency check, the paper solves the $N=1$ case corresponding to the massless Schwinger model, demonstrating equivalence in both spectrum and dispersion relations to continuum expectations via explicit diagonalization.

Implications and Future Outlook

The presented framework rigorously establishes the existence of exactly solvable anomaly-free chiral gauge theories on the lattice in 1+1 dimensions for abelian U(1) gauge groups. Quadratic Hamiltonians admit generalizations to arbitrary numbers of flavors and charge assignments, with exact lattice realization of chiral symmetries, including anomaly structures and their cancellation.

While the work provides a decisive step toward the non-perturbative definition of chiral gauge theories, the extension to non-abelian chiral symmetries and higher dimensions (notably 3+1d, relevant for the full Standard Model) remains open and is identified as an active area of future research. The modular T-duality analysis hints at further connections to lattice sigma models and the study of moduli spaces.

The construction also informs the lattice framework necessary for future studies in quantum simulation and non-perturbative calculations, including numerical validation and the design of quantum algorithms for chiral gauge theories.

Conclusion

This paper sets forth a mathematically precise and physically transparent method for constructing and exactly solving 1+1d anomaly-free chiral lattice gauge theories, using the modified Villain formulation and its bosonization/fermionization dualities. The approach achieves exact chiral symmetry realization and anomaly cancellation directly on the lattice, exemplified by the explicit solution of the 34-50 theory. These results constitute a strong theoretical foundation for the further development of lattice methods in chiral gauge theory, with practical and conceptual impact on both condensed matter and high-energy physics contexts.