Quantizing Chern-Simons Theory on a Spatial Lattice

This article discusses the process of quantizing a specific type of theoretical physics called Chern-Simons theory on a spatial lattice. This theory is significant in both condensed matter and high-energy physics and has various applications, such as the study of anomalies, dualities, and the behavior of certain materials.

Chern-Simons theory focuses on how certain fields behave in a space with special properties. It has many variations and formulations, particularly when put on a lattice – which resembles a grid or a mesh. Existing literature shows many different ways to express this theory on a lattice, indicating a lack of a single, universal method.

In previous work, a version of Chern-Simons theory was created as a lattice gauge theory. This means that the theory is defined using a grid structure, which may capture important features at a finite scale that usually appear only when looking at larger, continuous space. This work serves as a foundation for the current discussion, which revolves around the quantization of abelian Chern-Simons theory on a spatial lattice using a specific Hamiltonian approach.

The authors find that this lattice Hamiltonian model helps clarify key features of Chern-Simons theory that relate to its behavior at a finite scale. This includes aspects like the compactness of the gauge group, the quantization of levels, the framing of Wilson Lines, and more.

A major benefit of working with the Hamiltonian formulation is easy access to the Hilbert space, which is the mathematical space of possible states in the quantized theory. The building blocks of the lattice Chern-Simons theory consist of gauge fields located at the edges of the lattice and integer values at the vertices. These components are linked to infinite-dimensional Hilbert Spaces.

In continuous versions of Chern-Simons theory, however, the Hilbert space is finite-dimensional, directly linked to specific properties of the theory. This discrepancy vanishes in the lattice model, where the physical Hilbert space becomes a projection of the infinite-dimensional space down to a finite dimension due to various constraints.

The constraints arise from two types of symmetries in the theory. One set relates to local Gauge Transformations, while another set involves discrete symmetries associated with larger transformations. These constraints mean that only certain types of operators, specifically framed Wilson loops, are non-trivial and influential in the theory.

The article also addresses how to formulate odd-level Chern-Simons Theories, which require defining aspects like spin structures. These fermionic theories, involving half-integer values, depend on a specific choice of spin structure.

The physical observables in the compact Chern-Simons theory on a lattice relate to framed Wilson lines, which connect two points through a surface operator that is gauge-invariant and dependent on the structure of the lattice.

Previous literature has focused on extra degrees of freedom in simpler discretizations of the Chern-Simons term. The presence of zero modes, which are additional degrees of freedom, was initially viewed as problematic. However, they are critical, as they enforce the requirement of framing, demonstrating that physical Wilson lines are ribbons and not simple lines.

The article begins by discussing a simpler non-compact version of Chern-Simons theory, using a rectangular grid where time is real and continuous. The authors clarify the concept using well-defined terms and variable notations so that the lattice structure resembles a toroidal shape.

In this context, the gauge transformation is introduced, showing how the action remains unchanged under certain transformations. The canonical momentum is defined, revealing a particular constraint that deviates from what is expected in the continuous version of Chern-Simons theory.

As the work progresses, the authors delve into the quantization process, beginning with momentum space fields. They derive commutation relations that uphold the structure of the lattice model, ensuring that the resulting physical states adhere to quantization principles.

A significant aspect of the discussion is the importance of the framing constraint. This constraint ensures that only specific kinds of Wilson loops, termed framed Wilson loops, operate within the theory. The authors illustrate how this framing is crucial to the theory’s physical operators and their significance.

Next, the article transitions to discussing both symmetries present in the theory, clarifying how they act on Wilson lines. The authors emphasize that understanding these symmetries is essential for recognizing the underlying properties of Chern-Simons theory on a lattice.

The development leads to the compact Chern-Simons theory, expressing how to incorporate new variables and constraints in a way that maintains the overall structure and behavior of the model. This is done by introducing plaquette fields and adjusting gauge transformations to ensure gauge invariance is satisfied.

Through this process, the authors unveil the relationship between large gauge transformations and Wilson loops, clarifying how constraints on these transformations reveal the theory’s intricacies.

The next section focuses on defect Hilbert spaces, a concept that allows for exploring states that do not adhere to the standard conditions imposed by previous constraints. This opens up new avenues for understanding how Chern-Simons theory interacts with probe particles.

The interactions between these probe particles and the gauge structure provide insight into the nature of topological features in the theory. The authors illustrate that temporal Wilson lines serve as defects, linking back to the broader understanding of Chern-Simons theory as constructed on a lattice.

The latter part of the article discusses odd-level fermionic Chern-Simons theory, where certain anomalies present challenges in defining a consistent gauge-invariant Hilbert space. The authors introduce fermionic degrees of freedom to cancel these anomalies, emphasizing the importance of spin structure in the context of odd-level theories.

In this section, the relationship between fermions and the underlying gauge structure is clarified by introducing operators that satisfy specific relations akin to those observed in previous discussions. The careful structure of these operators encapsulates the nuanced interactions between bosonic and fermionic degrees of freedom.

In summary, the authors conclude that the lattice framework preserves core properties of the continuum theory, making it a useful tool for analysis. The methods discussed herein enhance the understanding of Chern-Simons theory, establishing connections across diverse topics within theoretical physics.

Future lines of inquiry include examining edge modes within the lattice theory, the role of Chern-Simons theory in higher dimensions, and the exploration of dualities in the context of charged matter. The article wraps up by acknowledging the collaboration and support received throughout the research process, inviting further exploration of the rich landscape of Chern-Simons theory on Lattices.