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> level quantization of chern-simons theory
In Chern-Simons theory, the level quantization refers to the fact that the level parameter $k$, which appears in the action of the theory, must take on integer values in order for the theory to be well-defined.
Chern-Simons theory is a topological quantum field theory that describes the behavior of certain gauge fields, such as electromagnetism, in three dimensions. The action of the theory involves a term that is proportional to the integral of the trace of a product of gauge potentials and their differentials. This term is known as the Chern-Simons term and is given by:
$S[A] = \frac{k}{4\pi} \int \Tr \left( A \wedge dA + \frac{2}{3} A \wedge A \wedge A \right) $
where $A$ is the gauge potential, $dA$ is its differential, $\wedge$ denotes the wedge product, $\Tr$ denotes the trace, and $k$ is the level parameter.
The level parameter $k$ plays an important role in the theory, as it determines the strength of the Chern-Simons term relative to other terms in the action. In particular, the Chern-Simons term is a topological term that does not depend on the metric of spacetime, and is therefore not renormalized by quantum corrections. However, the coefficient $k$ can be renormalized, and this leads to level quantization.
The level quantization arises from the fact that the path integral formulation of the theory requires that the action be well-defined. In particular, the path integral must converge, which requires that the integral of the exponential of the action be finite. This, in turn, requires that the level parameter $k$ be quantized. Specifically, the level must take on integer values in order for the theory to be well-defined.
This level quantization has important physical consequences, as it leads to the appearance of fractional statistics in certain physical systems. For example, in the fractional quantum Hall effect, electrons in a two-dimensional system are described by an effective Chern-Simons theory with a fractional value of the level parameter, which leads to the emergence of anyons with fractional statistics.
Overall, the level quantization of Chern-Simons theory is an important aspect of the theory that has implications for both mathematics and physics. It is an example of the deep connection between topology and quantum field theory, and has led to important insights into the behavior of physical systems at low temperatures and in the presence of strong magnetic fields.
> tell me about abelian chern-simons theory on the torus
Abelian Chern-Simons theory on the torus is a quantum field theory that is defined on a two-dimensional torus, which is a flat, doughnut-shaped surface. It is a special case of Abelian Chern-Simons theory, which is a three-dimensional topological field theory that describes the behavior of certain gauge fields, such as electromagnetism.
In Abelian Chern-Simons theory on the torus, the physical fields are the gauge potential $A(x,y)$ and a scalar field $\phi (x,y)$, where $(x,y)$ are the coordinates on the torus. The action for this theory takes the form:
$S[A, \phi ] = \frac{k}{4\pi} \int dxdy ( \ep^{\mu\nu\la} A_\mu \d_\nu A_\la + \phi \ep^{\mu \nu \la} \d_\mu A_\nu \d_\la \phi )$
where $\ep^{\mu \nu \la}$ is the Levi-Civita tensor in two dimensions, and $k$ is a constant known as the level of the theory.
The equations of motion for this theory are given by the Maxwell equations with an additional term due to the presence of the scalar field:
$ \d_\mu F_{\mu \nu} = k \ep^{\mu \nu \la} \d_\mu \phi \d_\la \phi$
$\d_\mu \d_\mu \phi = k \ep^{\mu \nu \la} F_{\mu\nu}$
where $F_{\mu\nu} = \d_\mu A_\nu - \d_\nu A_\mu$ is the field strength tensor.
One interesting feature of Abelian Chern-Simons theory on the torus is that it leads to the emergence of topological excitations known as magnetic flux tubes. These flux tubes are two-dimensional analogues of the anyons that arise in three dimensions, and are characterized by a topological invariant known as the magnetic flux, which is quantized in units of the level $k$.
Abelian Chern-Simons theory on the torus has applications in condensed matter physics, where it is used to describe the behavior of certain two-dimensional materials known as fractional quantum Hall systems. It is also of interest in string theory, where it arises as an effective field theory in certain compactifications of the theory.
