Why is it interesting?

The applications [of Chern-Simons terms] range from the mathematical characterization of knots to the physical description of electrons in the quantum Hall effect [5], vivid evidence for the deep significance of the Chern-Simons structure and of its antecedent, the chiral anomaly.Collaborating with David Gross; Descendants of the Chiral Anomaly by R. Jackiw

Electroweak baryogenesis proceeds via changes in the non-Abelian Chern-Simons number.Estimate of the primordial magnetic field helicity by Tanmay Vachaspati

Layman

Student

For a great summary see section 2 in Collaborating with David Gross; Descendants of the Chiral Anomaly by R. Jackiw and the Freiburg Lecture Notes by H.K.Moffatt

In Fluid Mechanics

In the beginning people tried to make a mechanical model of electrodynamics. For example, Maxwell though of Faraday's electric and magnetic field lines as "fine tubes of variable section carrying an incompressible fluid".

This is motivated by the observation that the equations of vacuum electrostatics

$$ \Delta \cdot \vec E =0, \quad \Delta \times \vec E =0,$$ are exactly the same as the equations for the flow of an incompressible frictionless fluid with no visosity and no "vorticity", or curl:

$$ \Delta \cdot \vec v =0, \quad \Delta \times \vec v =0,$$

where $\vec v$ is the velocity field. In addition, even in situations where the vorticity $\Delta \times \vec v$ is not zero, Helmholtz showed in 1858 that the vortex lines, which means the lines of $\Delta \times \vec v$, move in the direction of $\vec v$ as if they had an existence of their own. (Take note that these observation was also what led Kelvin to propose his theory of vortex atoms). (Source: Baez, Munian; Gauge Fields, Knots and Gravity, page 292)

This idea is made precise by the notion "kinetic helicity".

In fluid dynamics, the kinetic helicity is a measure of the degree of knottedness and/or linkage of the vortex lines of the flow.

Fluid helicity is one of the most important conserved quantities in ideal fluid flows, being an invariant of the Euler equations, and a robust quantity of the dissipative Navier-Stokes equations [13]. In ideal conditions its topological interpretation in terms of Gauss linking number was provided by Moffat [23] and extended by Moffat & Ricca [24]. In the context of vortex dynamics (kinetic) helicity is defined by

$$ H \equiv \int_\Omega u \cdot \omega d^3x, $$ where $u$ is the velocity field, $\omega = \Delta \times u$, is the vorticity, defined on $\Omega$, and $x$ is the position vector.

page 96 in How Nature Works: Complexity in Interdisciplinary Research and Applications by Ivan Zelinka

[H]elicity, is an integral over the fluid domain that expresses the correlation between velocity and vorticity, and an invariant of the classical Euler equations of ideal (inviscid) fluid flow. […] It is precisely because vortex lines are frozen in the fluid, thus conserving their topology, that helicity is conserved too. As Kelvin recognized, if two vortex tubes are linked, then that linkage survives in an ideal fluid for all time; if a vortex tube is knotted, then that knot survives in the same way for all time. Helicity is the integral manifestation of this invariance: For two linked tubes, it is proportional to the product of the two circulations (each conserved by Kelvin’s circulation theorem), whereas for a single knotted tube, or a deformed unknotted tube, it is proportional to its “writhe plus twist,” as encountered in differential geometry—a property of knotted ribbons that is invariant under continuous deformation (4, 5). Thus, for example, if an untwisted ribbon that goes twice round a circle before closing on itself is unfolded and untwisted back to circular form, then its writhe will decrease continuously from 1 to 0, and its twist will increase continuously from 0 to 1, by way of compensation. Such conversion of writhe to twist is familiar to anyone seeking to straighten out a coiled garden hose.

Helicity and singular structures in fluid dynamics by H. Keith Moffatt

: Can one write the Chern-Simons term as a total derivative, so that (as befits a topological quantity) the spatial volume integral becomes a surface integral. An argument that this should be possible is the following: The Chern-Simons term is a 3-form on 3-space, hence it is maximal and its exterior derivative vanishes because there are no 4-forms on 3-space. This establishes that on 3-space the Chern-Simons term is closed, so one can expect that it is also exact, at least locally, that is, it can be written as a total derivative. Of course, such a representation for the Chern-Simons term requires expressing the potentials in terms of “pre-potentials”, since the formulas (7), (8) show no evidence of derivative structure. [Recall that the total derivative formulas (5), (6) for the axial anomaly also require using potentials to express F.] There is a physical, practical reason for wanting the Abelian Chern-Simons term to be a total derivative. It is known in fluid mechanics that there exists an obstruction to constructing a Lagrangian for Euler’s fluid equations, and this obstruction is just the kinetic helicity $\int d^3r \vec v \cdot \vec \omega$, that is, the volume integral of the Abelian Chern-Simons term, constructed from the velocity 3-vector $\vec v$. This obstruction is removed when the integrand is a total derivative, because then the kinetic helicity volume integral is converted to a surface integral by Gauss’ theorem. When the integral obtains contributions only from a surface, the obstruction disappears from the 3-volume, where the fluid equation acts [4].John Bell’s Observations on the Chiral Anomaly and Some Properties of Its Descendants by R. Jackiw

For more information on this obstruction to construct the Lagrangian for Euler's fluid equations, see page 9ff in https://arxiv.org/pdf/hep-th/0004084.pdf and the great review:

HAMILTONIAN FLUID MECHANICS by Rick Salmon

For an experimental proof that knotted vortices exist indeed in nature, see Creation and dynamics of knotted vortices by Dustin Kleckner & William T. M. Irvine

In Electrodynamics

The magnetic helicity is the flux of the magnetic field through the surface bounding the volume, with θ acting as a modulating factor.https://arxiv.org/pdf/hep-th/0103017.pdf

In electrodynamics the Chern-Simons term is known as helicity and described the linking of magnetic flux lines. This can be seen by interpreting the magnetic field as an incompressible fluid flow, with vector potential $\vec A$: $B= \Delta \times A$.

For more on this, see Helicity and Electomagnetic Field Topology by Gerald E. Marsh

It has been suggested that primordial magnetic fields can develop large correlation lengths provided they carry nonvanishing “magnetic helicity” R d 3 r a · b, a quantity known to particle physicists as the Abelian, Euclidean Chern-Simons term. Here a is an Abelian gauge potential for the magnetic field b = ∇ × a. If there exists a period of decaying turbulence in the early universe, which can occur after a first-order phase transition, a magnetic field with nonvanishing helicity could have relaxed to a large-scale configuration, which enjoys force-free dynamics (source currents for the magnetic fields proportional to the fields themselves) thereby avoiding dissipation [1]. Creation and evolution of magnetic helicity by R. Jackiw et. al.

In Non-Abelian Gauge Theories

"anomalous currents are sourced by gauge field configurations with nonzero Chern-Simons number. The sphaleron reactions are therefore mediated by non-perturbative, extended gauge field configurations. In the SU(3)c sector the strong sphaleron mediates a vacuum-to-vacuum transition which induces a chirality-violating process among the colored particles [32]. We say that the transition is “vacuum-to-vacuum” because the SU(3)c gauge field strength tensor vanishes asymptotically. […] An anomalous reaction associated with the U(1)Y hypercharge sector is not usually included in leptogenesis calculations. This may be related to the fact that the hypercharge sphaleron, unlike the strong and weak sphalerons, is not a vacuum-to-vacuum transition. This is a consequence of the trivial topology of the vacuum manifold of the Abelian gauge theory, and for this case the ChernSimons number is proportional to the field strength tensor [38]. As a result, if the Abelian ChernSimons number changes in a process, then either the initial or the final state (or both) cannot be the vacuum. This exchange of energy between the particle sector and the gauge sector provides the basis of magnetogenesis. As we will see, the energy exchange is minimal, perhaps justifying the neglect of the hypercharge sphaleron in leptogenesis calculations. However, we should emphasize that field configurations with nonzero Chern-Simons number are known to exist, e.g., hypermagnetic knots [20, 21] and linked magnetic flux tubes [11]. We use the term “hypercharge sphaleron” to refer to any U(1)Y field configuration that interpolates between a vacuum configuration and a configuration with non-zero Chern-Simons number, which in MHD corresponds to magnetic field “helicity” (see below). Such a field configuration will source the anomaly equation for each Standard Model fermion field since they all carry hypercharge. Particle production due to the anomaly can also be understood as the emergence of filled states from the Dirac sea into the positive energy sector, just as in the case of the Abelian chiral anomaly in 3+1 dimensions [39]." Source

Ref 39 is C. Adam, B. Muratori, and C. Nash, Particle creation via relaxing hypermagnetic knots, Phys.Rev. D62 (2000) 105027, [hep-th/0006230].

Ref 21 is Creation and evolution of magnetic helicity by R. Jackiw, So-Young Pi

Researcher

On a manifold it is necessary to use covariant differentiation; curvature measures its noncommutativitiy. Its combination as a characteristic form measures the nontriviality of the underlying bundle. This train of ideas is so simple and natural that its importance can hardly be exaggerated. Shiing-shen Cern

• For a nice discussion, see Chapter 4 in Gauge Fields, Knots and Gravity by Baez and Muniain

Examples

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Example2:

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