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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
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
Chern-Simons terms are known under different names in different branches of physics. In fluid mechanics it is usually called "fluid helicity", in plasma physics and magnetohydrodynamics "magnetic helicity". In the context of field theories it is usually called Chern-Simons term.
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
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.
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
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