Chern–Simons theory

The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and James Harris Simons, who introduced the Chern–Simons 3-form. In the Chern–Simons theory, the action is proportional to the integral of the Chern–Simons 3-form.

In condensed-matter physics, Chern–Simons theory describes composite fermions and the topological order in fractional quantum Hall effect states. In mathematics, it has been used to calculate knot invariants and three-manifold invariants such as the Jones polynomial. which is generally used in the literature today and leads to a well-defined linking number. With the canonical framing the above phase is the exponential of 2πi/(k + N) times the linking number of L with itself.

;Problem (Extension of Jones polynomial to general 3-manifolds)

"The original Jones polynomial was defined for 1-links in the 3-sphere (the 3-ball, the 3-space R3). Can you define the Jones polynomial for 1-links in any 3-manifold?"

See section 1.1 of this paper for the background and the history of this problem. Kauffman submitted a solution in the case of the product manifold of closed oriented surface and the closed interval, by introducing virtual 1-knots. It is open in the other cases. Witten's path integral for Jones polynomial is written for links in any compact 3-manifold formally, but the calculus is not done even in physics level in any case other than the 3-sphere (the 3-ball, the 3-space R<sup>3</sup>). This problem is also open in physics level. In the case of Alexander polynomial, this problem is solved.

Relationships with other theories

Topological string theories

In the context of string theory, a U(N) Chern–Simons theory on an oriented Lagrangian 3-submanifold M of a 6-manifold X arises as the string field theory of open strings ending on a D-brane wrapping X in the A-model topological string theory on X. The B-model topological open string field theory on the spacefilling worldvolume of a stack of D5-branes is a 6-dimensional variant of Chern–Simons theory known as holomorphic Chern–Simons theory.

WZW and matrix models

Chern–Simons theories are related to many other field theories. For example, if one considers a Chern–Simons theory with gauge group G on a manifold with boundary then all of the 3-dimensional propagating degrees of freedom may be gauged away, leaving a two-dimensional conformal field theory known as a G Wess–Zumino–Witten model on the boundary. In addition the U(N) and SO(N) Chern–Simons theories at large N are well approximated by matrix models.

Chern–Simons gravity theory

In 1982, S. Deser, R. Jackiw and S. Templeton proposed the Chern–Simons gravity theory in three dimensions, in which the Einstein–Hilbert action in gravity theory is modified by adding the Chern–Simons term. ()

In 2003, R. Jackiw and S. Y. Pi extended this theory to four dimensions () and Chern–Simons gravity theory has some considerable effects not only to fundamental physics but also condensed matter theory and astronomy.

The four-dimensional case is very analogous to the three-dimensional case. In three dimensions, the gravitational Chern–Simons term is

:<math>\operatorname{CS}(\Gamma)=\frac{1}{2\pi^2}\int d^3x\varepsilon^{ijk}\biggl(\Gamma^p_{iq}\partial_j\Gamma^q_{kp}+\frac{2}{3}\Gamma^p_{iq}\Gamma^q_{jr}\Gamma^r_{kp}\biggr).</math>

This variation gives the Cotton tensor

:<math>=-\frac{1}{2\sqrt{g\bigl(\varepsilon^{mij}D_i R^n_j+\varepsilon^{nij}D_i R^m_j).</math>

Then, Chern–Simons modification of three-dimensional gravity is made by adding the above Cotton tensor to the field equation, which can be obtained as the vacuum solution by varying the Einstein–Hilbert action.

Chern–Simons matter theories

In 2013 Kenneth A. Intriligator and Nathan Seiberg solved these 3d Chern–Simons gauge theories and their phases using monopoles carrying extra degrees of freedom. The Witten index of the many vacua discovered was computed by compactifying the space by turning on mass parameters and then computing the index. In some vacua, supersymmetry was computed to be broken. These monopoles were related to condensed matter vortices. ()

The N = 6 Chern–Simons matter theory is the holographic dual of M-theory on <math>AdS_4\times S_7</math>.

Four-dimensional Chern–Simons theory

In 2013 Kevin Costello defined a closely related theory defined on a four-dimensional manifold consisting of the product of a two-dimensional 'topological plane' and a two-dimensional (or one complex dimensional) complex curve. He later studied the theory in more detail together with Witten and Masahito Yamazaki, demonstrating how the gauge theory could be related to many notions in integrable systems theory, including exactly solvable lattice models (like the six-vertex model or the XXZ spin chain), integrable quantum field theories (such as the Gross–Neveu model, principal chiral model and symmetric space coset sigma models), the Yang–Baxter equation and quantum groups such as the Yangian which describe symmetries underpinning the integrability of the aforementioned systems.

The action on the 4-manifold <math>M = \Sigma \times C</math> where <math>\Sigma</math> is a two-dimensional manifold and <math>C</math> is a complex curve is

<math display = block>S = \int_M \omega \wedge CS(A)</math>

where <math>\omega</math> is a meromorphic one-form on <math>C</math>.

Chern–Simons terms in other theories

The Chern–Simons term can also be added to models which aren't topological quantum field theories. In 3D, this gives rise to a massive photon if this term is added to the action of Maxwell's theory of electrodynamics. This term can be induced by integrating over a massive charged Dirac field. It also appears for example in the quantum Hall effect. The addition of the Chern–Simons term to various theories gives rise to vortex- or soliton-type solutions Ten- and eleven-dimensional generalizations of Chern–Simons terms appear in the actions of all ten- and eleven-dimensional supergravity theories.

One-loop renormalization of the level

If one adds matter to a Chern–Simons gauge theory then, in general it is no longer topological. However, if one adds n Majorana fermions then, due to the parity anomaly, when integrated out they lead to a pure Chern–Simons theory with a one-loop renormalization of the Chern–Simons level by −n/2, in other words the level k theory with n fermions is equivalent to the level k − n/2 theory without fermions.

References

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