Chern–Simons theory
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The Chern–Simons theory is a 3-dimensional
In
Particularly, Chern–Simons theory is specified by a choice of simple
It is also the central mathematical object in theoretical models for topological quantum computers (TQC). Specifically, an SU(2) Chern–Simons theory describes the simplest non-abelian anyonic model of a TQC, the Yang–Lee–Fibonacci model.[2][3]
The dynamics of Chern–Simons theory on the 2-dimensional boundary of a 3-manifold is closely related to fusion rules and conformal blocks in conformal field theory, and in particular WZW theory.[1][4]
The classical theory
Mathematical origin
In the 1940s
In 1974 S. S. Chern and
where T is the Chern–Weil homomorphism. This form is called Chern–Simons form. If df(ω) is closed one can integrate the above formula
where C is a (2k − 1)-dimensional cycle on M. This invariant is called Chern–Simons invariant. As pointed out in the introduction of the Chern–Simons paper, the Chern–Simons invariant CS(M) is the boundary term that cannot be determined by any pure combinatorial formulation. It also can be defined as
where is the first Pontryagin number and s(M) is the section of the normal orthogonal bundle P. Moreover, the Chern–Simons term is described as the eta invariant defined by Atiyah, Patodi and Singer.
The gauge invariance and the metric invariance can be viewed as the invariance under the adjoint Lie group action in the Chern–Weil theory. The
Configurations
Chern–Simons theories can be defined on any topological 3-manifold M, with or without boundary. As these theories are Schwarz-type topological theories, no metric needs to be introduced on M.
Chern–Simons theory is a
Dynamics
The
The constant k is called the level of the theory. The classical physics of Chern–Simons theory is independent of the choice of level k.
Classically the system is characterized by its equations of motion which are the extrema of the action with respect to variations of the field A. In terms of the field curvature
the field equation is explicitly
The classical equations of motion are therefore satisfied if and only if the curvature vanishes everywhere, in which case the connection is said to be flat. Thus the classical solutions to G Chern–Simons theory are the flat connections of principal G-bundles on M. Flat connections are determined entirely by holonomies around noncontractible cycles on the base M. More precisely, they are in one-to-one correspondence with equivalence classes of homomorphisms from the fundamental group of M to the gauge group G up to conjugation.
If M has a boundary N then there is additional data which describes a choice of trivialization of the principal G-bundle on N. Such a choice characterizes a map from N to G. The dynamics of this map is described by the Wess–Zumino–Witten (WZW) model on N at level k.
Quantization
To canonically quantize Chern–Simons theory one defines a state on each 2-dimensional surface Σ in M. As in any quantum field theory, the states correspond to rays in a Hilbert space. There is no preferred notion of time in a Schwarz-type topological field theory and so one can require that Σ be a Cauchy surface, in fact, a state can be defined on any surface.
Σ is of codimension one, and so one may cut M along Σ. After such a cutting M will be a manifold with boundary and in particular classically the dynamics of Σ will be described by a WZW model. Witten has shown that this correspondence holds even quantum mechanically. More precisely, he demonstrated that the Hilbert space of states is always finite-dimensional and can be canonically identified with the space of conformal blocks of the G WZW model at level k.
For example, when Σ is a 2-sphere, this Hilbert space is one-dimensional and so there is only one state. When Σ is a 2-torus the states correspond to the integrable representations of the affine Lie algebra corresponding to g at level k. Characterizations of the conformal blocks at higher genera are not necessary for Witten's solution of Chern–Simons theory.
Observables
Wilson loops
The
More concretely, given an irreducible representation R and a loop K in M, one may define the Wilson loop by
where A is the connection 1-form and we take the
HOMFLY and Jones polynomials
Consider a link L in M, which is a collection of ℓ disjoint loops. A particularly interesting observable is the ℓ-point correlation function formed from the product of the Wilson loops around each disjoint loop, each traced in the fundamental representation of G. One may form a normalized correlation function by dividing this observable by the partition function Z(M), which is just the 0-point correlation function.
In the special case in which M is the 3-sphere, Witten has shown that these normalized correlation functions are proportional to known
times the HOMFLY polynomial. In particular when N = 2 the HOMFLY polynomial reduces to the Jones polynomial. In the SO(N) case, one finds a similar expression with the Kauffman polynomial.
The phase ambiguity reflects the fact that, as Witten has shown, the quantum correlation functions are not fully defined by the classical data. The
- 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[6] 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.[7] 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 R3). 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. (Deser, Jackiw & Templeton (1982))
In 2003, R. Jackiw and S. Y. Pi extended this theory to four dimensions (Jackiw & Pi (2003)) 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
This variation gives the Cotton tensor
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
The N = 6 Chern–Simons matter theory is the holographic dual of M-theory on .
Four-dimensional Chern–Simons theory
In 2013
The action on the 4-manifold where is a two-dimensional manifold and is a complex curve is
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
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.
See also
- Gauge theory (mathematics)
- Chern–Simons form
- Topological quantum field theory
- Alexander polynomial
- Jones polynomial
- 2+1D topological gravity
- Skyrmion
References
- PMID 10021215.
- JSTOR 1971013.
- Deser, Stanley; Jackiw, Roman; Templeton, S. (1982). "Three-Dimensional Massive Gauge Theories" (PDF). S2CID 122537043.
- Intriligator, Kenneth; Seiberg, Nathan (2013). "Aspects of 3d N = 2 Chern–Simons Matter Theories". S2CID 119106931.
- S2CID 2243511.
- Kulshreshtha, Usha; Kulshreshtha, D.S.; Mueller-Kirsten, H. J. W.; Vary, J. P. (2009). "Hamiltonian, path integral and BRST formulations of the Chern-Simons-Higgs theory under appropriate gauge fixing". S2CID 120594654.
- Kulshreshtha, Usha; Kulshreshtha, D.S.; Vary, J. P. (2010). "Light-front Hamiltonian, path integral and BRST formulations of the Chern-Simons-Higgs theory under appropriate gauge fixing". S2CID 54602971.
- Lopez, Ana; PMID 9998334.
- S2CID 6207500.
- Marino, Marcos (2005). Chern–Simons Theory, Matrix Models, And Topological Strings. International Series of Monographs on Physics. Oxford University Press.
- S2CID 43230714.
- Witten, Edward (1995). "Chern–Simons Theory as a String Theory". Bibcode:1992hep.th....7094W.
- Specific
- ^ S2CID 14951363.
- arXiv:quant-ph/0101025.
- ^ Wang, Zhenghan. "Topological Quantum Computation" (PDF).
- ^ .
- ISSN 0040-9383.
- ^
Kauffman, L.H; Ogasa, E; Schneider, J (2018). "A spinning construction for virtual 1-knots and 2-knots, and the fiberwise and welded equivalence of virtual 1-knots". arXiv:1808.03023 [math.GT].
- ^
Kauffman, L.E. (1998). "Virtual Knot Theory". arXiv:math/9811028.
- ].
- .
- S2CID 119592177.
- arXiv:1908.02289 [hep-th].
- S2CID 9916364.
- S2CID 62882649.
External links
- "Chern-Simons functional". Encyclopedia of Mathematics. EMS Press. 2001 [1994].