A j 3 there are several comments to make about this chern simons lagrangian. Largen reduced models of supersymmetric quiver, chern. Fermionization and convergent perturbation expansions in chern simons gauge theory 331 346. Chernsimons theory is a quantum gauge theory involving a rather subtle action principle. One hundred years later, in the early 1970s, chern, and simons, proposed to study.
Some comments on canonical gauge theories with boundaries alejandro corichi center for mathematical sciences, unam, morelia, m exico ilqgs, lsu september, 2019. The action of the theory is the integral of the chern simons form associated to a gauge connection a corresponding to a gauge. Some comments on chernsimons gauge theory springerlink. One of the most important examples of this correspondence relates chern simons theory, a topological gauge theory in three dimensions which describes knot and.
The nature of this 11dim e8 gauge theory remains unknown. It was conjectured 2 that some form of three or fourdimensional gauge theory would be the key to understanding the threedimensiona l invariances of the particular braid traces that lead to the jones polynomial. E8 yangmills theory can naturally be embedded into a cl16 algebra gauge theory 33 and the 11d chern simons super gravity 4 is a very. A short comment on the supersymmetric structure of chern. Intersection pairings on spaces of connections and chern simons theory on seifert manifolds 317 332. Chern simons theory is a quantum gauge theory involving a rather subtle action principle. Gauge theories with an application to chernsimons theory. The chern simons theory can have different definitions for different people. While it is impractical to give complete references on chern simons theory, some contributions comparing the asymptotic or in some cases exact behavior of the chern simons partition function to geometry, but not focusing on analytic continuation, are 2234. Talk given at the workshop chernsimons gauge theory. Some comments on chernsimons gauge theory project euclid. In quantum mechanics, the gauge transformation acts on the wavefunction of the particle as. What makes the cs forms useful in physics is that under gauge transformations, they change like an abelian connection. First, it does not look gauge invariant, because it involves the gauge field a itself.
The usual di culties of quantum eld theory are exchanged for subtle questions in topology, but the latter turn out to be fairly accessible. Chernsimons theory, matrix models, and topological strings. These results may shed some light into the origins behind the hidden e 8 symmetry of 11d supergravity and reveal more important features of a cli. Chern simons theories with auxiliary vector multiplets yosuke imamura and keisuke kimura chern simons gauge theory and anyons at finite temperature shuxue ding, heshan song and ing an comments on holographic gravity dual of superconformal chern simons gauge theory changhyun ahnrecent citations hermitian freudenthalkantor triple. Chiral phases in frustrated 2d antiferromagnets and. Large kasymptotics of wittenreshetikhinturaevs invariants of 3. However, there are some situations where the total. Several years ago, in examining the proof of a rather surprising. It leads to quantum eld theory in which many, many, natural questions can be explicitly answered. It is in fact the sensible approach if the dimensionless chern simons term is o1 while e 2 chern simons 3form is precisely an analogue in higher gauge theory of the familiar relation two degrees down of how the electromagnetic potential 1form which is not globally defined in general has a curvature 2form that is globally well defined. There are several comments to make about this chernsimons lagrangian. Introduction to chernsimons theories rutgers physics.
It satis es our usual criteria for a sensible gauge theory it is lorentz invariant, gauge invariant, and local. Well highlight some famous and interesting gauge theories through concrete computation including. Secretly this higher principal connection structure also governs the first, seemingly simpler case. This is the form of the action functional first given as witten96 3. In general terms, however, it refers to the 3d topological quantum field theory. Schweda institut f theoretische physik, technische universt wien, wiedner hauptstra 810, a1040 vienna, austria o.
Witten w, we consider the space of flat connections on the trivialsu2 bundle over a surfacem, modulo the space of gauge transformations. In particular, i have not been systematic about trying to include references. Chern simons theories for nonabelian gauge fields 85 3. The colored jones polynomial jnq of a knot is interpreted in chern simons gauge. Witten w, we consider the space of flat connections on the trivial su2 bundle over a surface m, modulo the space of gauge transformations. Yangmills in two dimensions and chern simons in three 307 322. The massdeformed n2, 4 chern simons matter theories can be. For instance, threedimensional chern simons gauge theory can arise as a string theory. We will brie y mention yangmills theory as an example of a gauge theory, but will not go into any details.
A certain holomorphic analog of chernsimons theory can also arise as a string theory. Pdf we discuss a special case of the gauge theory to string theory correspondence where exact results are available. Chern simons and string theory marathe, kishore, journal of geometry and symmetry in physics, 2006. Its describes the field dynamics through the socalled chern simons form, hence its name. In recent years, the old idea that gauge theories and string theories are equivalent has been implemented and developed in various ways, and there are by now various models where the string theory gauge theory correspondence is at work.
When the gauge theory permits homotopically nontrivial gauge transformations, the strength of the addition must be quantized, so that the gauge change of the chern simons term be an integral multiple of 2. The action functional of chern simons theory is always the volume holonomy of a 3connection, the chern simons circle 3connection. Apart from electrodynamics chern simons theory is the only gauge theory we rigorously treat. Chern simons theory is an example of a topological quantum field theory. The simple connectedness of the group has the convenient effect that bundles over an oriented 3manifold are trivial so one neednt worry. The fundamental data in chern simons gauge theory are the following. The holographic nonsupersymmetric renormalization group flows in four dimensions are found. In the language of twisted differential cstructures, we may summarize this sitation as follows in order for the action functional of higher abelian chern simons theory to be correctly divisible, the images of the fields in. Physics letters b 300 1993 9295 northholland a short comment on the supersymmetric structure of chernsimons theory in the axial gauge a. General reativity is a nonabelian gauge theory for the group so3,1 in 4d. Aspects of chernsimons theory cern document server.
Chern simons theory can be studied in a few different ways based on what your interests and background is. Instanton counting and chern simons theory iqbal, amer and kashanipoor, amirkian, advances in theoretical and mathematical physics, 2003. What are some good resources for learning about the chern. This is possible since the chern simons term is finite though gauge dependent when evaluated for a monopole. The treatment of this chern simons theory is what is done in the fourth and nal chapter. Chern simons on the kagome lattice three inequivalent sites, a, b, and c, in the unit cell of the kagome lattice three plaquettes. It is a sigmamodel tqft which has target space in the smooth moduli stack and a background gauge field is a circular 3. Aj 3 there are several comments to make about this chern. In quantum eld theory, we have the same transformation but now with interpreted as the eld.
Pdf chernsimons theory, the 1n expansion, and string theory. Recently it has been shown 41 that threedimensional chern simons gauge theory for a compact gauge group. The mathematical machinery developed by maxwell to formulate the principles of faradays electromagnetism has been, subsequently, applied to pure mathematics, namely to describe bundles over even dimensional manifolds, using characteristic classes. Particularly, chern simons theory is specified by a choice of simple lie group g known as the gauge group of the theory and also a number referred to as the level of the theory, which is a constant that multiplies the action. An abstract algebraic model of the algebra of chern s characteristic classes and chern simons secondary characteristic classes and of the gauge group action on this algebra which also enables some. Nonabelian localization for chern simons theory beasley, chris and witten, edward, journal of differential geometry, 2005. Witten w, we consider the space of flat connections on the trivialsu2 bundle over a surfacem, modulo the.