Intersection pairings on spaces of connections and chernsimons theory on seifert manifolds 317 332. The chern simons gauge theory is studied using a functional integral quantization. Knot invariants and chernsimons theory 5 the vector space of chord diagrams can be characterized in an equivalent way using trivalent diagrams an introducing a series of new relations. Methods of computing the expectation value of wilson loop operators for framed unoriented knots and links will be outlined. Theory the chern simons action for a gauge theory with gauge group g and gauge. The treatment of this chernsimons theory is what is done in the fourth and nal chapter. Chernsimons theory and knot invariants oxford scholarship.
Knot homology from refined chernsimons theory request pdf. Knot invariants and chernsimons theory springerlink. Indeed, it is closely related to yangmills theory with purely the chernsimons action, which recently has turned out to define a soluble quantum field theory. Second, from a more pragmatic perspective, the supersymmetric wilson loop operators in chernsimons theory give nice examples in gauge theory for which the wilson loop path integral can be computed exactly, in a moreorless direct fashion. Knot operators are constructed by performing the calculation of matrix elements of wilson. Localization for wilson loops in chernsimons theory. At each stage the results obtained in the context of chern simons gauge theory are interpreted in the context of knot theory from a mathematical point of view. Kauffman knot invariant from son or spn chernsimons. Knot operators are constructed by performing the calculation of matrix elements of wilson line. Threedimensional chern simons gauge theories are quantized in a functional coherent state formalism. The knot invariants of refined chern simons theory are conjectured to coincide with the knot superpolynomials poincare polynomials of the triply graded knot homology theory. Notes on topological field theory harvard university.
The topological operators are the metric independent wilson loop knot. Proofs and derivations, veronica errasti diez, 1702. Cassons invariant and gauge theory classical chernsimons theory learning outcomes and competences at the end of the course the student will have a solid understanding of the theory of lie groups and connections in principal bundles as well as other prerequisites for studying gauge theory. Y is obtained by gluing two 4manifolds with corners. The quantisation of poisson structures arising in chernsimons theory with. It was discovered firstly by a mathematical physicist albert schwarz. This leads to a differential equation for expectations of wilson lines. Wilson line t hooft line in chernsimons theory surface operators in 4d gauge theory vector space mtheory on m5brane on lagrangian conifold space of bps states bps 5 3 gauge theory and categori. Pdf in this chapter, we present the basic elements of chernsimon theory and. The connection b et w een chern simons gauge theory and the theory o f knot and link in v arian ts w as established b y edw ard witten ten years ag o 1. Knot theory from a chernsimons gauge theory point of view. We shall give another method from the perturbative method of the su n chernsimons gauge theory in the lightcone gauge, which is more direct than already known methods.
A brief summary of the development of perturbative chernsimons gauge theory related to the theory of knots and links is presented. The presentation is done guided by a dictionary which relates knot theory concepts to quantum field theory ones. Threedimensional chernsimons gauge theories are quantized in a functional coherent state formalism. The normalized wave functionals are identified as generating functionals for the chiral blocks of twodimensional current algebra. Scott axelrod, geometric quantization of chern simons gauge theory phd thesis 1991 and for the generalization to complex lie groups in. I have read the wikipedia entry, but its pretty physicsy and i wasnt really able to get any sense for what chern simons theory really is in terms of mathematics. Talk given at the workshop chernsimons gauge theory. Knot invariants and chern simons theory 5 the vector space of chord diagrams can be characterized in an equivalent way using trivalent diagrams an introducing a series of new relations. The normalized wave functionals are identified as generating functionals for the. These are generalizations of the methods for su2 chernsimons theory presented in ref. The chernsimons gauge theory is studied using a functional integral quantization. The solution of this differential equation is shown to be simply related to the twovariable jones polynomial of the corresponding link, in the case where the gauge group issun. This chapter covers knot theory and its invariants, including, especially, homflypt polynomials.
We will brie y mention yangmills theory as an example of a gauge theory, but will not go into any details. There have been several methods to show that the expectation values of wilson loop operators in the su n chernsimons gauge theory satisfy the homfly skein relation. As a result, a formula for the homfly polynomial for arbitrary torus links is presented. Hitchin equations, chernsimons theory and surface operators, k. Hitchin equations, chernsimons actions, and surface operators. This easily leads to the generalization of homflypt polynomials to arbitrary gauge groups and. Chern simons theory is supposed to be some kind of tqft. The chernsimons theory, named after shiingshen chern and james harris simons, is. The operator formalism for chernsimons gauge theory with gauge group sun is presented.
The coecient k is sometimes called the level of the chernsimons term. Yangmills in two dimensions and chernsimons in three 307 322. This characterization is very important because it corresponds to the one that naturally arises from the point of view of chern simons gauge theory. Emphasis is made on the progress achieved towards the determination of a general combinatorial expression for vassiliev invariants. I have read the wikipedia entry, but its pretty physicsy and i wasnt really able to get any sense for what chernsimons theory really is in terms of mathematics. Pdf knot theory from a chernsimons gauge theory point. Chernsimons theory with gauge group based on the lie algebra and effective chernsimons level. The study of chern simons gauge theory is an unusual one because it was rst an. Chernsimons theory project gutenberg selfpublishing. The partition function and correlation functions of wilson loops in chern. Instanton counting and chernsimons theory iqbal, amer and kashanipoor, amirkian, advances in theoretical and mathematical physics, 2003. Gaugeinvariant operators for singular knots in chernsimons gauge theory j. A j 3 there are several comments to make about this chernsimons lagrangian.
The connection with twodimensional conformal field theory is found to emerge naturally. Three dimensional chernsimons theory as a theory of knots and links ii. The transformations between possible gauges, called gauge transformations, form a lie groupreferred to as the. The study of chernsimons gauge theory is an unusual one because it. The connection with rational conformal field theory is shown explicitly by identifying a basis for the hilbert space of the theory with the set of characters corresponding to a wesszuminowitten model for sun. Gauge theories with an application to chernsimons theory. Edward witten, quantization of chern simons gauge theory with complex gauge group, comm. In a groundbreaking paper, witten 1989 showed that chernsimons gauge theory, which is a quantum field theory in three dimensions, provides a physical description of a wide class of invariants of threemanifolds and of knots and links in threemanifolds. A consistent gauge invariant coupling for any group g and for higher dimensional. Three dimensional chernsimons theory as a theory of knots and links, nucl.
Three dimensional chern simons theory as a theory of knots and links ii. Second, from a more pragmatic perspective, the supersymmetric wilson loop operators in chern simons theory give nice examples in gauge theory for which the wilson loop path integral can be computed exactly, in a moreorless direct fashion. Using these operators a representation of the verlinde operators in the context of chern simons gauge theory is obtained. Fermionization and convergent perturbation expansions in chernsimons gauge theory 331 346. The quantum field theory approaches that have been developed to compute. Elsevier nuclear physics b 527 pmi 1998 499530 gaugeinvariant operators for singular knots in chernsimons gauge theory j. The expectation value of wilson loop operators in threedimensional son chernsimons gauge theory gives a known knot invariant. This characterization is very important because it corresponds to the one that naturally arises from the point of view of chernsimons gauge theory. Analytic continuation of chern simons theory 347 362. This chapter presents various aspects of chernsimons theory.
Topological string theory methods of computeraided drug. Pdf knot theory from a chernsimons gauge theory point of view. The presentation is done guided by a dictionary which relates knot theory concepts to quantum. A brief summary of the development of perturbative chern simons gauge theory related to the theory of knots and links is presented. Intersection pairings on spaces of connections and chern simons theory on seifert manifolds 317 332.
The theory is a variant of the theory givental put forward earlier, but uses crucially the fact that these are holomorphicsymplectic varieties. Quantum k theory of nakajima quiver varieties, generalizing their gromovwitten theory was developed a few years ago by okounkov with maulik and smirnov. Chernsimons theory is one of the most renowned quantum eld theory. Aspects of chernsimons theory cern document server. Modern theories describe physical forces in terms of fields, e. Perez, gaugeinvariant operators for singular knots in chernsimons gauge theory, nucl. Pdf an introduction to gauge theories researchgate. In physics, a gauge theory is a type of field theory in which the lagrangian does not change is invariant under local transformations from certain lie groups the term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom in the lagrangian.
Hyperkahler prequantization of the hitchin system and chernsimons theory with complex gauge group dey, rukmini, advances in theoretical and mathematical physics, 2007 chernsimons and string theory marathe, kishore, journal of geometry and symmetry in physics, 2006. Knot operators in chernsimons gauge theory sciencedirect. Pdf chernsimons theory, the 1n expansion, and string theory. The chernsimons functional consider a g bundle e over compact oriented three manifold m. The operator formalism for chern simons gauge theory with gauge group su n is presented. The chernsimons theory and knot polynomials springerlink. The connection with rational conformal field theory is shown explicitly by identifying a basis for the hilbert space of the theory with the set of characters corresponding to a wesszuminowitten model for su. Apart from electrodynamics chernsimons theory is the only gauge theory we rigorously treat. Yangmills in two dimensions and chern simons in three 307 322.
Chernsimons theory, knot invariants, vertex models and three. Chernsimons theory is an example of topological quantum. The operator formalism for chern simons gauge theory with gauge group sun is presented. Elsevier nuclear physics b 527 pmi 1998 499530 gauge invariant operators for singular knots in chern simons gauge theory j. Abstract we construct gauge invariant operators for. Knot operators in chernsimons gauge theory inspire. Nonabelian localization for chernsimons theory beasley, chris and witten, edward, journal of differential geometry, 2005. A similar equation has been used before to get the jones. Chernsimons theory is a gauge theory, which means that a classical configuration in the chernsimons theory on m with gauge group g is described by a principal gbundle on m. The chernsimons theory is a 3dimensional topological quantum field theory of schwarz type developed by edward witten. It satis es our usual criteria for a sensible gauge theory it is lorentz invariant, gauge invariant, and local. Furthermore it has remarkable links with conformal eld theory in two dimensions. Polynomial invariants corresponding to the fundamental representation of the gauge group sun are computed for arbitrary torus knots and links in the framework of chernsimons gauge theory making. Knot invariants 3d chernsimons theory on l gromovwittendeform.
It is the wilson loop along oriented knot k in representation r of group g. It is named after mathematicians shiingshen chern and james harris simons who invented chernsimons 3form. We put on e a connection 1form a which takes value in the lie algebra. The connection of this bundle is characterized by a connection oneform a which is valued in the lie algebra g of the lie group g. Chernsimons theory with gauge group based on the lie algebra. Integrating the strings in l k \s3 k out corresponds to inserting det1 holke x 1 in the chernsimons path integral, whereholk. The connection with rational conformal field theory is shown explicitly by identifying a basis for the hilbert space of the theory with the set of characters corresponding to a wesszuminowitten model for su n. Chernsimons and string theory marathe, kishore, journal of geometry and symmetry in physics, 2006. World heritage encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled. The treatment of this chern simons theory is what is done in the fourth and nal chapter. Locally exact then, under a gauge transformation, changes by a closed form. Integrating the strings in l k \s3 k out corresponds to inserting det1 holke x 1 in the chern simons path integral, whereholk. This chapter presents various aspects of chern simons theory. Fermionization and convergent perturbation expansions in chern simons gauge theory 331 346.
Analytic continuation of chernsimons theory 347 362. In the chernsimons theory, the action is proportional to the integral of the chernsimons 3. Gaugeinvariant operators for singular knots in chernsimons. In a groundbreaking paper, witten 1989 showed that chern simons gauge theory, which is a quantum field theory in three dimensions, provides a physical description of a wide class of invariants of threemanifolds and of knots and links in threemanifolds. Large nexpansion of chernsimons theory dual gauge theory string theory backgrounds. From the basic objects in both contexts the quantities leading to knot and link invariants are.
Topological sigma model, amodel open string interpretation of chernsimons gauge theory, and the duality between topological closed string and chernsimons theory. At each stage the results obtained in the context of chernsimons gauge theory are interpreted in the context of knot theory from a mathematical point of view. Gaugeinvariant operators for singular knots in chern. Knot operators are constructed by performing the calculation of matrix elements of wilson line operators on this hilbert space. Threemanifold invariants from chernsimons field theory with. The word gauge means a measurement, a thickness, an inbetween distance as in railroad tracks, or a resulting number of units per certain parameter a number of loops in an inch of fabric or a number of lead balls in a pound of ammunition.