This new type of gauge theory is known as a chernsimons theory the origin of this name is discussed below in section 2. In his seminal paper 2 witten has shown how su2 chern simons theory gave rise to the jones polynomial for links in s3, to new 3manifold invariants, and. Topological quantum field theories can be used as a powerful tool to probe geometry and topology in low dimensions. Physics the chernsimons theory and knot polynomials. Wit89 on quantum chernsimons theory and the jones polynomial jon85, jon87. Im typing as we go so please forgive all typos and unclear parts etc. General reativity is a nonabelian gauge theory for the group so3,1 in 4d. The two main steps are to reinterpret threedimensional chern simons gauge theory in four dimensional terms and then to apply electricmagnetic duality. Nonabelian localization for chernsimons theory beasley, chris and witten, edward, journal of differential geometry, 2005. In chernsimons theory, the operators are wilson loop operators.
I just havent kept up im still trying to understand. It is the simplest kind of knot, one lacking any kind of knotting. Dec 19, 2017 in the case of finite gauge groups, chernsimons theory is also known as dijkgraafwitten theory. Witten looking anew at the jones polynomial the ncategory cafe. The presentation is done guided by a dictionary which relates knot theory concepts to quantum. In this version, the jones polynomial can be generalized from s 3 to arbitrary three manifolds, giving invariants of three. It is the wilson loop along oriented knot k in representation r of group g. For oriented knotted vortex lines, t i satisfies the skein relations of the kauffman rpolynomial. Sl2,c chernsimons theory and the asymptotic behavior 81of the knot complement. Three dimensional gauge theories with the chernsimons term added to the usual action 1. 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. Lets look at the physics captured by the chernsimons term using 5. In the first of these two lectures i describe a gauge theory approach to understanding quantum knot invariants as laurent polynomials in a complex variable q.
The usual di culties of quantum eld theory are exchanged for subtle questions in topology, but the latter turn out to be fairly accessible. The two main steps are to reinterpret threedimensional chernsimons gauge theory in four dimensional terms and then to apply electricmagnetic duality. These chernsimons theories are interesting both for their theoretical novelty, and for. Quantum field theory and the jones polynomial inspire. We clarify and refine the relation between the asymptotic behavior of the colored jones polynomial and chernsimons gauge theory with complex gauge group. Sl 2 c chernsimons theory and the asymptotic behavior of. Look at the jones polynomial of a knot, clay conference, oxford, october 1, 20 pdf. In which s is the stransformation that we mentioned before. Sl2,c chern simons theory and the asymptotic behavior 81of the knot complement. For a discussion of just the simple special case of 3d chernsimons theory see costello 11, chapter 5.
This argument has later been made more precise in the language of tcft. In it was shown that the jones polynomial as a polynomial in q q is equivalently the partition function of su 2 su2chernsimons theory with a wilson loop specified by the given knot as a function of the exponentiated level of the chernsimons theory. Sl 2 c chernsimons theory and the asymptotic behavior of the. In his seminal paper 2 witten has shown how su2 chern simons theory gave rise to the jones polynomial for links in s3, to new 3manifold invariants, and to new jones invariants of links in manifolds. Aspects of chernsimons theory cern document server. The chernsimons theory and knot polynomials springerlink. Three dimensional gauge theories with the chern simons term added to the usual action 1. The main theme is the connection between this tqft and chernsimons theory cs74, cs85. Since the complement of a cartier divisor in an af. Edward wittem quantum field theory and the jones polynomial 1989 here the quantum space of states of chern simons theory was first analyzed and the relation to the conformal blocks of wzw theory observed. A similar equation has been used before to get the jones.
A topological invariant t i l is constructed for a link l, where i is the abelian chernsimons action and t a formal constant. Oct 03, 20 witten gave his second talk on khovanov homology and gauge theory. The proposition then follows from stokes theorem for the. This leads to a differential equation for expectations of wilson lines. Geometric quantization of chern simons gauge theory 789 are associated with the jones polynomial, from the point of view of the three dimensional quantum field theory. G be a smooth map, viewed as a change of coordinates. Chernsimons theory is a quantum gauge theory involving a rather subtle action principle. This new type of gauge theory is known as a \ chern simons theory the origin of this name is discussed below in section 2. This new type of gauge theory is known as a \chernsimons theory the origin of this name is discussed below in section 2. A related article is this one paywall by elitzur, moore, schwimmer and seiberg. In his lectures, among other things, witten explained his intrinsic threedimensional construction of jones polynomials via chernsimons gauge theory. Observables in topological quantum chernsimons theory correspond to link invariants known as sun colored jones polynomials wit89. Chern simons theories, which are examples of such field theories, provide a field theoretic framework for the study of knots and links in three dimensions. This thesis contains the results which were obtained during my phd studies at centre for quantum geometry of moduli spaces qgm at aarhus university with professor jorgen.
In particular also all theories of infinitychernsimons theorytype coming from binary invariant polynomials are perturbatively of this form, notably ordinary chernsimons theory. Chernsimons theory is an example of topological quantum. In this note we make the first step toward verifying this relation beyond the semiclassical approximation. Given a manifold and a lie algebra valued 1form, over it, we can define a family of pforms. Moreover, chernsimons theory is solvable explicitly, so it provides a physical way to not only interpret the knot polynomial, but to calculate it as well. It is well known that a g bundle with connection is a critical point of the chernsimons action functional a classical solution of the equations of motion if and only if the connection is flat. We clarify and refine the relation between the asymptotic behavior of the colored jones polynomial and chernsimons gauge theory with complex gauge group sl2,c. Jones polynomial aknotisanembeddingofacircleinmostclassicallys3. It leads to quantum eld theory in which many, many, natural questions can be explicitly answered. Reidemeister torsion the alexander polynomial and u1 1. The theory is named for shiingshen chern and james harris simons, coauthors of a 1974 paper entitled characteristic forms and geometric invariants, from which the theory arose. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a laurent polynomial in the variable with integer coefficients. Chern simons theory on s 3 is related via conifold transition to the allgenus generating function of the topological string amplitudes on a calabiyau manifold. Chernsimons theory project gutenberg selfpublishing.
At that time witten unified several important mathematical works in terms of quantum field theory, most notably the donaldson polynomial, the gromovfloer homology and the jones polynomials. But by far the best answer to your question is in wittens paper quantum field theory and the jones polynomial, communications in mathematical physics, 1989 vol. Witten gave his second talk on khovanov homology and gauge theory. Moreover, the interpretation via sl2,c chern simons theory predicts the structure of the subleading terms in the asymptotic expansion of the colored jones polynomial, where each term in. I get the impression that one can get such sections in a very elementary manner by using just the classical chernsimons functional.
Moreover, the interpretation via sl2,c chernsimons theory predicts the structure of the subleading terms in the asymptotic expansion of. 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. So, we see that the axioms of an ndimensional tqft automatically spit out a lot of interesting information about. The study of chern simons gauge theory is an unusual one because it was rst an. Kauffman knot polynomial invariants are discovered in classical abelian chernsimons field theory. The abelian gauge theory with only a chern simons term was studied by schwarz 18 and in unpublished work by i. In it was shown that the jones polynomial as a polynomial in q q is equivalently the partition function of su 2 su2 chern simons theory with a wilson loop specified by the given knot as a function of the exponentiated level of the chern simons theory. Looking back at the kinds of systems we met in section 2 which exhibit a hall conductivity, we see that they all break parity, typically because of a background magnetic. Edward wittem quantum field theory and the jones polynomial 1989 here the quantum space of states of chernsimons theory was first analyzed and the relation to the conformal blocks of wzw theory observed more details on the computations appearing there appeared shortly afterwards in.
Classical and quantum chernsimons theory mathoverflow. The variable q is associated to instanton number in the dual description in four. 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. On the jones polynomial and its applications alan chang abstract. Kauffman knot polynomials in classical abelian chern. Moreover, chern simons theory is solvable explicitly, so it provides a physical way to not only interpret the knot polynomial, but to calculate it as well. Ive also heard that witten has interpretted various knot invariants like the jones polynomial in terms of chern simons theory. 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. World heritage encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled. It has been proposed that the asymptotic behavior of the colored jones polynomial is equal to the perturbative expansion of the chernsimons gauge theory with complex gauge group \sl2, \mathbbc\ on the hyperbolic knot complement. The abelian gauge theory with only a chernsimons term was studied by schwarz 18 and in unpublished work by i. How do i take some sort of chern simons computation on a 3or 4. The jones polynomial of a knot and generally a link with an odd number of components is a laurent polynomial in t. The colored jones polynomial jnq of a knot is interpreted in chern simons gauge.
The observation which got all this started is the one in. The chern simons gauge theory is studied using a functional integral quantization. These are rare examples of quantum field theories which can be exactly nonperturbatively and explicitly solved. Gauge theory and the jones polynomial 5 it is a fact that the vector spaces zm for mare always nitedimensional.
These chern simons theories are interesting both for their theoretical novelty, and for. In mathematics, the chernsimons forms are certain secondary characteristic classes. We clarify and refine the relation between the asymptotic behavior of the colored jones polynomial and chern simons gauge theory with complex gauge group. Quantum eld theory plays an increasing role in the study of topological invariants in low dimensions 1. Chernsimons and string theory marathe, kishore, journal of geometry and symmetry in physics, 2006. Fortunately chernsimons theory is a success story in quantum. Finally, well sketch the construction of wittens chern. Chirality of knots 942 and 1071 and chernsimons theory. Chernsimons theory on s 3 is related via conifold transition to the allgenus generating function of the topological string amplitudes on a calabiyau manifold. Quantum field theory and the jones polynomial signal lake. I will not be as precise as mathematicians usually want. Chern simons theory is a gauge theory, which means that a classical configuration in the chern simons theory on m with gauge group g is described by a principal gbundle on m.
The most elementary ways to calculate vlt use the linear skein theory ideas of 7. In fact, i understand now that his two talks were really an exposition of his recent past work on a gauge theory explanation for khovanov homology and the jones polynomial as contained in analytic continuation of chernsimons theory and khovanov homology and gauge theory. There is first of all the original article by witten quantum field theory and the jones polynomial. The goal is to associate a hubert space to ev ery closed oriented 2 manifold. The chernsimons gauge theory is studied using a functional integral quantization.
Two lectures on the jones polynomial and khovanov homology. Knot theory, the jones polynomial and chern simons theory. The chernsimons term in three dimensional gauge theory has a relatively long history. Using canonical quantization of chernsimons theory on the torus, in the case of group su2, we can derive metric elements the calculation can do in geometrical quantization and also in conformal. Topological quantum field theories a meeting ground for. This paper is a selfcontained introduction to the jones polynomial that assumes no background in knot theory. In particular, in section 2 we show that the function s u can be interpreted as the semiclassical action in the sl2,c chernsimons theory, and differs from su by a choice of polarization. Gauge theory and the jones polynomial math berkeley. The chern simons term in three dimensional gauge theory has a relatively long history. The first fundamental property of the chernsimons theory is the quanti. 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.
The chernsimons theory, named after shiingshen chern and james harris simons, is. Factorization in quantum planes coulibaly, romain and price, kenneth, missouri journal of mathematical sciences, 2006. Quantum super a polynomial plan 1 knots, chernsimons theory and homological invariants knot invariants and chernsimons theory homological knot invariants 2 quantum super a polynomial supervolume conjectures, examples quantizability 327. We clarify and refine the relation between the asymptotic behavior of the colored jones polynomial and chern simons gauge theory with complex gauge group sl2,c. In the mathematical field of knot theory, the jones polynomial is a knot polynomial discovered by vaughan jones in 1984. I cant say anything about donaldson theory or floer homology, but ill mention some resources for chernsimons theory and its relation to the jones polynomial. In fact, i understand now that his two talks were really an exposition of his recent past work on a gauge theory explanation for khovanov homology and the jones polynomial as contained in analytic continuation of chern simons theory and khovanov homology and gauge theory. What are the details of the renormalization of chern. Quantum riemann surfaces in chernsimons theory dimofte, tudor, advances in theoretical and. The study of chernsimons gauge theory is an unusual one because it.
In this version, the jones polynomial can be generalized froms3 to arbitrary three manifolds. The original paper by witten presented a series of nonperturbative methods which led him to establish the equivalence between vacuum expectation values vevs of wilson loops and polynomial invariants like the jones polynomial 16 and its generalizations. I seem to find the same thing in more recent treatments, such as papers of andersen which ive hardly looked into at all. In the case of finite gauge groups, chernsimons theory is also known as dijkgraafwitten theory. Chernsimons theory this material is based on wittens qft and the jones polynomial, as well as the notes by himpel on lie groups and chernsimons theory. Perturbative algebraic quantum field theory and the renormalization groups brunetti, r. 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. Quantum riemann surfaces in chern simons theory dimofte, tudor, advances in theoretical and mathematical physics, 20.
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