Turaev Quantum Invariants Of Knots And 3-manifolds Pdf Writer

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These observables are traces of holonomies in a non-commutative Yang-Mills theory where the gauge symmetry is ensured by a quantum group. We show that these observables are link invariants taking values in a non-commutative algebra, the so-called Moduli Algebra.

Nevertheless, many of the difficult conceptual problems of quantizing gravity are still present. The task of quantizing general relativity is one of the outstanding problems of modern theoretical physics. Attempts to reconcile quantum theory and general relativity date back to the s see [ ] for a historical review , and decades of hard work have yielded an abundance of insights into quantum field theory, from the discovery of DeWitt-Faddeev-Popov ghosts to the development of effective action and background field methods to the detailed analysis of the quantization of constrained systems. But despite this enormous effort, no one has yet succeeded in formulating a complete, self-consistent quantum theory of gravity [ 83 ].

Bibliography of Vassiliev Invariants

We introduce Deligne cohomology that classifies fibre bundles over 3 manifolds endowed with connections. We show how the structure of Deligne cohomology classes provides a way to perform exact nonperturbative computations in Chern-Simons theory BF theory, resp.

The partition functions and observables of these theories are strongly related to topological invariants well known to the mathematicians. Consider the following actions: where and are connections. Here, the coupling constant is any real number. The gauge transformation , where is a function that leaves the actions 1 invariant.

Since in the quantum context we consider the complex exponential of the action, the invariance required is less restrictive.

Indeed, we can consider an invariance of up to an integer: which implies that is quantized. Studying the gauge invariance properties of the holonomies, which are the observables of Chern-Simons and BF theories, it turns out that the most general gauge transformation is , where is a closed 1-form with integral periods.

In particular, this is the case when the theory is defined in which is a contractible space. However, this generalized gauge transformation enables defining a theory on any closed i.

The classical gauge transformation appears thus to be a particular case of the quantum one. In this paper we will consider the equivalence classes according to this quantum gauge transformation. These classes classify fibre bundles over endowed with connections and their collection is the so-called first Deligne cohomology group of.

We will show that this structure enables performing exact computations in the framework of Chern-Simons and BF theories. The most general statement we can start from is a collection of local gauge fields in open sets that cover the manifold we are considering. To define a global field, we need to explain how and stick together in the intersection. This, by definition, is done thanks to a gauge transformation: The antisymmetry of this relation in and implies that , making a constant in that is an integer since 4 is nothing but the cocycle condition for a fibre bundle : The symmetry in , , and of this last relation implies that.

Thus, the generalization of our gauge potential on any closed 3-manifold imposes considering a collection constituted of a family of potentials defined in open sets , a family of functions defined in the double intersections , and a family of integers defined in the triple intersections all those open sets and intersections being contractible.

Elements of those collections are related by These statements define a Deligne cocycle. We need now to describe how this collection transforms when we perform a gauge transformation of the : where the family of is a family of functions defined in the. This implies that have to transform according to where the family consists in integers, mainly because do.

Finally, transform thus according to. Hence, the collection where are functions defined in the and are integers defined in the intersections together with the set of rules generalizes the idea of gauge transformation. These rules define the addition of a Deligne coboundary to a Deligne cocycle.

The quotient set of Deligne cocycles by Deligne coboundaries is the first Deligne cohomology group. It can be described in particular through two exact sequences. The first one is where is the quotient of the 1 form by the closed 1 form with integral periods and is the space of cohomology classes of the manifold.

This is an abelian group, which can thus be decomposed as a direct sum of a free part and a torsion part. This exact sequence shows that the space of Deligne cohomology classes can be thought as a set of fibres over the discrete net constituted by and inside which we can move thanks to elements of see Figure 1. The second exact sequence that enables representing is where is the first cohomology group -valued and is the set of closed forms with integral periods. This exact sequence leads to the representation shown in Figure 2.

Those two exact sequences contain the same information. Given two Deligne cohomology classes and with respective representatives and , we define a Deligne cohomology class with representative:. The integral of a Deligne cohomology class with representative over a cycle is defined by where means that the equality is satisfied in , that is, up to an integer. This integral is nothing but a holonomy, that is, a typical observable of Chern-Simons and BF quantum field theories. This definition ensures gauge invariance in the sense described in the introduction.

We can define in the same way the integral over of which provides a generalization of Chern-Simons and BF abelian actions: Let us point out that the first term is nothing but the local classical action, the other terms ensuring the gluing of local expressions up to an integer. Note that defines a bilinear pairing from the space of cycle and the space of Deligne cohomology classes both considered as -moduli in as well as.

Starting from that remark and for later convenience, we will consider Pontrjagin dual of a group. Considering as a functor, we can show that the following sequences are exact: Moreover, the information of the first two exact sequences is included in those two new ones. The Pontrjagin dual is a generalization to distributional objects. Finally, we see that in the sense of The structure of Deligne cohomology classes is such that each class can be decomposed as the sum of an origin indexed on the cohomology of basis of the discrete fibre bundle of Deligne cohomology classes and a translation taken in : The result of functional integrals over the space of Deligne cohomology classes will not depend on the choice of the origins, but the complexity of the computations will.

Thus, our goal is to find convenient origins with algebraic properties that will enable performing computations easily. Concerning the translations, we can decompose noncanonically as where denotes the set of closed form. Furthermore being the first Betti number. We will call zero modes the elements.

With this decomposition, we obtain. Let us consider generators of the free part of the homology of. Then, by Pontrjagin duality, we can associate to it a unique element.

Thus, for a fibre over we will consider as origin the element Note that since it represents a linking number which is necessarily an integer. We impose as a convention the so-called zero regularisation : which is ill-defined as self-linking. Finally, if we decompose as with , then we obtain. Let us consider now a generator of the component of the torsion part of the homology of.

This means that is the boundary of no surface, but is. Consider now defined by where. Thus, for a fibre over we will consider as origin the element This choice has several advantages since we can show that where is the so-called linking form, which is a quadratic form over the torsion of the cohomology.

Also for any free origin and for any translation. Chern-Simons abelian action is generalized as Since , then has to be quantized here: The partition function is defined as being a normalization that has to cancel the intrinsic divergence of the functional integral. The functional measure we use is then Assume that this measure verifies the so-called Cameron-Martin property; that is, for a fixed connection and a translation , then, for a translation in associated with a cycle : Using the algebraic properties given before, we can compute exactly the Chern-Simons abelian partition function.

As a convention, for the normalization we choose which corresponds to the trivial fibre of Deligne bundle for our theory defined over a manifold. This trivial fibre is the only one that constitutes Deligne bundle if we consider a theory over. This choice enables establishing a link with Reshetikhin-Turaev abelian invariant see [ 1 ].

Note that usually the normalization of Reshetikhin-Turaev invariant is chosen to be related to. However if the normalization is done with respect to then one recovers in the abelian case the invariants obtained with convention This way, we find Analogous considerations apply to BF abelian theory whose generalized action is being here also quantized which leads to a partition function written as.

Computations of expectation values of observables can also be performed thanks to this method in both Chern-Simons and BF abelian theories see [ 1 , 2 ]. Several correspondences in the nonabelian case, mainly , have been established formally, that is, with manipulations of ill-defined quantities: 1 Chern-Simons partition function is related to Reshetikhin-Turaev topological invariant [ 3 ]. This is summed up on the following diagram: the only result perfectly rigorously established being the one of Turaev, Reshetikhin, and Viro see [ 6 — 8 ].

In the abelian case, we saw that Deligne cohomology approach enables defining rigorously functional integration in the specific case of Chern-Simons and BF theories. Using this tool, we show that the previous diagram is no longer correct and has to be replaced by the following one: where the hypothesis of Turaev are not necessarily satisfied with abelian representations, leading to an inequality in general. This shows that the abelian theories, contrary to what could be expected, are not a simple trivial subcase of the nonabelian ones.

However, we expect to find some traces of this abelian case in the nonabelian one, which is the aim of present works. The author declares that there are no conflicts of interest regarding the publication of this paper.

This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Journal overview. Special Issues. Academic Editor: Ralf Hofmann. Received 15 May Accepted 18 Jun Published 30 Aug Abstract We introduce Deligne cohomology that classifies fibre bundles over 3 manifolds endowed with connections.

Introduction Consider the following actions: where and are connections. Deligne Cohomology The most general statement we can start from is a collection of local gauge fields in open sets that cover the manifold we are considering. This, by definition, is done thanks to a gauge transformation: The antisymmetry of this relation in and implies that , making a constant in that is an integer since 4 is nothing but the cocycle condition for a fibre bundle : The symmetry in , , and of this last relation implies that Thus, the generalization of our gauge potential on any closed 3-manifold imposes considering a collection constituted of a family of potentials defined in open sets , a family of functions defined in the double intersections , and a family of integers defined in the triple intersections all those open sets and intersections being contractible.

Finally, transform thus according to Hence, the collection where are functions defined in the and are integers defined in the intersections together with the set of rules generalizes the idea of gauge transformation.

Structure of the Space of Deligne Cohomology Classes is naturally endowed with a structure of -modulus. Figure 1. Figure 2. References P. Mathieu and F. Quantum Field Theory and Statistical Systems , vol.

Ponzano and T. Bloch, Ed. View at: Google Scholar A. Cattaneo, P. Cotta-Ramusino, J. Reshetikhin and V.

Exact Computations in Topological Abelian Chern-Simons and BF Theories

We introduce Deligne cohomology that classifies fibre bundles over 3 manifolds endowed with connections. We show how the structure of Deligne cohomology classes provides a way to perform exact nonperturbative computations in Chern-Simons theory BF theory, resp. The partition functions and observables of these theories are strongly related to topological invariants well known to the mathematicians. Consider the following actions: where and are connections. Here, the coupling constant is any real number.

Exact Computations in Topological Abelian Chern-Simons and BF Theories

Other results concern complexity theory of smooth maps, real algebraic geometry, generalized hypergeometric functions, topology of Lie groups, dynamical systems, geometrical combinatorics, potential theory, etc. Graduated from the Department of Mathematics and Mechanics of M. Arnold ; D. More than scientific publications.

Она была его помощницей, прекрасным техником лаборатории систем безопасности, выпускницей Массачусетс кого технологического института. Она часто работала с ним допоздна и, единственная из всех сотрудников, нисколько его не боялась. Соши посмотрела на него с укором и сердито спросила: - Какого дьявола вы не отвечаете. Я звонила вам на мобильник.

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