Advanced Search. Privacy Copyright. Title Skein theory and topological quantum field theory. Abstract Skein modules arise naturally when mathematicians try to generalize the Jones polynomial of knots. Document Availability at the Time of Submission Release the entire work immediately for access worldwide. Recommended Citation Cai, Xuanting, "Skein theory and topological quantum field theory" Included in Applied Mathematics Commons. Search Enter search terms:. New articles related to this author's research.
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Journal of High Energy Physics 09 , , Communications in Mathematical Physics 3 , , International Journal of Modern Physics A 16 16 , , The short answer is: by placing the enumerative problem in the right physical context. Within quantum theory it makes perfect sense to combine all the numbers N d into a single generating function.
In fact, this function has a straightforward physical interpretation.
It can be seen as a probability amplitude for a string to propagate in the Calabi—Yau space X. In quantum theory one has to operate under the fundamental principle of summing over all possible histories with a weight given by the classical action. In that spirit a quantum string can be thought to probe all possible rational curves of every possible degree d at the same time, with weight q d.
But there was a second ingredient that helped to find the physical solution. There exists an equivalent formulation of this physical process using a so-called mirror Calabi—Yau manifold Y. As far as classical geometry is concerned, the spaces X and Y are very different; they do not even have the same topology.
But in the realm of quantum theory, they share many properties. In particular, with a suitable identification, the string propagation in spaces X and Y are identical. It is a typical example of a quantum symmetry. Moreover, the difficult computation of the function F on the manifold X known by physicists as the A-model turned out to translate into a much simpler expression on the mirror manifold Y , where it could be captured by a so-called period integral known as the B-model.
These integrals are a well-known element in the theory of variation of complex structures in algebraic geometry. It is mirror symmetry that easily leads to the above table of numbers. Mirror symmetry is an example of a much broader area of mathematics influenced by physics: symplectic geometry—the branch of differential geometry based on the underlying structure of Hamiltonian mechanics.
The A-model and the B-model for Calabi—Yau manifolds were simply two sides of a common quantum theory. Finding the framework for proving this mathematically is currently a topic of great interest. Mirror symmetry is a good example of a more fundamental influence of physics on geometry, one which involves a significant change of viewpoint on the part of the pure mathematician. The functional integral and the partition function are the bread and butter of the theoretical physicist, but they go counter to the traditional mathematical approach to a problem.
Take the algebraic geometer studying algebraic curves in an algebraic variety, like the quintic threefold we have just seen. Traditionally, the mathematician restricts to a fixed topological invariant of the curve, for example its degree d. The physicist, by contrast, studies all curves at once—the degree being some indexing of terms in a series expansion of the partition function—and by doing so sees relations that are invisible in a step-by-step analysis.
The formulae above count rational curves, curves of genus zero, but one may also use the topological invariant of the genus as a parameter in a generating function, and there the physicist sees further relations. The very idea as occurs in Gromov—Witten theory that there is a link between counting higher-genus curves and those of genus zero is a very radical one for the mathematician to comprehend.
Supersymmetry, the link between bosons and fermions, is a closely related concept from physics that has also influenced differential geometry. As first noted by Edward Witten, supersymmetry applied within quantum mechanics is an elegant way to derive the basic principles of Morse theory Witten Although the definition has been in the differential-geometric literature since the s, it was 30 years later, as a result of the infiltration of ideas from the supersymmetric sigma model, that a mechanism for constructing good examples was found. There are many spaces under current investigation that have a dual interpretation—one in terms of the differential geometry of moduli spaces and the other derived from supersymmetric field theories.
These have now penetrated pure mathematics in algebraic and number-theoretic areas such as the geometric Langlands programme and representation theory.
Witten, Edward. Quantum field theory and the Jones polynomial. Comm. Math. Phys. (), no. 3, giuliettasprint.konfer.eu It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural .
Supersymmetry has also generated further differential-geometric structures, sometimes long before the mathematicians were ready to listen. One example, that of a generalized complex structure, is a conventional structure that unifies both symplectic and complex geometry and retains in classical form some of the consequences of supersymmetry.
For our second class of examples we move to three-dimensional topology. The prime example of a topological problem, where length is irrelevant because we allow deformation or stretching but not cutting , is that of knots in three-dimensional space. These are familiar to all of us and we easily recognize their complexity. For example, if we consider two closed knots with no loose ends it is not easy to tell whether we can deform one to look like the other.
A useful tool to distinguish knots is provided by a numerical invariant: the knot invariant.
This is a number that can be computed by a formula from the picture of a knot, as a piece of string laid on a table, but which is unchanged if we turn the knot around and place it back in a different pattern. One such invariant, known since the s and for a long time the only one, is the Alexander polynomial. This expression in one variable t is of the form.
These a k actually provide a string of invariants, but it is sensible to turn them into a polynomial Z t. In the world of knot theory underwent a remarkable new development when Vaughan Jones discovered a polynomial knot invariant now named after him that was different from the Alexander polynomial Jones Crucially, it was chiral, that is, it could distinguish knots from their mirror images, which the Alexander polynomial could not.
It soon emerged that this new invariant was part of a grand family of invariants based on Lie algebras and their representations. The Jones polynomial emerged from a mysterious background of mathematics and physics and seemed to fit into no general framework. But shortly afterwards, Witten showed how to interpret the Jones polynomial in terms of a quantum field theory in three dimensions Witten In fact, this relation between knot invariants and particles goes to the very beginning of relativistic quantum field theory as developed by Feynman and others in the s.
The basic idea is that, if we think of a classical particle moving in space—time, it will move in the direction of increasing time. However, within quantum theory the rules are more flexible. Now a particle is allowed to travel back in time. Such a particle going backwards in time can be interpreted as an anti-particle moving forwards in time. Once it is allowed to turn around, the trajectory of a particle can form, as it were, a complicated knot in space—time.
Feynman was the first to realize that the processes of particle physics can be captured by these diagrams.
If one produces an actual closed loop, as in a mathematical knot, such a diagram is called a vacuum amplitude. The particle travelling along the loop cannot be detected experimentally—the corresponding Feynman diagram describes a so-called virtual particle. Yet, these diagrams describe a clear physical effect: the quantum contributions to the vacuum energy.
In order to establish contact between this formulation of particle physics and knot theory, Witten had to replace the usual four dimensions of space—time with three dimensions and work with a special type of gauge theory based on the Chern—Simons topological invariant. The full theory allowed for many choices, such as the gauge group, the representation of the particles associated to the knot and coupling constants. Relating quantum field theory to knot invariants along these lines had many advantages.
First, it was fitted into a general framework familiar to physicists. Second, it was not restricted to knots in three-dimensional Euclidean space and could be defined on general three-dimensional manifolds. One could even dispense with the knot and get an invariant of a closed three-dimensional manifold. Such topological invariants are now called quantum invariants and have been extensively studied.