Leiden University

Mini-course on topological interactions

For those who are interested, and the "Stripe Club" in particular, I will give an introduction into the concepts of topological interactions and Hopf symmetry. This will be presented in several sessions of one-and-a-half hours each.

On this page:

Date and time

Each session will be held in the coffee room of the Instituut-Lorentz, at the following times:

A second series will be given in April:

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Very short summary

Apart from regular particles, whose behaviour under gauge transformations is characterized by a certain representation of the gauge group, a system can also possess topological excitations. Due to topological interactions, which are in essence analogues of the Aharonov–Bohm effect, such excitations may interact with each other and with the gauge particles.

In this setting, we can assign the correct quantum numbers to both the topological and the regular degrees of freedom by describing the excitations as repesentations of the so-called quantum double of the gauge group. The quantum double is a special kind of Hopf algebra, which one can think of as a generalization of a group. This formalism captures many features of these systems, such as braid statistics, fusion rules and also condensate phases.

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Topics

Here is a list of topics that I would like to discuss, probably in this order.

discrete gauge theories
Theories where a continuous gauge group is spontaneously broken down to a finite group. The Coulomb forces will be screened by the Higgs medium. We only look at 2+1-dimensional systems.
topological defects
When topological defects are present, they are labelled by elements of the residual gauge group. Such a label may be considered to be its topological charge. The way to describe topological defects in ordered media is by an order parameter, which is isomorphic to the residual gauge group in our case.
topological interactions
Topological defects may have Aharonov–Bohm-type interactions with fundamental particles which encircle a defect. These interactions survive the screening by the Higgs particles.
flux metamorphosis
The topological charge of a topological defect carrying group element h which moves around a defect g will change to ghg-1. This can be easily seen by looking a the long-range properties of a two-defect composite. Therefore, defects should be organized into conjugacy classes instead of elements of a group. This is also a topological interaction, between two defects.
braid statistics
Statistics determines the behaviour of the system under interchange of two identical particles. For regular particles, such an interchange will yield factors 1 (bosons) or -1 (fermions). This corresponds to the two one-dimensional actions of the permutation group of n particles Sn.
In 2+1-dimensional physics, the permutation group must be replaced by the braid group Bn, an infinite group. The action of this group takes into account the topological interactions which particles undergo when they move around each other and interchange their position. Particles showing such interchange properties are said to be governed by braid statistics.
anyons and dyons
The braid group has infinitely many one-dimensional representations, which correspond to phase factors eiθ under interchange of two particles. Such particles are called anyons, and show Abelian braid statistics.
A dyon is a particle having both non-trivial topological and non-trivial fundamental charge. It is therefore neither a flux nor a charge when turning to the terminology of QED. When the interchange between such particles is not commutative, they are said to show non-Abelian braid statistics. It will yield a q-number under interchange, for example a matrix.
centralizer charge
The (fundamental) charge of a particle can be determined by interference or scattering experiments with other defects. When such a charge also carries flux (it is a dyon), such experiments only give useable results when the flux of the incoming defect commutes (it is a group element) with the flux of the dyon. In other words, we are only able to determine the charge under the centralizer group of the topological charge, instead of the full residual symmetry group.
fusion
By the fusion of two particles, we mean that we consider a two-particle state as a single particle by only looking at the long-range properties of the composite. The possible single-particle states that such a composite will correspond to are determined by the fusion rules, which are similar to the Clebsch–Gordan decomposition of multi-particle states.
quantum double
Out of the residual symmetry group H, we construct the quantum double of this group, denoted by D(H). This is no longer a group but a so-called Hopf algebra, which can be viewed as a generalization of a group. It is shown that the irreducible representations of this quantum double correspond exactly to the quantum numbers of the possible excitations in our system: conjugacy classes of H as topological charge, and irreducible representations of the centralizers of these classes as fundamental charge.
Furthermore, this construction also provides a prescription for the braiding of particles, and the fusion rules can be determined as well. In fact, fusion and braiding will appear to be connected.
Hopf algebra
The quantum double is a special case of a Hopf algebra. A Hopf algebra is a vector space equipped with a multiplication (a group is a set with a multiplication); a comultiplication to create tensor products (a group acting on a tensor product space has a trivial comultiplication in this sense); an antipode which leads to anti-particles (corresponding more or less to the group inverse).
The additional structure in a braided Hopf algebra is called the R-matrix, which determines the interchange of particles, leading to braid statistics. In fact, it yields solutions to the Yang–Baxter equation for all its irreducible representations. The R-matrix is also provided by the quantum double construction.
Hopf symmetry breaking
We start out from a system showing symmetry of a Hopf algebra, for example quantum double symmetry. We now imagine that this system assumes a new ground state, which is filled with particles in a certain state, which is a vector in the representation space of an irreducible representation of that Hopf algebra.
Not all elements of the Hopf algebra will still correspond to valid symmetry transformations. We state that a transformation leaves the condensate state invariant when it acts ‘in the same way’ on the condensate state as it would on the vacuum state.
The Hopf symmetry is now broken to a subalgebra, and with some additional demands to a sub-Hopf algebra. Many times the residual symmetry algebra takes some general form, but several exceptions have been found, for which it must be calculated by hand.
confinement
We consider what happens when a particle enters a region with a condensate phase from a vacuum region. This particle will then behave as one of the condensate excitations, i.e. an irreducible representation of the residual symmetry algebra. It may be that this particle braids non-trivially with the background condensate particles. In that case, travelling further and further through this region will introduce a domain wall, costing more energy as this wall ‘grows’. In any case, such a particle cannot travel/exist freely in the condensate region, and is said to be confined.
Now, in our formalism, non-trivial braiding is reduced to an algebraic property. So when assuming that non-trivial braiding with the condensate particles leads to confinement, we are able to classify by algebraic means which condensate excitations are confined or unconfined respectively. It turns out that the unconfined particles are characterized by another Hopf algebra, the unconfined algebra. When relating to physical systems with condensate phases, this is the algebra that one should use, as confined particles do not exist, or form unconfined composites.

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Tables and pictures

Comparison of group and Hopf algebra concepts
Comparison of group and Hopf algebra concepts (gif, pdf 34 KB)

Hopf algebra symmetry breaking scheme
Hopf algebra symmetry breaking scheme (gif, pdf 54 KB)

Duality of algebra and coalgebra structures
Duality of algebra and coalgebra structures (gif, pdf 20 KB)

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References

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