Constructing and studying topological invariants

What is it about?

Chern–Simons theory is a topological quantum field theory. That is, a quantum field theory that depends only on the shape, or topology, of the space it’s defined on. In this work, we extend earlier studies of Chern–Simons theory with a gauge group (U(1)) to more general cases involving multiple copies of U(1). We then define a mathematical object called the partition function, which captures topological features of the space. Finally, using a so-called reciprocity formula, we establish a duality between two such U(1)^n theories.

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Photo by Philip Oroni on Unsplash

Photo by Philip Oroni on Unsplash

Why is it important?

U(1) and U(1)^n Chern-Simons (CS) theories have the characteristic of not always admitting a global section (since the gauge group is not simply connected). This means that we have to use other tools to handle these cases as opposed to SU(2) CS theories. This work generalizes results from U(1) CS and demonstrates the connection between CS and BF theories by showcasing that U(1)^n BF is just a subcase of U(1)^2n CS theory. Furthermore, the duality we exhibit between two such CS partition functions, helps reveal the structure of the theory itself. Moreover, it opens up paths for further progress such as calculating the expectation values of observables in our theory and the connection to other topological invariants such as the Reshetikhin-Turaev invariant. Finally, CS theories have been shown to model topological anyons in condensed matter physics so the theory can have applications in quantum computing.

Perspectives