Notes on Lattice Topological Field Theory in Three Dimensions, Part I

I have recently been interested in constructing topological field theories for loop quantum gravity, with a specific eye towards the topspin construction. With this in mind I was lead to the classic paper by Dijkgraaf and Witten, “Topological Gauge Theories and Group Cohomology” (1990). I am working my way through it to try and understand – which usually requires taking notes on the paper and working on things I don’t understand. I thought maybe I would post my notes here.

Introduction

For an oriented, smooth 3-manifold M we can specify a principal bundle E=(M,\pi,G) with a gauge group G. An action on this bundle must be constructed from some kind of invariant, to make sure it is somehow independent of the choice of connection A. If E is trivial, the connection is a Lie-algebra valued one-form A\in \Omega^1(M)\otimes \mathfrak{g} and the action is the Chern-Simons functional

S(A)=\frac{k}{8pi ^2}\int_M Tr(A\wedge dA + \frac{2}{3}A\wedge A\wedge A).

The path integral is Z(M)=\int\mathcal{D}A\exp (2\pi i S(A). The integer k is called the level of the theory, and must be an integer. This is because when we take G to be non-abelian, under a gauge transformation the action goes to S(A^g)=S(A)-2\pi k N for some integer N (http://arxiv.org/abs/hep-th/9902115). Thus in order for the amplitude \exp(S(A)) to remain a gauge-invariant, k must at least be an integer.

If E is not a trivial bundle, the action is not a global gauge-invariant. However, since we know curvatures are good gauge invarients (in 4 dimensions), we can use them to define a global Chern-Simons action. So, assume we can extend the bundle E over a smooth four-manifold B. Then we could write

S(A)=\frac{k}{8\pi^2}\int_B Tr(F\wedge F),

where F is the usual curvature of the connection A. If we have a trivial extension of the connection A(x)=\partial_t A dt+A(\vec{x}) then the curvature is only a function of the coordinates \vec{x} on M:

F(x)=dA+A\wedge A=dA(\vec{x})+A(\vec{x})\wedge A(\vec{x}). The question remains how to do this in general, since one cannot always find such an extension of the connection to the boundary (I basically chose B=\mathbb{R}\times M, but we cannot guarantee that E can be extended in the way that I extended A).

Advertisements

7 thoughts on “Notes on Lattice Topological Field Theory in Three Dimensions, Part I

  1. Pingback: Notes on Lattice Topological Field Theory, Part II | Christopher Duston's Home

  2. Pingback: Notes on Lattice Topological Field Theory in Three Dimensions, Part III | Christopher Duston's Home

  3. Pingback: Torsion for Dummies (Physicists) | Christopher Duston's Home

  4. Pingback: Notes on Lattice Topological Field Theory in Three Dimensions, Part IV | Christopher Duston's Home

  5. Pingback: Notes on Lattice Topological Field Theory in Three Dimensions, Part V | Christopher Duston's Home

  6. Pingback: Constructing spacetime from the quatum tetrahedron: Spacetime as a Bose-Einstein Condensate | quantumtetrahedron

  7. Pingback: Notes on Lattice Topological Field Theory in Three Dimensions, Part VI | Christopher Duston's Home

Add A Comment:

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s