Chern-Simons for 2n-dimensional manifolds - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:52:18Z http://mathoverflow.net/feeds/question/119353 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119353/chern-simons-for-2n-dimensional-manifolds Chern-Simons for 2n-dimensional manifolds Daniel 2013-01-19T18:42:58Z 2013-01-21T10:27:25Z <p>In the literature I can only find Chern-Simons terms for odd-dimensional manifolds. For example, for a $G$-bundle over a 3-dimensional manifold we have $A \wedge dA + A \wedge A \wedge A$ with $A$ being a $\mathfrak{g}$-valued 1-form. Why can't I write such forms for even-dimensional manifolds?</p> http://mathoverflow.net/questions/119353/chern-simons-for-2n-dimensional-manifolds/119359#119359 Answer by Liviu Nicolaescu for Chern-Simons for 2n-dimensional manifolds Liviu Nicolaescu 2013-01-19T19:59:46Z 2013-01-21T10:27:25Z <p>It has to do with the fact that the characteristic classes (over the reals) of a principal $G$-bundle have <em>even</em> degree. We can associate Chern-Simons-like theory to each characteristic class of degree $2k$ together with a $G$-bundle $P$ over a manifold of dimension $2k-1$.</p> <p>To be a bit more technical a Chern-Simons-like form is asssociated to the following data</p> <p><strong>1.</strong> A homogeneous polynomial $\Phi$ of degree $k$ on the Lie algebra of $G$ invariant under the action of $G$ by conjugation.</p> <p><strong>2.</strong> A principal $G$-bundle $P\to M$ over $M$.</p> <p><strong>3.</strong> A pair of connections $\nabla^0, \nabla^1$ on $P\to M$.</p> <p>The Chern-Weil theory produces two <strong>closed</strong> forms</p> <p>$$\Phi(\nabla^0),\Phi(\nabla^1)\in \Omega^{2k}(M)$$</p> <p>and a form </p> <p>$$T\Phi(\nabla^1,\nabla^0)\in \Omega^{2k-1}(M),$$</p> <p>such that </p> <p>$$d T\Phi(\nabla^1,\nabla^0)= \Phi(\nabla^1)-\Phi(\nabla^0).$$</p> <p>(For details see Chapter 8 of <a href="http://www3.nd.edu/~lnicolae/Lectures.pdf" rel="nofollow">these notes</a>.)</p> <p>The <em>transgression</em> form $T\Phi(\nabla^1,\nabla^0)$ is the one used in Chern-Simons theories. It depends on two connections, but usually $\nabla^0$ is some fixed connection.</p>