25
$\begingroup$

Hi all. The question I have should be a rather simple one, but I just can't think it through.

So the Chern-Simons action reads \begin{equation} S = \int_M {\rm tr} (A\wedge dA + \frac{2}{3} A\wedge A \wedge A) \end{equation} where $M$ is 3-fold, and similarly for higher dimensional manifold.

Now, my question is:

*since $A$, the connection 1-form is only defined patch by patch, what do we really mean by doing the integration? *

It would be understandable if I write \begin{equation} S = \int_M {\rm tr} \left[(A-A_0)\wedge d(A-A_0) + \frac{2}{3} (A-A_0)\wedge (A-A_0) \wedge (A-A_0)) \right ] \end{equation} where $A_0$ is some reference connection, since $A-A_0$ is globally defined 1-form valued in ${\rm Lie}G$.

I see that under gauge transformation (or different chart), \begin{equation} CS(A^g) - CS(A) = d\alpha(A,g) + Q(g) \end{equation} where $Q(g)$ is closed. But I don't know how I can infer the validity of doing the integration from this gauge transformation.

Thank you!

$\endgroup$
2
  • 6
    $\begingroup$ I have no complaints about Urs's answer below, but if you want a less high-brow discussion, I recommend Freed's articles "Classical Chern-Simons Theory Part I" arxiv.org/abs/hep-th/9206021 and "Part II" ace1.ma.utexas.edu/users/dafr/cs2.pdf . $\endgroup$ Apr 19, 2013 at 1:30
  • $\begingroup$ Thanks~Actually I came across Freed's notes some time ago, but I stopped reading as I found myself further and further away from "real physics", in some sense:).But no doubt the 2 are very good notes and I shall go back to them. $\endgroup$
    – Lelouch
    Apr 19, 2013 at 14:02

2 Answers 2

22
$\begingroup$

Often in the literature by "Chern-Simons theory" is meant by default $G$-Chern-Simons theory whose gauge group is a connected and simply connected semisimple compact group $G$, such as $G = SU$. In this case it so happens that all $G$-principal bundles on a 3-manifold $\Sigma_3$ are trivializable, and hence one can identify the space of G-principal connections on $\Sigma_3$ just with that of $\mathfrak{g}$-valued differential forms. So one gets away with the naive formula that you recall above.

In stark contrast to this is what may seem to be a simpler example, namely $U(1)$-Chern-Simons theory. Since $U(1)$ is not simply connected, clearly, there are of course non-trivial $U(1)$-principal bundles on $\Sigma_3$, in general, and hence the above naive approach fails, as you notice.

In this case the correct Chern-Simons action is instead obtained this way: given a field configuration $\nabla$ which is a circle-principal connection, we can form its differential cup-product square in ordinary differential cohomology. This yields a $\mathbf{B}^2 U(1)$-principal 3-connection $\nabla \cup \nabla$, often known as a bundle 2-gerbe with connection or else as a degree-4 cocycle in Deligne cohomology. This now has a connection 3-form and hence has a volume holonomy over $\Sigma_3$. And this now is the correct action functional for Chern-Simons theory. For more on this see at nLab:higher dimensional Chern-Simons theory.

Secretly this higher principal connection structure also governs the first, seemingly simpler case. The action functional of Chern-Simons theory is always the volume holonomy of a 3-connection, the Chern-Simons circle 3-connection.

This is in fact the general abstract characterization of Chern-Simons theories and all its higher (and lower) dimensional variants. A Chern-Simons-type action functional is always the volume holonomy of a refinement of a universal characteristic class to ordinary differential cohomology. Further remarks along these lines are for instance in

Domenico Fiorenza, Hisham Sati, Urs Schreiber, A higher stacky perspective on Chern-Simons theory.

$\endgroup$
2
  • 1
    $\begingroup$ Though I do not fully understand the categorical generalization you provided, I do now understand why higher, say 5-dimensional CS theory is much less studied in physics community: the action is difficult to write down with enough generality. But now 5d gauge theory is attracting more attentions, and I see in physics papers we are still using the most simple CS-action. Maybe we should write down a sensible 5d Chern-Simons(-like) theory, with some obvious ("easy" but general enough) modification to 3d one. Does anything like this exists already? Thanks. $\endgroup$
    – Lelouch
    Apr 18, 2013 at 20:53
  • 2
    $\begingroup$ Using the differential cup product that I mentioned, one can build higher dimensional theories by coupling lower dimensional ones. For instance one gets a 5d CS-type theory on a U(1)-field by cup-cubing the differential first Chern class, and one on pairs consisting of an SU- and of a U(1)-gauge field by forming the cup product of the differential second and first Chern-class, respectively. This is described here: ncatlab.org/schreiber/show/… $\endgroup$ Apr 18, 2013 at 21:50
18
$\begingroup$

Here is a mid 1970s point of view, courtesy of Atiyah-Patodi-Singer.

Suppose you have a complex vector bundle $E$ of rank $r$ over a smooth manifold $M$. A polynomial function $P$ on the space of $r\times r$ matrices is called invariant if $P(T AT^{-1})=P(A)$ for any $r\times r$ complex matrix $A$ and any invertible $r\times r$ matrix $T$. If you look at

$$ \Delta(x)=\det(1+ xA)=\sum_{k=0}^r c_k(A) x^k, $$

then the coefficient $c_k(A)$ is a homogeneous invariant polynomial function of degree $k$. For example

$$c_1(A)={\rm tr}\; A,\;\;c_r(A)=\det A. $$

To a connection $\nabla$ on $E$ with curvature $F(\nabla)$, we can associate the degree $2k$ form on $E$

$$ c_k(\nabla) = c_k\bigl(\; F(\nabla)\;\bigr), $$

where in the above equality one thinks of $F(\nabla)$ as an $r\times r$-matrix whose entries are $2$-forms. For example

$$ c_1(\nabla)= {\rm tr}\; F(\nabla)= F_{11}(\nabla)+\cdots +F_{rr}(\nabla). $$

Chern-Weil theory proves two things:

  • The form $c_k(\nabla)$ is closed.

  • If $\nabla^1$, $\nabla^0$ are two connections on $E$, then there exists a canonical form of degree $(2k-1)$, called the transgression form and denoted by $Tc_k(\nabla^1,\nabla^0)$, which satisfies

$$ d Tc_k(\nabla^1, \nabla^0)= c_k(\nabla^1)-c_k(\nabla^0). $$

In other words, the cohomology class determined by $c_k(\nabla)$ is independent of $\nabla$. This cohomology class is the $k$-th Chern class of $E$.

Suppose now that $\dim M= 2k-1$. Then, on account of dimension, $c_k(\nabla)=0$, yet $Tc_k(\nabla^1,\nabla^0)$ is a top degree form well defined for any choices of $\nabla^0,\nabla^1$.

Suppose additionally that $E$ is trivial and we have fixed a trivialization. Then we can choose $\nabla^0$ to be the trivial connection on $E$ and then we set

$$ CS_k(\nabla):= Tc_k(\nabla,\nabla^0). $$

The usual Chern-Simmons theory is a special case of this construction when $k=2$, i.e., $E$ is a trivial complex vector bundle of rank $r\geq 2$ over a $3$-manifold.

$\endgroup$
3
  • 1
    $\begingroup$ Thanks for the explanation. To me there is a hierarchy between odd and even dimension: even dimensional characteristic classes are much easier to understand/write down than the secondary ones, while the latter needs a "reference connection $\nabla_0$" to be well-defined. I can understand that since there is no elementary gauge invariant odd-forms (all we can use is $F_{\mu\nu}$ with traces) on the base manifold. It seems mathematical notion "bundle" care/like even dimensional cohomology more than odd ones. Is there another notion that could care more about odd-dimensional cohomology? $\endgroup$
    – Lelouch
    Apr 19, 2013 at 14:20
  • 1
    $\begingroup$ That bundles "care" only about even dimensions is a reflection of the fact that the (real) cohomology rings of compact Lie groups are generated by odd degree classes. But the point of of Chern-Simmons theory is that it is an action functional directly relevant to physics. One can write many action functionals no physical relevance. I guess my question would be, what are you looking for? $\endgroup$ Apr 19, 2013 at 14:43
  • 5
    $\begingroup$ By the way, strictly speaking this is not "another point of view", but is part of what it means to refine a universal characteristic class to differential cohomology! And it holds more generally than for traditional Chern-Weil theory, too, notably it holds also for invariant polynomials not just on matrix Lie algebras, but generally on Lie algebroids and higher Lie algebroids. This then identifies "AKSZ sigma-models" as Chern-Simons type theories (ncatlab.org/schreiber/show/…) and in fact generalizes them to globalized field data. $\endgroup$ Apr 19, 2013 at 19:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.