# Group Cohomology and Topological Field Theories

I have a two-part question:

1. First and foremost: I have been going through the paper by Dijkgraaf and Witten "Group Cohomology and Topological Field Theories". Here they give a general definition for the Chern-Simons action for a general $$3$$-manifold $$M$$. My question is if anyone knows of any follow-up to this, or notes about their paper?

2. To those who know the paper: They say that they have no problem defining the action modulo $$1/n$$ (for a bundle of order $$n$$) as $$n\cdot S = \int_B Tr(F\wedge F)$$ $$(mod 1)$$, but that this has an $$n$$-fold ambiguity consisting of the ability to add a multiple of $$1/n$$ to the action - What do they mean here? Also, later on they re-define the action as $$S = 1/n\left(\int_B Tr(F\wedge F) - \langle \gamma^\ast(\omega),B\rangle\right)$$ $$(mod 1)$$ - How does this get rid of the so-called ambiguity?

Basically my question is if anyone can further explain the info between equations 3.4 and 3.5 in their paper. Thanks.

First, the full paper is here:

Second, the paper has 150 citations. See all this information at INSPIRE (updated SPIRES):

http://inspirebeta.net/record/278923?ln=en

Third, the text between 3.4 and 3.5 looks totally comprehensible. At that point, they are able to define $n\cdot S$ modulo 1, which is equivalent to defining the action $S$ modulo $1/n$. The goal is to define the action $S$ itself modulo 1; I suppose that their normalization of the path integral has to have $\exp(2\pi i S)$ with the atypical $2\pi$ factor. Yes, confirmed, it's equation 1.2.

If you shift the action by 1 - or $2\pi$ in the ordinary conventions - it doesn't change the integrand of the path integral; it doesn't change the physics. So quite generally, if one is able to say that the action $S$ is equal to $S_0+n$ (or $2\pi n$ normally) for some integer $n$, he knows everything about the physics of the action he needs; shifting it by an integer doesn't change anything. That's why, in fact, the action is often defined modulo 1 only (up to the addition of an integer multiple of 1).

So it's enough to know the "fractional part" of the action; the integer part is irrelevant. However, at the point of the equation 3.4, their uncertainty is larger than that: they only know the action modulo $1/n$. For example, if the action is $9.37$ modulo $1/2$, it means that the fractional part may be $0.37$ but it may also be $0.87$. These two values of $S$ would change the physics because the contribution of the configuration to the path integral changes the sign if one changes $S$ by $1/2$ (in normal conventions, by $\pi$).

If one only knows $S$ modulo $1/n$, and if he thinks it's $S_0$ - in this case, the $F\wedge F$ expression - it means that the real action is $$S = S_0 + K/n$$ and the integer $K$ has to be determined. Because the change of the action $S$ by an integer doesn't change physics, it doesn't matter if $K$ in the equation above is changed by a multiple of $n$. So the goal is to find the right $K$ to define the action - and $K$ is an unknown integer defined (or relevant) modulo $n$, i.e. up to the addition of an irrelevant and arbitrary multiple of $n$.

At some point, they find the right answer and it is $$K = -\langle \gamma^*(\omega),B\rangle$$ which removes the ambiguity of $S$ - the missing knowledge whether $S$ should be the original $S$ or higher or smaller by a particular multiple of $1/n$. If you don't understand the text above, then apologies, I have no way to find out why, so I can't give you a better answer unless you improve your question.

• No, I understand. So, do you have any ideas/motivation on how they came to adding $-\langle \gamma^\ast(\omega),B\rangle$? I agree that it works, just have no idea it would be that. Thanks for clearing things up! Apr 22, 2011 at 15:52
• Also, just to be completely clear, removing the ambiguity in $S$ is equivalent to finding $K$? Thanks again for the answer! Apr 22, 2011 at 20:29
• Yup, removing the ambiguity of $S$ is equivalent to finding $K$, more precisely finding $K$ mod $n$. But in the full quantum theory, $S$ only matters mod $1$ (in normal normalizations of physics, $2\pi$), because it appears in $\exp(2\pi i S)$ in the path integral only. May 27, 2012 at 5:27

Dijkgraaf and Witten used $\mathcal H^3[G,U(1)]$ to define CS theory for gauge group $G$. Recently, group cohomology has found applications in condensed matter physics. It may classify the so called "symmetry protected topological phases" of interacting bosons:

The $d$-dimensional symmetry protected topological phases of interacting bosons with symmetry group $G$ has a subclass, which can be one-to-one labeled by elements in $\mathcal H^{d+1}[G,U(1)]$. ($d$ is the space dimensions.)

(The symmetry protected topological phases are for interacting systems, which are similar to the topological insulators of non-interacting fermions. They are short-range entangled states with symmetry.)

The integer part of this is a cocycle condition, which is a measure of the winding number for a gauge transformation. The Chern-Simons (CS) theory is a $2~+~1$ dimensional quantum field theory for a non-dynamical gauge field$A_\mu$. The action for such a theory is $$S_{CS}~=~\frac{k}{4\pi}\int A\wedge dA~+~\frac{2}{3}A\wedge A\wedge A$$ Where $A_\mu$ is a component of the one form ${\underline A}~=~A_\mu{\underline e}^\mu$ for a non-abelian gauge field transforming in the adjoint representation of the gauge group $U(N)$.

The theory to make sense must be well behaved under gauge transformations. While it is relatively easy to show invariance in the abelian case, the non-abelian case is a little more subtle. In this case $$S_{CS}~\rightarrow~S_{CS}~+~2\pi kN$$ Where $N$ is a integer for the winding number of the gauge transformation performed. Quantization of the theory using Feynman’s path integral formalism requires that$e^{iS_{CS}}$ be gauge invariant. This leads to the condition that $k~\in~{\mathbb Z}$. The integer $k$ is the Chern-Simons level $A_\mu$. Typically every gauge group in the Chern-Simons theory has a level associated to it.

This form of the Chern-Simons theory is not supersymmetric. However it is possible to make the gauge field $A_\mu$ a component of an ${\cal N}~=~2$ vector multiplet. This necessarily introduces two scalar fields $A_\mu$ $F$, an auxiliary field, and a 2-component Dirac spinor $\psi$ to the theory in a superfield $$\Psi~=~\psi~+~\theta \sigma^\mu A_\mu~+~H.C.~+~{\bar\theta}\theta F.$$ It is possible to extend this theory to admit the full ${\cal N}~=~8$ SUSY (16 supercharges).

• Since I know very little about supersymmetry, I have a few simple question. How does the $\cal N = 2$ SUSY Chern-Simons action look like in terms of the superfield $\Psi$? Is it possible to write this down without the superspace/superfield notation? Finally, does this SUSY extension add new mathematical features to the CS theory (relation to knot theory, quantum groups, modular tensor categories, CFT and so on)? Apr 23, 2011 at 12:27
• The generic form of the CS is the same as the above. The CS Lagrangian has that cubic form, which means that in $2~+~1$ spacetime a fermion can have bosonic statistics and visa versa. The exchange statistics which apply in $3$ space gets "pushed" into the time part --- so to speak. This gives anyonic behavior. In string theory this describes the $M_2$ brane, and in condensed matter physics graphene. Apr 23, 2011 at 12:57
• This is not an answer to the question, which is not about the more elementary relation which restricts k to an integer, but about fixing up the action when the group gauge group is a quotient or discrete. You are answering a different (much simpler) question. May 27, 2012 at 10:05