TQFT characterization of braiding statistics

Question

In the TQFT language, quasiparticles correspond to Wilson loop operators. It is well-known that quasiparticles can have non-trivial braiding statistics. 

Take $2+1$ dimensional Abelian Chern-Simons theory with $U(1)$ gauge group as an example, 
\begin{equation}
S_{CS} = \int \frac{k}{4\pi}A\wedge dA - q A\wedge *j
\end{equation}
where $j$ is a source term representing quasiparticle currents (which carries $q$ units of $A-$gauge charge). 

The equation of motion reads 
\begin{equation}
\frac{2\pi q}{k}*j = dA 
\end{equation}

This implies that each quasiparticle not only carries $q$ units of $A-$gauge charge, but are dressed with $\frac{2\pi q}{k}$ units of $A-$gauge flux. So braid a charge$-q$ particle around a charge$-q^{\prime}$ particle induces (by the Aharonov-Bohm effect) a non-trivial $U(1)-$phase $\theta = \frac{2\pi}{k}q q^{\prime}$. 

Wilson lines in TQFT captures the above statistics nicely. In order to see this, one can calculate the regulated expectation value of two line operators,
\begin{equation}
<\exp(iq\oint_{C_1}A)\exp(iq^{\prime}\oint_{C_2}A)> = <\exp(\frac{2\pi i}{k}q q^{\prime}L(C_1,C_2))>
\end{equation}
where $L(C_1,C_2)$ is the linking number of the two lines. The process of braiding one charge around another corresponds to $L=1$. The Chern-Simons theory thus automatically calculates the linking number of chosen line operators and multiplies the wavefunction by the appropriate phase.

Another example is $2+1$ dimensional BF theory, 
\begin{equation}
S_{BF} = \int \frac{k}{2\pi}B\wedge dA - A\wedge *j - B\wedge *J
\end{equation}
where both $A$ and $B$ are $1-$form gauge fields, and $j$ and $J$ are quasiparticle current carrying $A$ and $B$-gauge charges respectively. 

The equation of motion reads 
\begin{align}
dA &= \frac{2\pi}{k}*J \nonumber \\
dB &= \frac{2\pi}{k}*j
\end{align}
The magnetic flux of $A$ is attached to the charges of $B$ and vice versa, and braiding $q$ units of $A-$charge around $m$ units of $B-$charge induces a phase $\frac{2\pi q m}{k}$. This is again nicely captured by the following equation
\begin{equation}
<\exp(iq\oint_{C_1}A)\exp(im\oint_{C_2}B)> = <\exp(\frac{2\pi i}{k}qmL(C_1,C_2))>
\end{equation}

My question is, are there any similar characterizations for braiding statistics in $3+1$ dimension? In $3+1$ dimension, generic excitations are quasiparticles as well as quasistrings. So we need to be able to characterize braidings between Wilson loops and Wilson surfaces. I think equation (2.13) on page 8 of http://arxiv.org/abs/1011.5120 defines something similar. 

However, besides braidings between quasiparticles and quasistrings. The really fundamental braiding process in $3+1$ dimensional gauge theories are the so-called three-loop braiding process, as argued in http://arxiv.org/abs/1403.7437. The statistics is between 3 loops carrying gauge fluxes, where two loops braid with each other while both are linked to a third base loop. I'm wondering if there is an analogous characterization of the above process in terms of Wilson surfaces. Is there something called "linking number" of Wilson surfaces? If yes, how is it defined?

Answer 1

There is the "triple linking number" of surfaces in four dimensions,which corresponds to the three-loop braiding process. This is discussed in http://arxiv.org/abs/1404.1062. ;

In field theory, braiding statistics in 2+1d can be computed from the commutation relation of Wilson loops. Similarly, in 3+1d the three-loop braiding is given by some sort of commutation algebra of three Wilson surface operators, as given by eq. 36 in http://arxiv.org/abs/1404.1062. This can be actually computed in continuous field theories.

Comments

Thanks for the reference Meng, I'll take a look at it.

Answer 2

The description of worldline and worldsheet linkings are explored in this preprint: by Juven Wang, Xiao-Gang Wen, Shing-Tung Yau. 

Quantum Statistics and Spacetime Surgery

(1) The short answer is that there are systematical ways to construct worldline and worldsheet linkings by inserting nontrivial homology group generators of submanifolds and glue the submanifolds together via geometric-topology surgery theory. The approach there in [1602.05951](http://arxiv.org/abs/1602.05951) does not require using field theory (quantum or classical) description. The approach in [1602.05951](http://arxiv.org/abs/1602.05951) only needs the basic quantum mechanics and the definition of path integral on the spacetime manifold. There are more details of an introduction of braiding as linking, the surgery theory and the spacetime path integral using the lattice or field theory in [1602.05569](http://arxiv.org/abs/1602.05569).

(2) If you are looking for the path integral with worldline/worldsheet linking computed with a given topological field theory (TQFT) action, you can find it in the TABLE of [1602.05951](http://arxiv.org/abs/1602.05951), such as:

(2) The descriptions of those field theories can be found in:

1. J Wang, X-G Wen, S-T Yau, http://arxiv.org/abs/1602.05951

2. A. Kapustin and R. Thorngren, (2014), arXiv:1404.3230 [hep-th].

3.  J. C. Wang, Z.-C. Gu, and X.-G. Wen, [Phys. Rev. Lett. 114, 031601], http://arxiv.org/abs/1405.7689 [cond-mat.str-el]. 

4.   Z.-C. Gu, J. C. Wang, and X.-G. Wen, (2015), http://arxiv.org/abs/1503.01768 [cond-mat.str-el]. [PhysRevB.93.115136](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.115136)

5.  P. Ye and Z.-C. Gu, (2015), arXiv:1508.05689 [cond-mat.str-el]. 

6. etc..

And reference therein.

(3)  In [1404.1062](http://arxiv.org/abs/1404.1062) and [1602.05951](http://arxiv.org/abs/1602.05951) , it is mentioned that how the spacetime process of 3-loop braiding as the Spun[Hopf link] of two $T^2$-tori linked with the third torus is realized in the 3-worldsheet linking in $S^4$. In [1602.05951](http://arxiv.org/abs/1602.05951), it is remarked that how the spacetime process of 4-loop braiding as the Spun[Borromean rings] of three $T^2$-tori linked with the fourth torus is realized in the 4-worldsheet linking in $S^4$. If you are interested in the spinning of closed strings and its topological spin quantum number, you may refer to [1404.7854](http://arxiv.org/abs/1404.7854) [PhysRevB.91.035134](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.91.035134) and [1409.3216](http://arxiv.org/abs/1409.3216) [PhysRevB.92.045101](http://journals.aps.org/prb/abstract/10.1103/PhysRevB.92.045101). There in  [1404.7854](http://arxiv.org/abs/1404.7854), the possibility of spinning of trefoil knots or Borromean rings are also mentioned.

(4) In [1602.05951](http://arxiv.org/abs/1602.05951), some generalized Verlinde formulas, as certain new quantum surgery kind of formulas combining quantum statistics and spacetime surgery constraints are given. For examples, 

• Verlinde formulas in 2+1D topological orders TQFT / 1+1D CFT:

• quantum surgery formulas  in 3+1D topological orders (4d TQFT) / 2+1D boundary states (CFT, etc) are given in [1602.05951](http://arxiv.org/abs/1602.05951):
  
  So yes, the original post had hinted that it seems to be a lot more fun to study higher dimensional braiding process of particles and strings/loops with worldline and worldsheet linkings in 3-manifolds and 4-manifolds of spacetime.