Dear prof. Wen, are there ways to compute the ground state degeneracy of a Chern-Simons theory on torus involving more than two non-abelian gauge fields coupled with each other? For example CS theory like SU(2)_n \times SU(3)_m on torus.

Can we develop a path integral formalism for amplitude of general states not just amplitude for position eigenstates? I.e start from some general state , evolve with time , then calculate amplitude for ending in some final state. Can this be formulated in path integral formalism?

Is your description, the same reason why people use similar forms in parton decomposition of spin?

Thanks for the reference.

@physshyp , woluld you please give some reference papers where above mentioned steps are worke out

Now my confusion is that if we completely fill the some Landau level, lets consider lowest Landau level, then there is a finite current in the system. In this scenario it seems to me that bulk of the material also conducts current , since all eigenstates contribute to current. But since we have completely filled level and there is a finite gap to next level , so it is an band insulator. It shouldn't carry current. In this sense it seems contradictory to me . would you please clarify on this?

Thanks for the answer. Lets consider a finite size , so it will restrict the number of transverse momentum value (Let's say $k_{y}$). But don't consider any edge potential. Then also if we calculate expectation value of velocity in an eigenstate of Hamiltonian lets say lowest Landau level eigenstates in the presence of electric field we get non zero answer. It is same for all lowest Landau level states (in fact for all states.). So there is a finite current.

@FraSchelle , is not the particle-hole symmetry tells that Majorana zero modes must come in pairs?

(cont..) When we consider many particle ground state , many particle- 1st excited state ,2nd excited state, etc ,it will consists of different particle in each excited state. But how can be definition of excited state be meaningful when particles number changes. I think in condensed matter , people view energy spectrum in terms of only single particle states. Any comments on this will be usefull.

I think the above Hamiltonian "H" gives single particle eigenstates , which have same energy "t". The number of such eigenstates is "N". From this we can construct multiparticle states, by filling various states with a fermionic particle. There will be degeneracy in these multi-particle states as well.

'd' is a fermionic operator.