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Topological ground state degeneracy of SU(N), SO(N), Sp(N) Chern-Simons theory

+ 7 like - 0 dislike
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We know that level-k Abelian 2+1D Chern-Simons theory on the $T^2$ spatial torus gives ground state degeneracy($GSD$): $$GSD=k$$

How about $GSD$ on $T^2$ spatial torus of:

SU(N)$_k$ level-k Chern-Simons theory?

SO(N)$_k$ level-k Chern-Simons theory?

Sp(N)$_k$ level-k Chern-Simons theory?

What are the available methods to compute them? such (i) algebraic geometry; (ii) Lie algebra; (iii) topological theory or (iv) quantum hall fluids parton construction?

an example of SU(2)$_k$ using this approach shows $GSD$ on $T^2$ is $$GSD=(k+1)$$

SU(3)$_k$ using this approach shows $GSD$ on $T^2$ is $$GSD=(k+1)(k+2)/2$$.

If there are examples of $G_2,F_4,E_6,E_7,E_8$ (level-k?) Chern-Simons theory and its $GSD$ on $T^2$, it will be even nicer. Reference are welcome.

This post imported from StackExchange Physics at 2014-06-04 11:34 (UCT), posted by SE-user Idear
asked Jan 8, 2014 in Theoretical Physics by wonderich (1,420 points) [ no revision ]
It is possible that some points what you are asking are related in some way to the formulae $(1.3)$ ( for $SU(N)$), $(1.6)$ ( for $Sp(2N)$), $(1.7)$ (for $G_2$) of this paper

This post imported from StackExchange Physics at 2014-06-04 11:34 (UCT), posted by SE-user Trimok
@ Trimok, thanks, those eqs are useful. How about SO(N)?

This post imported from StackExchange Physics at 2014-06-04 11:34 (UCT), posted by SE-user Idear
Hi Idear, I think you should consider to use the research-level tag for this very nice question ;-). Cheers

This post imported from StackExchange Physics at 2014-06-04 11:34 (UCT), posted by SE-user Dilaton
@ Dilaton, thanks a lot. I thought it is not yet "real" research-level.

This post imported from StackExchange Physics at 2014-06-04 11:34 (UCT), posted by SE-user Idear

1 Answer

+ 6 like - 0 dislike

After stating the solution, I'll try to give some physical insights to the best of my knowledge and some more references.

The dimension of the required state space is given by the Verlinde formula, having the following form for a general compact semisimple Lie group $G$ on a Riemann surface with genus $g$ corresponding to the level $k$:

$$ \mathrm{dim} V_{g,k} =(C (k+h)^r)^{g-1} \sum_{\lambda \in \Lambda_k}\prod_{\alpha \in \Delta}(1-e^{i\frac{\alpha .(\lambda+\rho)}{k+h}})^{(1-g)}$$

(Please see Blau and Thompson equation 1.2.). Here, $C$ is the order of the center, $h$ is dual Coxeter invariant, $\rho$ is half the sum of the positive roots, and $r$ is the rank of $G$. $g$ is the genus, $\Delta$ is the set of roots and $\Lambda_k$ is the set of integrable highest weights of the Kac-Moody algebra $G_k$.

For the torus ($g=1$), this formula simplifies to:

$$ \mathrm{dim} V_{\mathrm{Torus},k} = \# \Lambda_k $$

i.e., the dimension is equal to the number of integrable highest weights of the Kac-Moody algebra $G_k$.

The integrable highest weights of a level-$k$ Kac-Moody algebra are given by the following constraints:

$$ \lambda - \mathrm{dominant}, 0 \leq \sum_{i=1}^r \frac{2 \lambda. \alpha^{(i)}}{ \alpha^{(i)}. \alpha^{(i)}}\leq k$$

Where $ \alpha^{(i)}$ are the simple roots, please see, for example, the following review by Fuchs on Kac-Moody algebras.

(My favorite reference for the representation theory of Kac-Moody algebras is the Goddard and Olive review which seems not available on line)

For example for $SU(3)_k$ whose dominant weights are $2$-tuples of nonnegative numbers $(n_1, n_2)$, the above condition reduces to:

$$\mathrm{dim} V^{SU(3)}_{\mathrm{Torus},k} = \# (n_1\geq 0, n_2\geq 0, 0\leq n_1 + n_2 \leq k )= \frac{(k+1)(k+2)}{2}$$

To perform the computations for the more general cases, one can use the seminal review by Slansky.

The Verlinde formula was discovered before the Chern-Simons theory came into the world. Originally it is the dimension of the space of conformal blocks for the WZW model. This formula has been derived in a large variety of ways, please, see footnote 26 in the Fuchs review. It is still an active research topic, please see for example a new derivation in this recent article by Gukov.

The Chern-Simons theory may be the most sophisticated example in which the Dirac quantization postulates can be carried out in spirit. (More precisely their generalization in geometric quantization). I mean starting from a phase space and utilizing a specified set of rules to associate a Hilbert space to it. In the case of the Chern-Simons theory, the phase space is the set of solutions of the classical equations of motion. The classical equations of motion require the field strength to vanish, in other words the connection to be flat.

This phase space (the moduli space of flat connections) is finite dimensional, it has a Kähler structure and it can geometrically quantized as a Kähler manifold, just like the case of the harmonic oscillator. Thus the problem can be reduced in principle to a problem in quantum mechanics.

The case of the torus is the easiest because everything can be carried out explicitly in the Abelian and the non-Abelian case, please see the following explicit construction by Bos and Nair, (a more concise treatment appears in Dunne's review).

In the case of the torus, the moduli space of flat connections in the Abelian case is also a torus and in the non-Abelian case it is:

$$\mathcal{M} = \frac{T \times T}{W}$$

where $T$ is the maximal torus of $G$. Basically, a Fock quantization can be carried away, but there is a further restriction on the admissible wave functions coming from the invariance requirement under the large gauge transformations (please see for example, the Dunne's review ). The invariant wave functions are called non-Abelian theta functions and they are just in a one to one correspondence with the Kac-Moody algebra integrable highest weights. (In the Abelian case, the wave functions are the Jacobi theta functions).

In the higher genus case, although the quantization program leading to the Verlinde formula can be carried out in principle, few explicit results are known, please see the following article by Lisa Jeffrey (and also the following lecture notes). The dimension of these moduli spaces is known. In addition. Witten in an ingenious work computed their symplectic volumes and their cohomology ring in some cases.

Witten's idea is that as in the case of a simple spin, the dimension of the Hilbert space in the semiclassical limit ($k \rightarrow \infty$) becomes proportional to the volume and the leading exponent of $k$ is the complex dimension of the moduli space (please observe for example, that in the case of $SU(3)$ on the torus, the leading exponent is $2$ which is the rank of $SU(3)$ which is the dimension of the maximal torus $T$).

This post imported from StackExchange Physics at 2014-06-04 11:34 (UCT), posted by SE-user David Bar Moshe
answered Jan 8, 2014 by David Bar Moshe (3,875 points) [ no revision ]
Dear David, +1, you are certainly an expert in the field. Thank you so much, let me take a further look, hopefully get back to you soon.

This post imported from StackExchange Physics at 2014-06-04 11:34 (UCT), posted by SE-user Idear
@ David, now I understand your Eq.(4), you are saying: $$GSD=\#(n1≥0,n2≥0,0≤n1+n2≤k)=(k+1)(k+2)/2$$; i.e. counting the volume. Perhaps adding a bracket is more clear.

This post imported from StackExchange Physics at 2014-06-04 11:34 (UCT), posted by SE-user Idear
@ David, if there are explicit formulas for g=1 T2 torus of SO(N) level-k Chern-Simons theory is just good enough. (in any Ref) Trimok's ref seems to provide SU(N), Sp(N) already.

This post imported from StackExchange Physics at 2014-06-04 11:34 (UCT), posted by SE-user Idear
@David, By saying "leading exponent is 2 which is the rank of SU(3)" I know exponent is 2, but you mean "the rank of SU(3) is 2" is for Cartan matrix (correct ?)

This post imported from StackExchange Physics at 2014-06-04 11:34 (UCT), posted by SE-user Idear
@Idear For the case of $SO(N)$ You can use equation (1b) together with the coefficients given in sections 3.B and 3.D of Gannon's article arxiv.org/abs/hep-th/0106123 to compute the integrable highest weights. I'll correct and update my answer soon. I used the symbol # for the cardinality of the set. Yes, the rank is the dimension of the Cartan matrix, it is also the dimension of the maximal torus $T$.

This post imported from StackExchange Physics at 2014-06-04 11:34 (UCT), posted by SE-user David Bar Moshe
@DavidBarMoshe : I am certainly wrong, but it seems to me that your formula $0 \leq \sum_{i=1}^r \frac{2 \lambda. \alpha^{(i)}}{ \alpha^{(i)}. \alpha^{(i)}}\leq k$, for instance in a rank $2$ algebra, always give the constraint $0\leq n_1 + n_2 \leq k$, while I have the feeling that it is not correct for $G_2$ (well, I based my (maybe false) intuition) on the formula $1.7$ of this paper.

This post imported from StackExchange Physics at 2014-06-04 11:34 (UCT), posted by SE-user Trimok
@ DavidBarMoshe (or Trimok), it will be really nice if you can shed light on the relation between C-S theory's GSD and the Cartan matrix of a given gauge group. Please.

This post imported from StackExchange Physics at 2014-06-04 11:34 (UCT), posted by SE-user Idear

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