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The graphene system is fascinating for quantum field on the sheet are not restricted into being strictly fermionic or bosonic. Under an exchange of particles with state vectors |Ψ_1> and |Ψ_2> there is |Ψ_1Ψ_2> = e^{iθ}|Ψ_2Ψ_1>, which in dimensions > 2 has θ = {0, π} so the exchange is 1 for bosons and -1 for fermions. The spin statistic is defined by π_1(SO(n,1)) which for n > 2 is π_1(SO(n,1)) = Z_2, but for n = 2 is π_1(SO(2,1)) = Z. The first homotopy group indicates that orientation of the spin, which is infinitely cyclic in n = 2. Further this means the standard sequence
1 --> Z_2 --->spin(n) ---> SO(n) ---> 1
does not apply, so spin(2,1) is not a universal covering group. The projective system over SO(2,1) is the braid group B_2, which describes the anyon.
The analogue of a string in graphene is the occurrence of an effective magnetic monopole with the quantum Hall effect. The quantized currents on a graphene sheet with the quantum Hall effect can result in an effective magnetic monopole. The quantum flux of B-field due to currents around the sheet has an effect of setting one of the magnetic poles ---> infinity like the Dirac solonoid monopole. This sets up a duality between the electric and magnetic charges, e and b respectively, so that eb = nħ. This is the T-duality which a Type IIA string has with respect to D2, D4, D6, D8 and D10 branes. In 10 dimensions the T-duality obtains on a D8-brane, and the 11-dimensional duality obtains on a D9-brane, which gives a correspondence between Type IIB and Type IIA strings.
This physics is generalized into Chern-Simons theory. The 2+1 C-S theory is given by a Lagrangian
L = A Λ dA + (2/3)A Λ A Λ A,
Where this Lagrangian defines a cyclicity S = ∫d^3x L, with S --> S + 2πN, for N a winding number. By Poincare duality this is carried over to the 8 dimensional surface as the boundary of the 9 dimensional surface, here the surfaces in fact being D-branes and the two surface a D2-brane bounding a 3-dimensional region. The duality carries this information over to a supersymmetric WZW Lagrangian. The D2-branes exist in a foliation that is in the bulk transverse to C^4/Z^8, for Z^8 a set of discrete points (Z^8 a lattice of charges or roots of the E_8). The transversality condition preserves the anionic structure on the D2 as supersymmetry (N=8 SUSY) on the C^4/Z^8.
Lawrence B. Crowell