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February 20, 2018

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

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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

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For those interested in the subject of black holes, I have started a three part series called, Black Holes Through Outer Space, on the subject. The first paper, Predictable Barriers Precluding Any Consideration of Transversable Black Holes Through Outer Space, is online at scientificblogging,com.

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God bless you physicists--I am unable to comprehend even your most rudimentary explanations.

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