Energy Relations

The total energy functional for all our fields is

$$E_{\mbox{\tiny {TOTAL}}} = \int \, d^2\, r \left\{\frac{1}{2\, e^2}\ B^2\mid {\bf D}\, \varphi\mid^2 + V(\varphi^\ast\, \varphi) + \frac{1}{2\, e^2}\ B^2\right\}$$

$$+ \int \, d^2 \, r \, \bar{\Psi}^\ast \, K_A \, \Psi$$ (20)

Varying this with respect to $\Psi^\ast$ produces our Dirac equation (16) at zero eigenvalue. Varying with respect to the Bose fields $\varphi^\ast \$   and${\bf A}$ derives (13) and (14), but with a back reaction form the Dirac fields

$${\bf D}\cdot {\bf D}\, \varphi = V^\prime \ (\varphi^\ast \varphi) + \frac{g}{2}\ \bar{\boldsymbol \psi } \ (1+ \gamma_5) \ \Psi$$ (21)

$$\frac{1}{e^2}\ \varepsilon^{ij} \, \partial_j\, B = J^i_{\mbox{\tiny {BOSE}}} + \bar{\boldsymbol \psi }\, \gamma^i\, \gamma_5\, \Psi$$ (22)

But with a zero mode that is an eigenstate of M , the back reaction Dirac bilinears vanish. Thus our Dirac zero mode, together with the scalar field/gauge field NO/AG vortex, is a self consistent solution of the coupled system.

Chiral gauge theories have periodically entered physics, but in even-dimensional space-time, where the chiral anomaly influences the structure and physical utility of these models. In the present work, we have chiral gauge theory in odd-dimensional space-time, and the structure is mathematically very elegant, owing its self-consistency. It remains to be determined whether a microscopic description for graphene can lead to the chiral gauge field that enters our theory.