Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== Take spacetime MMM with boundary ∂M\partial M∂M (for our physical picture think of a small timelike tube around a single atom/electron sphere; at a fixed time slice the spatial boundary is a 2-sphere S2S^2S2 of radius RRR). ===

Bulk electromagnetic action (SI-like, covariant form)

SEM  =  −14μ0∫Md4x  FμνFμν  −  ∫Md4x  JbulkμAμ.S_{\rm EM} \;=\; -\frac{1}{4\mu_0}\int_M d^4x \; F_{\mu\nu}F^{\mu\nu} \;-\; \int_M d^4x \; J^\mu_{\rm bulk} A_\mu .SEM=−4μ01∫Md4xFμνFμν−∫Md4xJbulkμAμ.
(Here Fμν=∂μAν−∂νAμF_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\muFμν=∂μAν−∂νAμ.)

Boundary (surface) matter / spinor action — 2D surface (on the sphere) with a charged spinor field ψ(xa)\psi(x^a)ψ(xa) living on ∂M\partial M∂M. Use surface coordinates xax^axa (with a=0,1,2a=0,1,2a=0,1,2 for time + two angles on sphere), metric γab\gamma_{ab}γab induced on the boundary.
A compact form of the surface Dirac-like action is

Ssurf  =  ∫∂Md3σγ  [ ih2π  ψˉγaDaψ  −  msψˉψ  +  ⋯],S_{\rm surf} \;=\; \int_{\partial M} d^3\sigma \sqrt{\gamma}\;
\Bigg[\, i\frac{h}{2\pi}\; \bar\psi \gamma^a D_a \psi \;-\; m_s \bar\psi\psi \;+\; \cdots \Bigg],Ssurf=∫∂Md3σγ[i2πhψˉγaDaψ−msψˉψ+⋯],
where Da=∇a−iqAaD_a = \nabla_a - i q A_aDa=∇a−iqAa is the covariant derivative with the pullback of the bulk gauge potential Aa:=AμeμaA_a := A_\mu e^\mu{}_aAa:=Aμeμa to the boundary, qqq is the boundary charge (for electron q=−eq=-eq=−e) and γa\gamma^aγa are 2+1D surface gamma matrices (or Pauli matrices for a non-relativistic reduction). The dots indicate possible surface self-interaction or curvature couplings.

Interaction term already appears in DaD_aDa (minimal coupling). If you prefer to isolate it:

Sint  =  −∫∂Md3σγ  AaJsurfa,Jsurfa  =  q ψˉγaψ.S_{\rm int} \;=\; -\int_{\partial M} d^3\sigma \sqrt{\gamma}\; A_a J^a_{\rm surf},
\qquad
J^a_{\rm surf} \;=\; q\,\bar\psi \gamma^a \psi.Sint=−∫∂Md3σγAaJsurfa,Jsurfa=qψˉγaψ.
So the full action is

Stot=SEM+Ssurf+(optional bulk matter).S_{\rm tot} = S_{\rm EM} + S_{\rm surf} + \text{(optional bulk matter)}.Stot=SEM+Ssurf+(optional bulk matter).