Kaluza-Klein theory ($d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }+R^2(dy+A_{\mu }d{x}^{\mu }{)}^2$)

Kaluza-Klein theory obtains lower-dimensional fields from a higher-dimensional metric and other fields on compact dimensions. In the displayed fixed-radius circle ansatz, the off-diagonal metric supplies a vector. The coordinate change $y\mapsto y+\lambda (x)$ is compensated by $A\mapsto A-d\lambda$, producing an Abelian gauge symmetry. Allowing the radius to vary adds a scalar modulus. Kaluza-Klein modes also include nonzero internal momenta, giving a tower of massive fields.

// Target: quantum-field-theory.bigb

For the Poisson equation ${\nabla }_4^2\varphi =-4\pi Rq{\delta }^3(\mathbf{x}){\delta }_{2\pi R}(z)$ on ${\mathbb{R}}^3\times S^1$, expand the periodic delta as $(2\pi R{)}^{-1}{\sum }_n{e}^{inz/R}$. The $n$th Kaluza-Klein mode has radial potential $q{e}^{-|n|r/R}/(2\pi r)$, since $({\nabla }_3^2-m^2)e^{-mr}/r=-4\pi {\delta }^3(\mathbf{x})$. Summing the geometric series gives the displayed Green function. At equal compact coordinate it reduces to $q\coth (r/(2R))/(2\pi r)$. Thus the interaction energy changes from $q_1{q}_2/(2\pi r)$ at large separation to $q_1{q}_{2}R/(\pi r^2)$ at small separation, with normalization fixed by the stated Poisson equation.

For a massless field on Minkowski space times a circle of circumference $R$, the compact delta distribution expands as $R^{-1}{\sum }_n{e}^{2\pi in\Delta w/R}$. The noncompact retarded Green function in each Fourier sector is that of a four-dimensional field with mass $m_n$. Equivalently, the time-frequency denominator is ${\mathbf{k}}^2+m_n^2-(\omega +i0{)}^2$, putting both poles below the real axis. At equal compact positions the sum has one zero mode and a doubly counted nonzero mass tower.

At zero frequency, sum the Kaluza-Klein tower of three-spatial-dimensional Yukawa kernels: $[4\pi Rr{]}^{-1}(1+2{\sum }_{n\ge 1}{e}^{-2\pi nr/R})$. The geometric series gives the displayed function. It behaves as $1/(4{\pi }^2{r}^2)$ for $r\ll R$, the four-spatial-dimensional Coulomb kernel, and as $1/(4\pi Rr)$ for $r\gg R$, the three-spatial-dimensional kernel with a zero-mode normalization factor.

A real five-dimensional Maxwell field on a circle of radius $r$ has a Fourier series with coefficients $A_M^{(n)}$, $A_M^{(-n)}=(A_M^{(n)}{)}^{\ast }$. Its mixed field-strength component is $F_{\mu 5}^{(n)}={\partial }_{\mu }{A}_5^{(n)}-in{A}_{\mu }^{(n)}/r$. The nonzero modes transform as $\delta A_{\mu }^{(n)}={\partial }_{\mu }{\Lambda }^{(n)}$, $\delta A_5^{(n)}=in{\Lambda }^{(n)}/r$. Thus $A_5^{(n)}$ is a Stueckelberg mechanism field and the gauge-invariant massive vector is $A_{\mu }^{(n)}-r{\partial }_{\mu }{A}_5^{(n)}/(in)$. Its mass is $|n|/r$ and it has three physical polarizations. The zero modes instead consist of a massless four-dimensional Maxwell field and a real massless scalar field; the latter cannot be removed by a periodic infinitesimal gauge transformation. A positive Fourier level and its negative level are conjugate, not independent real fields.

A gravitational magnetic monopole for a compact circle, whose fibre is nontrivially twisted over a surrounding two-sphere. With flat spectator directions its nonsingular local geometry is a Taub–NUT space. Reducing along the M-circle yields the D6-brane. The magnetic dual of the IIB circle vector is instead a five-brane in nine noncompact dimensions.

// Target: geometry-and-topology.bigb