Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 53 / 3

Regard the antisymmetric tensor as a real differential-form potential $A_p$. For a free massless p-form gauge field, its field strength is $F_{p+1}=d{A}_p$ and its gauge transformation is $A_p\mapsto A_p+d{\Lambda }_{p-1}$. The Bianchi identity for an Abelian p-form and the source-free field equation areThe Hodge star makes $\ast F$ a closed $(D-p-1)$-form. On a contractible patch, the Poincaré lemma gives $\ast F=d{B}_q$, henceConversely, the original Bianchi identity becomes the dual field equation $d\ast dB=0$, up to the harmless signature sign in $\ast \ast$. The original equation becomes the dual Bianchi identity. This establishes massless p-form duality between gauge-equivalence classes of local solutions. It requires $0\le p\le D-2$ for an ordinary dual potential; global flux sectors and sources require extra data.

For a massive p-form field of mass $m>0$, let $\delta$ denote the formal codifferential and take its free equation to beApplying $\delta$ yields $\delta A=0$, since ${\delta }^2=0$. DefineHere the dual field is the Hodge dual of the field strength itself, rather than a potential for it. For definiteness, use one timelike direction and $\ast \ast {\omega }_r=(-1{)}^{r(D-r)+1}{\omega }_r$. With $\delta {\omega }_r=(-1{)}^{D(r+1)}\ast d\ast {\omega }_r$, write $\eta =(-1{)}^{D(p+2)}$. The massive equation impliesThis gives an invertible differential map at nonzero mass. Also $\delta B=0$ follows from $dF=0$, and substitution yields $\delta dB+m^{2}B=0$. Indeed the two codifferential signs multiply to $(-1{)}^{D(p+q+4)}=(-1{)}^{D(D+3)}=1$. Thus massive p-form duality exchanges the equations of two massive tensors of the displayed ranks.

The polarization counts confirm the distinction: a massive tensor in its rest frame has $\left(\genfrac{}{}{0ex}{}{D-1}{p}\right)$ states, equal to $\left(\genfrac{}{}{0ex}{}{D-1}{D-p-1}\right)$, whereas a massless gauge field has $\left(\genfrac{}{}{0ex}{}{D-2}{p}\right)$, equal to $\left(\genfrac{}{}{0ex}{}{D-2}{D-p-2}\right)$. The ordinary massive dual applies for $0\le p\le D-1$. A massless $(D-1)$-form has no local wave but may carry a constant top-form flux; its formal negative dual rank is a warning that it is not described by an ordinary dual potential. A degree-$D$ potential likewise has no local propagating field strength. These nondynamical endpoint cases do not change the duality of propagating fields.

For a null momentum, use light cone gauge to remove potential components along one null direction; the free field equations remove the remaining longitudinal components. The physical polarizations of a massless p-form gauge field are therefore antisymmetric tensors on the $D-2$ transverse directions, transforming under the rotational little group $SO(D-2)$. Choosing $p$ distinct transverse indices givesThis already takes account of the reducibility of the higher-form gauge parameter; naïvely subtracting just one gauge-parameter component count would not give the right answer. No self-duality constraint is being imposed.

For dimensional reduction to four dimensions, retain the zero modes on a flat $(D-4)$-torus, without fluxes or projections. Write $d=D-4$ and split tensor indices into spacetime indices $\mu$ and internal indices $i$. A component with $r$ spacetime indices is a four-dimensional $r$-form, with $p-r$ internal indices and hence multiplicity $\left(\genfrac{}{}{0ex}{}{d}{p-r}\right)$. The complete field decomposition isUse the convention that a binomial coefficient vanishes when its lower entry lies outside its range. In four dimensions the scalar, vector and two-form have respectively $1,2,1$ local polarizations, while the three- and four-form potentials have none. Hence toroidal reduction of a p-form gauge field givesThe middle identity follows by comparing the coefficient of $t^p$ in $(1+t{)}^2(1+t{)}^d=(1+t{)}^{d+2}$. ThereforeA four-dimensional two-form may instead be counted as one scalar by two-form scalar duality, with the same result. Higher-form zero modes can describe nondynamical flux data, which are distinct from these propagating degrees of freedom. Nonzero Kaluza-Klein modes form massive lower-dimensional multiplets with longitudinal components supplied by the associated gauge fields; the displayed matching concerns the massless zero-mode sector.

A massless graviton in $D$ dimensions has a symmetric traceless transverse polarization tensor. Its state count isThus the metric of eleven-dimensional supergravity has $44$ polarizations, while its three-form has $\left(\genfrac{}{}{0ex}{}{9}{3}\right)=84$. The total bosonic count is $128$.

For circle reduction to ten dimensions, split $M=(\mu ,10)$. The metric supplies $g_{\mu \nu }$, a Kaluza-Klein vector from $G_{\mu ,10}$, and a scalar from $G_{10,10}$, conventionally the dilaton. The three-form supplies $C_{\mu \nu \rho }=A_{\mu \nu \rho }$ and $B_{\mu \nu }=A_{\mu \nu ,10}$. Their ten-dimensional counts areThis is the bosonic content of type IIA supergravity: $g$, $B_2$ and the dilaton together with $C_1$ and $C_3$. Consequently

For flat seven-torus reduction directly to four dimensions, split $M=(\mu ,i)$, $i=1,\dots ,7$. The metric yields one four-dimensional metric, seven vectors $G_{\mu i}$ and $7\cdot 8/2=28$ scalars $G_{ij}$. Their polarizations are $2+7\cdot 2+28=44$. The three-form givesThe three-form count is $0+7+42+35=84$, as required by toroidal reduction of a p-form gauge field. Applying two-form scalar duality to the seven two-forms, the propagating four-dimensional spectrum isThis is the bosonic content of four-dimensional N=8 supergravity. The remaining four-dimensional three-form has no local polarization and can be omitted from the propagating spectrum, though a flux choice can matter in other backgrounds. The toroidal reduction of eleven-dimensional supergravity here retains all zero modes; fixing moduli, introducing fluxes or using an orbifold projection would change the spectrum and would not be this unprojected counting problem.