Type II supercharge count on periodic flat T6

At [A2], $R_{MN}=0$, $R=0$, $d\mathrm{\Phi }=0$, and every NS/RR field strength is zero. In [F6] the Einstein variation is proportional to the Einstein tensor plus derivatives of $e^{-2\mathrm{\Phi }}$, both zero; the dilaton variation is proportional to $R$, $d\mathrm{\Phi }$, and $H_3$, all zero; the two-form variation is proportional to $H_3$. The additional IIA and IIB kinetic terms in [F7] and [F8] are quadratic in field strengths, so their first variations vanish at zero fields. Each displayed Chern–Simons term and its first variation also vanishes because every potential and field strength is zero. Thus all bulk Euler–Lagrange equations in [F10] and [F11] vanish under compactly supported variations. The potential Bianchi identities hold by [A2], and the IIB self-duality equation [F9] is $\ast 0=0$. Therefore the background is an on-shell source-free bosonic solution in both models, conditionally on the current supplier interfaces.

Write $w_j=u_j\wedge$ and $q_j={\iota }_{{\bar{u}}_j}$ on $S_6$. Their defining exterior formulas give $w_j{w}_k+w_k{w}_j=q_j{q}_k+q_k{q}_j=0$ and $q_j{w}_k+w_k{q}_j={\delta }_{jk}I$. Thus ${\rho }_{2j-1}=i(w_j-q_j)$ and ${\rho }_{2j}=-(w_j+q_j)$ have square $I$ and distinct generators anticommute. If $n_j=w_j{q}_j$, then $n_j$ is the occupation projection for $u_j$ and ${\rho }_{2j-1}{\rho }_{2j}=i(I-2n_j)$. Hence ${\mathrm{\Gamma }}_6=i{\prod }_{j=1}^{3}i(I-2n_j)=(-1{)}^{\text{exterior degree}}$. Its even and odd eigenspaces each have complex dimension $4$, since their exterior bases have respectively $1+3$ and $3+1$ elements. The four-dimensional operator ${\mathrm{\Gamma }}_4$ squares to $I$, and an invertible ${\gamma }_{\mu }$ exchanges its two eigenspaces, each therefore of dimension $2$. The product generators in [A3] satisfy $\{{\widehat{\mathrm{\Gamma }}}_A,{\widehat{\mathrm{\Gamma }}}_B\}=2{\eta }_{AB}I$: mixed terms vanish because ${\mathrm{\Gamma }}_6$ anticommutes with each ${\rho }_a$. Thus $F$ is a $32$-complex-dimensional representation of the same ordered physical Clifford relations as the pinned module in [F3]. This establishes a candidate representation, not yet its identification with the physical modules.

Here is a finite construction that applies to any $32$-complex-dimensional representation $M$ of those ordered ten-generator relations. Put $D_0=i{\mathrm{\Gamma }}_0$ and $D_a={\mathrm{\Gamma }}_a$ for $1\le a\le 9$, so $\{D_r,D_s\}=2{\delta }_{rs}I$. For $1\le j\le 5$ define $a_j=(D_{2j-2}+i{D}_{2j-1})\mathrm{/}2$ and $b_j=(D_{2j-2}-i{D}_{2j-1})\mathrm{/}2$. Expansion gives $\{a_j,a_k\}=\{b_j,b_k\}=0$ and $\{a_j,b_k\}={\delta }_{jk}I$. In particular $N_j=b_j{a}_j$ is idempotent, and the five $N_j$ commute. For $ϵ\in \{0,1{\}}^5$ set $P_{ϵ}={\prod }_j{N}_j^{{ϵ}_j}(I-N_j{)}^{1-{ϵ}_j}$. These are mutually annihilating idempotents with sum $I$, so $M={⨁}_{ϵ}{M}_{ϵ}$, where $M_{ϵ}=P_{ϵ}M$. No Hermitian-adjoint convention or Clifford-classification theorem is used.

If ${ϵ}_j=1$, then $a_j:M_{ϵ}\to M_{ϵ-e_j}$ has inverse $b_j$; indeed $b_j{a}_j=N_j=I$ on the first space and $a_j{b}_j=I-N_j=I$ on the second. Both operators commute with each $N_k$ for $k\mathrm{\ne }j$. Thus all $32$ joint spaces have the same dimension, and their direct sum has dimension $32$, so each has dimension $1$. Choose $0\mathrm{\ne }v\in M_0$ and set $v_{ϵ}=b_1^{{ϵ}_1}\cdots b_5^{{ϵ}_5}v$. These vectors are nonzero and form a basis, because they lie in the distinct one-dimensional joint spaces. On this basis $a_j$ kills a vector with ${ϵ}_j=0$ and otherwise lowers that bit with sign $(-1{)}^{{\sum }_{k<j}{ϵ}_k}$; $b_j$ raises a zero bit with the same sign and kills a one bit. These formulas follow by commuting $a_j$ or $b_j$ past the earlier $b_k$ and using $a_{j}v=0$ and $a_j{b}_{j}v=v$. Matching such bases for two representations gives an explicit complex-linear intertwiner of every $a_j,b_j$, hence of every $D_r$ and ${\mathrm{\Gamma }}_A$. An operator commuting with all ${\mathrm{\Gamma }}_A$ preserves $M_0$ and is scalar there; commuting with the $b_j$ makes it the same scalar on the entire basis. This also proves the scalar-commutant assertion needed below, without invoking Schur's lemma as an additional input.

Apply step 2.1 first to the pinned complex module $S_0=({\mathbb{C}}^2{)}^{\otimes 5}$ in [F3] and to $F$ in step 1.2, obtaining an intertwiner $T:S_0\to F$. The pinned involution $J_{0}v=b_1^{(0)}{b}_2^{(0)}{b}_3^{(0)}{b}_4^{(0)}\bar{v}$ is the actual matrix involution of [F3]; the superscript distinguishes those pinned Pauli matrices from the creation operators in step 1.3. The supplier proves $J_0^2=I$ and $J_0{\mathrm{\Gamma }}_A={\mathrm{\Gamma }}_A{J}_0$. Put $R=T{J}_0{T}^{-1}$ on $F$. For either real rank-$32$ Clifford module supplied by [F4] or the full bundle in [F5], complexify its real representation and denote its conjugation by $C$. Its complex dimension is $32$, and $C$ is an antilinear involution commuting with every real generator. Step 2.1 gives a complex Clifford intertwiner $U$ from that complexification to $S_0$. Then $C^{\mathrm{\prime }}:=UC{U}^{-1}$ and $J_0$ commute with every generator, so the complex-linear map $C^{\mathrm{\prime }}{J}_0$ is scalar by step 2.1, say $C^{\mathrm{\prime }}{J}_0=\lambda I$. Thus $C^{\mathrm{\prime }}=\lambda J_0$, and $C^{\mathrm{\prime }2}=I$ gives $\lambda \bar{\lambda }=1$. Choose a unit complex number $z$ with $z^2={\lambda }^{-1}$. Replacing $U$ by $zU$ gives $(zU)C(zU{)}^{-1}=(z\mathrm{/}\bar{z})\lambda J_0=J_0$. Consequently $TU$ after this replacement intertwines the actual Clifford representations and their Majorana real structures. These are constant finite-dimensional maps. They intertwine every even Clifford product, hence the external and internal Spin actions, and intertwine the exact ordered chirality products. There is no assumed representation compatibility and no identification based on dimension equality alone.

In the product orientation fixed above, the ordered ten-dimensional chirality is exactly ${\widehat{\mathrm{\Gamma }}}_0\cdots {\widehat{\mathrm{\Gamma }}}_9=({\gamma }_0{\gamma }_1{\gamma }_2{\gamma }_3)\otimes ({\rho }_1\cdots {\rho }_6)=-{\mathrm{\Gamma }}_4\otimes {\mathrm{\Gamma }}_6\mathrm{.}$ Thus the positive eigenspace is $({\mathrm{\Delta }}_4^{+}\otimes S_6^{-})\oplus ({\mathrm{\Delta }}_4^{-}\otimes S_6^{+})$, and the negative one is $({\mathrm{\Delta }}_4^{+}\otimes S_6^{+})\oplus ({\mathrm{\Delta }}_4^{-}\otimes S_6^{-})$. Each summand has complex dimension $8$. The external Spin action on either space is the action on ${\mathrm{\Delta }}_4$ with internal multiplicity $4$, since ${\widehat{\mathrm{\Gamma }}}_{\mu }{\widehat{\mathrm{\Gamma }}}_{\nu }={\gamma }_{\mu }{\gamma }_{\nu }\otimes I$. The induced internal action agrees with the six-dimensional Clifford convention: its positive-square rotation generator is ${\rho }_a{\rho }_b\mathrm{/}2=-c(e_a)c(e_b)\mathrm{/}2$, the negative-square generator with the same orthogonal infinitesimal action. The Majorana involution $R$ in step 3.1 commutes with this external Spin action and with the real chirality product. Its restrictions therefore give the actual sixteen-real-dimensional Majorana–Weyl modules under the proven intertwiners. Reversing the total orientation interchanges the two ten-dimensional labels and does not change either multiplicity.

The product metric is the exact rectangular geometry in [F1]. In the global product orthonormal frame its coefficients are constant and the Levi-Civita connection coefficients vanish. The periodic lift [A1], using [F12] and the IIB bundle data in [F5], gives trivial associated spinor bundles with zero spin connections. The constant intertwiners of step 3.1 commute with the identity deck actions of this lift, so they define global parallel bundle identifications. Pulling any Killing spinor to the connected universal cover ${\mathbb{R}}^4\times {\mathbb{R}}^6$, the equations ${\mathrm{\partial }}_Mϵ=0$ make it constant; conversely each constant fiber spinor descends by periodicity. Thus no additional global solutions or holonomy constraints enter the count. Substitution of [A2] into the actual IIA system [F4] and IIB system [F5] reduces both gravitino equations to ${\mathrm{\partial }}_Mϵ=0$ and both dilatino equations to $0=0$. Every constant spinor is periodic, and there are no projections, so every fiber spinor gives a global Killing-spinor parameter.

Write $H={\mathrm{Spin}}^{+}(1,3)$ and let $W=S_4^{+}$ be its complex two-dimensional Weyl module. In Weyl coordinates $H=S{L}_2(\mathbb{C})$ acts on $W$ by $g$ and on the opposite Weyl module by $(g^{†}{)}^{-1}$. With $\epsilon =\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, the identity $\epsilon \bar{g}{\epsilon }^{-1}=(g^{†}{)}^{-1}$ identifies the latter module with $\bar{W}$. These coordinates follow equivalently by the action $X\mapsto gX{g}^{†}$ on Hermitian two-by-two matrices, whose determinant is the Lorentz quadratic form, and its induced Weyl actions. There is no nonzero $H$-equivariant antilinear map $W\to W$. Indeed, such a map is $z\mapsto C\bar{z}$, and equivariance for $g=\mathrm{diag}(t,t^{-1})$ gives $C\mathrm{diag}(\bar{t},{\bar{t}}^{-1})=\mathrm{diag}(t,t^{-1})C$. Taking $t=2$ kills the off-diagonal entries of $C$, and taking $t=i$ kills its diagonal entries. The same argument applies to $\bar{W}$ and, entry by entry, to arbitrary finite multiplicities. Consequently every equivariant antilinear involution on $W^{\oplus m}\oplus {\bar{W}}^{\oplus m}$ exchanges the two summands. Let $R$ be any such involution and put $U=W^{\oplus m}$. The map $U_{\mathbb{R}}⟶\mathrm{Fix}(R),u⟼u+Ru$ is real-linear and $H$-equivariant, with inverse the projection to $U$: if $u+v$ is fixed, then $v=Ru$, and conversely $R(u+Ru)=u+Ru$. Choose these Weyl coordinates also on the supplied complex Dirac module in [F2], fixing a complex equivariant identification of its positive-Weyl summand with $W$. This choice is legitimate for any supplied irreducible Dirac module: the Pauli block matrices ${\gamma }_0=\left(\begin{array}{cc}0& I\\ -I& 0\end{array}\right)$ and ${\gamma }_j=\left(\begin{array}{cc}0& {\sigma }_j\\ {\sigma }_j& 0\end{array}\right)$ obey the stated Clifford relations. To verify the generated algebra, write ${\gamma }_I$ for the ordered monomial indexed by $I\subseteq \{0,1,2,3\}$. Every nonempty monomial has trace zero: if $\mathrm{\mid }I\mathrm{\mid }$ is even, conjugation by a generator indexed in $I$ negates it; if $\mathrm{\mid }I\mathrm{\mid }$ is odd, conjugation by one outside $I$ negates it. For $I\mathrm{\ne }J$, ${\gamma }_I^{-1}{\gamma }_J$ is a nonzero scalar multiple of the monomial indexed by $I\mathrm{△}J$, so $\mathrm{tr}({\gamma }_I^{-1}{\gamma }_J)=0$, whereas $\mathrm{tr}({\gamma }_I^{-1}{\gamma }_I)=4$. The sixteen monomials are therefore linearly independent. The Clifford relations reduce every word to their span, so the complex Clifford algebra is $M_4(\mathbb{C})$ in this representation. In any nonzero irreducible four-dimensional module $M$, the matrix units decompose $M={⨁}_{i=1}^4{E}_{ii}M$; the maps $E_{i1}$ and $E_{1i}$ identify each summand with $E_{11}M$, so each has dimension one. Choosing $0\mathrm{\ne }v\in E_{11}M$ gives the defining-module basis $E_{i1}v$. This transports the even Spin action and the orientation-fixed Weyl summands. Here ${\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$, ${\sigma }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$, and ${\sigma }_3=\mathrm{diag}(1,-1)$. Apply the same argument to the independently supplied involution $J$ in [F2] in these coordinates. The restriction ${\pi }_{+}:S_4^{\mathrm{M}}\to W_{\mathbb{R}}$ is an equivariant real isomorphism, inverse $z\mapsto z+Jz$. Choose a complex basis $e_1,\dots ,e_m$ of the internal multiplicity space, on which $H$ acts trivially. An explicit identification with the specified module, without equating $R$ with $J$, is $\mathrm{Fix}(R)⟶(S_4^{\mathrm{M}}{)}^{\oplus m},u+Ru⟼(z_1+J{z}_1,\dots ,z_m+J{z}_m),u={\sum }_{j=1}^m{z}_j\otimes e_j\mathrm{.}$ Its inverse sends the tuple to $u+Ru$ with $z_j={\pi }_{+}(s_j)$. This construction allows $R$ to mix internal multiplicities. On either ten-dimensional chirality in step 4.1, take $R$ to be the Majorana involution constructed from the actual modules in step 3.1 and $m=4$. It is $H$-equivariant by the Majorana convention, so one Majorana–Weyl parameter is equivariantly isomorphic to four copies of the particular $S_4^{\mathrm{M}}$ in [F2]. On an $R$-invariant pair with one internal invariant line per chirality, the identical construction has $m=1$.

Type IIA has two opposite-chirality ten-dimensional Majorana–Weyl parameters and Type IIB has two same-chirality parameters; in either case each parameter contributes four four-dimensional Majorana parameters by steps 3.1, 4.1, and 5.1. Thus each reduction has $4+4=8$ independent four-dimensional Majorana parameters, and [F2] identifies the result as $N=8$. This proves the stated supersymmetry count, conditional on the two-derivative KSE interfaces.