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Type II on a flat six-torus and enhanced four-dimensional supersymmetry

Physical scope

The N=8 result is for source-free massless Type IIA or Type IIB supergravity at two-derivative order on Minkowski space times a rectangular flat T6 with periodic spin structure and no projection. The comparison is conditional on a separate compact Ricci-flat six-manifold with exact SU(3) spin holonomy. The Kaluza–Klein formula is conditional on the stated free component wave equation, which includes the minimally coupled scalar sector; it is not the full massive string spectrum.

Example

Let $M_{4} = R^{1, 3}$ with signature $(-, +, +, +)$ , and let $T_{R}^{6} = \prod_{i = 1}^{6} R / (2 π R_{i} Z), R_{i} \in (0, \infty),$ with angular coordinates $θ^{i} \sim θ^{i} + 2 π$ and metric $\sum_{i} R_{i}^{2} (d θ^{i})^{2}$ . On the product use the flat ten-dimensional metric, constant dilaton $Φ_{0}$ , vanishing NS and RR potentials and field strengths, and no localized sources, fluxes, Wilson lines, orbifold action, orientifold projection, or supersymmetry-breaking deformation. Choose the periodic internal spin structure: spinor components in the global orthonormal frame are periodic under every $θ^{i} \mapsto θ^{i} + 2 π$ . Work in the massless source-free Type IIA or Type IIB supergravity model with the two-derivative Hamilton string-frame conventions.

For either theory, the Killing-spinor equations reduce to $\nabla_{M} ϵ = 0$ because all fluxes and $d Φ$ vanish. The internal spin connection is flat, and the periodic spin structure admits all constant internal spinors. Thus the unprojected four-dimensional massless theory has 32 independent real supersymmetry parameters, which is $N = 8$ under the stated four-dimensional counting convention. This count is a two-derivative supergravity deduction conditional on the named physical and Killing-spinor inputs; it is not a statement about moduli stabilization.

For comparison, let $Y$ be a compact connected complex threefold with a Ricci-flat Riemannian metric whose Levi-Civita holonomy is exactly the standard $S U (3) \subset S O (6)$ and whose chosen spin lift has exactly the corresponding $S U (3) \subset Spin (6)$ holonomy. This is the full-holonomy Calabi–Yau regime in this example. On $M_{4} \times Y$ , use the same source-free massless Type IIA or Type IIB two-derivative model, constant dilaton, zero NS and RR fields, and no localized sources or projections. The four-dimensional supersymmetry count is then $N = 2$ . The comparison is conditional on exact $S U (3)$ holonomy; flat tori and lower-holonomy limits are excluded from it.

For any free field component $Ψ$ in a parallel product frame that satisfies $(□_{4} + Δ_{T_{R}^{6}} - m_{0}^{2}) Ψ = 0$ , a Fourier mode $e^{i \sum_{i} n_{i} θ^{i}}$ has $m_{n}^{2} = m_{0}^{2} + \sum_{i = 1}^{6} \frac{n_{i}^{2}}{R_{i}^{2}}, n \in Z^{6} .$ For the minimally coupled scalar sector this is the equation and convention defined on the A page; massless parent components have $m_{0} = 0$ . The formula is a free two-derivative Kaluza–Klein contribution and does not give the massive string oscillator or winding spectrum.

Example references

Rectangular flat-torus string compactification data

Four-dimensional supersymmetry counting convention

SU(3)-fixed chiral spinor lines in six dimensions

Type II supercharge count on periodic flat T6

Supergravity Conventions And Normalizations

Type Iia Two Derivative Supergravity Action

Type Iia Bosonic Field Equations

Type Iia Supergravity Killing Spinor Equations

Type IIB Two Derivative Supergravity Pseudoaction

Type IIB Field Equations with Five Form Self Duality

Type IIB Supergravity Killing Spinor Equations

Lorentzian Type IIB Spinor Bundle Data

Given Spin and Spin-c structures and their spinor bundles

Facts & Assumptions

Given: The flat product geometry and periodic spin structure; the separate exact- $S U (3)$ -holonomy product background; the zero-field configurations; and the exact supplier interfaces below. Any supplier with a current escalated Step-1 record is used only at its inspected conditional statement.

[F1]

Rectangular flat-torus string compactification data: the internal metric is $\sum_{i} R_{i}^{2} (d θ^{i})^{2}$ , and in the minimally coupled scalar sector the Kaluza–Klein masses are $m_{n}^{2} = m_{0}^{2} + \sum_{i} n_{i}^{2} / R_{i}^{2}$ .

[F2]

Four-dimensional supersymmetry counting convention: $N$ counts copies of the specified real ${Spin}^{+} (1, 3)$ -module $S_{4}^{M}$ ; a parameter space that is a direct sum of $N$ such modules has real dimension $4 N$ .

[F3]

Type II supercharge count on periodic flat T6: under the periodic spin structure and zero-field source-free background, both unprojected Type-II reductions have 32 real global supersymmetry parameters and $N = 8$ . Its proof is conditional on the current IIA/IIB equations and Killing-spinor interfaces; its Step-1 record is escalated on the IIB suppliers.

[F4]

SU(3)-fixed chiral spinor lines in six dimensions: with exact lifted $S U (3)$ holonomy, each internal chiral spinor bundle has one complex line of global parallel sections, and its holonomy-fixed ten-dimensional Majorana–Weyl subspace is a real form of one opposite-Weyl pair. The Clifford intertwiners in [F3], Proof 3.1, identify the actual IIA/IIB modules with these product representations; its ordered chirality calculation allows either internal-parity pairing. The equivariant isomorphism of the invariant real form with the particular $S_{4}^{M}$ in [F2] is constructed in [F3], Proof 5.1, not inferred from dimension alone.

[F5]

Supergravity Conventions And Normalizations: Type IIA has two opposite-chirality ten-dimensional Majorana–Weyl parameters and Type IIB has two same-chirality such parameters; each parameter has 16 real components.

[F6]

Type Iia Bosonic Field Equations: the source-free massless IIA equations follow from the adopted two-derivative action, with the displayed potential Bianchi identities. On a Ricci-flat product with constant dilaton and all field strengths zero, every displayed equation reduces to $0 = 0$ . Its current Step-1 record is ready.

[F7]

Type IIB Field Equations with Five Form Self Duality: the current IIB theorem gives the string-frame metric, dilaton, form, Bianchi, and separate five-form equations. On a Ricci-flat product with constant dilaton and every field strength zero, all displayed terms vanish, including $* {\tilde{G}}_{5} = {\tilde{G}}_{5}$ . Its current Step-1 record is escalated; the consuming comparison remains conditional on this exact interface.

[F8]

Type Iia Supergravity Killing Spinor Equations: on a source-free IIA bosonic solution, the gravitino and dilatino systems define the supersymmetry condition. When all fluxes and $d Φ$ vanish, they reduce to $\nabla^{S} ϵ = 0$ and $0 = 0$ . Its current Step-1 record is ready.

[F9]

Type IIB Supergravity Killing Spinor Equations: the current IIB doublet equations are imposed on a source-free bosonic solution with the separate five-form constraint. When all fluxes and $d Φ$ vanish, they reduce to $\nabla^{S} ϵ = 0$ and $0 = 0$ . Its current Step-1 record is escalated, so the comparison is conditional.

[F10]

Lorentzian Type IIB Spinor Bundle Data: a supplied ten-dimensional Spin lift defines the positive-chirality Majorana–Weyl spinor bundle $S_{+}$ , and the IIB parameter is a smooth doublet in $S_{+} \oplus S_{+}$ . The spinor-bundle interface supplies the full Clifford bundle and its chiral subbundles; these are the supplied data used here.

[F11]

Given Spin and Spin-c structures and their spinor bundles: a spin structure is a principal Spin lift of the oriented frame bundle, and the spinor bundle is associated to the specified Clifford representation.

[F12]

Type Iia Two Derivative Supergravity Action: the adopted IIA model is the massless source-free leading two-derivative string-frame action with $F_{0} = 0$ .

[F13]

Type IIB Two Derivative Supergravity Pseudoaction: the adopted IIB model is the massless source-free leading two-derivative string-frame pseudoaction; the five-form constraint is imposed separately after variation. Its current Step-1 record is escalated.

[A1]

The flat-torus internal spin structure is periodic in all six angular coordinates. All potentials and field strengths vanish, and the dilaton is constant.

[A2]

No quotient, orientifold, Wilson line, or other projection acts on the flat-torus spinors or scalar modes.

[A3]

The displayed mode formula assumes a free component in a parallel product frame satisfying $(□_{4} + Δ_{T_{R}^{6}} - m_{0}^{2}) Ψ = 0$ , with $Δ = div grad$ nonpositive. For the minimally coupled scalar sector this is the equation in [F1]. No flux, curvature, higher-derivative, loop, string oscillator, or winding contribution is included.

[A4]

For the comparison, $Y$ is a compact connected complex threefold with a Ricci-flat Riemannian metric whose Levi-Civita holonomy and chosen spin lift are exactly the standard $S U (3)$ actions specified in the Example.

[A5]

On $M_{4} \times Y$ all NS and RR potentials and field strengths vanish and $Φ = Φ_{0}$ is constant. The external metric is flat Minkowski, and there are no localized sources, fluxes, or projections.

Proof

Given: [F1]–[F13], [A1]–[A5], the positive finite radii $R_{i}$ , and the two source-free product backgrounds in the Example. The current IIB equation and Killing-spinor inputs remain conditional.

1.1A1A2F2F3F12F13given

The geometry, periodic spin structure, zero-field background, and absence of projections in [A1]–[A2] satisfy the hypotheses of [F3]. Thus the flat-torus reductions have 32 real global supersymmetry parameters, and [F2] records this as $N = 8$ . This is a conditional deduction within the adopted two-derivative models [F12]–[F13], not a claim about a full string vacuum.

1.2A4A5F6F7given

On $M_{4} \times Y$ , the product Ricci tensor and scalar curvature vanish because the external metric is flat and $Ric (g_{Y}) = 0$ . Also $d Φ = 0$ and every NS and RR field strength is zero by [A4]–[A5]. Substitution in the current IIA equations [F6] and the current IIB equations [F7] makes every metric, dilaton, form, Bianchi, and five-form equation vanish. Therefore this is an on-shell bosonic background in both models, conditional on the current supplier interfaces.

1.3F1A3algebra

For the mode in [A3], the rectangular metric gives $- Δ_{T_{R}^{6}} e^{i \sum_{i} n_{i} θ^{i}} = (\sum_{i} n_{i}^{2} / R_{i}^{2}) e^{i \sum_{i} n_{i} θ^{i}}$ by direct differentiation. Substitution into $(□_{4} + Δ_{T_{R}^{6}} - m_{0}^{2}) Ψ = 0$ gives the external equation $(□_{4} - m_{0}^{2} - \sum_{i} n_{i}^{2} / R_{i}^{2}) ψ_{n} = 0$ . Thus the mass-squared shift is exactly $\sum_{i} n_{i}^{2} / R_{i}^{2}$ for every component satisfying the declared free wave equation.

2.1A4A5F4F8F9F10F11step 1.2algebra

Substituting [A5] into the IIA systems [F8] and the IIB doublet systems [F9]–[F10] removes every flux and dilaton term, leaving the parallel-spinor equation $\nabla^{S} ϵ = 0$ and a zero dilatino equation. The product spin connection splits into the flat external connection and the internal spin connection; the global internal parallel-spinor spaces are exactly those covered by [F4]. The no-projection hypothesis in [A5] retains all of them.

3.1F2F3F4F5step 2.1constructalgebra∎

By the explicit Clifford identification in [F3] and the two internal-parity cases of [F4], the invariant complex space for either actual ten-dimensional chirality is $U \oplus V$ , where $U = S_{4}^{+} \otimes L_{+}$ and $V = S_{4}^{-} \otimes L_{-}$ for one-dimensional internal invariant lines, or with the two internal lines interchanged according to the ordered chirality sign. This interchange does not change the following real-module construction. The ten-dimensional Majorana involution $R$ commutes with internal holonomy, so preserves this fixed space, and is externally Spin-equivariant. By the explicit no-antilinear-self-map calculation in [F3], Proof 5.1, it exchanges $U$ and $V$ . Choose a nonzero vector $e$ in the internal invariant line defining $U$ . Every real fixed vector is uniquely $z \otimes e + R (z \otimes e)$ . The map to $z + J z \in S_{4}^{M}$ , with inverse obtained by its positive-Weyl projection, is a real Spin-equivariant isomorphism by that same construction. Thus either ten-dimensional chirality contributes exactly one copy of the specified module, not merely four real dimensions. By [F5], each of Type IIA and Type IIB has two such ten-dimensional parameters. Therefore each compactification has $2 \cdot 4 = 8$ independent real four-dimensional supersymmetry parameters, and [F2] identifies this count as $N = 2$ . The result is conditional on the source-free two-derivative equations and Killing-spinor systems, particularly the current escalated IIB inputs.

Remarks

The Type-II $N = 8$ count is supplied by the local A-page theorem [F3]. The exact- $S U (3)$ comparison is derived conditionally from the local mathematical representation lemma [F4] and the actual IIA/IIB field-equation and Killing-spinor interfaces. The IIB field-equation and Killing-spinor inputs currently have escalated Step-1 records, so this example remains escalated until those exact upstream records are resolved. The Graña–Triendl notes and Hamilton independently support the Type-II toroidal inputs. A Guarino thesis source entry was dropped after the author and university-hosted PDF URLs exhausted fetch attempts; the local proof and lecture notes preserve coverage. The KK statement is conditional on the declared free component wave equation, including the minimally coupled scalar sector; no moduli stabilization or exact full-string spectrum is claimed.

Type II on a flat six-torus and enhanced four-dimensional supersymmetry

Example

Example references

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources