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Flat Type IIB Vacuum with Constant Axiodilaton

Physical scope

A local classical source-free vacuum of the adopted two-derivative type IIB model. It is not a compactification, flux vacuum, stability result, or empirical claim.

Example

Let $M = R^{1, 9}$ with coordinates $(x^{0}, \dots, x^{9})$ , metric $g_{s} = η = diag (- 1, 1, \dots, 1)$ , orientation $d x^{0} \land \dots \land d x^{9}$ , and time orientation defined by the future timelike cone containing $\partial_{x^{0}}$ . Take constant finite real fields $Φ = Φ_{0}$ and $C_{0} = C_{00}$ , and set $B_{2} = C_{2} = C_{4} = 0$ . The associated complex axiodilaton in this convention is $τ : = C_{0} + i e^{- Φ}$ , so $τ = C_{00} + i e^{- Φ_{0}}$ is constant. Let $u_{x} : V \to T_{x} M$ be the global frame sending the standard basis to $(\partial_{x^{0}}, \dots, \partial_{x^{9}})$ . Give $P = M \times {S p i n}^{+} (9, 1)$ the right action $(x, s) \cdot t = (x, s t)$ and the covering map $\tilde{λ} (x, s) = u_{x} \circ λ (s)$ ; by the supplied spinor-bundle definition this is the product spin structure. Choose $0 \neq δ = (δ_{1}, δ_{2}) \in Δ_{9, 1}^{+} \oplus Δ_{9, 1}^{+}$ from the supplied Clifford module and define the constant section $ϵ (x) = [(x, 1), δ] \in Γ (S_{+} \oplus S_{+})$ using the right-principal associated-bundle convention of Lorentzian Type IIB Spinor Bundle Data.

Then this is a flat source-free solution of the bosonic equations in Type IIB Field Equations with Five Form Self Duality and is supersymmetric in the sense of Type IIB Supergravity Killing Spinor Equations. It is an exact solution of the stated classical truncation for every finite real $Φ_{0}$ and $C_{00}$ ; no weak-coupling expansion or perturbative stability claim is made.

Example references

Supergravity Conventions And Normalizations

Type IIB Two Derivative Supergravity Pseudoaction

Type IIB Five Form Self Duality Constraint

Type IIB Field Equations with Five Form Self Duality

Type IIB Supergravity Killing Spinor Equations

Lorentzian Type IIB Spinor Bundle Data

Hamilton, §7.1–§7.2

Facts & Assumptions

Given: The smooth fields, metric, orientation, time orientation, and spin structure in the Example; the field equations and spinor condition referenced above.

[F1]

Type IIB Two Derivative Supergravity Pseudoaction: $H_{3} = d B_{2}$ , $G_{1} = d C_{0}$ , $G_{3} = d C_{2}$ , $G_{5} = d C_{4}$ , ${\tilde{G}}_{3} = G_{3} - C_{0} H_{3}$ , and ${\tilde{G}}_{5} = G_{5} - \frac{1}{2} H_{3} \land C_{2} + \frac{1}{2} G_{3} \land B_{2}$ .

[F2]

Type IIB Field Equations with Five Form Self Duality: the required system consists of the Einstein, dilaton, RR, NS, Bianchi, and self-duality equations stated there.

[F3]

Lorentzian Type IIB Spinor Bundle Data: the spinor bundle is the associated bundle $S = P \times_{ρ} Δ$ ; a section is a smooth map to $S$ over the identity on $M$ . In a local spin frame $s$ , the lifted connection has component expression $\nabla_{X}^{S} [s, ψ] = [s, X (ψ) + d ρ (A_{s} (X)) ψ]$ .

[F4]

Type IIB Supergravity Killing Spinor Equations: supersymmetry means that a nonzero smooth spinor doublet solves the gravitino and dilatino equations with the displayed string-frame flux operators.

[A1]

Supergravity Conventions And Normalizations: use Lorentzian signature $(-, +, \dots, +)$ , natural units, real dimensionless dilaton and axion, $2 κ_{0}^{2} = (2 π)^{7} α^{' 4}$ , and the stated string-frame curvature convention.

Verification

1.1A1F1algebra

Under [A1], the constant scalars $Φ_{0}, C_{00}$ are admissible real dimensionless fields. The potentials give $H_{3} = G_{1} = G_{3} = G_{5} = 0$ by [F1], so both tilded strengths vanish as well. The metric coefficients are constant; every Christoffel coefficient and curvature component is zero, hence $R_{M N} = R = 0$ . The constant dilaton has $d Φ = 0$ , $\nabla d Φ = 0$ , and $□ Φ = 0$ .

2.1F1F2step 1.1algebra

In [F2], every term in the Einstein, dilaton, RR, and NS equations is therefore zero; every Bianchi identity has zero left and right sides; and $* {\tilde{G}}_{5} = 0 = {\tilde{G}}_{5}$ . Thus the bosonic configuration solves the complete source-free system. This conclusion concerns the stated classical truncation only.

3.1F1F3F4step 1.1algebra∎

By [F3], the chosen constant doublet defines a smooth section of the associated bundle. In the global frame $u_{x}$ , the metric coefficients are constant, so every Christoffel coefficient vanishes and the lifted spin-connection form $A_{s}$ is zero; the connection formula in [F3] therefore gives $\nabla^{S} ϵ = 0$ . By [F1], every flux-dependent term in both operators of [F4] vanishes, and $d Φ = 0$ ; consequently both Killing-spinor equations reduce to $0 = 0$ . The chosen nonzero doublet proves supersymmetry of this example.

Flat Type IIB Vacuum with Constant Axiodilaton

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources