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Type IIB Field Equations with Five Form Self Duality

Physical scope

Conditional local source-free classical field equations of the selected massless two-derivative type IIB bosonic pseudoaction. Potentials and variations are smooth on a contractible patch and variations are compactly supported. No brane currents, loop or higher-derivative corrections, global flux quantization, or empirical claim is included.

Statement

Let $U \subseteq R^{10}$ be a contractible oriented coordinate patch, and use the field strengths, string-frame metric, norms, and curvature convention of Type IIB Two Derivative Supergravity Pseudoaction and Supergravity Conventions And Normalizations. A compactly supported independent variation means a jointly smooth family $(g_{s} (t), Φ (t), B_{2} (t), C_{0} (t), C_{2} (t), C_{4} (t))$ for $t$ near $0$ , equal to its value at $0$ outside one common compact set $K \subset U$ , with $g_{s} (t)$ Lorentzian of inertia $(9, 1, 0)$ . Each field is varied separately while the other five are held fixed; the self-duality constraint is imposed only on the stationary field configuration at $t = 0$ , not along the unconstrained variation. Stationarity means that each first-variation density, which is compactly supported in $K$ , has zero coordinate integral. Every smooth compactly supported tangent variation is admissible by taking a linear path for the scalar, potential, or inverse metric and restricting $t$ to a sufficiently small interval. A configuration is stationary precisely when, after the separate constraint Type IIB Five Form Self Duality Constraint, the following source-free equations hold:

$\begin{aligned} R_{M N} - \frac{1}{2} g_{M N} R = & - 2 \nabla_{M} \nabla_{N} Φ + 2 (□ Φ - ∣ d Φ ∣^{2}) g_{M N} \\ + \frac{1}{2} (⟨ i_{M} H_{3}, i_{N} H_{3} ⟩ - \frac{1}{2} g_{M N} ∣ H_{3} ∣^{2}) \\ + \frac{1}{2} e^{2 Φ} (G_{1 M} G_{1 N} - \frac{1}{2} g_{M N} ∣ G_{1} ∣^{2}) \\ + \frac{1}{2} e^{2 Φ} (⟨ i_{M} {\tilde{G}}_{3}, i_{N} {\tilde{G}}_{3} ⟩ - \frac{1}{2} g_{M N} ∣ {\tilde{G}}_{3} ∣^{2}) + \frac{1}{4} e^{2 Φ} ⟨ i_{M} {\tilde{G}}_{5}, i_{N} {\tilde{G}}_{5} ⟩, \\ □ Φ = & 2 ∣ d Φ ∣^{2} - \frac{1}{2} ∣ H_{3} ∣^{2} + e^{2 Φ} ∣ G_{1} ∣^{2} + \frac{1}{2} e^{2 Φ} ∣ {\tilde{G}}_{3} ∣^{2}, \\ d * G_{1} = & - H_{3} \land * {\tilde{G}}_{3}, d * {\tilde{G}}_{5} = H_{3} \land {\tilde{G}}_{3}, \\ d * {\tilde{G}}_{3} = & - H_{3} \land {\tilde{G}}_{5}, \\ d (e^{- 2 Φ} * H_{3}) = & {\tilde{G}}_{3} \land {\tilde{G}}_{5} + G_{1} \land * {\tilde{G}}_{3}, \\ d H_{3} = & 0, d G_{1} = 0, d {\tilde{G}}_{3} = H_{3} \land G_{1}, d {\tilde{G}}_{5} = H_{3} \land {\tilde{G}}_{3}, * {\tilde{G}}_{5} = {\tilde{G}}_{5} . \end{aligned}$

Here $\nabla$ and $□ = g^{M N} \nabla_{M} \nabla_{N}$ use the string-frame Levi-Civita connection; $∣ F_{p} ∣^{2} = F_{M_{1} \dots M_{p}} F^{M_{1} \dots M_{p}} / p!$ ; and $⟨ i_{M} F, i_{N} F ⟩ = F_{M A_{2} \dots A_{p}} F_{N}^{A_{2} \dots A_{p}} / (p - 1)!$ . The five-form norm vanishes under self-duality because its degree is odd; the metric equation retains the separate rank-two term $\frac{1}{4} e^{2 Φ} ⟨ i_{M} {\tilde{G}}_{5}, i_{N} {\tilde{G}}_{5} ⟩$ , which can vanish for particular fields, including ${\tilde{G}}_{5} = 0$ . Bianchi identities and self-duality are separate from pseudoaction variation.

Statement references

Supergravity Conventions And Normalizations

Type IIB Two Derivative Supergravity Pseudoaction

Type IIB Five Form Self Duality Constraint

Type IIB Source Free Bianchi Identities

Lorentzian Hodge Star in Ten Dimensions

Existence and Five-Form Properties of the Lorentzian Hodge Star

Hamilton, §7.1, Theorem 7.2

Sterckx, §3.2.3, equations (3.33)–(3.35)

Facts & Assumptions

Given: Smooth source-free fields and potentials on a contractible oriented Lorentzian ten-dimensional patch, the exact pseudoaction and norm convention in [F1], compactly supported independent variations of all six bosonic fields, and the separate five-form constraint.

[F1]

Type IIB Two Derivative Supergravity Pseudoaction: $S_{N S} = (2 κ_{0}^{2})^{- 1} \int {vol}_{g_{s}} e^{- 2 Φ} (R + 4 ∣ d Φ ∣^{2} - \frac{1}{2} ∣ H_{3} ∣^{2})$ , the RR terms have coefficient $- (4 κ_{0}^{2})^{- 1}$ and bracket $∣ G_{1} ∣^{2} + ∣ {\tilde{G}}_{3} ∣^{2} + \frac{1}{2} ∣ {\tilde{G}}_{5} ∣^{2}$ , and the Chern–Simons term is $- (4 κ_{0}^{2})^{- 1} \int C_{4} \land H_{3} \land G_{3}$ .

[F2]

Type IIB Five Form Self Duality Constraint: impose $* {\tilde{G}}_{5} = {\tilde{G}}_{5}$ only after variation; in ten-dimensional Lorentzian signature $∣ {\tilde{G}}_{5} ∣^{2} = 0$ but the contraction $⟨ i_{M} {\tilde{G}}_{5}, i_{N} {\tilde{G}}_{5} ⟩$ need not be zero.

[F3]

Weighted Einstein Hilbert Variation in Ten Dimensions: for $f = e^{- 2 Φ}$ and $q^{M N} = δ g^{M N}$ , the weighted curvature coefficient is $f G_{M N} + g_{M N} □ f - \nabla_{M} \nabla_{N} f$ .

[F4]

Metric Variation of a Differential Form Kinetic Term: a term ${vol}_{g} A ∣ F_{p} ∣^{2}$ with metric-independent $A, F_{p}$ varies with coefficient $A (⟨ i_{M} F_{p}, i_{N} F_{p} ⟩ - \frac{1}{2} g_{M N} ∣ F_{p} ∣^{2})$ .

[F5]

Localized Ten Dimensional Variations Detect Their Coefficients: if a continuous coordinate coefficient pairs to zero under every compactly supported test variation on an open ten-dimensional chart, it vanishes pointwise.

[F6]

Type IIB Source Free Bianchi Identities: the four Bianchi identities in the Statement follow from $d^{2} = 0$ and the potential definitions; they are not Euler–Lagrange equations.

[F7]

Lorentzian Hodge Star in Ten Dimensions and Existence and Five-Form Properties of the Lorentzian Hodge Star: the star is the metric Lorentzian Hodge operator, and $*^{2} = 1$ on five-forms.

[F8]

The exterior derivative squares to zero: for every smooth form $ω$ , $d^{2} ω = 0$ ; in particular, the potential definition $G_{3} = d C_{2}$ gives $d G_{3} = 0$ .

[F9]

The exterior derivative is a graded derivation and The wedge product is associative and graded commutative: exterior differentiation obeys the graded product rule and degree exchanges have sign $(- 1)^{r s}$ .

[F10]

Supergravity Conventions And Normalizations: use signature $(-, +, \dots, +)$ , the factorial p-form norm, $2 κ_{0}^{2} = (2 π)^{7} α^{' 4}$ , and the declared string-frame metric and curvature conventions; $κ_{0}$ is held fixed in field variations.

[F11]

Compactly Supported Metric Variation of a Weighted Scalar-Curvature Action states and proves the exact chart identity $\int_{U} \partial_{M} J^{M} d^{10} x = 0$ for every smooth compactly supported vector-density $J^{M}$ on an open chart $U$ . Its proof 4.1 extends by zero to a box and uses Fubini and the one-dimensional fundamental theorem of calculus. We use that identity below for the coefficient densities of compactly supported exact top forms, not a global boundaryless Stokes theorem.

Proof

1.1F6

By [F6], $d H_{3} = 0$ , $d G_{1} = 0$ , $d {\tilde{G}}_{3} = H_{3} \land G_{1}$ , and $d {\tilde{G}}_{5} = H_{3} \land {\tilde{G}}_{3}$ .

1.2F2F7F9algebra

The final equation $* {\tilde{G}}_{5} = {\tilde{G}}_{5}$ is the independent postulate [F2], with the Hodge operator fixed by [F7]. Its odd degree and [F9] give ${\tilde{G}}_{5} \land {\tilde{G}}_{5} = 0$ ; the Hodge-pairing definition in [F7] then gives $∣ {\tilde{G}}_{5} ∣^{2} = 0$ . This does not imply the rank-two tensor $⟨ i_{M} {\tilde{G}}_{5}, i_{N} {\tilde{G}}_{5} ⟩$ vanishes.

1.3F1F3F4F10algebra

Let $q^{M N} = δ g^{M N}$ . Apply [F3] to the $e^{- 2 Φ} R$ term and [F4] to $4 ∣ d Φ ∣^{2}$ , $- ∣ H_{3} ∣^{2} / 2$ , and each RR kinetic term, with the norm and string-frame curvature convention fixed by [F10]. Expanding $\nabla_{M} \nabla_{N} (e^{- 2 Φ}) = e^{- 2 Φ} (4 \partial_{M} Φ \partial_{N} Φ - 2 \nabla_{M} \nabla_{N} Φ)$ and $□ (e^{- 2 Φ}) = e^{- 2 Φ} (4 ∣ d Φ ∣^{2} - 2 □ Φ)$ , the NS metric coefficient, after removing its common $e^{- 2 Φ}$ , is $G_{M N} + 2 \nabla_{M} \nabla_{N} Φ - 2 (□ Φ - ∣ d Φ ∣^{2}) g_{M N} - \frac{1}{2} (⟨ i_{M} H_{3}, i_{N} H_{3} ⟩ - \frac{1}{2} g_{M N} ∣ H_{3} ∣^{2})$ . The RR coefficient in the same normalization is $- \frac{1}{2} e^{2 Φ}$ times the $G_{1}$ and ${\tilde{G}}_{3}$ stress brackets, and $- \frac{1}{4} e^{2 Φ} ⟨ i_{M} {\tilde{G}}_{5}, i_{N} {\tilde{G}}_{5} ⟩$ . The Chern–Simons integral has no metric and contributes zero.

1.4F1F9F11algebra

First make the compact-support identity needed by all the following integrations explicit. In an oriented chart with coordinates $x^{0}, \dots, x^{9}$ , every smooth compactly supported nine-form has the unique expression $β = \sum_{M = 0}^{9} (- 1)^{M} J^{M} d x^{0} \land \dots \land \hat{d x^{M}} \land \dots \land d x^{9}$ . Coordinate differentiation gives $d β = (\partial_{M} J^{M}) d x^{0} \land \dots \land d x^{9}$ , so [F11] proves $\int d β = 0$ . In particular, for an $r$ -form variation $a$ of compact support and a smooth $(9 - r)$ -form $B$ , [F9] gives $\int d a \land B = (- 1)^{r + 1} \int a \land d B .$ Every exact term discarded below contains a compactly supported variation (or its derivative), so its nine-form primitive has compact support even if the background fields do not. For the scalar kinetic term, write $w = \sqrt{∣ \det g ∣}$ and use the compact vector-density $J^{M} = w e^{- 2 Φ} φ g^{M N} \partial_{N} Φ$ . Its zero divergence integral and $\partial_{M} (w V^{M}) = w \nabla_{M} V^{M}$ give $\int w e^{- 2 Φ} ⟨ d Φ, d φ ⟩ = - \int w φ \nabla_{M} (e^{- 2 Φ} \nabla^{M} Φ)$ . Thus no compact-support boundary term is being assumed away. For a compactly supported scalar variation $φ = δ Φ$ , the NS density variation before integration by parts is proportional to $e^{- 2 Φ} [- 2 (R + 4 ∣ d Φ ∣^{2} - \frac{1}{2} ∣ H_{3} ∣^{2}) φ + 8 ⟨ d Φ, d φ ⟩]$ . Integrating the last term once gives the scalar coefficient $- 2 R + 8 ∣ d Φ ∣^{2} + ∣ H_{3} ∣^{2} - 8 □ Φ$ , or equivalently $R + 4 □ Φ - 4 ∣ d Φ ∣^{2} - \frac{1}{2} ∣ H_{3} ∣^{2} = 0$ . The RR terms and Chern–Simons term contain no explicit $Φ$ in this unshifted string-frame convention.

2.1F1F5F9F10F11step 1.4algebra

Vary $C_{0}$ by a compactly supported scalar $u$ . Then $δ G_{1} = d u$ and $δ {\tilde{G}}_{3} = - u H_{3}$ . The RR variation is $- c \int [2 d u \land * G_{1} - 2 u H_{3} \land * {\tilde{G}}_{3}]$ , where $c = (4 κ_{0}^{2})^{- 1}$ is held fixed by [F10]. Since $d u \land * G_{1} = d (u * G_{1}) - u d * G_{1}$ by [F9], compact support removes the exact term and leaves $2 c u (d * G_{1} + H_{3} \land * {\tilde{G}}_{3})$ . Applying [F5] proves the first RR equation.

2.2F1F5F9F11step 1.4algebra

Vary $C_{4}$ by a compactly supported four-form $a_{4}$ . Then $δ {\tilde{G}}_{5} = d a_{4}$ and the varying action is $- c \int [d a_{4} \land * {\tilde{G}}_{5} + a_{4} \land H_{3} \land G_{3}]$ . Integration by parts uses $d (a_{4} \land * {\tilde{G}}_{5}) = d a_{4} \land * {\tilde{G}}_{5} + a_{4} \land d * {\tilde{G}}_{5}$ , so stationarity and [F5] yield $d * {\tilde{G}}_{5} = H_{3} \land G_{3} = H_{3} \land {\tilde{G}}_{3}$ , the second equality following from $H_{3} \land H_{3} = 0$ .

2.3F1F9F11step 1.4algebra

Vary $C_{2}$ by a compactly supported two-form $a_{2}$ . The field definitions give $δ {\tilde{G}}_{3} = d a_{2}$ and $δ {\tilde{G}}_{5} = - \frac{1}{2} H_{3} \land a_{2} + \frac{1}{2} d a_{2} \land B_{2}$ . Using [F9] to integrate the RR and Chern–Simons terms by parts yields $δ S = c \int a_{2} \land [2 d * {\tilde{G}}_{3} + H_{3} \land * {\tilde{G}}_{5} + \frac{1}{2} B_{2} \land d * {\tilde{G}}_{5} - G_{5} \land H_{3}]$ . The last term is from $δ (C_{4} \land H_{3} \land G_{3})$ and keeps raw $G_{5} = d C_{4}$ .

2.4F1F8F9F11step 1.4algebra

Vary $B_{2}$ by a compactly supported two-form $b_{2}$ . For the Chern–Simons term, $G_{3} = d C_{2}$ and [F8] give $d G_{3} = d^{2} C_{2} = 0$ . Thus $d (C_{4} \land b_{2} \land G_{3}) = G_{5} \land b_{2} \land G_{3} + C_{4} \land d b_{2} \land G_{3} + C_{4} \land b_{2} \land d G_{3} = G_{5} \land b_{2} \land G_{3} + C_{4} \land d b_{2} \land G_{3},$ whose integral vanishes because $b_{2}$ is compactly supported. Consequently $- c \int C_{4} \land d b_{2} \land G_{3} = + c \int b_{2} \land G_{5} \land G_{3} = - c \int b_{2} \land G_{3} \land G_{5}$ , where exchanging the degrees $5$ and $3$ contributes $(- 1)^{15} = - 1$ . The NS variation is $2 c \int b_{2} \land d (e^{- 2 Φ} * H_{3})$ . The ${\tilde{G}}_{3}$ kinetic variation, after one integration by parts, is $2 c \int b_{2} \land [- G_{1} \land * {\tilde{G}}_{3} - C_{0} d * {\tilde{G}}_{3}]$ . The five-form kinetic variation together with this Chern–Simons contribution is $2 c \int b_{2} \land [- \frac{1}{2} G_{3} \land * {\tilde{G}}_{5} - \frac{1}{4} C_{2} \land d * {\tilde{G}}_{5} - \frac{1}{2} G_{3} \land G_{5}]$ . The kinetic terms follow by differentiating ${\tilde{G}}_{3} = G_{3} - C_{0} H_{3}$ and ${\tilde{G}}_{5} = G_{5} - \frac{1}{2} H_{3} C_{2} + \frac{1}{2} G_{3} B_{2}$ , then applying [F9].

2.5F5step 1.3algebra

Stationarity makes the integral of this continuous symmetric coefficient against every compactly supported symmetric $q^{M N}$ vanish; [F5] makes the coefficient zero pointwise. Rearranging gives exactly the metric equation in the Statement, with the factor $1 / 4$ for the five-form coming from both its action coefficient $1 / 2$ and the RR prefactor.

3.1F2F5step 2.2step 2.3F9algebra

Use step 2.2, the self-duality [F2], and the identity $G_{5} = {\tilde{G}}_{5} + \frac{1}{2} H_{3} \land C_{2} - \frac{1}{2} G_{3} \land B_{2}$ . Since $- G_{5} \land H_{3} = H_{3} \land G_{5}$ , the bracket in step 2.3 becomes $2 d * {\tilde{G}}_{3} + 2 H_{3} \land {\tilde{G}}_{5}$ : the $- \frac{1}{2} H_{3} \land G_{3} \land B_{2}$ term from $H_{3} \land G_{5}$ cancels $+ \frac{1}{2} B_{2} \land H_{3} \land G_{3}$ . By [F5] this is zero exactly when $d * {\tilde{G}}_{3} = - H_{3} \land {\tilde{G}}_{5}$ .

3.2F2step 2.5step 1.4algebra

Trace the metric equation from step 2.5. In ten dimensions $g^{M N} ⟨ i_{M} F_{p}, i_{N} F_{p} ⟩ = p ∣ F_{p} ∣^{2}$ by summing the $p$ antisymmetric slots. Using [F2] and $∣ {\tilde{G}}_{5} ∣^{2} = 0$ gives $- 4 R = 18 □ Φ - 20 ∣ d Φ ∣^{2} - ∣ H_{3} ∣^{2} - 2 e^{2 Φ} ∣ G_{1} ∣^{2} - e^{2 Φ} ∣ {\tilde{G}}_{3} ∣^{2}$ . Substitute $R = 4 ∣ d Φ ∣^{2} + \frac{1}{2} ∣ H_{3} ∣^{2} - 4 □ Φ$ from step 1.4, collect terms, and divide by $- 2$ to obtain the dilaton equation in the Statement.

4.1F2F5step 2.2step 3.1step 2.4F9algebra

Substitute steps 2.2 and 3.1 into the coefficient in step 2.4. The terms $- C_{0} d * {\tilde{G}}_{3}$ and $- G_{1} \land * {\tilde{G}}_{3}$ combine to $- G_{1} \land * {\tilde{G}}_{3} + C_{0} H_{3} \land {\tilde{G}}_{5}$ . In the remaining terms use $* {\tilde{G}}_{5} = {\tilde{G}}_{5}$ and $G_{5} = {\tilde{G}}_{5} + \frac{1}{2} H_{3} C_{2} - \frac{1}{2} G_{3} B_{2}$ ; $G_{3} \land G_{3} = 0$ and $C_{2} \land H_{3} \land G_{3} = - G_{3} \land H_{3} \land C_{2}$ make the $C_{2}$ terms cancel, leaving $- G_{3} \land {\tilde{G}}_{5}$ . Thus the total coefficient is $d (e^{- 2 Φ} * H_{3}) - G_{1} \land * {\tilde{G}}_{3} - (G_{3} - C_{0} H_{3}) \land {\tilde{G}}_{5}$ . By [F5] it vanishes exactly when the NS equation in the Statement holds.

5.1F1F2F5step 1.1step 1.2step 2.5step 3.2step 2.1step 2.2step 3.1step 4.1∎

Steps 1.1, 1.2, 1.3, 1.4, 2.1, 2.2, 2.3, 2.4, 2.5, 3.1, 3.2, 4.1 prove all Bianchi, metric, dilaton, RR, and NS equations under the stated compact-support hypotheses; step 1.2 keeps self-duality separate from variation. The pointwise converse from the displayed equations to stationarity follows by substituting their zero coefficients into the first variations, while the forward direction used [F5]. No source term or higher-order term was varied.

Type IIB Field Equations with Five Form Self Duality

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources