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Type Iia Bosonic Field Equations

Physical scope

Local Euler–Lagrange equations of the named massless source-free two-derivative action. Variations are smooth and compactly supported in an oriented coordinate patch. The equations are model deductions, not empirical claims, and include no localized currents or higher-derivative corrections.

Statement

On a smooth contractible oriented Lorentzian coordinate patch $U \subset R^{1, 9}$ with no sources, stationarity under arbitrary smooth compactly supported variations of $g_{s}, Φ, B_{2}, C_{1}, C_{3}$ for the separately adopted massless type IIA action gives the following equations. Use $□ Φ = \nabla^{M} \nabla_{M} Φ$ , $∣ F_{p} ∣^{2} = F_{M_{1} \dots M_{p}} F^{M_{1} \dots M_{p}} / p!$ , and $⟨ i_{M} F_{p}, i_{N} F_{p} ⟩ = \frac{1}{(p - 1)!} F_{M A_{2} \dots A_{p}} F_{N}^{A_{2} \dots A_{p}}, T_{M N}^{(p)} (F_{p}) = ⟨ i_{M} F_{p}, i_{N} F_{p} ⟩ - \frac{1}{2} (g_{s})_{M N} ∣ F_{p} ∣^{2} .$ With $G_{M N} = R_{M N} - \frac{1}{2} R (g_{s})_{M N}$ , the metric equation is $G_{M N} = - 2 \nabla_{M} \nabla_{N} Φ + (2 □ Φ - 2 ∣ d Φ ∣^{2}) (g_{s})_{M N} + \frac{1}{2} T_{M N}^{(3)} (H_{3}) + \frac{1}{2} e^{2 Φ} (T_{M N}^{(2)} (G_{2}) + T_{M N}^{(4)} ({\tilde{G}}_{4})) .$ The raw dilaton equation is $R + 4 □ Φ - 4 ∣ d Φ ∣^{2} - \frac{1}{2} ∣ H_{3} ∣^{2} = 0.$ Under the metric equation displayed above, this raw equation is equivalent to the trace-reduced equation $□ Φ = 2 ∣ d Φ ∣^{2} - \frac{1}{2} ∣ H_{3} ∣^{2} + \frac{3}{4} e^{2 Φ} ∣ G_{2} ∣^{2} + \frac{1}{4} e^{2 Φ} ∣ {\tilde{G}}_{4} ∣^{2} .$ The form equations are $d (* G_{2}) = H_{3} \land * {\tilde{G}}_{4}, d (* {\tilde{G}}_{4}) = - H_{3} \land {\tilde{G}}_{4},$ $d (e^{- 2 Φ} * H_{3}) = \frac{1}{2} {\tilde{G}}_{4} \land {\tilde{G}}_{4} - G_{2} \land * {\tilde{G}}_{4} .$ The potential Bianchi identities are $d H_{3} = 0$ , $d G_{2} = 0$ , $d {\tilde{G}}_{4} = - H_{3} \land G_{2}$ , as in the source-free potential-Bianchi theorem. The Lorentzian Hodge star is characterized by $α \land * β = ⟨ α, β ⟩_{g_{s}} {vol}_{g_{s}}$ for $p$ -forms, where ${vol}_{g_{s}}$ is the oriented Lorentzian volume form and the inner product is induced by $g_{s}$ , including its one negative direction. No positive-definite Hodge-star convention is used. $R_{M N}$ uses Hamilton’s curvature sign, equivalently the sign fixed by variation identity [F5] below.

Theorem references

Type Iia Two Derivative Supergravity Action

Type Iia RR Bianchi From Potentials

Facts & Assumptions

Given: A smooth contractible oriented Lorentzian coordinate patch $U \subset R^{1, 9}$ with signature $(- + \dots +)$ , smooth fields $g_{s}, Φ, B_{2}, C_{1}, C_{3}$ , $d H_{3} = 0$ , and arbitrary compactly supported smooth field variations. Here $g_{s} = G^{(s)}$ is the string-frame metric, coordinates have length dimension, the metric and $Φ$ are dimensionless, the potential components are dimensionless, and the field-strength components have dimension length $^{- 1}$ , as fixed in Type Iia Two Derivative Supergravity Action. All actions and p-form norms are those of that item.

[A1]

Supergravity Conventions And Normalizations (current batch-56 carrier and current ready Step 1 record inspected; its stated hypotheses match this use): the signature is $(- + \dots +)$ , $∣ F_{p} ∣^{2} = F^{2} / p!$ , $α^{'} > 0$ , and $2 κ_{0}^{2} = (2 π)^{7} α^{' 4} > 0$ ; the string and Einstein metrics are distinct and this variation uses $g_{s}$ .

[A2]

Ten Dimensional NS NS Supergravity Action (current batch-56 carrier and current ready Step 1 record inspected; its stated hypotheses match this use): the NS action is $(2 κ_{0}^{2})^{- 1} \int \sqrt{- g_{s}} e^{- 2 Φ} (R + 4 ∣ d Φ ∣^{2} - \frac{1}{2} ∣ H_{3} ∣^{2}) d^{10} x$ , with $H_{3} = d B_{2}$ .

[A3]

Type Iia Two Derivative Supergravity Action adopts the added terms $- \frac{1}{4 κ_{0}^{2}} \int \sqrt{- g_{s}} (∣ G_{2} ∣^{2} + ∣ {\tilde{G}}_{4} ∣^{2}) d^{10} x - \frac{1}{4 κ_{0}^{2}} \int B_{2} \land G_{4} \land G_{4}$ , with $G_{2} = d C_{1}$ , $G_{4} = d C_{3}$ , and ${\tilde{G}}_{4} = G_{4} - C_{1} \land H_{3}$ .

[F1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume ${A C}_{ω}$ , the axiom that every countable family of nonempty sets has a choice function.

[F2]

The general Stokes theorem: under ${A C}_{ω}$ , on an oriented smooth manifold without boundary, $\int_{U} d η = 0$ for every compactly supported smooth 9-form $η$ . Thus all integrations by parts below have no boundary contribution.

[F3]

The exterior derivative is a graded derivation and The exterior derivative squares to zero: $d (α \land β) = d α \land β + (- 1)^{\deg α} α \land d β$ , and $d^{2} = 0$ .

[F4]

The norm convention in [A1] implies that variation of $- \frac{1}{2} ∣ F_{p} ∣^{2} \sqrt{- g_{s}}$ with $F_{p}$ held fixed contributes $- \frac{1}{2} \sqrt{- g_{s}} T_{M N}^{(p)} (F_{p}) δ g^{M N}$ . This follows by varying each of the $p$ inverse metrics in $F_{A_{1} \dots A_{p}} F^{A_{1} \dots A_{p}} / p!$ ; each equal term gives the stated $(p - 1)!$ -normalized contraction, and the determinant contributes $+ \frac{1}{4} \sqrt{- g_{s}} g_{M N} ∣ F_{p} ∣^{2} δ g^{M N}$ .

[F5]

For $f = e^{- 2 Φ}$ , the curvature variation in the convention of Hamilton §2 is $δ \int \sqrt{- g_{s}} f R = \int \sqrt{- g_{s}} [f G_{M N} + (g_{s})_{M N} □ f - \nabla_{M} \nabla_{N} f] δ g^{M N} .$ It follows from $δ \sqrt{- g_{s}} = - \frac{1}{2} \sqrt{- g_{s}} (g_{s})_{M N} δ g^{M N}$ , $δ R_{M N} = \nabla_{P} δ Γ^{P}_{M N} - \nabla_{N} δ Γ^{P}_{M P}$ , $δ Γ^{P}_{M N} = \frac{1}{2} g^{P Q} (\nabla_{M} δ g_{Q N} + \nabla_{N} δ g_{Q M} - \nabla_{Q} δ g_{M N})$ , and [F2] applied to the compactly supported divergence terms.

[F6]

Type Iia RR Bianchi From Potentials: $d H_{3} = 0$ , $d G_{2} = 0$ , and $d {\tilde{G}}_{4} = - H_{3} \land G_{2}$ for the potentials in [A3].

[F7]

Hamilton, §6.1, Theorem 6.2(a) prints a metric equation with dilaton-gradient metric coefficient $- ∣ d Φ ∣^{2} g_{M N}$ . The curvature sign used here is the one for which [F5] holds; the printed coefficient is checked against the action rather than imported as a premise.

[F8]

Compactly Supported Tests Determine Continuous Local Coefficients: on an open Euclidean chart, if a finite tuple of continuous coefficient functions has zero integral pairing against every smooth compactly supported tuple of tests with a positive continuous weight, every coefficient vanishes pointwise.

[F9]

Compactly Supported Metric Directions Stay Lorentzian for Small Amplitude: for a smooth Lorentzian metric on a smooth ten-manifold, each smooth compactly supported symmetric inverse-metric direction is realized by the derivative of a smooth family of Lorentzian metrics for all sufficiently small parameter values.

Proof

1.1givenA1A2F1F5algebra

Vary $g_{s}^{M N}$ with all forms and $Φ$ fixed. For $f = e^{- 2 Φ}$ , $\nabla_{M} f = - 2 f \nabla_{M} Φ$ and $□ f = f (4 ∣ d Φ ∣^{2} - 2 □ Φ)$ . Substitution in [F5] gives $f [G_{M N} + 2 \nabla_{M} \nabla_{N} Φ + (4 ∣ d Φ ∣^{2} - 2 □ Φ) g_{M N} - 4 \nabla_{M} Φ \nabla_{N} Φ]$ . The variation of $4 f ∣ d Φ ∣^{2} \sqrt{- g_{s}}$ is $4 f [\nabla_{M} Φ \nabla_{N} Φ - \frac{1}{2} g_{M N} ∣ d Φ ∣^{2}] \sqrt{- g_{s}} δ g^{M N}$ . Adding these contributions cancels the rank-two gradient term and leaves $f [G_{M N} + 2 \nabla_{M} \nabla_{N} Φ + (2 ∣ d Φ ∣^{2} - 2 □ Φ) g_{M N}]$ .

1.2givenA1A2F1F2F8algebra

Vary $Φ$ while holding the other fields fixed. The terms are $- 2 f (R + 4 ∣ d Φ ∣^{2} - \frac{1}{2} ∣ H_{3} ∣^{2}) δ Φ$ and $8 f \nabla^{M} Φ \nabla_{M} δ Φ$ . By [F2] and $\nabla_{M} f = - 2 f \nabla_{M} Φ$ , the integral of the second term becomes $- 8 f (□ Φ - 2 ∣ d Φ ∣^{2}) δ Φ$ . The sum is $- 2 f [R + 4 □ Φ - 4 ∣ d Φ ∣^{2} - \frac{1}{2} ∣ H_{3} ∣^{2}] δ Φ$ . This coefficient is smooth; the coordinate weight is $\sqrt{- g_{s}} / (2 κ_{0}^{2}) > 0$ by [A1] and $f = e^{- 2 Φ} > 0$ . Therefore [F8], applied to arbitrary compactly supported scalar tests, gives the raw dilaton equation pointwise.

1.3givenA1A3F1F2F8algebra

Vary $C_{1}$ , so $δ G_{2} = d δ C_{1}$ and $δ {\tilde{G}}_{4} = - δ C_{1} \land H_{3}$ . The kinetic variation is $- \frac{1}{2 κ_{0}^{2}} \int [d δ C_{1} \land * G_{2} - δ C_{1} \land H_{3} \land * {\tilde{G}}_{4}] .$ By [F2], $\int d δ C_{1} \land * G_{2} = \int δ C_{1} \land d (* G_{2})$ . The resulting coefficient is smooth; its wedge pairing with the independent one-form components is a finite tuple of coordinate test pairings with positive weight $\sqrt{- g_{s}} / (2 κ_{0}^{2})$ by [A1]. The coordinate wedge pairing is nondegenerate, so [F8] makes the equation's coefficient vanish pointwise. The Chern–Simons term is independent of $C_{1}$ .

1.4givenA1A3F1F2F3F8algebra

Vary $C_{3}$ , so $δ G_{4} = δ {\tilde{G}}_{4} = d δ C_{3}$ . The kinetic term contributes $- \frac{1}{2 κ_{0}^{2}} \int δ C_{3} \land d (* {\tilde{G}}_{4})$ by [F2]. The Chern–Simons variation is $- \frac{1}{2 κ_{0}^{2}} \int B_{2} \land d δ C_{3} \land G_{4}$ . Using $d (B_{2} \land δ C_{3} \land G_{4}) = H_{3} \land δ C_{3} \land G_{4} + B_{2} \land d δ C_{3} \land G_{4}$ , $d G_{4} = 0$ by [F3] and $G_{4} = d C_{3}$ , and compact support, this equals $- \frac{1}{2 κ_{0}^{2}} \int δ C_{3} \land H_{3} \land G_{4}$ . The resulting coefficient is smooth; its wedge pairing with independent three-form components is a finite tuple of coordinate test pairings with positive weight $\sqrt{- g_{s}} / (2 κ_{0}^{2})$ . Nondegeneracy of the coordinate wedge pairing and [F8] give the equation pointwise. Since $H_{3} \land C_{1} \land H_{3} = 0$ , $H_{3} \land {\tilde{G}}_{4} = H_{3} \land G_{4}$ , giving the displayed equation.

2.1givenA1A3F4F8F9step 1.1algebra

Apply [F4] to the NS $H_{3}$ term and the two RR kinetic terms. The coefficient of $δ g^{M N}$ in the full variation, after removing the common $\sqrt{- g_{s}} / (2 κ_{0}^{2})$ , is $f [G_{M N} + 2 \nabla_{M} \nabla_{N} Φ + (2 ∣ d Φ ∣^{2} - 2 □ Φ) g_{M N} - \frac{1}{2} T_{M N}^{(3)} (H_{3})] - \frac{1}{2} [T_{M N}^{(2)} (G_{2}) + T_{M N}^{(4)} ({\tilde{G}}_{4})] .$ The Chern–Simons term has no metric dependence, since it is an integral of a top-degree form. By [F9], every compactly supported symmetric inverse-metric test direction is realized through Lorentzian metrics, so stationarity makes the pairing of each independent symmetric component of this coefficient vanish against every such test. The coefficient components are smooth and the volume weight $\sqrt{- g_{s}} / (2 κ_{0}^{2})$ is positive by [A1]. Apply [F8] to the finite list of independent symmetric components to conclude the coefficient vanishes pointwise. Since $f > 0$ , division by $f$ gives the displayed metric equation.

2.2givenA1A3F1F2F8step 1.4algebra

Vary $B_{2}$ . The NS term contributes $+ \frac{1}{2 κ_{0}^{2}} \int δ B_{2} \land d (e^{- 2 Φ} * H_{3})$ . Since $δ {\tilde{G}}_{4} = - C_{1} \land d δ B_{2}$ , the RR kinetic term contributes $\frac{1}{2 κ_{0}^{2}} \int δ B_{2} \land [G_{2} \land * {\tilde{G}}_{4} - C_{1} \land d (* {\tilde{G}}_{4})],$ using [F2] and compact support. The explicit Chern–Simons variation contributes $- \frac{1}{4 κ_{0}^{2}} \int δ B_{2} \land G_{4} \land G_{4}$ . Substitute step 1.4, namely $d (* {\tilde{G}}_{4}) = - H_{3} \land G_{4}$ , to obtain $d (e^{- 2 Φ} * H_{3}) = \frac{1}{2} G_{4} \land G_{4} - C_{1} \land H_{3} \land G_{4} - G_{2} \land * {\tilde{G}}_{4} .$ Because $(C_{1} \land H_{3}) \land (C_{1} \land H_{3}) = 0$ , the first two terms on the right equal $\frac{1}{2} {\tilde{G}}_{4} \land {\tilde{G}}_{4}$ . The resulting smooth equation pairs with the independent two-form variation components through the nondegenerate coordinate wedge pairing and positive weight $\sqrt{- g_{s}} / (2 κ_{0}^{2})$ ; [F8] therefore makes it hold pointwise.

3.1givenstep 2.1step 1.2algebra

Trace the metric equation of step 2.1 in ten dimensions. For each $p$ -form, $g^{M N} ⟨ i_{M} F_{p}, i_{N} F_{p} ⟩ = p ∣ F_{p} ∣^{2}$ , so $tr T^{(p)} = (p - 5) ∣ F_{p} ∣^{2}$ . Since $g^{M N} G_{M N} = - 4 R$ , the trace gives $R = - \frac{9}{2} □ Φ + 5 ∣ d Φ ∣^{2} + \frac{1}{4} ∣ H_{3} ∣^{2} + \frac{3}{8} e^{2 Φ} ∣ G_{2} ∣^{2} + \frac{1}{8} e^{2 Φ} ∣ {\tilde{G}}_{4} ∣^{2} .$ Equating this with $R = - 4 □ Φ + 4 ∣ d Φ ∣^{2} + \frac{1}{2} ∣ H_{3} ∣^{2}$ from step 1.2 and rearranging yields the displayed trace-reduced dilaton equation.

3.2givenA3F6step 1.3step 1.4step 2.2

The potential definitions imply $d H_{3} = 0$ and [F6] gives $d G_{2} = 0$ , $d {\tilde{G}}_{4} = - H_{3} \land G_{2}$ . Thus all field equations and Bianchi identities listed in the theorem have been obtained from the action or its potential definitions. The equations are conditional model deductions under the declared action and source-free hypotheses.

4.1givenF7step 1.1step 2.1step 1.2step 3.1step 3.2∎

Hamilton’s displayed metric equation [F7] contains $- ∣ d Φ ∣^{2} g_{M N}$ in place of the $- 2 ∣ d Φ ∣^{2} g_{M N}$ in the corrected equation above; direct variation in steps 1.1–1.2 gives the latter, and only that coefficient makes the trace calculation in step 3.1 consistent with the displayed raw dilaton equation. Hamilton’s printed plus sign for $d {\tilde{G}}_{4}$ is likewise incompatible with the stated potential definition, as proved in Type Iia RR Bianchi From Potentials.

Remarks

Hamilton’s Theorem 6.2(a) is transcribed with a metric coefficient that fails direct variation and its own trace/dilaton consistency check; this item records the locally derived corrected coefficient. The sign defect in Theorem 6.2(e) is separate. Neither source defect is silently adopted.

The factor ${A C}_{ω}$ enters only through the chosen general Stokes supplier for compactly supported integration by parts. The physical deduction otherwise uses local differential-form calculus and compact support.

References

M. J. D. Hamilton, §6.1, Theorem 6.2 and its proof

Joseph Polchinski, §12.1