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Classical Type IIA vector prepotential from Calabi–Yau intersections

Physical scope

Source-reported leading homogeneous prepotential for the tree-level, large-volume, unfluxed, unorientifolded Type IIA vector sector. The adopted IIA action is current Step 1 ready. The local proof verifies the homogeneous lift and real quadratic ambiguity but does not derive the source-reported cubic coefficient from the ten-dimensional action; that derivation remains an explicit owner hold.

Definition

In the tree-level, large-volume vector-multiplet sector of Type IIA on a smooth compact Calabi–Yau threefold $X$ with no flux or orientifold projection, use the real closed $(1, 1)$ -form basis $[ω_{a}]$ and Kähler coefficients $[J] = v^{a} [ω_{a}]$ of the intersection and Kähler-volume datum, with complex-oriented intersections $κ_{a b c} = \int_{X} ω_{a} \land ω_{b} \land ω_{c}$ . Supply also a smooth real closed dimensionless two-form $B$ representing the retained internal NS–NS two-form sector, with $[B] \in H^{1, 1} (X; R)$ , and define its unique real coefficients by $[B] = b^{a} [ω_{a}]$ . Thus $B = B_{2, p h y s} / α^{'}$ in this local string-unit normalization, and its internal field strength is zero. Choose real representatives $b^{a}$ on a local patch; no global gauge-period identification is asserted. Define the complexified Kähler coordinates $t^{a} : = b^{a} + i v^{a}$ and choose homogeneous special coordinates $(X^{0}, X^{a})$ on a patch with $X^{0} \neq 0$ and $X^{a} = X^{0} t^{a}$ . In a local symplectic frame admitting a homogeneous prepotential, the source-reported leading classical term is $F^{(0)} (X) = - \frac{1}{6} κ_{a b c} \frac{X^{a} X^{b} X^{c}}{X^{0}},$ up to the stated real quadratic symplectic-polynomial ambiguity. Worldsheet-instanton terms and higher-derivative or string-loop corrections are excluded from this classical definition.

Definition references

Calabi–Yau threefold intersection and Kähler-volume coordinates

Facts & Assumptions

Given: The fixed intersection and Kähler-volume data of Calabi–Yau threefold intersection and Kähler-volume coordinates, the chosen closed two-form $B$ and its real coefficients $b^{a}$ , a local symplectic frame with special coordinates $X^{0}, X^{a}$ satisfying $X^{0} e 0$ and $X^{a} = X^{0} (b^{a} + i v^{a})$ , and the tree-level large-volume unfluxed Type IIA regime.

[F1]

Calabi–Yau threefold intersection and Kähler-volume coordinates defines $κ_{a b c} = \int_{X} ω_{a} \land ω_{b} \land ω_{c}$ and uses dimensionless $α^{'} = 1$ Kähler coordinates.

[F2]

Special Kähler period data in Calabi–Yau threefold reductions defines a local degree-two homogeneous prepotential only in a symplectic frame where it exists, and states the patchwise special-Kähler norm convention.

[F3]

Supergravity Conventions And Normalizations defines the Lorentzian signature, string-frame form norm, $α^{'}$ , $g_{s t r}$ , frame relation, and $κ_{0}$ normalization used here. Its current Step 1 record is ready.

[F4]

The current carrier Four-Dimensional $N = 2$ Matter-Coupled Supergravity adopts the local two-derivative four-dimensional $N = 2$ framework, special-Kähler vector sector, and selected regular electric frame. Its current Step 1 record remains escalated; its recorded reason cites the conventions carrier as absent although that live carrier is now ready. The framework interface is used conditionally, and the stale escalation is preserved.

[F5]

The current postulate Type Iia Two Derivative Supergravity Action adopts the source-free massless tree-level two-derivative IIA action in string frame, with $F_{0} = 0$ , local potentials, $d H_{3} = 0$ , and no localized or magnetic sources, Romans mass, loops, or $α^{'}$ corrections. Its adopted action and current Step 1 record are ready; the source-reported prepotential is still used only in the stated tree-level, large-volume regime.

[A1]

The closed NS–NS two-form sector and its combination $t^{a} = b^{a} + i v^{a}$ are supplied physical coordinate data, not additional conclusions of [F1]. The explicit coefficient and sign of the classical cubic are adopted as the source-reported effective-field-theory interface at large volume. The local calculations below verify its homogeneous lift and frame ambiguity; they do not derive the ten-dimensional reduction. The inspected Aspinwall lecture displays a different normalization, so its equation (37) is not used for this coefficient.

Proof

technique · homogeneous-coordinate verification and symplectic ambiguity check

Given: The intersection form and Kähler coordinates, a local prepotential frame, and the source-reported large-volume Type IIA cubic interface stated above.

1.1F1F2F3F4F5A1givenalgebra

The quotient $t^{a} = X^{a} / X^{0}$ is invariant under simultaneous scaling $X^{I} \mapsto λ X^{I}$ , while $(X^{0})^{2}$ scales by $λ^{2}$ . Thus $F^{(0)} = (X^{0})^{2} [- \frac{1}{6} κ_{a b c} t^{a} t^{b} t^{c}]$ is homogeneous of degree two. In the gauge $X^{0} = 1$ , it reduces to the source-reported affine cubic $- \frac{1}{6} κ_{a b c} t^{a} t^{b} t^{c}$ of [A1]. The physical identification of this term with the leading Type IIA reduction remains conditional on the $N = 2$ framework [F4] and action [F5].

2.1F2A1step 1.1algebra∎

Add a real quadratic polynomial $Q (X) = \frac{1}{2} A_{I J} X^{I} X^{J}$ with $A_{I J} = A_{J I} \in R$ . Then $\partial_{I} Q = A_{I J} X^{J}$ , and symmetry gives ${\overline{X}}^{I} \partial_{I} Q - X^{I} \overline{\partial_{I} Q} = {\overline{X}}^{I} A_{I J} X^{J} - X^{I} A_{I J} {\overline{X}}^{J} = 0$ . Hence this addition leaves the symplectic norm and its scalar metric unchanged. The period vector transforms by $(\binom{X}{F}) \mapsto (\begin{array}{cc} I & 0 \\ A & I \end{array}) (\binom{X}{F})$ ; symmetry of $A$ makes this matrix symplectic. In a regular prepotential frame, put $F_{I J} = \partial_{I} \partial_{J} F$ and define $N_{I J} = {\overline{F}}_{I J} + 2 i \frac{(Im F)_{I K} X^{K} (Im F)_{J L} X^{L}}{X^{M} (Im F)_{M N} X^{N}}$ on the regular patch where the denominator is nonzero. Adding the real $Q$ leaves every imaginary second derivative unchanged and adds $A_{I J}$ to the first term, so the vector period matrix transforms exactly as $N_{I J} \mapsto N_{I J} + A_{I J}$ : its imaginary kinetic matrix is unchanged, while its real theta matrix shifts by $A$ . Equivalently the Abelian vector action changes by exactly $c A_{I J} F^{I} \land F^{J}$ for constant adopted normalization $c$ if its theta term is written $c Re N_{I J} F^{I} \land F^{J}$ . On a local patch with $F^{I} = d A^{I}$ this is the total derivative $d (c A_{I J} A^{I} \land F^{J})$ , so it leaves the local field equations unchanged. It need not leave globally normalized theta data unchanged. This is the stated local symplectic-polynomial ambiguity, not literal equality of all vector couplings. It does not derive the adopted cubic coefficient from ten dimensions.

Classical Type IIA vector prepotential from Calabi–Yau intersections

Definition

Definition references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources