Matter Field Kähler Potential

• The paper establishes precise analytic criteria using bundle extension structures to prove the vanishing of mixed kinetic terms in the matter field Kähler potential.
• Explicit examples, including rank-2 and nontrivial SU(3) extensions, demonstrate that the vanishing persists even under Higgsing and deformations.
• The robust analytic results impact moduli stabilization and supersymmetric phenomenology by imposing nontrivial constraints on effective four-dimensional theories.

The matter field Kähler potential is a central object in supersymmetric field theories and string compactifications, capturing the kinetic couplings and scalar field geometry of chiral multiplets in four-dimensional effective actions. In both pure mathematics and theoretical physics, the structure of the matter field Kähler potential strongly reflects the underlying geometry and topology of the compactification manifold and its vector bundles. Recent developments have revealed that, beyond constraints imposed by effective symmetries, the Kähler potential can exhibit vanishing structures rooted in the geometry of extension bundles, with broad implications for model building, moduli stabilization, and the phenomenology of supersymmetric vacua (Gray, 18 Aug 2025).

1. Analytic Criteria for Vanishing Terms in the Matter Field Kähler Potential

The paper establishes precise analytic conditions under which certain terms in the matter field Kähler potential vanish due to the structure of bundle extensions. Consider a holomorphic vector bundle V defined as an extension: $0 \to A \xrightarrow{i} V \xrightarrow{m} B \to 0$ Cohomology classes in $H^1(V)$, which correspond to matter fields, can be represented as images from $H^1(A)$ via $i$ or as pullbacks from $H^1(B)$ via the dual (with respect to the Hodge star) map $m^*$. The inner product on the space of $V$-valued forms, used to construct the Kähler metric, satisfies the adjoint property: $$\langle m^* \beta, i \alpha\rangle_V = -\langle \alpha, m \circ i \beta\rangle_B$$ Due to the long exact sequence in cohomology, $m \circ i = 0$, so cross terms in the Kähler metric arising between $i(H^1(A))$ and $m^*(H^1(B))$ must vanish: $\langle m^* \beta, i \alpha\rangle_V = 0$ This result is highly nontrivial: mixed terms in the Kähler metric (and thus the Kähler potential) that are naively allowed by four-dimensional symmetries can vanish exactly due to bundle extension structure, independent of any manifest gauge symmetry [(Gray, 18 Aug 2025), eqs. (3.6), (4.5)].

2. Explicit Geometric and Model-Theoretic Examples

Three classes of explicit examples demonstrate the analytic result:

• Rank-2 bundle with trivial extension: On a complete intersection Calabi–Yau threefold (CICY), the sum $$V = \mathcal{O}(-2,2,0,0) \oplus \mathcal{O}(2,-2,0,0)$$ realizes V as an extension with a trivial class. The matter fields from each summand are charged under an anomalous $U(1)$, and kinetic mixing terms vanish by both the analytic argument and $U(1)$-invariance.
• Nontrivial $SU(3)$ extension bundle: With $0 \to \mathcal{AF} \to V_3 \to \mathcal{K} \to 0$, matter arises from both sub-bundles. The vanishing of the mixed inner product $\langle i \alpha, m^* \beta \rangle$ is not enforced by any known symmetry but follows from the extension and Hodge-theoretic structure, and the corresponding Kähler potential terms vanish even though they could be allowed by gauge invariance.
• Higgsing transitions: Starting from a bundle $V_3$ representing unbroken $E_6$, implementing a Higgsing via singlet vevs (to $SO(10)\times U(1)$) corresponds to a transition to another extension bundle $V_4$. The analytic cancellation persists: cross-coupling terms, including entire infinite series involving insertions of the Higgsed vev, remain absent in the Kähler potential due to the extension sequence structure.

These examples underscore that analytic bundle topology, rather than only field-theoretic symmetry, controls the vanishing of potentially generic terms [(Gray, 18 Aug 2025), Sec. 4].

3. Persistence of Structure Under Higgsing and Bundle Deformations

Higgsing transitions—corresponding to symmetry breaking via vacuum expectation values of bundle-valued matter fields—could, in principle, generate new nonvanishing Kähler potential couplings at higher order. However, the extension structure is robust under standard Higgsing deformations: the sequence of maps and the vanishing of composite chain maps ($m \circ i = 0$) survive, leading to the vanishing of cross terms at all orders. Explicitly, for couplings of the schematic form $\tilde{f} \bar{f} (f \bar{f})^n$, all orders in a perturbative expansion in the matter fields vanish if the analytic extension criteria are satisfied [(Gray, 18 Aug 2025), Sec. 4.3].

The general result is that Higgsing does not “regenerate” forbidden kinetic mixings or higher-point Kähler couplings when they are analytically constrained by bundle extension structure, even as the gauge bundle and low-energy gauge symmetry change.

4. Extension to All Orders in the Matter Field Expansion

The analytic conditions yielded by the extension sequence and the adjoint property of the relevant maps guarantee that the vanishing is not restricted to quadratic order (kinetic terms). Higher-point terms in the Kähler potential, e.g.,

$$\tilde{f} \bar{f} (f \bar{f})^n \quad \text{for all}\ n \geq 0$$

must also vanish. This follows from iterative application of the extension sequence and Hodge theory, ensuring the persistent cancellation of the corresponding inner products and thus the absence of these terms in the full perturbative series [(Gray, 18 Aug 2025), eq. (4.19)].

5. Theoretical Implications and Constraints on Effective Theories

The finding that entire sectors of the effective Kähler potential can be structurally absent—even when no explicit four-dimensional symmetry mandates their vanishing—has multiple implications:

• Model-building and phenomenology: Non-genericity in the Kähler metric can impact flavor structure, soft terms, and moduli stabilization, modifying expected couplings and mass hierarchies.
• Moduli dependence: Since Kähler potential terms often depend on both bundle and complex structure moduli, their systematic absence significantly constrains dynamics in the moduli space and the structure of supersymmetry breaking effects.
• Guidance for numerical calculations: The analytic results indicate which numerical Kähler potential coefficients must vanish exactly, streamlining both computation and interpretation.
• New classes of selection rules: The extension sequence structure acts as an “analytic selection rule,” separate from conventional gauge or discrete symmetries.

6. Broader Context and Future Research Directions

The observed vanishing parallels similar phenomena long-known for superpotential couplings in heterotic theories, where certain Yukawa couplings vanish due to deeper geometric or group-theoretic reasons not captured by visible symmetries. The analytic methods developed, based on bundle extension sequences and Hodge-theoretic adjoints, open further directions to:

• Investigate the systematic origin of such vanishing, potentially uncovering new hidden symmetries or dualities.
• Examine implications for global properties of moduli spaces and landscape statistics.
• Explore whether these structures appear in other string compactifications, including F-theory and type II scenarios, where bundles and associated exact sequences control matter content.

The results emphasize that the geometry of vector bundle extensions can impose nontrivial, physically relevant constraints on the matter field Kähler potential, with widespread consequences for heterotic compactifications and their phenomenology (Gray, 18 Aug 2025).