Fractional Shift-Symmetric Kähler Potentials

• Fractional shift-symmetric Kähler potentials are defined as potentials invariant under discrete fractional shifts, playing a key role in string compactifications and intersecting brane models.
• They stabilize moduli by encoding orbifold, discrete torsion, and fractional R-charge effects, preventing unwanted continuous directions in effective field theories.
• Applications in inflationary attractors and effective theories demonstrate that these potentials support robust predictions for observables like the spectral index and tensor-to-scalar ratio.

Fractional shift-symmetric Kähler potentials refer to a class of Kähler potentials whose analytic structure and transformation properties are invariant under discrete or fractional shifts of moduli, with particular relevance in string compactifications, effective field theories derived from branes on orbifolds, and cosmological inflationary models. These potentials encode nontrivial geometry and couplings in moduli space with symmetry under finite, non-continuous shifts, typically reflecting underlying orbifold, discrete torsion, or field-theoretic mechanisms that project out phenomenologically unfavorable directions.

1. Definition and Physical Origin

A Kähler potential $K$ on a complex manifold encodes the information of the metric $g_{i\bar{j}} = \partial_i \partial_{\bar{j}} K$ and determines the kinetic terms of scalar fields in effective theories. The "fractional shift-symmetry" property entails that under a finite (often discrete or fractional) shift of open- or closed-string moduli (such as Wilson lines, local displacements, or orbifold fixed-point labels), the Kähler potential, and consequently the metric, remain invariant (up to a Kähler transformation). This symmetry can be explicitly realized in several contexts:

• Intersecting D-brane models on orbifolds where matter arises at brane intersections and the moduli correspond to geometric, discrete parameters (Honecker, 2011).
• Calabi–Yau moduli spaces in GLSM approaches where noncompact directions and orbifold sectors introduce fractional R-charges, leading to invariance under fractional shifts (Honma et al., 2013, Erkinger et al., 2020).
• Cosmological inflationary attractors where fractional pole-like Kähler potentials are endowed with a shift symmetry in the inflaton field (Pallis, 3 Jul 2025).

2. Realization in Intersecting Brane Effective Theories

In the perturbatively exact computation for toroidal orbifolds, the analytic form of the open-string Kähler metric for matter fields localized at the intersection of D6-branes is tightly connected to the underlying orbifold and intersection data (Honecker, 2011). For sectors with three non-vanishing intersection angles, the Kähler metric is

$$K_{n_a, N_b} = f(S, U_\ell) \cdot \sqrt{ \prod_{i=1}^3 \frac{1}{v_i} \left( \frac{ \Gamma(|\varphi^{(i)}_{ab}|) }{ \Gamma(1 - |\varphi^{(i)}_{ab}|) } \right)^{ \frac{ -\varphi^{(i)}_{ab} }{ I_{ab} } } }$$

where $v_i$ are two-torus volumes, $\varphi^{(i)}_{ab}$ are intersection angles, and $I_{ab}$ is the intersection number. The prefactor $f(S, U_\ell)$ depends on the four-dimensional dilaton and complex structure moduli. Discrete Wilson lines and brane displacements enter as arguments of lattice sum functions, leading to invariant metric structure under their discrete shifts — this fractional symmetry ensures that only discrete, physically meaningful choices influence the metric. In the presence of discrete torsion, branes become rigid with the cycles given by:

$$\Pi_{a} = \frac{1}{4} [ \Pi_{a}^{\text{bulk}} + \sum_{i=1}^3 \Pi_{a}^{(\mathbb{Z}_2^{(i)})} ]$$

thus changing normalization factors and threshold corrections, but keeping the "fractional shift-symmetry" intact. This symmetry is essential for moduli stabilization, avoiding unwanted continuous moduli, and controlling soft term structures.

3. Fractional Shift Symmetry in Hybrid GLSM and Calabi-Yau Moduli Spaces

The quantum-corrected Kähler potential for Calabi-Yau fourfolds and related GLSM models can be written as

$$e^{-K(t, \bar{t})} = -\sum_{i,j,k,l} K_{ijkl} [(t^i - \bar{t}^i) - (F_i(t) - \overline{F}_i(\bar{t}))]\cdots$$

where $K_{ijkl}$ are intersection numbers and $F_i(t)$ are series whose coefficients are genus-0 Gromov-Witten invariants (Honma et al., 2013). The important feature is the invariance of $K$ (up to a Kähler transformation) under shifts $t^i \to t^i + \delta^i$ with $\delta^i$ fractional, reflecting the fractional charge assignments in the GLSM — the structure is robust under such shifts. In the sphere partition function formulation (Erkinger et al., 2020), fractional shift symmetry of the Kähler potential emerges from the Gamma class corrections and discrete data in hybrid phases (LG orbifold over a base manifold), where the action of fractional R-charges manifests in the transformation properties of the exact quantum-corrected $K$.

4. Mathematical Formulation and Symmetry Actions

Fractional shift symmetry is realized mathematically as invariance (or covariance up to a Kähler transformation) under:

$$\phi \to \phi + \delta,\quad \text{with}\ \delta = \text{fractional value}$$

where $\phi$ is a modulus (e.g., inflaton or open string modulus). In the context of isometries on the space of Kähler potentials equipped with the Mabuchi metric, the only rigidity-breaking transformations allowed are fractional shifts of the constant mode:

$$F_*(\xi) = \alpha\,\varphi^*\xi - b \int_X \xi\,\omega_u^n$$

with $b$ potentially a fractional parameter depending on normalization (Lempert, 2017). Legendre duality and Calabi diastasis function symmetries can also incorporate shift invariance in the potentials (Berndtsson et al., 2016).

Fractional mirror symmetry, introduced via asymmetric orbifold constructions in CFT (e.g., Gepner model simple current extensions), effects a partial exchange of chiral and twisted chiral fields, resulting in moduli invariances under discrete transformations of fractional nature (Israel, 2014). In such cases, the Kähler potentials governing twisted chiral multiplets are invariant only under partial/fractional shift symmetry, leading to novel non-geometric phases.

5. Applications in Inflationary Model Building

Fractional Kähler potentials with pole-like structure appear in attractor models for inflation (Pallis, 3 Jul 2025):

$K \simeq \frac{N}{(1 - \phi^{q_M})^p}$

where quantum corrections to the kinetic term ("pole") induce shift symmetry in the inflaton sector. Endowing $K$ with a shift symmetry in $\phi^{q_M}$ — i.e., $K(\phi^{q_M} + a)$ invariance for $a$ fractional — ensures robustness against higher-order corrections and supports inflation for subplanckian values of $\phi$. The kinetic pole enhances the field range after canonical normalization, resulting in attractor behavior for inflationary observables (spectral index $n_s$, tensor-to-scalar ratio $r$) that are largely insensitive to microphysical details, with $r$ increasing with e-fold number $N$.

In supergravity, fractional shift-symmetric Kähler potentials are constructed by supplementing the holomorphic Kähler sector with a symmetric piece, combining chiral superfields $\Phi$ and a stabilizing $S$ with a linear (in $S$) monomial superpotential, achieving controlled inflationary dynamics and stabilization.

6. Geometric and Physical Consequences

Fractional shift-symmetric Kähler potentials play a key role in controlling moduli space geometry and cosmological physics:

• In string model building, they prevent unwanted continuous moduli, enforce discrete selections vital for phenomenology, and enable correct reproduction of gauge and Yukawa couplings.
• In moduli stabilization, the symmetry ensures that only physically acceptable directions remain flat or stabilized, aiding in the construction of realistic low-energy effective supergravity theories.
• In black hole and gravitational instanton solutions, fractional shift symmetry quantifies invariance under nontrivial coordinate transformations (e.g., Newman–Janis shift), tying together integrability and double copy relations in complex geometry (Aksteiner et al., 2022).

Geometric symmetries such as constant holomorphic sectional curvature, semisymmetry, and holomorphic pseudosymmetry can result from — or imply — fractional shift-symmetries in potentials, as the latter restricts the possible structures of curvature tensors (e.g., vanishing complex Tachibana tensor) (Albujer et al., 2022).

7. Summary of Key Formulas Associated with Fractional Shift Symmetry

Formula	Context	Role of Fractional Shift Symmetry
$K_{n_a, N_b}$	D6-brane intersections (Honecker, 2011)	Invariance under Wilson line/displacement
$e^{-K(t, \bar{t})}$	Calabi-Yau fourfold (Honma et al., 2013)	Symmetry under $t^i \to t^i + \delta^i$
$K \simeq N/(1 - \phi^{q_M})^p$	Inflationary attractor (Pallis, 3 Jul 2025)	Shift symmetry in $\phi^{q_M}$
$$F_*(\xi) = \alpha\varphi^*\xi - b \langle \xi, 1 \rangle$$	Mabuchi metric isometries (Lempert, 2017)	Fractional shifts of constant mode

Fractional shift-symmetric Kähler potentials encapsulate both profound geometric invariance and practical constraints on the moduli and couplings in string-derived and inflationary effective field theories. Their mathematical form and physical properties are dictated by discrete orbifold choices, pole-like kinetic enhancements, and duality symmetries, ensuring robustness and phenomenological viability across a spectrum of high-energy models.