Kaleidoscopes, Waves and the Prepotential

Abstract: Isomorphic flops are topology-changing transitions connecting two diffeomorphic families of Calabi-Yau threefolds. They correspond to the generators of certain Coxeter groups acting on the moduli space. As a consequence of these symmetries, the prepotential of 4D $\mathcal{N} = 2$ Type IIA compactifications on such varieties must assemble into Coxeter-invariant functions. We construct a database of all Coxeter symmetries from isomorphic flops in Kähler-favorable CICYs. The action of the Coxeter group on the Kähler moduli space leaves a symmetric bilinear form invariant, which we interpret as a metric and construct its associated Laplace-Beltrami operator. We argue that the Coxeter-invariant functions featured in the prepotential solve the Helmholtz equation with this Laplacian, and that the prepotential can then be resummed into a decomposition in terms of eigenfunctions of the Laplace-Beltrami operator. The convergence rate of the raw orbit sums of worldsheet instanton contributions and the resummed expressions are complementary, with the latter sharply localizing around the first few terms in the interior of the moduli space.

• The paper demonstrates that isomorphic flops generate Coxeter symmetries which impose nontrivial constraints on the Type IIA prepotential in Calabi–Yau compactifications.
• It employs exhaustive computations on Kähler-favorable CICYs and leverages harmonic analysis to decompose instanton expansions into Coxeter-invariant spectral blocks.
• The findings establish novel computational frameworks linking birational geometry, group theory, and quantum corrections, guiding future string theory research.

Constraints on the Type IIA Prepotential from Isomorphic Flops and Coxeter Symmetries

Introduction and Context

This work addresses the geometric and physical consequences of isomorphic flops—topology-changing but diffeomorphism-preserving transitions—between chambers in the Kähler moduli space of Calabi-Yau threefolds, particularly those represented as complete intersections in products of projective spaces (CICYs). In compactifications of Type IIA string theory, these transitions act as simple reflections and generate Coxeter group symmetries, imposing nontrivial invariances on physical quantities, chief among which is the 4D $\mathcal{N}=2$ prepotential. The prepotential governs vector multiplet couplings in the low-energy effective supergravity theory and encapsulates both classical and quantum (worldsheet instanton) corrections.

The authors undertake an exhaustive computation of Coxeter symmetries generated by isomorphic flops across all Kähler-favorable CICYs, analyze the decomposition of prepotential instanton expansions into Coxeter-invariant structures, and interpret these constraints using harmonic analysis and the representation theory of reflection (Coxeter) groups.

Isomorphic Flops and Coxeter Group Structure

Isomorphic flops (iso-flops) relate pairs of Calabi–Yau threefolds in the same diffeomorphism class via birational transformations contracting and replacing rational curves. The passage between Kähler cone chambers across such iso-flop walls corresponds physically to a reflection symmetry in the moduli space, and mathematically to an involutive automorphism of the Kähler (and Mori) cone lattices.

Multiple iso-flops combine to form a discrete symmetry group generated by these reflections; these are Coxeter groups, abstractly characterized by generating involutions $s_i$ subject to relations $(s_i s_j)^{m_{ij}} = 1$.

On the set of 4874 Kähler-favorable CICYs, the authors classify all such Coxeter groups arising from iso-flops, providing a comprehensive database. Notably, 2182 models present nontrivial Coxeter actions, with 590 exhibiting rank $\geq 2$ cases (i.e., involving more than one independent iso-flop generator). A total of 19 distinct Coxeter group types are attained, with the (possibly infinite) dihedral groups $I_2(m)$—acting as basic local symmetries between pairs of flop walls—being most prevalent.

Constraints on the Prepotential

The structure of the 4D prepotential is decomposed as

$$F(T) = F_\mathrm{class}(T) + F_\mathrm{flop}(T) + F_\mathrm{non-flop}(T),$$

where $F_{\mathrm{non-flop}}(T)$ incorporates corrections from curve classes not directly associated with the flopping rays. It is in this sector that the Coxeter invariance manifests: curve classes are organized into orbits under the Coxeter group, and the Gromov-Witten invariants are constant along these orbits. Therefore, the non-flop instanton sum must yield a Coxeter-invariant function on the Kähler moduli.

Explicitly, the invariant functions take the form

$$\Upsilon_W(T) = \sum_{w \in W/Stab_W(d)} e^{2\pi i (w d, T)},$$

where $W$ is the Coxeter group, $d$ is a chosen curve class, and $s_i$0 the complexified Kähler parameters.

Explicit Spectral Decomposition: Dihedral Case Analysis

For the case of dihedral Coxeter symmetry ($s_i$1), the authors provide a detailed spectral analysis leading to compact, resummed expressions for the invariant functions $s_i$2. The dihedral group representations are classified into three types, depending on geometric realization in moduli space:

• Elliptic (finite): yields a finite sum expressible via ordinary Bessel functions.
• Parabolic (affine/infinite, unipotent case): yields objects involving Jacobi theta functions, associated with additive periodization of a Gaussian, and strictly modular transformation properties.
• Hyperbolic (infinite, loxodromic case): leads to infinite sums that are resummed as superpositions of modified Bessel functions of the second kind, corresponding to periodization over multiplicative lattices.

These spectral decompositions have significant convergence and computational advantages: the large-volume limit (deep in the Kähler cone) is captured efficiently by the "raw" instanton sum, but deep in the interior, the dual (resummed) spectral decomposition in terms of Bessel or theta functions converges rapidly.

Numerically, explicit examples are given for threefolds with $s_i$3 (notably, CICY 6771 and 6971), demonstrating the occurrence of infinite dihedral symmetry, the structure of the Mori and Kähler cones, stabilizer loci, and the explicit form of the prepotential decompositions.

Harmonic Analysis and Laplacian Interpretation

The appearance of Bessel and theta functions is shown to have a geometric origin: the action of the Coxeter group defines a symmetric bilinear form (interpreted as a metric) on the moduli space, leading to a Laplace-Beltrami operator whose eigenfunctions underpin the spectral decomposition of the prepotential expansion. Specifically, the Coxeter-invariant functions are solutions to the Helmholtz equation

$s_i$4

where $s_i$5 is the Laplace-Beltrami operator for the invariant metric. In the dihedral cases, separation of variables produces Bessel and theta functions as explicit eigenmodes. This harmonizes the duality between exponential (instanton sum) and spectral (eigenmode) expansions and explains the complementary convergence properties.

General Coxeter Groups: Automata and Dihedral Block Decomposition

For arbitrary Coxeter groups, the authors adapt techniques from geometric group theory and automata—particularly finite-state automata traversing the group's Cayley graph—to avoid overcounting in the orbit sum and build resolvent-type linear algebraic expressions for the full instanton expansion. To exploit the solvability of the dihedral case, a decomposition into "dihedral blocks" associated to edges in the Coxeter graph is devised, capturing the local symmetry along two-flop subsystems and resolving their mixing via linear operators. This approach makes the general structure computationally tractable and elucidates the interplay between global group theory and local geometric data.

Implications and Outlook

The explicit identification of Coxeter symmetries from isomorphic flops imposes exacting constraints on the structure of the prepotential—and therefore on the effective supergravity couplings—arising in compactifications on Calabi–Yau threefolds. The decomposition into Coxeter-invariant functions, their spectral properties, and their geometric interpretation via Laplacian eigenmodes uncover a rigid mathematical structure underlying the string vacuum degeneracy and quantum corrections.

Key claims include:

• The non-flop sector of the prepotential is forced to assemble into Coxeter-invariant (spectrally decomposed) functions, determined by the combinatorics of flop transitions.
• The Laplace-Beltrami operator, constructed from the Coxeter-invariant metric, governs the relevant Helmholtz/heat equations whose solutions encode instanton corrections.
• The duality between the raw and resummed expansions is systematic, with spectral convergence properties aligning with physical expectations about moduli dependence.

Future directions include:

• Extension of the Coxeter symmetry classification to larger datasets (e.g., Kreuzer-Skarke), possibly discovering new infinite or exceptional groups.
• Explicit construction of Laplace-Beltrami operators for higher-rank and non-dihedral Coxeter symmetries and the study of their eigenmodes.
• Analysis of higher-genus Gromov-Witten invariants and their organization under Coxeter symmetries, with ramifications for topological string theory and the swampland conjectures.
• Exploration of flux compactifications and $s_i$6 settings where these rigidities might survive or constrain moduli stabilization.

Conclusion

This work rigorously formulates and answers the question of how isomorphic flops generate nontrivial Coxeter symmetries in the moduli space of Calabi-Yau threefolds, and how these symmetries act as powerful constraints forcing the 4D $s_i$7 prepotential into a sharply defined, spectrally decomposed structure. The results clarify both the geometry of string vacua and the algebraic structure of quantum corrections, providing new computational tools and suggesting deep connections between birational geometry, representation theory, and quantum field theory. The construction of the CICY Coxeter database and the explicit harmonic analysis both establish a platform for further investigations into the symmetry properties of compactification moduli spaces and their physical implications.