Cosmological Hyperplane Arrangements

• Cosmological hyperplane arrangements are frameworks that apply the theory of hyperplane arrangements to model and classify cosmic structures using combinatorial and topological invariants.
• They employ techniques like Milnor fibrations, D-module theory, and perverse sheaves to analyze singularities, cyclic structures, and topological defects in cosmological models.
• This approach enables researchers to derive explicit invariants and analytic criteria for cosmic phenomena, enhancing our ability to study phase transitions and cosmic topology.

A cosmological hyperplane arrangement is the application of the mathematical theory of hyperplane arrangements—deeply rooted in combinatorics, topology, and algebraic geometry—to the modeling, classification, and analysis of cosmic structures and phenomena that arise in advanced cosmological theories. This domain synthesizes techniques from the topology of complex arrangements, algebraic D-module theory, perverse sheaves, mixed Hodge theory, and recent developments in the analysis of cosmological correlators through the geometry of singularities governed by hyperplane configurations. The core of the subject is the paper of how configurations of hyperplanes (or more general hypersurfaces) and their complements encode, via explicit combinatorial and cohomological data, the topology, singularity content, and analytic properties of spaces of physical and cosmological interest.

1. Hyperplane Arrangements and Topological Invariants

A finite collection of hyperplanes in a vector space $V$ (with a primary focus on $V \cong \mathbb{C}^{d+1}$) constitutes a hyperplane arrangement $\mathcal{A}$. The complement $M = V \setminus \bigcup_{H \in \mathcal{A}} H$ is a central object, equipped with invariants sensitive to both the combinatorial structure of intersections (encoded by the intersection poset or lattice $L(\mathcal{A})$) and the topology of $M$. The Orlik–Solomon algebra provides an algebraic presentation of the cohomology ring $H^*(M, \mathbb{Z})$ based purely on the lattice $L(\mathcal{A})$.

Characteristic varieties (or cohomology jump loci),

$$V^q_s(M, \Bbbk) = \left\{ \rho \in \operatorname{Hom}(\pi_1(M), \Bbbk^*)\ \Big|\ \dim_{\Bbbk} H^q(M, {}_\rho \Bbbk) \ge s \right\},$$

parametrize the ways in which rank‑1 local system cohomology can have specified ranks, revealing deep topological structures not visible from the Betti numbers alone (Suciu, 2013). These loci can be interpreted, in a cosmological analogy, as capturing "phases" or resonance phenomena of the underlying cosmic topology.

Resonance varieties, defined via the Aomoto complex associated to elements $a \in H^1(M, \Bbbk)$,

$$R^q_s(M, \Bbbk) = \left\{ a \in H^1(M, \Bbbk) \mid \dim_{\Bbbk} H^q(A, \cdot a) \ge s \right\},$$

provide a linearized shadow of the characteristic varieties and are often easier to compute, with potential cosmological relevance for selecting or classifying cohomological or physical modes.

2. Milnor Fibrations, Monodromy, and Cyclic Structures

Given a hyperplane arrangement $\mathcal{A}$ defined by a (homogeneous) polynomial $Q(z) = \prod_{H \in \mathcal{A}} f_H(z)$, the map $Q: M \to \mathbb{C}^*$ yields a Milnor fibration whose fiber $F = Q^{-1}(1)$ supports a monodromy operator $h: F \to F$, $h(z) = e^{2\pi i/N} z$, with $N = \deg Q$ (Suciu, 2013). The action of $h$ on homology, via the characteristic polynomial

$$\Delta_{h,q}(t) = \det(t \cdot \operatorname{id} - h_*),$$

is a topological invariant dictated by the intersection lattice. Monodromy eigenvalues reflect cyclic or periodic structures, echoing physical phenomena such as phase transitions or cyclic cosmological scenarios. In practice, these invariants classify the possible global periodicities or "windings" the universe admits.

3. Algebraic Models: Orlik–Solomon dga, Logarithmic Forms, and Mixed Hodge Theory

The Orlik–Solomon model generalizes to arrangements of hypersurfaces in smooth projective varieties, yielding a differential graded algebra $M^\bullet(X, L)$ whose cohomology computes $H^*(X \setminus L)$ and is compatible with the mixed Hodge structure (Dupont, 2013). The construction hinges on local systems of logarithmic forms,

$$\Omega^\bullet_{\langle X, L \rangle} \subset j_*\Omega^\bullet_{X \setminus L},$$

with a weight filtration $W_k\Omega^\bullet_{\langle X, L \rangle}$ encoding the complexity of pole structure. The isomorphism

$$H^n(\Omega^\bullet_{\langle X, L \rangle}) \cong H^n(X \setminus L, \mathbb{C})$$

provides an explicit, functorial tool for cohomology computation. Blow-ups leading to "wonderful compactifications" reduce the arrangement locally to normal crossings, where Deligne's theory ensures tractable mixed Hodge structures.

In cosmological modeling, this framework encodes the combinatorics of intersecting branes, domain walls, or event horizons within the ambient cosmic geometry, and the mixed Hodge structure offers an algebraic-geometric stratification reflective of physical hierarchies (e.g., scale separation or anomaly levels).

4. D-module Theory, Local Cohomology, and Perverse Sheaves

D-module theory provides algebraic structures for studying modules of differential operators supported on arrangements. For a hyperplane arrangement $A$ in $K^n$ given by $f \in R = K[x_1,\dotsc,x_n]$,

$H^1_{(f)}(R) = R[f^{-1}]/R,$

is a holonomic $D_n$-module (Oaku, 2015). The length and multiplicity of $H^1_{(f)}(R)$ as a $D_n$-module coincide and equal $T(A,1) - 1$, where $T(A,t)$ is the Poincaré polynomial,

$$T(A, t) = \sum_{X\in L(A)}\mu(X)(-t)^{\operatorname{codim} X},$$

with $\mu$ the Möbius function on the intersection lattice $L(A)$.

Furthermore, in the category of perverse sheaves, the direct image $Rj_*(\mathbb{C}_U[n])$ for the inclusion $j: U \hookrightarrow \mathbb{C}^n$ decomposes into irreducibles with length

$$c(Rj_*(\mathbb{C}_U[n])) = \sum_{F \in L(A)}|\mu(F)|,$$

and each perverse sheaf corresponds to local cohomology along a flat $F$ (Bøgvad et al., 2017). These invariants control the decomposition of cohomological objects into irreducible "modes," giving a combinatorial blueprint for the possible local and global behaviors of field-theoretical quantities defined on cosmological models constructed from such arrangements.

5. Homotopically Non-trivial Embedded Spheres and Twisted Topological Defects

Recent constructions realize explicitly embedded spheres in the complement of real hyperplane arrangements, formed by shifting the real unit sphere into the imaginary direction specified by a locally consistent system of half-spaces (Yoshinaga, 30 May 2024). The embedded sphere,

$$\mathcal{S}(\epsilon) = \{x + i(-\theta(x))\,|\, x \in S\},$$

is homotopically non-trivial if and only if the system of half-spaces is only locally (but not globally) consistent. Non-triviality is detected via a twisted intersection number,

$$I(f([Z] \otimes \tau), i([W] \otimes \sigma)) = \sum_{p \in f^{-1}(W)} I_p(f(Z), W) \langle \tau(f(p)), \sigma(f(p)) \rangle,$$

with coefficients in a local system. These spheres represent "defects" or non-contractible cycles, analogy to topological defects such as cosmic strings or domain walls in the structure of the universe. This technique provides a topological measure of global obstructions invisible from local observations, a feature relevant for the paper of observable imprints of cosmic topology.

6. Hyperplane Arrangements in Cosmological Integrals and Analytic Structures

Cosmological correlators, encoding primordial statistics, are represented as Mellin integrals of rational functions associated to graphs—the "flat space wavefunction"—whose singularities are dictated by binary hyperplane arrangements: $$\psi_{\epsilon}(X, Y) = \int_{\mathbb{R}_+^n} \psi_{\text{flat}}(X+\alpha, Y)^{\epsilon} d\alpha,$$ with the "binary" arrangement defined by linear forms with coefficients 0 or 1, partitioning the integration domain (Fevola et al., 18 Oct 2024). The singular locus and analytic structure are studied via the Euler discriminant of the arrangement, and the geometry of the arrangement mirrors the time-ordering and causal structure of interactions in cosmology.

Algebraic techniques, including D-module and Weyl algebra approaches, yield systems of differential and (via Mellin transform) recurrence equations—often in the framework of GKZ A-hypergeometric systems—facilitating analytic continuation and recursion for the evaluation of cosmological correlators. A notable advance is a graph-based multivariate partial fraction algorithm: for a given Feynman graph, the flat space wavefunction is decomposed along spanning subgraphs, rendering the analytic structure and time ordering explicit and tractable.

7. Physical Interpretations and Future Directions

The amalgamation of hyperplane arrangement theory, D-module invariants, perverse sheaf decomposition, and combinatorial stratification provides a robust mathematical apparatus for classifying and analyzing the topology of space–time-like models in cosmology. Application areas include:

• Classification of possible cosmic topologies via characteristic and resonance varieties.
• Rigorous encoding of the moduli of field configurations and the impact of domain wall or brane intersections on cohomology and singularity structure.
• Detection and classification of global topological defects using twisted homotopy invariants.
• Analysis of singularities and analytic continuations of cosmological integrals via combinatorial and geometric invariants of hyperplane arrangements.
• Computation of physical observables (such as inflationary correlators) through algebraic and combinatorial decompositions related to hyperplane-intersection posets.

Further research is directed toward extending these tools to arrangements in curved or non-affine (cosmologically realistic) settings, interaction with mirror symmetry and string-theoretic degenerations, and the algebraic-geometric analysis of configuration spaces relevant for multi-particle and multinodal cosmic models.

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Table: Core Mathematical Constructs in Cosmological Hyperplane Arrangements

Construct	Mathematical Definition	Physical/Cosmological Analogy
Intersection lattice $L(\mathcal{A})$	Poset of non-empty intersections of hyperplanes	Combinatorial blueprint of cosmic region connectivity
Orlik–Solomon algebra	Exterior algebra modulo relations from $L(\mathcal{A})$	Algebraic encoding of field/charge cohomology
Characteristic varieties $V^q_s$	Jump loci of local system cohomology	Moduli of "phases" or resonance in topology
Milnor fibration $Q: M \to \mathbb{C}^*$	Fibration with monodromy $h$ acting on Milnor fiber homology	Cyclic/periodic cosmic transitions or windings
Poincaré polynomial $T(A,t)$	$T(A, t) = \sum \mu(X)(-t)^{\operatorname{codim} X}$	Counts irreducible field-theory modes per symmetry-breaking
Embedded spheres $\mathcal{S}(\epsilon)$	Shifted sphere in complexified complement, non-trivial for globally inconsistent systems	Topological defects, non-contractible cycles
Mellin integral domains (arrangement)	Complement to binary hyperplane arrangement defined by singular loci of $\ell_i$	Regions of analytic continuation for cosmological integrals

Cosmological hyperplane arrangements, bridging combinatorics, topology, D-module theory, and geometric analysis, provide a unified mathematical platform for probing both the observable structure and the hidden topological complexity of the universe. This synthesis opens direct avenues for the calculation of invariants, analysis of cosmic topology, and the classification of possible "phases" and defect structures in advanced cosmological models.