Spin Parity of k-Differentials

• Spin parity of k-differentials is a key invariant that distinguishes connected components in the moduli space based on zeros and poles of the differential.
• It is computed via methods including Arf invariants, explicit arithmetic formulas, and canonical cyclic covers, linking geometric and combinatorial aspects.
• These insights have implications in flat geometry, Teichmüller theory, and enumerative geometry, influencing the study of tautological classes and intersection theory.

A $k$-differential on a compact Riemann surface $X$ of genus $g$ is a (possibly meromorphic) section of $K_X^{\otimes k}$, the $k$-th tensor power of the canonical bundle. The moduli space of $k$-differentials stratifies according to the number and multiplicities of zeros and poles, leading to an intricate topological and geometric structure. The notion of spin parity, which for $k=1$ coincides with the parity of theta-characteristics, extends to general $k$ and serves as a powerful invariant distinguishing components of strata of $k$-differentials, determining deep connections with the geometry of moduli spaces, their tautological classes, and refined intersection-theoretic structures.

1. Foundations: $k$-Differentials and Spin Structures

Let $X$ be a compact Riemann surface of genus $g$, and fix integers $k>0$ and a partition $\mu=(m_1,\ldots,m_n)$ of $k(2g-2)$, i.e., $\sum_i m_i=k(2g-2)$. A $k$-differential is a section $\xi\in H^0(X,K_X^{\otimes k})$ whose divisor consists of zeros and poles of orders $m_1,\ldots,m_n$ at marked points $p_1,\ldots,p_n$. The corresponding moduli locus $\Omega^k\mathcal M_g(\mu)$ forms a natural stratum in the moduli space of $k$-differentials, central for the study of flat surfaces, Teichmüller theory, and algebraic geometry (Chen et al., 2021).

When $k=1$, the classical concept of spin structures (theta-characteristics) appears: a line bundle $L$ satisfying $L^{\otimes 2}\cong K_X$. The parity of such a theta-characteristic is defined as $\dim H^0(X,L)\bmod2$. For higher $k$—especially for $k$ odd—this structure lifts via the canonical cyclic cover $\pi: \widehat X\to X$ of degree $k$ determined by $\pi^*\xi=\widehat\omega^k$, and the parity is transferred to $\widehat\omega$ on $\widehat X$ by requiring that the zeros and poles are all even, i.e., the stratum is of parity type (Chen et al., 3 Feb 2026, Wong, 2022).

2. Definition and Computation of Spin Parity for $k$-Differentials

For $k>1$, spin parity is defined for "parity-type" $k$-differentials, i.e., those for which the canonical cover has abelian differentials with all even singularities. Explicitly, for $(X,\xi)\in \Omega^k\mathcal M_g(\mu)$, the spin parity is

$$\mathrm{parity}(X,\xi) = \dim H^0\left(\widehat X, \frac{1}{2}\,\mathrm{div}(\widehat\omega)\right)\bmod2,$$

where $\widehat\omega$ is the abelian differential satisfying $\pi^*\xi = \widehat\omega^k$. This generalizes the parity of theta-characteristics for $k=1$ (Chen et al., 2021, Wong, 2022).

Equivalently, in the tautological description, for $\eta$ a $k$-differential with $\mathrm{div}(\eta)=\sum m_i p_i$ and all $m_i$ even (even type), one attaches the line bundle

$$L = \omega_X^{\otimes(1-k)/2}\otimes\mathcal O_X\left(\frac{1}{2}\sum m_i p_i\right), \quad L^{\otimes2}\cong \omega_X,$$

and defines the spin parity as the parity of $h^0(X,L)$ (Wong, 2022, Chen et al., 3 Feb 2026).

In flat geometric terms, the spin parity can be computed as the Arf invariant of the associated quadratic form on $H_1(X,\mathbb{Z}/2)$, given algebraically by

$$\mathrm{parity} = \sum_{i=1}^g (Ind_\omega(\alpha_i)+1)(Ind_\omega(\beta_i)+1)\bmod2,$$

for a symplectic basis $\{\alpha_i,\beta_i\}$ and $Ind_\omega(\gamma)$ the tangent vector winding index (Chen et al., 2021, Wong, 2022).

3. Classification and Component Structure by Spin Parity

The role of spin parity is fundamentally different for even and odd $k$:

• Even $k$: Theorem 5.3 of (Chen et al., 2021) states that for even $k$ and signatures $\mu$ of parity type, every connected component of the primitive stratum $\Omega^k(\mu)^{prim}$ has the same spin parity—hence parity does not further split the stratum.
• Odd $k$: Theorem 5.7 of (Chen et al., 2021) proves that for odd $k\geq3$, and $\mu$ of parity type, except for special cases, each primitive stratum $\Omega^k(\mu)^{prim}$ contains both even and odd parity components, so spin parity distinguishes at least two connected components.

For quadratic differentials ($k=2$), the behavior aligns with Lanneau’s classification; for $k>2$ spin parity provides the main invariant distinguishing non-hyperelliptic components (Chen et al., 2021).

4. Explicit Formulas and Arithmetic Characterization

In genus zero and one, closed formulas determine spin parity via elementary arithmetic:

• Genus 0 (odd $k$): Given $\mu=(m_1,\dots,m_n)$ all even, define $\mathcal Q$ as the set of primes $q$ dividing $k$ with $\lfloor(q+1)/4\rfloor\equiv1\bmod2$, and set

$$n_k(\mu) = \#\left\{i:\;\nu_{\mathcal Q}(m_i)\not\equiv\nu_{\mathcal Q}(k)\bmod2\right\},$$

with $$\nu_{\mathcal Q}(m)=\sum_{q_j\in\mathcal Q}\min(\nu_{q_j}(m),\ell_j)$$. Then the parity is $n_k(\mu)\bmod2$ (Chen et al., 3 Feb 2026, Chen et al., 2021).

• Genus 1 (odd $k$): For rotation number $d\mid k$,

$\mathrm{parity} = (n_k(\mu)+d+1)\bmod2.$

An equivalent form is given in terms of Jacobi symbols:

$$n_k(\mu) = \#\Bigl\{\,i:\;\left(\frac{2}{\gcd(m_i,k)}\right)\neq\left(\frac{2}{k}\right)\Bigr\},$$

where $\left(\frac{2}{a}\right)$ is the Jacobi symbol (Chen et al., 3 Feb 2026).

Recent work formalizes the underlying number-theoretic conjectures, linking the parity computation to combinatorial sums governed by Eisenstein’s lemma and quadratic reciprocity for the Jacobi symbol (Chen et al., 3 Feb 2026).

5. Tautological Classes, Refined Double Ramification Cycles, and Intersection Theory

Spin parity enters the intersection theory of strata of differentials in the moduli space $\overline{\mathcal M}_{g,n}$ via the construction of tautological Chow and cohomology classes:

• The class $$[\mathcal D r_g(a,k)]^\pm\in A^*(\overline{\mathcal M}_{g,n})$$ for the stratum of $k$-differentials with fixed parity is tautological and uniquely determined by explicit star-graph sums, known as the spin star-graph formula (Holmes et al., 3 Sep 2025). The star-graph expansion decomposes the boundary stratification respecting spin:

$$[\mathcal D r_g(a,k)]^\pm = \sum_{(\Gamma,I)\in SStar_g^{odd}(a,k)}c_{\Gamma,I}\,\zeta_{\Gamma*}\Bigl([\mathcal D r_{g(v_0)}(I(v_0),k)]^\pm \otimes \bigotimes_{v\in V_{out}}[\mathcal D r_{g(v)}(I(v)/k)]^\pm\Bigr),$$

where coefficients $c_{\Gamma,I}$ encode the combinatorics and $SStar_g^{odd}(a,k)$ is the set of odd-twisted simple star graphs (Holmes et al., 3 Sep 2025).

• The refined double ramification (DR) cycle $DR_g^{spin}(a,k)$, constructed via Pixton-type graph sums restricted to odd edge weights, captures the spin-variant stratification in $A^g(\overline{\mathcal M}_{g,n})$, and satisfies $DR_g^{\pm}=DR_g\pm DR_g^{spin}/2$ (Costantini et al., 2021, Holmes et al., 3 Sep 2025).

A unifying theorem asserts $$DR_g^{\pm}(a,k) = [\mathcal D r_g(a,k)]^\pm = \mathcal P_g^{\pm}(a,k)$$ (Pixton's spin class) in the Chow ring (Holmes et al., 3 Sep 2025).

Integrals of tautological classes split by parity admit explicit generating functions involving exponential and hyperbolic cosine factors, reflecting the deep connection between the topology of these strata and their spin refinement (Costantini et al., 2021).

6. Computational Aspects and Algorithmic Approaches

The computation of cycle classes for spin components is algorithmically accessible. The primary workflow involves:

1. Boundary restriction via clutching maps: Classes of spin strata are recursively determined by restrictions to the boundary, via 1-edge clutching morphisms and contraction formulas, reducing calculations to lower-genus, lower-complexity cases (Wong, 2022).
2. Multi-scale differentials: The compactification by the moduli of multi-scale $k$-differentials provides a normal-crossing boundary stratification, crucial for tracing spin parity through degeneration [BCCGM19 in (Wong, 2022)].
3. Linear algebraic reconstruction: For each tautological degree, boundary pullbacks assemble into a linear system whose unique solution is the spin class in question (Wong, 2022).
4. Sage/admcycles implementation: Explicit computations of spin strata and DR cycles, including checking conjectural formulas, are implemented in admcycles and verified for $g\leq4,\,n\leq3$ (Wong, 2022).

These methods are robust for all $k$ and arbitrary signatures $\mu$ of even type.

7. Examples, Applications, and Interactions with Hyperelliptic Structures

Explicit examples in low genus illustrate the arithmetic and geometric richness:

• In genus $0$ (odd $k$), the parity of a stratum $\Omega^k_0(2m_1,2m_2,2m_3)$ is even if any $m_i\equiv 0\pmod{k}$, otherwise determined by the parity of an explicit lattice count or the function $n_k(\mu)\bmod2$ (Chen et al., 2021, Chen et al., 3 Feb 2026).
• In genus $1$ (odd $k$), the connected components labeled by rotation $d$ have parity $(n_k(\mu)+d+1)\bmod2$, recovering and confirming earlier conjectural descriptions (Chen et al., 3 Feb 2026).
• Hyperelliptic components correspond to configurations grouped at Weierstrass points and are always contained in a single spin parity class, with the corresponding component being either "even" or "odd" depending on the configuration (Chen et al., 2021).

Closed formulas exist for the Euler characteristic and intersection numbers with $\psi$-classes on even/odd components, allowing for direct computation of Masur–Veech volumes, Siegel–Veech constants, sums of Lyapunov exponents, and other enumerative invariants stratified by parity (Costantini et al., 2021).

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References:

• (Chen et al., 2021)
• (Wong, 2022)
• (Chen et al., 3 Feb 2026)
• (Holmes et al., 3 Sep 2025)
• (Costantini et al., 2021)