Kontsevich–Zorich sum of Lyapunov exponents

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Numbers

$g$

$\mu$

$c$

$L_{g,\mu,c}$

hyp:

4/3

1, 1

hyp:

3/2

hyp:

9/5

odd:

8/5

3, 1

connected:

7/4

2, 2

hyp:

2, 2

odd:

5/3

2, 1, 1

connected:

11/6

1, 1, 1, 1

connected:

53/28

hyp:

16/7

even:

odd:

13/7

5, 1

connected:

4, 2

even:

32/15

4, 2

odd:

29/15

4, 1, 1

connected:

1137/550

3, 3

hyp:

5/2

3, 3

nonhyp:

3, 2, 1

connected:

25/12

3, 1, 1, 1

connected:

66/31

2, 2, 2

even:

737/333

comment: The value $737/333$ in [1], Appendix A, follows from the cylinder configurations in [3], Appendix B.2; it differs from $166/75$ printed in [4], Section 6.9.

2, 2, 2

odd:

2, 2, 1, 1

connected:

5045/2358

2, 1, 1, 1, 1

connected:

131/60

1, 1, 1, 1, 1, 1

connected:

839/377

Definition

The Kontsevich–Zorich sum $L_{g,\mu,c}=\sum_{j=1}^g\lambda_j$ is the sum of nonnegative Lyapunov exponents of the real Hodge bundle on the area-one stratum component $c\subset\mathcal H(\mu)$, for its Masur–Veech probability measure [1].

Parameters

$g$

— genus ($g\geq2$)

$\mu$

— zero profile ($\mu=(m_1,\ldots,m_n)$ is a non-increasing partition of $2g-2$ into positive integers)

$c$

— connected component (A connected component allowed by the classification in [2], with the labels defined in Comments)

Formulas

(1)

For the area Siegel–Veech constant $c_{\mathrm{area}}(g,\mu,c)$ with the same component probability measure, $\displaystyle L_{g,\mu,c}=\frac1{12}\sum_{i=1}^n\frac{m_i(m_i+2)}{m_i+1}+\frac{\pi^2}{3}c_{\mathrm{area}}(g,\mu,c)$ [1].

(2)

For $g\geq2$, $\displaystyle L_{g,(2g-2),\mathrm{hyp}}=\frac{g^2}{2g-1}$ [1].

(3)

For $g\geq2$, $\displaystyle L_{g,(g-1,g-1),\mathrm{hyp}}=\frac{g+1}{2}$ [1].

Comments

(4)

The zero profile $\mu=(m_1,\ldots,m_n)$ specifies the positive orders of all zeros of a nonzero holomorphic differential on a compact connected genus-$g$ Riemann surface. There are no poles or added regular marked points. A component carries its own probability measure; a measure supported on a Teichmüller curve inside it can have a different Lyapunov sum [4].

(5)

The Masur–Veech measure is defined by the relative-period lattice $H^1(X,\{z_i\};\mathbb Z\oplus i\mathbb Z)$ of covolume one. All zeros are labelled, including equal-order zeros. The area is $A=(i/2)\int_X\omega\wedge\bar\omega=1$, and the phase of $\omega$ is retained. For $D=2g+n-1$ and $\omega=r\omega_1$ with $A(\omega_1)=1$, the radial convention is $d\nu=r^{2D-1}dr\,d\nu_1$. The restriction to each component is divided by that component's volume to obtain its probability measure [3].

(6)

The label hyp denotes the hyperelliptic component; for $\mathcal H(g-1,g-1)$ its involution interchanges the two zeros. The labels even and odd denote the nonhyperelliptic components of even and odd spin parity; they exclude the hyperelliptic component even when it has the same parity. The label nonhyp denotes the nonhyperelliptic component of $\mathcal H(g-1,g-1)$ for even $g\geq4$. The label connected denotes the sole component of any other connected stratum. Both genus-two strata use hyp [2].

(7)

The Kontsevich–Zorich cocycle is Gauss–Manin parallel transport on $H^1(X;\mathbb R)$, with Lyapunov exponents measured using the Hodge norm [1]. The Teichmüller flow is $\operatorname{diag}(e^t,e^{-t})$, so $\lambda_1=1$. Its spectrum is $\lambda_1\geq\cdots\geq\lambda_g\geq-\lambda_g\geq\cdots\geq-\lambda_1$. Each entry sums the first $g$ exponents, including $\lambda_1=1$.

Programs

(P1)

Sage

# Download the attached generate.py.
from generate import lyapunov_sum, hyperelliptic_sum
lyapunov_sum(4, (2, 2, 2), 'even')  # exact 737/333
hyperelliptic_sum(5, zeros=1)       # exact 25/9, beyond the stored range

References

[1]

A. Eskin, M. Kontsevich and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow. Publications Mathématiques de l’IHÉS 120 (2014), 207–333. Section 1.6, Theorem 1, Corollaries 1–2 and Appendix A. (arXiv) (doi)

[2]

M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Inventiones Mathematicae 153 (2003), 631–678. Theorems 1–2. (arXiv) (doi)

[3]

A. Eskin, H. Masur and A. Zorich, Moduli spaces of Abelian differentials: the principal boundary, counting problems, and the Siegel–Veech constants. Publications Mathématiques de l’IHÉS 97 (2003), 61–179. Sections 2.1–2.2 and 5.2, Examples 14.12–14.13 and Appendix B.2. (arXiv)

[4]

D. Chen and M. Möller, Nonvarying sums of Lyapunov exponents of Abelian differentials in low genus. Geometry & Topology 16 (2012), 2427–2479. Propositions 4.5 and 4.8; Sections 5 and 6. (arXiv) (doi)

[5]

G. Farkas, Koszul divisors on moduli spaces of curves. American Journal of Mathematics 131 (2009), 819–867. Theorem 4.6.

Links

[6]

Eskin, Kontsevich and Zorich: full text

[7]

Farkas: Koszul divisors on moduli spaces of curves, published full text

Similar tables

T561 — area Siegel–Veech constants of the same components determine these sums through the Eskin–Kontsevich–Zorich formula

T560 — Masur–Veech volumes of the same components normalize the measures used for their Lyapunov exponents

Data properties

Entries are of type: rational number

How well the digits are known: exact

Table is complete: no (it holds all 25 components of all 18 positive zero profiles in genera $2\leq g\leq4$, covering minimal, principal and intermediate strata and every component type in this range; this practical cutoff gives complete low-genus breadth supported by exact source tables, rather than extending only the hyperelliptic family, and genus one is excluded because it requires a regular marked point)

How they were obtained:

The two genus-two values follow from [1], Corollary 2. The other 23 exact rationals are transcribed from the exact sum column of Appendix A, published page 326, available in [6]; the experimental individual exponents are not used. Every value is constructed with exact rational arithmetic, without a precision or rounding claim.

All 25 entries agree with a separate extraction of the source column and the genus-two corollary. The default enumeration is checked against all partitions of $2g-2$ and the component classification. Six values are reconstructed from the sphere double-cover formula, and ten nonhyperelliptic values from the geometric divisor arguments in [4]. For $\mathcal H(3,3)^{\mathrm{nonhyp}}$, the divisor coefficients are taken from [5], Theorem 4.6, available in [7]; its positive marked-point coefficients resolve the minus signs printed in CM equation (8).

Three entries also agree with cylinder-weighted configuration sums from [3], Appendix B.2. These recover $737/333$ for $\mathcal H(2,2,2)^{\mathrm{even}}$, resolving its conflict with $166/75$ in CM Section 6.9. EKZ uses EMZ data, so these reconstructions are checks of the underlying source rather than independent published derivations. Additional mathematical checks cover 17 distinct entries; the remaining eight rely on the exact EKZ source with transcription checks.

The relation to all 25 stored area constants is checked with certified 512-bit arithmetic and their stated decimal uncertainties. T561 itself uses the EKZ sums, so this is a consistency check. Known-answer controls and deliberate corruption checks are included. The reproducible generator, selection justification, source reconciliation and numerical checks are attached.