A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces

Abstract

The Kawazumi-Zhang invariant $\phi$ for compact genus-two Riemann surfaces was recently shown to be a eigenmode of the Laplacian on the Siegel upper half-plane, away from the separating degeneration divisor. Using this fact and the known behavior of $\phi$ in the non-separating degeneration limit, it is shown that $\phi$ is equal to the Theta lift of the unique (up to normalization) weak Jacobi form of weight $-2$. This identification provides the complete Fourier-Jacobi expansion of $\phi$ near the non-separating node, gives full control on the asymptotics of $\phi$ in the various degeneration limits, and provides a efficient numerical procedure to evaluate $\phi$ to arbitrary accuracy. It also reveals a mock-type holomorphic Siegel modular form of weight $-2$ underlying $\phi$. From the general relation between the Faltings invariant, the Kawazumi-Zhang invariant and the discriminant for hyperelliptic Riemann surfaces, a Theta lift representation for the Faltings invariant in genus two readily follows.

16 pages; v2: many improvements: the main conjecture is now a theorem, numerical checks and applications are performed, connections to Gromov-Witten invariants are discussed, various clarifications throughout, 3 extra pages, 10 extra references; v3: cosmetic changes, added details on the proof of (78), one new reference; v4: journal version

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• The SAGEX Review on Scattering Amplitudes, Chapter 10: Selected topics on modular covariance of type IIB string amplitudes and their $\mathcal{N}=4$ supersymmetric Yang-Mills duals
• Poincaré series for modular graph forms at depth two. I. Seeds and Laplace systems
• Infrared divergences and harmonic anomalies in the two-loop superstring effective action
• Exact ${\nabla }^4{R}^4$ couplings and helicity supertraces
• Eigenvalue equation for genus two modular graphs
• New explicit formulas for Faltings' delta-invariant
• Relations between elliptic modular graphs
• $D^6{R}^4$ curvature corrections, modular graph functions and Poincaré series
• 1/8-BPS Couplings and Exceptional Automorphic Functions
• Néron-Tate heights of cycles on jacobians
• Asymptotics of the $D^8{R}^4$ genus-two string invariant
• String theory integrands and supergravity divergences
• Identities among higher genus modular graph tensors
• Poisson equation for genus two string invariants: a conjecture
• Frobenius' theta function and Arakelov invariants in genus three
• Integrating simple genus two string invariants over moduli space
• One-loop matrix elements of effective superstring interactions: ${\alpha }^{\prime }$-expanding loop integrands