Asymptotics of the $D^8{R}^4$ genus-two string invariant

Abstract

We continue our investigation of the modular graph functions and string invariants that arise at genus-two as coefficients of low energy effective interactions in Type II superstring theory. In previous work, the non-separating degeneration of a genus-two modular graph function of weight $w$ was shown to be given by a Laurent polynomial in the degeneration parameter $t$ of degree $(w,w)$. The coefficients of this polynomial generalize genus-one modular graph functions, up to terms which are exponentially suppressed in $t$ as $t\to \infty$. In this paper, we evaluate this expansion explicitly for the modular graph functions associated with the $D^8{\mathcal{R}}^4$ effective interaction for which the Laurent polynomial has degree $(2,2)$. We also prove that the separating degeneration is given by a polynomial in the degeneration parameter $\ln (|v|)$ up to contributions which are power-behaved in $v$ as $v\to 0$. We further extract the complete, or tropical, degeneration and compare it with the independent calculation of the integrand of the sum of Feynman diagrams that contributes to two-loop type II supergravity expanded to the same order in the low energy expansion. We find that the tropical limit of the string theory integrand reproduces the supergravity integrand as its leading term, but also includes sub-leading terms proportional to odd zeta values that are absent in supergravity and can be ascribed to higher-derivative stringy interactions.

arXiv comment50+44 pages, 13 figures; v2: minor corrections, including an overall factor of 2 in Eq. 5.18, published version; v3: more factors of 2 corrected, see footnote 7

References in corpus (20)

• On the Relationship between Yang-Mills Theory and Gravity and its Implication for Ultraviolet Divergences
• Duality and higher derivative terms in M theory
• Two-Loop Superstrings VI: Non-Renormalization Theorems and the 4-Point Function
• Two loops in eleven dimensions
• The low energy expansion of the one-loop type II superstring amplitude
• Low energy expansion of the four-particle genus-one amplitude in type II superstring theory
• Equivalence of Two-Loop Superstring Amplitudes in the Pure Spinor and RNS Formalisms
• Two-loop superstrings and S-duality
• On the modular structure of the genus-one Type II superstring low energy expansion
• Super-Poincare Covariant Two-Loop Superstring Amplitudes
• Closed Superstring Amplitudes, Single-Valued Multiple Zeta Values and Deligne Associator
• Matching the $D^6{\mathcal{R}}^4$ interaction at two-loops
• Lectures on Two-Loop Superstrings
• Higher genus modular graph functions, string invariants, and their exact asymptotics
• Proving relations between modular graph functions
• Tropical Amplitudes
• Multiloop World-Line Green Functions from String Theory
• Worldline Green Functions for Arbitrary Feynman Diagrams
• Johnson's homomorphisms and the Arakelov-Green function
• A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces

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• Higher genus modular graph functions, string invariants, and their exact asymptotics
• All loop structures in Supergravity Amplitudes on $Ad{S}_5\times S^5$ from CFT
• Poisson equations for elliptic modular graph functions
• Relations between elliptic modular graphs
• 1/8-BPS Couplings and Exceptional Automorphic Functions
• Supergravity limit of genus two modular graph functions in the worldline formalism
• Integrating simple genus two string invariants over moduli space