Integrating simple genus two string invariants over moduli space

Anirban Basu

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Integrating simple genus two string invariants over moduli space

Anirban Basu

High Energy Physics - Theory · Submitted 27 Dec 2020

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that both the integrals vanish.

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We consider an Sp(4,Z) invariant expression involving two factors of the Kawazumi--Zhang (KZ) invariant each of which is a modular graph with one link, and four derivatives on the moduli space of genus two Riemann surfaces. Manipulating it, we show that the integral over moduli space of a linear combination of a modular graph with two links and the square of the KZ invariant reduces to a boundary integral. We also consider an Sp(4,Z) invariant expression involving three factors of the KZ invariant and six derivatives on moduli space, from which we deduce that the integral over moduli space of a modular graph with three links reduces to a boundary integral. In both cases, the boundary term is completely determined by the KZ invariant. We show that both the integrals vanish.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider an Sp(4,Z) invariant expression involving two factors of the Kawazumi--Zhang (KZ) invariant each of which is a modular graph with one link, and four derivatives on the moduli space of genus two Riemann surfaces.	contribution 0.85	0.66	0.04	asserted 0.88
s1	Manipulating it, we show that the integral over moduli space of a linear combination of a modular graph with two links and the square of the KZ invariant reduces to a boundary integral.	result 0.79	0.96	0.16	asserted 1.00
s2	We also consider an Sp(4,Z) invariant expression involving three factors of the KZ invariant and six derivatives on moduli space, from which we deduce that the integral over moduli space of a modular graph with three links reduces to a boundary integral.	result 0.52	0.95	0.11	asserted 1.00
s3	In both cases, the boundary term is completely determined by the KZ invariant.	result 0.98	0.90	0.10	asserted 1.00
s4	We show that both the integrals vanish.	result 1.00	0.96	0.75	asserted 1.00

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Original abstract

We consider an Sp(4,Z) invariant expression involving two factors of the Kawazumi--Zhang (KZ) invariant each of which is a modular graph with one link, and four derivatives on the moduli space of genus two Riemann surfaces. Manipulating it, we show that the integral over moduli space of a linear combination of a modular graph with two links and the square of the KZ invariant reduces to a boundary integral. We also consider an Sp(4,Z) invariant expression involving three factors of the KZ invariant and six derivatives on moduli space, from which we deduce that the integral over moduli space of a modular graph with three links reduces to a boundary integral. In both cases, the boundary term is completely determined by the KZ invariant. We show that both the integrals vanish.

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2 arXiv papers cite this in 4 sentences

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What it cites · 19 sentences without a confident label

Uses the method or tool of 2 papers

In obtaining these relations which are valid everywhere in the bulk of moduli space, we use the eigenvalue equation the KZ invariant satisfies, as well as the identities deduced in [ref] .

Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality · Introduction · 0.85
In order to evaluate the integrals of $ℬ_2 - ℬ_1^2$ and $ℬ_9$ over moduli space, we shall think of the boundary contributions as limits of contributions in the bulk [ref] .

Matching the $D^6 {\cal R}^4$ interaction at two-loops · Integrating simple string invariants over moduli space · 1.00

Uses data or values from 2 papers

On the other hand, the asymptotic expansions around the non–separating node are given by[In the asymptotic expansion of $ℬ_2$, apart from terms that are $O(e^-t)$, we have also ignored the $O(t^0)$, $O(t^-1)$ and $O(t^-2)$ terms [ref] , as they are not relevant for our analysis. ] ℬ_1 = -2πt/3 -2 g(v) -5F_2 (v)/πt +O(e^-t), ℬ_2 = 4π^2t^2/9+8πt g(v)/3 +O(t^0).

Higher genus modular graph functions, string invariants, and their exact asymptotics · Integrating simple string invariants over moduli space · 0.92
On the other hand, the asymptotic expansions around the non–separating node are given by[In the asymptotic expansion of $ℬ_2$, apart from terms that are $O(e^-t)$, we have also ignored the $O(t^0)$, $O(t^-1)$ and $O(t^-2)$ terms [ref] , as they are not relevant for our analysis. ] ℬ_1 = -2πt/3 -2 g(v) -5F_2 (v)/πt +O(e^-t), ℬ_2 = 4π^2t^2/9+8πt g(v)/3 +O(t^0).

Asymptotics of the $D^8 R^4$ genus-two string invariant · Integrating simple string invariants over moduli space · 0.92

Cites as background 36 papers

Let us consider the terms in the effective action that arise in the low momentum expansion of the four graviton amplitude at genus two [ref] .

Two-Loop Superstrings VI: Non-Renormalization Theorems and the 4-Point Function · Introduction · 0.92
The leading contribution which is the $D^4ℛ^4$ interaction involves simply integrating over the volume element of the fundamental domain of $Sp(4,ℤ)$ [ref] .

Two-loop superstrings and S-duality · Introduction · 0.81
Such integrals have been considered at genus one and two leading to various interactions in the effective action [ref] .

Multiparticle one-loop amplitudes and S-duality in closed superstring theory · Introduction · 0.88

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