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String theory dualities and supergravity divergences

Michael B. Green, Jorge G. Russo, Pierre Vanhove

TL;DR

This work shows that duality-invariant automorphic coefficient functions in the Type II string theory low-energy expansion fix the coefficients of ultraviolet divergences in maximal supergravity across dimensions. By analyzing the analytic part of the four-graviton amplitude and its Eisenstein-series structure, the authors relate logarithmic terms in moduli to UV divergences seen in dimensional regularisation, and demonstrate how nonperturbative instanton contributions overwhelm perturbative terms in the supergravity limit. The results establish precise correspondences between L-loop divergences in D=8,7,6 and specific derivative interactions (R^4, ∂^4R^4, ∂^6R^4), while revealing that higher-order terms may not be protected and could introduce new divergences at higher loops. Collectively, the work underscores the inextricable link between string dualities and the ultraviolet behavior of supergravity, and highlights the essential role of instantons in the low-energy limit.

Abstract

We demonstrate how duality invariance of the low energy expansion of the four-graviton amplitude in type II string theory determines the precise coefficients of multiloop logarithmic ultraviolet divergences of maximal supergravity in various dimensions. This is illustrated by the explicit moduli-dependence of terms of the form D^{2k}R^4, with k<= 3, in the effective action. Furthermore, we show that in the supergravity limit the perturbative contributions are swamped by an accumulation of non-perturbative effects of zero-action instantons.

String theory dualities and supergravity divergences

TL;DR

This work shows that duality-invariant automorphic coefficient functions in the Type II string theory low-energy expansion fix the coefficients of ultraviolet divergences in maximal supergravity across dimensions. By analyzing the analytic part of the four-graviton amplitude and its Eisenstein-series structure, the authors relate logarithmic terms in moduli to UV divergences seen in dimensional regularisation, and demonstrate how nonperturbative instanton contributions overwhelm perturbative terms in the supergravity limit. The results establish precise correspondences between L-loop divergences in D=8,7,6 and specific derivative interactions (R^4, ∂^4R^4, ∂^6R^4), while revealing that higher-order terms may not be protected and could introduce new divergences at higher loops. Collectively, the work underscores the inextricable link between string dualities and the ultraviolet behavior of supergravity, and highlights the essential role of instantons in the low-energy limit.

Abstract

We demonstrate how duality invariance of the low energy expansion of the four-graviton amplitude in type II string theory determines the precise coefficients of multiloop logarithmic ultraviolet divergences of maximal supergravity in various dimensions. This is illustrated by the explicit moduli-dependence of terms of the form D^{2k}R^4, with k<= 3, in the effective action. Furthermore, we show that in the supergravity limit the perturbative contributions are swamped by an accumulation of non-perturbative effects of zero-action instantons.

Paper Structure

This paper contains 13 sections, 69 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Summary of duality invariant coefficients in the low energy expansion
  3. Constraints on the coefficients
  4. Definition and properties of Eisenstein series.
  5. Logarithmic terms and ultraviolet divergences in supergravity
  6. Logarithmic thresholds in the Einstein frame
  7. The supergravity limit and instanton corrections
  8. The perturbative terms
  9. Supergravity limit including the instanton terms
  10. Comments on higher-order interactions and higher-loop supergravity.
  11. Summary and discussion of higher-order contributions
  12. The $\partial^6\mathcal{R}^4$ interaction in $D=6$ dimensions.
  13. Normalisations

Figures (1)

  • Figure 1: The Dynkin diagrams of the U-duality groups $E_{d+1(d+1)}$ ($0\leq d\leq 7$)