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The D^6 R^4 term in type IIB string theory on T^2 and U-duality

Anirban Basu

TL;DR

The paper addresses the D^6 ${\cal{R}}^4$ coupling in type IIB string theory compactified on $T^2$ by constructing a manifestly U-duality invariant modular form. It combines $SL(3,\mathbb{Z})_M$ and $SL(2,\mathbb{Z})_U$ Eisenstein-series contributions with nontrivial functions $f(T,\bar{T})$ and $f(U,\bar{U})$, and provides a nonperturbative completion incorporating D-instantons and $(p,q)$-string instantons; the structure is tightly constrained by perturbation theory, decompactification limits, and seven- and eleven-dimensional dualities. The authors verify parts of the perturbative content via eleven-dimensional supergravity on $T^3$ and derive an explicit Poisson/Laplace equation framework for the exact modular form, showing consistency with known ten- and nine-dimensional limits and making concrete predictions for higher-genus contributions to related higher-derivative interactions. This work advances understanding of how U-duality constrains higher-derivative couplings and provides testable nonperturbative predictions across M-/string-theory dualities.

Abstract

We propose a manifestly U-duality invariant modular form for the D^6 R^4 interaction in the effective action of type IIB string theory compactified on T^2. It receives perturbative contributions upto genus three, as well as non-perturbative contributions from D-instantons and (p,q) string instantons wrapping T^2. Our construction is based on constraints coming from string perturbation theory, U-duality, the decompactification limit to ten dimensions, and the equality of the perturbative part of the amplitude in type IIA and type IIB string theories. Using duality, parts of the perturbative amplitude are also shown to match exactly the results obtained from eleven dimensional supergravity compactified on T^3 at one loop. We also obtain parts of the genus one and genus k amplitudes for the D^{2k} R^4 interaction for arbitrary k > 3. We enhance a part of this amplitude to a U-duality invariant modular form.

The D^6 R^4 term in type IIB string theory on T^2 and U-duality

TL;DR

The paper addresses the D^6 coupling in type IIB string theory compactified on by constructing a manifestly U-duality invariant modular form. It combines and Eisenstein-series contributions with nontrivial functions and , and provides a nonperturbative completion incorporating D-instantons and -string instantons; the structure is tightly constrained by perturbation theory, decompactification limits, and seven- and eleven-dimensional dualities. The authors verify parts of the perturbative content via eleven-dimensional supergravity on and derive an explicit Poisson/Laplace equation framework for the exact modular form, showing consistency with known ten- and nine-dimensional limits and making concrete predictions for higher-genus contributions to related higher-derivative interactions. This work advances understanding of how U-duality constrains higher-derivative couplings and provides testable nonperturbative predictions across M-/string-theory dualities.

Abstract

We propose a manifestly U-duality invariant modular form for the D^6 R^4 interaction in the effective action of type IIB string theory compactified on T^2. It receives perturbative contributions upto genus three, as well as non-perturbative contributions from D-instantons and (p,q) string instantons wrapping T^2. Our construction is based on constraints coming from string perturbation theory, U-duality, the decompactification limit to ten dimensions, and the equality of the perturbative part of the amplitude in type IIA and type IIB string theories. Using duality, parts of the perturbative amplitude are also shown to match exactly the results obtained from eleven dimensional supergravity compactified on T^3 at one loop. We also obtain parts of the genus one and genus k amplitudes for the D^{2k} R^4 interaction for arbitrary k > 3. We enhance a part of this amplitude to a U-duality invariant modular form.

Paper Structure

This paper contains 18 sections, 189 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The perturbative part of the proposed modular form
  3. Constraints using string perturbation theory
  4. Constraints using U--duality and the decompactification limit
  5. Evidence using eleven dimensional supergravity at one loop on $T^3$
  6. The expression for the exact modular form
  7. Understanding the structure of $f(T,\bar{T})$
  8. Understanding the non--perturbative structure of ${\mathcal{E}}_{(3/2,3/2)} (M)$
  9. More predictions from eleven dimensional supergravity at one loop on $T^3$
  10. Discussion
  11. Appendix
  12. Expressions for the Eisenstein series
  13. The Eisenstein series for $SL(2,\mathbb{Z})$
  14. The Eisenstein series for $SL(3,\mathbb{Z})$
  15. Calculating $\hat{I}_1$ and $\hat{I}_2$
  16. ...and 3 more sections

Figures (4)

  • Figure 1: Schematics of the torus amplitude.
  • Figure 2: Degeneration limits of the genus two and genus three surfaces.
  • Figure 3: Schematics of the $D^4 R^4$ torus amplitude.
  • Figure 4: Schematics of part of the $D^8 R^4$ torus amplitude.