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Supersymmetry Constraints and String Theory on K3

Ying-Hsuan Lin, Shu-Heng Shao, Yifan Wang, Xi Yin

TL;DR

This work derives second-order differential constraints on the moduli dependence of 4- and 6-derivative tensor-multiplet couplings in six-dimensional $(2,0)$ supergravity, and uses Type II/heterotic duality to obtain exact, non-perturbative expressions for these couplings in Type IIB string theory on K3. The couplings $f^{(4)}$ and $f^{(6)}$ are shown to be sections over a 105-dimensional moduli space and satisfy Hessian-type equations with fixed constants, linking worldsheet and spacetime perspectives. In the weak coupling limit, the results reduce to integrated four-point functions of exactly marginal K3 CFT operators, revealing nontrivial moduli dependence and worldsheet instanton contributions, and highlighting singular behavior at ADE points. The analysis connects the tensor branch data of the 6d $(2,0)$ SCFT, 5d MSYM, and the K3 CFT moduli, offering a framework to study ADE physics and potential bootstrap implications for the K3 CFT.

Abstract

We study supervertices in six dimensional (2,0) supergravity theories, and derive supersymmetry non-renormalization conditions on the 4- and 6-derivative four-point couplings of tensor multiplets. As an application, we obtain exact non-perturbative results of such effective couplings in type IIB string theory compactified on K3 surface, extending previous work on type II/heterotic duality. The weak coupling limit thereof, in particular, gives certain integrated four-point functions of half-BPS operators in the nonlinear sigma model on K3 surface, that depend nontrivially on the moduli, and capture worldsheet instanton contributions.

Supersymmetry Constraints and String Theory on K3

TL;DR

This work derives second-order differential constraints on the moduli dependence of 4- and 6-derivative tensor-multiplet couplings in six-dimensional supergravity, and uses Type II/heterotic duality to obtain exact, non-perturbative expressions for these couplings in Type IIB string theory on K3. The couplings and are shown to be sections over a 105-dimensional moduli space and satisfy Hessian-type equations with fixed constants, linking worldsheet and spacetime perspectives. In the weak coupling limit, the results reduce to integrated four-point functions of exactly marginal K3 CFT operators, revealing nontrivial moduli dependence and worldsheet instanton contributions, and highlighting singular behavior at ADE points. The analysis connects the tensor branch data of the 6d SCFT, 5d MSYM, and the K3 CFT moduli, offering a framework to study ADE physics and potential bootstrap implications for the K3 CFT.

Abstract

We study supervertices in six dimensional (2,0) supergravity theories, and derive supersymmetry non-renormalization conditions on the 4- and 6-derivative four-point couplings of tensor multiplets. As an application, we obtain exact non-perturbative results of such effective couplings in type IIB string theory compactified on K3 surface, extending previous work on type II/heterotic duality. The weak coupling limit thereof, in particular, gives certain integrated four-point functions of half-BPS operators in the nonlinear sigma model on K3 surface, that depend nontrivially on the moduli, and capture worldsheet instanton contributions.

Paper Structure

This paper contains 24 sections, 149 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Supervertices in 6$d$ $(2,0)$ supergravity
  3. 6$d$ $(2,0)$ Super-spinor-helicity formalism
  4. Supervertices for tensor multiplets
  5. Supervertices for supergravity and tensor multiplets
  6. Four-point supervertices.
  7. Three-point supervertices.
  8. Differential constraints on $f^{(4)}$ and $f^{(6)}$ couplings
  9. An example of $f^{(4)}$ and $f^{(6)}$ from Type II/Heterotic duality
  10. Type II/Heterotic duality
  11. Heterotic string amplitudes and the differential constraints
  12. One-loop four-point amplitude
  13. Two-loop four-point amplitude
  14. Implications of $f^{(4)}$ and $f^{(6)}$ for the K3 CFT
  15. Reduction to the K3 CFT moduli space
  16. ...and 9 more sections

Figures (4)

  • Figure 1: Factorization channels for the $\varphi^2H^4$ superamplitude. The solid lines stand for the tensor multiplet states while the dotted lines stand for the supergravity multiplet states. The black circles represent the 4-derivative four-tensor-multiplet supervertex, and the trivalent vertices represent the 2-derivative supervertex involving one gravity and two tensor multiplets.
  • Figure 2: Factorization channels for the $D^2(\varphi^2H^4)$ superamplitude. The solid lines stand for the tensor multiplet states while the dotted lines stand for the supergravity multiplet states. The black and white circles represent the 4 and 6-derivative four-tensor-multiplet supervertices, respectively, and the trivalent vertices represent the 2-derivative supervertex involving one gravity and two tensor multiplets.
  • Figure 3: The reduction of the genus one $T^5$ compactified heterotic string amplitude $\mathcal{A}_1|_{F^4}$ to the one-loop amplitude $\mathcal{A}_1^{SYM}$ in 5$d$ maximal SYM.
  • Figure 4: The reduction of the genus two $T^5$ compactified heterotic string amplitude $\mathcal{A}_2|_{D^2F^4}$ to two-loop amplitudes $\mathcal{A}_2^{SYM}$ in 5$d$ maximal SYM.