# Four-derivative couplings and BPS dyons in heterotic CHL orbifolds

### Submission summary

 As Contributors: Boris Pioline Arxiv Link: http://arxiv.org/abs/1702.01926v3 (pdf) Date submitted: 2017-05-08 02:00 Submitted by: Pioline, Boris Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Approach: Theoretical

### Abstract

Three-dimensional string models with half-maximal supersymmetry are believed to be invariant under a large U-duality group which unifies the S and T dualities in four dimensions. We propose an exact, U-duality invariant formula for four-derivative scalar couplings of the form $F(\Phi) (\nabla\Phi)^4$ in a class of string vacua known as CHL $\mathbb{Z}_N$ heterotic orbifolds with $N$ prime, generalizing our previous work which dealt with the case of heterotic string on $T^6$. We derive the Ward identities that $F(\Phi)$ must satisfy, and check that our formula obeys them. We analyze the weak coupling expansion of $F(\Phi)$, and show that it reproduces the correct tree-level and one-loop contributions, plus an infinite series of non-perturbative contributions. Similarly, the large radius expansion reproduces the exact $F^4$ coupling in four dimensions, including both supersymmetric invariants, plus infinite series of instanton corrections from half-BPS dyons winding around the large circle, and from Taub-NUT instantons. The summation measure for dyonic instantons agrees with the helicity supertrace for half-BPS dyons in 4 dimensions in all charge sectors. In the process we clarify several subtleties about CHL models in $D=4$ and $D=3$, in particular we obtain the exact helicity supertraces for 1/2-BPS dyonic states in all duality orbits.

### Ontology / Topics

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Published as SciPost Phys. 3, 008 (2017)

### Author comments upon resubmission

We thank the referees for their valuable remarks, to which we have replied separately. In this revised version, we have made various changes in section 2.1 and 2.2, in order to clarify the relation between the automorphism group of the Narain lattice and the U-duality group. We have also corrected several misprints pointed out by the referee, made cosmetic changes to various equations, and added references [24,25]. In addition, we removed footnote 5 and the former reference [29] to Witten (1994). The reason is that the vector $Q$ in the extended Narain lattice cannot be interpreted as a charge vector for BPS particles in $D=3$, since the latter are instead classified by conjugacy classes of the U-duality group (as explained e.g. in [arXiv:1209.6056]). We hope that this revised version will be found suitable for publication.

### List of changes

- In abstract, replaced "NS5-brane, Kaluza-Klein monopole and H-monopole instantons" by "non-perturbative contributions".
- At end of abstract, added "in particular we obtain the exact helicity supertraces for 1/2-BPS dyonic states in all duality orbits."
- Clarified the discussion of the U-duality group $G_4(Z)$, starting 4 lines before eq (2.2) up until 2 lines above eq (2.6).
- Replaced $q$ by $q_S=e^{2\pi i S}$ in (2.5)
- Above (2.14), added references [24,25] to Dabholkar et al (2005) and Sen (2005)
- Defined $q=e^{2\pi i \tau}$ below (2.14)
- Clarified the discussion of the U-duality group $G_3(Z)$, starting below eq (2.19) up until eq (2.22).
- removed footnote 5, along with the sentence "(which can be viewed as a one-loop corrected mass formula)" and the former reference [29]
- Restored missing index $J$ in (3.11)
- Replaced $Q$ by $\widetilde{Q}$ in (4.8)
- Explicited the integration domain In Eq (4.24), (4.29), (5.18), (5.28), (5.42)
- Rewrote sentences around (4.36)
- Removed the unwanted sum of $\widetilde{Q}$ in (5.9), and replaced $\Lambda_{p-1,q-1}$ by $\Lambda_{p-2,q-2}$ in that same equation
- Third bullet on page 33 (starting with "The remaining contributions A with $(n_2,m_2)=(0,0)$...) moved into main text
- Rewrote the sentence below (5.33)
- Added "heterotic" in the title of Appendix A
- Corrected various misprints pointed out by the referees