Exact effective interactions and 1/4-BPS dyons in heterotic CHL orbifolds

Abstract

Motivated by precision counting of BPS black holes, we analyze six-derivative couplings in the low energy effective action of three-dimensional string vacua with 16 supercharges. Based on perturbative computations up to two-loop, supersymmetry and duality arguments, we conjecture that the exact coefficient of the ${\nabla }^2(\nabla \varphi {)}^4$ effective interaction is given by a genus-two modular integral of a Siegel theta series for the non-perturbative Narain lattice times a specific meromorphic Siegel modular form. The latter is familiar from the Dijkgraaf-Verlinde-Verlinde (DVV) conjecture on exact degeneracies of 1/4-BPS dyons. We show that this Ansatz reproduces the known perturbative corrections at weak heterotic coupling, including tree-level, one- and two-loop corrections, plus non-perturbative effects of order $e^{-1/g_3^2}$. We also examine the weak coupling expansions in type I and type II string duals and find agreement with known perturbative results, as well as new predictions for higher genus perturbative contributions. In the limit where a circle in the internal torus decompactifies, our Ansatz predicts the exact ${\nabla }^2{F}^4$ effective interaction in four-dimensional CHL string vacua, along with infinite series of exponentially suppressed corrections of order $e^{-R}$ from Euclideanized BPS black holes winding around the circle, and further suppressed corrections of order $e^{-R^2}$ from Taub-NUT instantons. We show that instanton corrections from 1/4-BPS black holes are precisely weighted by the BPS index predicted from the DVV formula, including the detailed moduli dependence. We also extract two-instanton corrections from pairs of 1/2-BPS black holes, demonstrating consistency with supersymmetry and wall-crossing, and estimate the size of instanton-anti-instanton contributions.

arXiv comment77+78 pages, 1 figure; v2: cosmetic changes, references added

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• Holographic description of Narain CFTs and their code-based ensembles
• 1/8-BPS Couplings and Exceptional Automorphic Functions
• Perturbative higher-derivative terms in N=6 asymmetric orbifolds
• Arithmetic of decay walls through continued fractions: a new exact dyon counting solution in $\mathcal{N}=4$ CHL models
• Two-loop superstring five-point amplitudes I: Construction via chiral splitting and pure spinors
• Rademacher expansion of a Siegel modular form for $\mathcal{N}=4$ counting
• Refinement and modularity of immortal dyons
• An approach to BPS black hole microstate counting in an N=2 STU model
• Some comments on symmetric orbifolds of K3
• Lost Chapters in CHL Black Holes: Untwisted Quarter-BPS Dyons in the ${\mathbb{Z}}_2$ Model