Heterotic strings on (K3 × T2)/ℤ3 and their dual Calabi-Yau threefolds

Published for SISSA by Springer Received: December 5, 2019 Accepted: April 8, 2020 Published: April 30, 2020 Andreas Banlaki,a Aradhita Chattopadhyaya,b Abhiram Kidambi,a,d Thorsten Schimannekc and Maria Schimpfa a Institute for Theoretical Physics, TU Wien, A-1040 Vienna, Austria b Department of Mathematics, Trinity College Dublin, Dublin 2, Ireland c Faculty of Physics, University of Vienna, A-1090, Vienna, Austria d Stanford Institute for Theoretical Physics, Stanford University, Palo Alto, U.S.A. E-mail: , , , , Abstract: In this paper we study compactifications of the N = 2 heterotic E8 × E8 string on (K3 × T 2 )/Z3 with various gauge backgrounds and calculate the topological couplings in the effective supergravity action that arise from one-loop amplitudes. We then identify candidates for dual type IIA compactifications on Calabi-Yau threefolds and compare the heterotic results with the corresponding topological string amplitudes. We find that the dual Calabi-Yau geometries are K3 fibrations that are also genus one fibered with threesections. Moreover, we show that the intersection form on the polarization lattice of the K3 fibration has to be three times the intersection form on the Narain lattice Γ 1,1 . Keywords: String Duality, Superstrings and Heterotic Strings, Topological Strings ArXiv ePrint: 1911.09697 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP04(2020)203 JHEP04(2020)203 Heterotic strings on (K3 × T 2)/Z3 and their dual Calabi-Yau threefolds Contents 1 2 The new supersymmetric index for CHL orbifolds 2.1 The twisted elliptic genus of K3 2.2 Calculating the massless spectrum 2.3 The new supersymmetric index 3 4 8 11 3 Heterotic computation of Gopakumar-Vafa invariants 3.1 Extraction of Gopakumar-Vafa invariants from Fg 3.2 Conifold singularities 14 15 17 4 Dual type IIA compactifications on Calabi-Yau threefolds 4.1 A brief review of heterotic/type II duality 4.2 Constructing Calabi-Yau duals 4.3 All genus checks via the modular bootstrap 18 18 20 27 5 Conclusions 30 A Conventions for modular forms 31 B Higgsing of the gauge group 32 1 Introduction The four-dimensional N = 2 effective supergravity actions that arise from heterotic compactifications on (K3 × T 2 )/ZN and type II compactifications on Calabi-Yau threefolds contain the well-known supersymmetry protected couplings Z 2 S = Fg (y, ȳ) · F+2g−2 R+ , (1.1) where F+ and R+ respectively denote the self-dual parts of the graviphoton field strength and the Riemann tensor. In particular, the coefficients Fg (y, ȳ) depend only on the vector moduli. On the heterotic side, one of the vector moduli corresponds to the axio-dilaton and all terms (1.1) that do not involve the dilaton arise already at one-loop level. The one-loop amplitudes have been evaluated for compactifications with various embeddings of the gauge connection on K3 × T 2 by [1–3] and receive contributions only from BPS states that are encoded in the so-called new supersymmetric index [4]. Subsequently this analysis has been extended to compactifications on (K3 × T 2 )/ZN [5] and again the result is essentially encoded in the new supersymmetric index. The indices in turn had been –1– JHEP04(2020)203 1 Introduction –2– JHEP04(2020)203 calculated for various compactifications with standard embedding of the spin connection into the gauge connection on (K3 × T 2 )/ZN and some non-standard embeddings when N = 2 [6]. Calabi-Yau duals for some of the N = 2 cases have been proposed in [7–9]. In this paper we extend the calculation of the new supersymmetric index to compactifications on (K3 × T 2 )/Z3 with non-standard embeddings and predict the geometric data of the dual Calabi-Yau manifold. To this end we take an explicit realization of the orbifold limit of a K3 as T 4 /Z3 . A list of the possible gauge embeddings is provided in table 3 on page 10. For two embeddings the effective theories are such that the gauge group on a generic point of the hypermultiplet moduli space is maximally broken to U(1) 3 . For those models one can hope to find a dual Calabi-Yau compactification with only three Kähler moduli and without a larger non-higgsable gauge group. Indeed we will be able to identify many candidates. At certain points in the moduli space there might be singularities as additional vector bosons become massless. The strength of these singularities and the extraction of the Gopakumar-Vafa invariants are discussed in section 3. On the type IIA side one can identify limy→∞ Fg (y, ȳ) with the complex conjugate of the topological string free energy at genus g in the holomorphic limit [2, 10–12]. Following arguments from [9] that we review in section 4.1, one expects that for heterotic compactifications on (K3 × T 2 )/Z3 the dual Calabi-Yau manifolds are genus one fibered with three sections. Moreover, based on the prepotential of the effective supergravity action, one can argue that the Calabi-Yau should also exhibit a K3 fibration [13]. We thus systematically construct all K3 fibered Calabi-Yau threefolds with h1,1 = 3 that exhibit a genus one fibration with three-sections and are realized as hypersurfaces in toric ambient spaces. We then apply the modular bootstrap that has been extended to genus one fibrations with multi-sections in [9] and obtain all-genus results for the topological string amplitudes. This provides all order checks of the duality. A list of the Calabi-Yau threefolds is provided in table 6 on page 23. Let us note that on the heterotic side, the quotient acts as an order three symplectic automorphism on K3 together with a one-third shift along T 2 . Symplectic automorphisms of K3 manifolds have been classified and form subgroups of the Mathieu group M23 [14]. In particular, there is up to conjugacy only one automorphism of order three and this corresponds to the so-called 3A class. It turns out that the automorphism group of a nonlinear sigma model into K3 is actually larger [15] and contains another element of order three. However, we will restrict to the geometric case. The paper is structured as follows. In section 2 we study heterotic compactifications on (K3 × T 2 )/Z3 where the K3 is realized in the orbifold limit as T 4 /Z3 . We first calculate the twisted elliptic genus of K3 with order three twists by directly evaluating the corresponding trace. This confirms earlier results from the literature that were based on indirect arguments and determines the new supersymmetric index for the standard embedding. We then evaluate the new supersymmetric index for various non-standard embeddings. In section 3 we describe how the new supersymmetric index encodes the topological couplings and extract predictions for the Gopakumar-Vafa invariants of the dual CalabiYau compactification spaces. We also discuss the singular behaviour of the generating functions at points where additional vector states become massless. Acknowledgments We thank Justin David, Rajesh (...truncated)