Mathieu moonshine and \( \mathcal{N}=2 \) string compactifications

Miranda C.N. Cheng 4 Xi Dong 2 3 John F.R. Duncan 0 Jeffrey A. Harvey 1 Shamit Kachru 2 Timm Wrase 2 3 Open Access 0 Department of Mathematics, Case Western Reserve University , Cleveland, OH 44106, U.S.A 1 Enrico Fermi Institute and Department of Physics, University of Chicago , Chicago, IL 60637, U.S.A 2 SITP, Department of Physics and Theory Group , SLAC, Stanford University , Stanford, CA 94305, U.S.A 3 Kavli Institute for Theoretical Physics , Santa Barbara, CA 93106, U.S.A 4 Universite Paris 7 , UMR CNRS 7586, Paris, France There is a 'Mathieu moonshine' relating the elliptic genus of K3 to the sporadic group M24. Here, we give evidence that this moonshine extends to part of the web of dualities connecting heterotic strings compactified on K3T 2 to type IIA strings compactified on Calabi-Yau threefolds. We demonstrate that dimensions of M24 representations govern the new supersymmetric index of the heterotic compactifications, and appear in the Gromov-Witten invariants of the dual Calabi-Yau threefolds, which are elliptic fibrations over the Hirzebruch surfaces Fn. 1 Introduction 2 3 4 Universal threshold corrections in heterotic K3 compactifications 2.1 Basic facts about N = 2 heterotic models 2.2 Threshold corrections and the new supersymmetric index 2.3 Discovering M24 representations 2.4 Compactifications with Wilson lines Type IIA compactifications on elliptic Calabi-Yau threefolds 3.1 Prepotentials of Calabi-Yau models 3.2 The higher genera 4.1 Simplest case 4.2 Other values of n 4.3 Gravitational threshold corrections 5 Discussion A Conventions Introduction Monstrous moonshine [1] is a mysterious relation between two very natural objects in mathematics: the largest of the sporadic simple finite groups (the Fischer-Griess Monster), and the simplest modular function (the J-function). More generally, it relates conjugacy classes of the Monster to Hauptmoduls for genus zero subgroups of SL(2, R). This relationship is (partially) explained by the physics of a particular string compactification. The chiral conformal field theory corresponding to a Z2 orbifold of the bosonic string on R24/ where is the Leech lattice has as its partition function the J-function with constant term set equal to zero, and admits Monster symmetry. The relation between moonshine and vertex algebras of chiral conformal field theories is discussed in the mathematics literature in [2, 3], and from an accessible physical point of view in [4]. A more general review of the story of moonshine appears in [5]. It is encouraging that the Monstrous moonshine relations between fundamental objects from two a priori distinct areas of mathematics find a natural home in string theory. However, it is fair to say that the string vacuum which appears here has not played a very central role in other developments in string theory and quantum gravity. For this reason, moonshine has not yet had significant impact on our present understanding of string theory. In 2010, Eguchi, Ooguri and Tachikawa (EOT) observed similar mysterious relations between the sporadic group M24 and the elliptic genus of the (4,4) superconformal field theory with K3 target [6]. This hints that generalizations of moonshine may be important in understanding more physically central string vacua. K3 serves as the simplest nontrivial example of Calabi-Yau compactification, and has played a central role in duality relations between string theories [7]. Further work has considerably elucidated, refined and generalized the EOT conjecture [819]. For fairly recent reviews, see [20, 21]. In this note, we provide evidence that the Mathieu moonshine of [6] extends to a richer structure visible also in 4d N = 2 string compactifications. Such compactifications arise simply in two different ways: from heterotic strings on K3T 2 (with suitable gauge bundles over the compactification manifold), or from type II strings on Calabi-Yau threefolds. These two sets of compactifications are related by string duality [22, 23]. We show here that in the heterotic theories, with either arbitrary choices of gauge bundles and no Wilson lines or with all instantons embedded in one E8 and only Wilson lines in the other E8, the one-loop prepotential universally exhibits a structure encoding degeneracies of M24 representations. We demonstrate that this structure is also visible in the Gromov-Witten invariants of the dual type IIA Calabi-Yau compactifications. Our results build on a large body of work on threshold corrections in heterotic strings [2433] as well as recent advances in understanding the Gromov-Witten theory of elliptic Calabi-Yau manifolds [34, 35]. Universal threshold corrections in heterotic K3 compactifications Basic facts about N = 2 heterotic models A heterotic compactification on a manifold X involves, in an obligatory way, data beyond the sigma-model metric on X. The heterotic string comes equipped with E8 E8 (or Spin(32)/Z2) gauge fields in ten dimensions. T (...truncated)