Non-perturbative selection rules in F-theory

Published for SISSA by Springer Received: July 22, 2015 Accepted: August 28, 2015 Published: September 29, 2015 Non-perturbative selection rules in F-theory a Dipartimento di Fisica e Astronomia ‘Galileo Galilei’, Università di Padova, and I.N.F.N. Sezione di Padova, via Marzolo 8, Padova, I-35131 Italy b Institut für Theoretische Physik, Ruprecht-Karls-Universität, Philosophenweg 19, Heidelberg, 69120 Germany E-mail: , Abstract: We discuss the structure of charged matter couplings in 4-dimensional Ftheory compactifications. Charged matter is known to arise from M2-branes wrapping fibral curves on an elliptic or genus-one fibration Y . If a set of fibral curves satisfies a homological relation in the fibre homology, a coupling involving the states can arise without exponential volume suppression due to a splitting and joining of the M2-branes. If the fibral curves only sum to zero in the integral homology of the full fibration, no such coupling is possible. In this case an M2-instanton wrapping a 3-chain bounded by the fibral matter curves can induce a D-term which is volume suppressed. We elucidate the consequences of this pattern for the appearance of massive U(1) symmetries in F-theory and analyse the structure of discrete selection rules in the coupling sector. The weakly coupled analogue of said M2-instantons is worked out to be given by D1-F1 instantons. The generation of an exponentially suppressed F-term requires the formation of half-BPS bound states of M2 and M5-instantons. This effect and its description in terms of fluxed M5-instantons is discussed in a companion paper. Keywords: F-Theory, D-branes ArXiv ePrint: 1506.06764 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP09(2015)198 JHEP09(2015)198 Luca Martuccia and Timo Weigandb Contents 2 2 Resolution homology, (massive) U(1)s and effective field theory 2.1 Massless U(1) gauge symmetries 2.2 Geometrical selection rules and massive U(1)s 2.3 Zk gauge symmetries and torsion in (non-)perturbative homology 4 5 7 10 3 Simple local examples 3.1 Local examples of perturbative relations 3.2 Local examples of non-perturbative relations 13 13 15 4 Application to a global SU(5) example 17 5 Z2 -symmetry and instantons 5.1 Fibration with two rational sections 5.2 Bisection versus torsional Weierstrass models 5.3 Perturbative U(1) symmetry and non-perturbative breakdown 21 21 23 26 6 Non-perturbative relations, massive U(1)s and D1-instantons at weak coupling 27 6.1 Axionic gauging and D1-instantons in Type IIB 27 6.2 D1 versus M2 instantons: local discussion 30 6.3 D1 versus M2 instantons: global discussion via the stable degeneration 34 6.4 The dual picture: (massive) U(1)s and axions via stable degeneration 39 6.5 Realisation of M2/D1-instantons in a concrete model 41 7 Conclusions 45 A Splitting curves in the SU(5) model 47 B Axionic gauging 48 C Mayer-Vietoris exact sequence C.1 Homology of the central fibre C.2 Cohomology of the central fibre 49 50 50 –1– JHEP09(2015)198 1 Introduction 1 Introduction –2– JHEP09(2015)198 Among the most intriguing aspects of F-theory [1–3] is the ingenuity with which the symmetries of the effective action are encoded in the topological and geometrical properties of the compactification space. For instance, non-abelian gauge symmetries are related to the fibre structure of the F-theory genus-one fibration in complex codimension-one, while the massless abelian gauge symmetries are in one-to-one correspondence with the MordellWeil group of rational sections. This endows abstract structures in algebraic and arithmetic geometry with a direct physics interpretation. From the perspective of 4-dimensional effective actions an equally rich and important concept is the notion of a massive U(1) symmetry. By this one means a global abelian symmetry of the effective action which becomes gauged at scales above a certain mass scale of the theory. Indeed it is conjectured that in any consistent quantum theory of gravity, global symmetries in the infrared must arise from a gauged symmetry in the ultra-violet (see e.g. [4] and references therein), and string theory is no exception. For instance, in perturbative string compactifications, abelian gauge symmetries can become massive via a Stückelberg coupling with an axionic field. At energies below the Stückelberg mass scale, the U(1) symmetry persists as a perturbative selection rule of the effective action. It is broken only by non-perturbative effects involving the exponential of the Stückelberg axion [5–9] to a discrete subgroup Zk [10–15]. This includes the case k = 1, in which the U(1) symmetry is completely broken even though the couplings breaking it in the effective action are exponentially suppressed. The non-perturbative breakdown of the massive U(1) to a discrete subgroup leads to a potentially interesting hierarchy of couplings in the effective action with many applications in string model building including the more recent [16–19] and references therein. Given this rich structure, a natural question is to what extent this intriguing pattern and a possible generalisation thereof is realised also in non-perturbative string compactifications. F-theory is our key laboratory to study such effects. In this article we address this and other related questions by carefully re-examining the conditions for a coupling involving charged matter fields to appear in the F-theory effective action. Since charged matter arises from the excitations of M2-branes along fibral curves (see, e.g., [20–22]), the structure of interactions is rooted in the possible joinings and splittings of such fibral curves [23]. We will argue that it is key to distinguish between a perturbative and a nonperturbative homological relation for fibral curves wrapped by M2-branes: a set of fibral curves satisfies a perturbative homological relation if, loosely speaking, their sum is trivial in the fibre homology. In the non-perturbative case, by contrast, triviality holds only in the homology ring of the full genus-one fibration. This implies that the states associated with M2-branes wrapping such fibral curves can only interact in conjunction with an interpolating M2-brane instanton wrapping a 3-chain Γ bounded by the respective curves. The 3-chain Γ in question has the important property that its volume does not vanish in the F-theory limit. Such couplings are therefore non-perturbatively suppressed by a factor e−2πvol3 (Γ) (in natural units). This allows us to make a clear distinction between the operators associated with the two types of homological relations. This behaviour is indeed 1 The geometry associated with a Zk symmetry is much richer, though, and involves a set of k − 1 additional genus-one fibrations classified by the Tate-Shafarevich group [29] of the original elliptic fibration, whose associated effective action becomes equivalent only in the F-theory limit [28, 30, 31]. –3– JHEP09(2015)198 e (...truncated)