Cobordism conjecture, anomalies, and the String Lamppost Principle

Published for SISSA by Springer Received: September 26, 2020 Accepted: November 24, 2020 Published: January 13, 2021 Miguel Montero and Cumrun Vafa Jefferson Physical Laboratory, Harvard University, Cambridge, MA 02138, U.S.A. E-mail: , Abstract: We consider consequences of triviality of cobordism classes and anomaly cancellation in supergravity theories in d > 6. We argue that this leads to the existence of certain defects which we call “I-folds” (a generalization of orientifolds). The requirement that compactifications to lower dimensions involving these defects be anomaly free leads to conditions on the higher dimensional theory. We show that in theories with 16 supercharges in d > 6 this leads to restrictions on the rank of the allowed gauge groups and thus provides an explanation for the observed restrictions in known string theory constructions. In particular, in eight and nine dimensions the only solutions to our constraints are precisely the ones realized in string theory compactifications. We also use these techniques to place constraints on the global structure of the gauge group in eight and nine dimensions. Keywords: Anomalies in Field and String Theories, String theory and cosmic strings, Gauge Symmetry, Extended Supersymmetry ArXiv ePrint: 2008.11729 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP01(2021)063 JHEP01(2021)063 Cobordism conjecture, anomalies, and the String Lamppost Principle Contents 1 2 Theories with 16 supersymmetries and string constructions for d > 6 2.1 d = 9 2.2 d = 8 2.3 d = 7 3 3 5 5 3 Rank restrictions and cobordism triviality 3.1 Parity symmetry 3.1.1 Parity symmetry in d = 9 3.1.2 Parity symmetry in d = 8 3.1.3 Parity symmetry in d = 7 3.2 Cobordism conjecture and I-folds 3.2.1 I-fold compactifications and charge quantization 3.3 Rank 8 periodicity in d = 9 3.4 Modulo 8 periodicity in d = 8 3.5 Modulo 2 periodicity in d = 7 6 6 7 9 10 11 17 19 21 24 4 Constraints on the gauge group of 8d and 9d theories 25 5 Comments on charge lattices 28 6 Conclusions 30 A Real fermions and Pin structures 33 B Pin− fermions in nine and six dimensions B.1 Parity action on the 9d multiplets B.2 Dimensional reduction of fermions on toroidal I-fold 35 35 36 C Supersymmetry and parity anomaly in one dimension 37 D 6d anomaly cancellation 39 1 Introduction One of the key questions in high-energy physics is what constitutes a consistent quantum theory of gravity. Understanding this systematically is the basic aim of the Swampland program [1] (see also the reviews [2, 3]). String theory constructions provide a laboratory –1– JHEP01(2021)063 1 Introduction –2– JHEP01(2021)063 of features that a consistent quantum gravitational theory should enjoy. To narrow down the search for Swampland principles it is natural to restrict to the most symmetric cases in string theory, and that would involve constructions with high supersymmetry in higher dimensions admitting Minkowski backgrounds. The low energy content of theories with the highest amount of supersymmetry (32 supercharges) is uniquely fixed by supersymmetry. The next case of interest are theories with 16 supercharges, which is the only other possibility in d > 6. Progress in this direction has been made recently. In particular it has been shown that the only possibility in d = 10 are the ones with gauge groups E8 × E8 and SO(32) [4, 5]. For d < 10 it has been shown that the rank of the gauge group is bounded by rG ≤ 26 − d [6], which is consistent with all known string constructions. This in particular establishes that only a finite number of matter contents are possible, at least for theories with 16 supercharges. However there is more structure in what is observed in string theory constructions. In particular, not all ranks in the allowed range seem to appear. For example in d = 9, the only observed ranks are 1, 9, 17. It is important to settle whether there are other theories with rank less than or equal to 17 in d = 9 in addition to the known string compactifications. If there are and string theory constructions cannot yield them, it shows the incompleteness of the string landscape. On the other hand if these rank restrictions are forced by consistency arguments of quantum gravity alone, it lends further support for the String Lamppost Principle (SLP): That the string landscape includes all consistent quantum gravitational backgrounds. One aim of this paper is to show that the rank restriction at least in higher dimensions is a necessary consequence of other well established swampland principles, such as the absence of global symmetries. One proposed Swampland principle we use is the Cobordism Conjecture [7]. This conjecture, which can be viewed as a generalization of the lack of global symmetries and completeness of spectrum in quantum theories of gravity (in d > 3), states that in a consistent theory of gravity all cobordism classes are trivial. In particular if there appears to be a non-trivial cobordism class at low energies, there must exist an object within the theory which trivializes that class. We use this conjecture to argue that in any consistent quantum theory of gravity there must exist defect singularities we call “I-folds” (inversion-folds), which are a generalization of orientifold planes. Using this we construct compactifications of a putative consistent quantum gravity theory with 16 supercharges to lower dimensions with 8 supercharges. Consistency of the lower dimensional theory, and in particular anomaly cancellation, turns out to require that the higher dimensional theory satisfy some rank conditions. In particular we show that in d = 9 the rank should be 1 mod 8, in d = 8 it should be 2 mod 8, and d = 7 should have odd rank. All these are consistent with rank restrictions arising in string theory constructions and for d = 8, 9 it yields exactly the observed ranks in string compactifications. The organization of this paper is as follows. In section 2 we review the rank restrictions in d > 6 theories from known string constructions. In section 3 we review P and CP symmetries of supergravity theories and explain how the cobordism conjecture leads to the existence of I-folds. We then use these ingredients in compactification to lower dimensions and use anomaly cancellation in the lower dimensional theory to obtain rank restrictions for the higher dimensional theory. In section 4 we discuss constraints on the global structure of the gauge groups that can appear at points of enhanced symmetry. In section 5 we discuss aspects of charge lattices that can or cannot arise in theories with 16 supercharges. We conclude in section 6 with some open questions. Some of the technical details of the arguments are presented in the appendices. 2 Theories with 16 supersymmetries and string constructions for d > 6 2.1 d=9 Consider a nine-dimensional N = 1 (16 real supercharges) supergravity theory. We work in Lorentzia (...truncated)