Tension Between a Vanishing Cosmological Constant and Non-Supersymmetric Heterotic Orbifolds

Abstract

We investigate under which conditions the cosmological constant vanishes perturbatively at the oneloop level for heterotic strings on non-supersymmetric toroidal orbifolds. To obtain model-independent results, which do not rely on the gauge embedding details, we require that the right-moving fermionic partition function vanishes identically in every orbifold sector. This means that each sector preserves at least one, but not always the same Killing spinor. The existence of such Killing spinors is related to the representation theory of finite groups, i.e. of the point group that underlies the orbifold. However, by going through all inequivalent (Abelian and non-Abelian) point groups of six-dimensional toroidal orbifolds we show that this is never possible: For any non-supersymmetric orbifold there is always (at least) one sector, that does not admit any Killing spinor. The underlying mathematical reason for this no-go result is formulated in a conjecture, which we have tested by going through an even larger number of finite groups. This conjecture could be applied to situations beyond symmetric toroidal orbifolds, like asymmetric orbifolds.

Nonexistence of locally but not globally supersymmetric orbifolds

“…The last paragraph of the previous section sketched an argument for the nonexistence of locally but not globally supersymmetric orbifolds by explicit construction of all spin representations associated to all (candidate) point groups P. To confirm this result, we also made use of representation theory of abstract finite groups [22]. In the following we provide some of the necessary ingredients of this analysis.…”

“…In this proceedings, based on [22], we first recall some necessary aspects of the space group description of toroidal orbifolds. After that we determine the conditions for the one-loop partition function to vanish 1 .…”

“…Among these close to 29 million toroidal orbifolds only 520 preserve at least N = 1 supersymmetry with 60 distinct underlying Qclasses [30,31]. This orbifold classification can be used to identify the Q-classes of the toroidal orbifolds, that admit Killing spinors in all sectors locally, but none globally [22]: Since most of them correspond to the 60 Q-classes of the globally supersymmetric orbifolds, this leaves three candidate Q-classes for toroidal orbifolds with the desired properties:…”

“…In all cases there is at least one point group element that does not preserve any Killing spinor! This leads to the conclusion, that there does not exist any non-supersymmetric orbifold for which all point group elements separately preserve some Killing spinors [22].…”

“…In addition, I would like to thank Johanna Knapp for the kind hospitality at the mathematical physics group at Vienna University. But most of all, I would like to express my gratitude to my collaborators Orestis Loukas, Andreas Mütter, Erik Parr and Patrick Vaudrevange for the project [22] on which this talk was based. Finally, I would like to thank Paul Fogarty, Orestis Loukas and Patrick Vaudrevange for proofreading the manuscript.…”