Sign Choices for Orientifolds
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Communications in
Mathematical Physics
Sign Choices for Orientifolds Pedram Hekmati1 , Michael K. Murray2 , Richard J. Szabo3,4,5 , Raymond F. Vozzo2 1 Department of Mathematics, University of Auckland, Auckland 1010, New Zealand
E-mail: [email protected]
2 School of Mathematical Sciences, University of Adelaide, Adelaide, SA 5005, Australia
E-mail: [email protected]; [email protected]
3 Department of Mathematics, Maxwell Institute for Mathematical Sciences and The Higgs Centre for
Theoretical Physics, Heriot-Watt University, Edinburgh EH14 4AS, UK E-mail: [email protected]
4 Dipartimento di Scienze e Innovazione Tecnologica and INFN Torino, Gruppo collegato di Alessandria,
Università del Piemonte Orientale, Alessandria, Italy
5 Arnold–Regge Centre, Turin, Italy
Received: 26 May 2019 / Accepted: 29 May 2020 Published online: 18 August 2020 – © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: We analyse the problem of assigning sign choices to O-planes in orientifolds of type II string theory. We show that there exists a sequence of invariant p-gerbes with p ≥ −1, which give rise to sign choices and are related by coboundary maps. We prove that the sign choice homomorphisms stabilise with the dimension of the orientifold and we derive topological constraints on the possible sign configurations. Concrete calculations for spherical and toroidal orientifolds are carried out, and in particular we exhibit a four-dimensional orientifold where not every sign choice is geometrically attainable. We elucidate how the K -theory groups associated with invariant p-gerbes for p = −1, 0, 1 interact with the coboundary maps. This allows us to interpret a notion of K -theory due to Gao and Hori as a special case of twisted K R-theory, which consequently implies the homotopy invariance and Fredholm module description of their construction. 1. Introduction In this paper we study the Real Brauer group and related structures on orientifolds, that is, pairs (M, τ ) consisting of a manifold M equipped with an involution τ : M → M. The fixed point set of the involution is M τ and its connected components will be called orientifold planes or O-planes for short. Orientifolds are more commonly refered to as Real manifolds in the mathematics literature; the present terminology is borrowed from string theory where these give backgrounds which are important for model building and understanding T-duality [1,6]. The authors acknowledge support under the Australian Research Council’s Discovery Projects funding scheme (project numbers DP120100106, DP130102578 and DP180100383), the Consolidated Grant ST/P000363/1 from the UK Science and Technology Facilities Council, the Marsden Foundation (project number 3713803), and the Action MP1405 QSPACE from the European Cooperation in Science and Technology (COST). We thank David Baraglia and Jonathan Rosenberg for helpful discussions.
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P. Hekmati, M. K. Murray, R. J. Szabo, R. F. Vozzo
O-planes can carry a positive or negative Ramond–Ramo
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