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Sign Choices for Orientifolds

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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 \(p\ge -1\), 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 \(KR\)-theory, which consequently implies the homotopy invariance and Fredholm module description of their construction.

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Notes

  1. This seems to be at odds with the result of [7] who used \(C^*\)-algebra methods to show that any sign choice could arise, whereas we have found constraints. This raises the possibility that the correspondence between continuous trace \(C^*\)-algebras and groupoids for the Real case is not a straightforward generalisation of the complex case.

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Authors and Affiliations

  1. Department of Mathematics, University of Auckland, Auckland, 1010, New Zealand

    Pedram Hekmati

  2. School of Mathematical Sciences, University of Adelaide, Adelaide, SA, 5005, Australia

    Michael K. Murray & Raymond F. Vozzo

  3. Department of Mathematics, Maxwell Institute for Mathematical Sciences and The Higgs Centre for Theoretical Physics, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    Richard J. Szabo

  4. Dipartimento di Scienze e Innovazione Tecnologica and INFN Torino, Gruppo collegato di Alessandria, Università del Piemonte Orientale, Alessandria, Italy

    Richard J. Szabo

  5. Arnold–Regge Centre, Turin, Italy

    Richard J. Szabo

Authors
  1. Pedram Hekmati
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  2. Michael K. Murray
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  3. Richard J. Szabo
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  4. Raymond F. Vozzo
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Correspondence to Pedram Hekmati.

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Communicated by X. Yin

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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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Hekmati, P., Murray, M.K., Szabo, R.J. et al. Sign Choices for Orientifolds. Commun. Math. Phys. 378, 1843–1873 (2020). https://doi.org/10.1007/s00220-020-03831-z

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  • DOI: https://doi.org/10.1007/s00220-020-03831-z

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