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A Unique Connection for Born Geometry

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Abstract

It has been known for a while that the effective geometrical description of compactified strings on d-dimensional target spaces implies a generalization of geometry with a doubling of the sets of tangent space directions. This generalized geometry involves an O(d,d) pairing \({\eta}\) and an O(2d) generalized metric \({\mathcal{H}}\). More recently it has been shown that in order to include T-duality as an effective symmetry, the generalized geometry also needs to carry a phase space structure or more generally a para-Hermitian structure encoded into a skew-symmetric pairing \({\omega}\). The consistency of string dynamics requires this geometry to satisfy a set of compatibility relations that form what we call a Born geometry. In this work we prove an analogue of the fundamental theorem of Riemannian geometry for Born geometry. We show that there exists a unique connection which preserves the Born structure \({(\eta,\omega,\mathcal{H})}\) and which is torsionless in a generalized sense. This resolves a fundamental ambiguity that is present in the double field theory formulation of effective string dynamics.

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Acknowledgements

F.J.R. would like to thank Chris Blair and Thomas Strobl for useful discussions. L.F. would like to thank long-time collaborator D. Minic for inputs and encouragements. D.S. would like to thank his co-supervisors Ruxandra Moraru and Shengda Hu for their supervision. The work of F.J.R. is supported by DFG Grant TRR33 “The Dark Universe”. D.S. is currently a Ph.D. student at Perimeter institute and University of Waterloo. His research is supported by NSERC Discovery Grants 378721. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science.

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Authors and Affiliations

  1. Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON, N2L 2Y5, Canada

    Laurent Freidel & David Svoboda

  2. Arnold Sommerfeld Center for Theoretical Physics, Department für Physik, Ludwig-Maximilians-Universität München, Theresienstr. 37, 80333, Munich, Germany

    Felix J. Rudolph

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  1. Laurent Freidel
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Correspondence to Felix J. Rudolph.

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Freidel, L., Rudolph, F.J. & Svoboda, D. A Unique Connection for Born Geometry. Commun. Math. Phys. 372, 119–150 (2019). https://doi.org/10.1007/s00220-019-03379-7

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