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Courant Algebroids, Poisson–Lie T-Duality, and Type II Supergravities

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Abstract

We reexamine the notions of generalized Ricci tensor and scalar curvature on a general Courant algebroid, reformulate them using objects natural w.r.t. pull-backs and reductions, and obtain them from the variation of a natural action functional. This allows us to prove, in a very general setup, the compatibility of the Poisson–Lie T-duality with the renormalization group flow and with string background equations. We thus extend the known results to a much wider class of dualities, including the cases with gauging (so called dressing cosets, or equivariant Poisson–Lie T-duality). As an illustration, we use the formalism to provide new classes of solutions of modified supergravity equations on symmetric spaces.

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Notes

  1. To be precise (i.e. to satisfy all the physical requirements), we need \(\dim V_+=10\), the signature of \(\langle ,\rangle |_{V_+}\) needs to be Lorentzian, and we also need the transversality condition \({\text {Ad}}_g{\mathfrak {h}}\cap ({\mathfrak {s}}+V_+)=0\) for all \(g\in {\mathsf {G}}\) (the last condition is needed to make \(V'_{+,\mathrm {red}}\subset E'_\mathrm {red}\) correspond to a pair (g, H), and it is needed because \(\langle ,\rangle |_{V_+}\) is indefinite).

  2. We exclude the non-interesting cases when the involutions are equal to the identity.

  3. This fixes the overall scaling freedom present in the SUGRA equations of motion.

  4. i.e. the real span of the coroots

  5. This corresponds (up to a constant) to the symplectic form on \({\mathbb {C}}\text {P}^M\).

  6. in the sense of Riemannian symmetric spaces

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Authors and Affiliations

  1. Section of Mathematics, University of Geneva, Geneva, Switzerland

    Pavol Ševera

  2. Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Fridrich Valach

Authors
  1. Pavol Ševera
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  2. Fridrich Valach
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Correspondence to Pavol Ševera.

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Communicated by H. T. Yau

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Supported by the NCCR SwissMAP of the Swiss National Science Foundation. F.V. was supported also by the GAČR Grant EXPRO 19-28628X.

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Ševera, P., Valach, F. Courant Algebroids, Poisson–Lie T-Duality, and Type II Supergravities. Commun. Math. Phys. 375, 307–344 (2020). https://doi.org/10.1007/s00220-020-03736-x

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