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Journal of High Energy Physics

, 2013:49 | Cite as

Multi-centered invariants, plethysm and grassmannians

  • Sergio L. Cacciatori
  • Alessio Marrani
  • Bert van Geemen
Open Access
Article

Abstract

Motivated by multi-centered black hole solutions of Maxwell-Einstein theories of (super)gravity in D = 4 space-time dimensions, we develop some general methods, that can be used to determine all homogeneous invariant polynomials on the irreducible (SL h (p, \( \mathbb{R} \)) ⊗ G 4)-representation (p , R), where p denotes the number of centers, and SL h (p, \( \mathbb{R} \)) is the “horizontal” symmetry of the system, acting upon the indices labelling the centers. The black hole electric and magnetic charges sit in the symplectic representation R of the generalized electric-magnetic (U -)duality group G 4.

We start with an algebraic approach based on classical invariant theory, using Schur polynomials and the Cauchy formula. Then, we perform a geometric analysis, involving Grassmannians, Plücker coordinates, and exploiting Bott’s Theorem.

We focus on non-degenerate groups G 4of type E 7” relevant for (super)gravities whose (vector multiplets’) scalar manifold is a symmetric space. In the triality-symmetric stu model of \( \mathcal{N} \) = 2 supergravity, we explicitly construct a basis for the 10 linearly independent degree-12 invariant polynomials of 3-centered black holes.

Keywords

Black Holes Supergravity Models Black Holes in String Theory Global Symmetries 

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Copyright information

© SISSA 2013

Authors and Affiliations

  • Sergio L. Cacciatori
    • 1
    • 2
  • Alessio Marrani
    • 3
  • Bert van Geemen
    • 4
  1. 1.Dipartimento di Scienze ed Alta TecnologiaUniversità degli Studi dell’InsubriaComoItaly
  2. 2.INFN, Sezione di MilanoMilanoItaly
  3. 3.Physics Department, Theory Unit, CERNGeneva 23Switzerland
  4. 4.Dipartimento di MatematicaUniversità di MilanoMilanoItaly

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