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 4 “of 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.
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Cacciatori, S.L., Marrani, A. & van Geemen, B. Multi-centered invariants, plethysm and grassmannians. J. High Energ. Phys. 2013, 49 (2013). https://doi.org/10.1007/JHEP02(2013)049
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DOI: https://doi.org/10.1007/JHEP02(2013)049