Two-center black holes duality-invariants for stu model and its lower-rank descendants

Journal of Mathematical Physics

Volumn 52, Issue 6, 2011, Pages

  Ferrara, Sergio ^a,b,c   Marrani, Alessio ^a   Orazi, Emanuele ^a   Stora, Raymond ^a,d   Yeranyan, Armen ^b,e  

^a CERN   (Switzerland)

^b INFN LABORATORI NAZIONALI DI FRASCATI   (Italy)

^c UNIVERSITY OF CALIFORNIA   (United States)

^d UNIVERSITÉ DE SAVOIE   (France)

^e YEREVAN STATE UNIVERSITY   (Armenia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 79959969213     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.3589319     Document Type: Article

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69. Here, U-duality is referred to as the "continuous" limit (valid for large values of the charges) of the non-perturbative string theory symmetries introduced by Hull and Townsend in Ref. 19.
70. Actually, in these cases the horizontal symmetry group is GL (p,R), where the additional scale symmetry with respect to GL (p,R) is encoded by the homogeneity of the G4-invariant polynomials in charges.
71. The Calabi-Vesentini basis for charges and holomorphic sections is discussed in Appendix.
72. For reasons of covariance, in Eqs. (3.9),(3.10)-(3.16),(4.3),(4.12),(4.13) and (7.16), "Tr" denotes the η-trace, namely, the trace in which the indices are raised and lowered by the pseudo-Euclidean metric η.
73. Through the "stu → st2 → t3reduction" procedure, the non-BPS ZH = 0 attractor flow of stu model consistently degenerates into the 1/2-BPS attractor flow of the t3 model.
74. For a discussion of the "stu → st2 → t3degeneration" in a different symplectic frame (relevant for applications to Quantum Information Theory, see also Ref. 66), see, e.g., the discussion in Sec. 5 of Ref. 42.
75. Note that the quartic invariant polynomial of the 1/2 (spin s = 3/2 ) irrepr. of SL (2, R) can be written as (see also Ref. 67)constrained by (B11), where "×" denotes the exterior product of the 3-vectors →r 1 and →r 2, and the square norms are computed with the SO(2, 1)-metric ηλσ. In the "special coordinates" symplectic frame used in Sec. VII, I4 is given by Eq. (7.6).
76. For p > n, Tr(Ap) can be expressed in terms of {Tr (Ap)}1≤p≤n since, by virtue of the Cayley-Hamilton theorem, A fulfills its characteristic equation det(λIn -A) = 0 (In denoting the n × n identity).
77. We thank Professor. Michel Dubois Violette for this argument.
78. Indeed, an even permutation of the indices {1, ..., n} can be represented by a suitable transformation of SO(n,R).