Universal BPS Structure of stationary supergravity solutions K.S. Stelle Imperial College London Inaugural Conference, Galileo Galilei Institute Firenze, 8 April 2009 G. Bossard, H. Nicolai & K.S.S. 0809.5218 and 0902.4438 [hep-th] 1 / 21
Outline ◮ Introduction: timelike dimensional reductions ◮ Examples: Einstein-Maxwell solution families ◮ Gravitational and vector charges ◮ Characteristic equation ◮ Supersymmetry ‘Dirac equation’ ◮ Almost Iwasawa decompositions ◮ Conclusions 2 / 21
Stationary solutions and timelike dimensional reduction The search for supergravity solutions with assumed Killing symmetries can be recast as a Kaluza-Klein problem. Consider a D = 4 theory with a nonlinear bosonic symmetry ¯ G ( e.g. E 7 for maximal N = 8 supergravity). Scalar fields take their values in a target space ¯ Φ = ¯ G / ¯ H , where ¯ H is the corresponding linearly realized subgroup, generally the maximal compact subgroup of ¯ G ( e.g. SU (8) for N = 8 SG). Searching for stationary solutions to such a theory amounts to assuming further that a solution possesses a timelike Killing vector field κ µ ( x ). • We assume that the solution spacetime is asymptotically flat or asymptotically Taub-NUT and that there is a ‘radial’ function r which is divergent in the asymptotic region, g µν ∂ µ r ∂ ν r ∼ 1 + O ( r − 1 ). • The Killing vector κ will be assumed to have W := − g µν κ µ κ ν ∼ 1 + O ( r − 1 ). 3 / 21
• We also assume asymptotic hypersurface orthogonality, κ ν ( ∂ µ κ ν − ∂ ν κ µ ) ∼ O ( r − 2 ). • In any vielbein frame, the curvature will fall off as R abcd ∼ O ( r − 3 ). • Lie derivatives with respect to κ are assumed to vanish on all fields. The D = 3 theory dimensionally reduced with respect to the timelike Killing vector κ will have an Abelian principal bundle structure, with a metric ds 2 = − W ( dt + ˆ B i dx i ) 2 + W − 1 γ ij dx i dx j where t is a coordinate adapted to the Killing vector κ and γ is the metric on the 3-dimensional hypersurface Σ 3 at constant t . If the D = 4 theory has Abelian vector fields A µ , they similarly reduce to D = 3 as √ 4 π G A µ dx µ = U ( dt + ˆ B i dx i ) + ˆ A i dx i 4 4 / 21
Comparison to spacelike dimensional reductions The timelike D = 3 reduced theory will have a G / H ∗ coset space structure similar to the G / H coset space structure of a D = 3 theory similarly reduced on a spacelike Killing vector. Thus, for a spacelike reduction of maximal supergravity one obtains an E 8 / SO (16) theory continuing on in the sequence of dimensional reductions originating in D = 11. As for the analogous Julia spacelike reduction, the D = 3 theory has the possibility of exchanging D = 3 Abelian vector fields for scalars by dualization, contributing to the appearance of an enlarged D = 3 bosonic ‘duality’ symmetry. The resulting D = 3 theory contains D = 3 gravity coupled to a G / H ∗ nonlinear sigma model. ◮ However, although the numerator group G is the same for a timelike reduction to D = 3 as that obtained for a spacelike reduction, the divisor group H ∗ is a noncompact form of the spacelike divisor group H . Breitenlohner, Gibbons & Maison 1988 ◮ The origin of this H → H ∗ change is the appearance of negative-sign kinetic terms for scalars descending from D = 4 vectors under the timelike reduction. 5 / 21
Some examples of G / H ∗ and G / H theories in D = 3 * The D = 3 classification of extended supergravity stationary solutions via timelike reduction generalizes the D = 3 supergravity systems obtained from spacelike reduction. de Wit, Tollsten & Nicolai 6 / 21
310 P. Breitenlohner, D. Maison, and G. Gibbons Equation (3.29) turns into an equation for 2, since the left-hand side vanishes for ~kt = hu- One finds 102 2- ~ = ~ (M-18eM, M-ld~M> , (3.32) 2-- 1~2 ~-- ~ ((M- lOoM , M- aSQM> -- (M- lgqzM , M- ~O~M>). From these equations ), can be computed by a simple integration once M is known. The integrability conditions are satisfied if Eq. (3.30) is fulfilled. 4. Spherically Symmetric Solutions The system of Eqs. (2.2-3) looks deceptively simple due to its elegant mathematical Stationary Maxwell-Einstein solutions description. But it has to be remembered that it describes rather complex and complicated physical situations and mathematical structures. Most of its explicitly known solutions are therefore distinguished by some symmetry properties of the Consider an initial theory comprising just D = 4 gravity together remaining 3-dimensional Riemannian space $3 and the a-model fields gbi(x) reducing the number of essential variables. The maximal symmetry group for S 3 is with an Abelian U (1) vector field, i.e. D = 4 Maxwell-Einstein the 6-parameter euclidean group of motions, which singles out the trivial theory. Search for stationary spherically symmetric solutions, with "vacuum" solution, 4-dimensional Minkowski space with vanishing vector field an isometry group SO (3). Using polar coordinates, the D = 3 strengths and constant scalar fields. A physically more interesting class of solutions are the spherically symmetric solutions with an isometry group S0(3) acting on metric on Σ 3 can then be parametrized as 2-dimensional orbits. Note that if the NUT-charge is non-zero the action of S0(3) ds 2 = γ ij dx i dx j = dr 2 + f ( r ) 2 ( d ϑ 2 + sin 2 ϑ d φ 2 ). The reduced on the 4-dimensional space-time has 3-dimensional orbits. Using polar coordi- D = 3 equations of motion become in this case nates the metric of X3 can be parametrized as ds 2 =habdxadxb=dr2+f(r)2(dO 2 + sin20drp2). The Eqs. (2.2-3) become under these circumstances 0, (4 a, Rrr = -2f -a d2f dqbi d4J (4.1b) d~-=~'J(~) dr dr' -2 d The last equation has the general solution f(r) z = (r-- ro) 2 + c. (4.2) 7 / 21 Introducing ~(r)=- ~ f-2(s)ds, which is a harmonic function on X3 equipped r with the metric hab, Eq. (4.1a) becomes dZq~' d~J d~k =0 (4.3) dz 2 + F~((o) dz dz with ~bi(r)=~)i(z(r)). This is the equation for a geodesic in the symmetric space G/H. The decomposition of q~: ~3 ~ G/H into a harmonic map z: S 3 ~R 1 and a geodesic
310 P. Breitenlohner, D. Maison, and G. Gibbons Equation (3.29) turns into an equation for 2, since the left-hand side vanishes for ~kt = hu- One finds 102 2- ~ = ~ (M-18eM, M-ld~M> , (3.32) 2-- 1~2 ~-- ~ ((M- lOoM , M- aSQM> -- (M- lgqzM , M- ~O~M>). From these equations ), can be computed by a simple integration once M is known. The integrability conditions are satisfied if Eq. (3.30) is fulfilled. 4. Spherically Symmetric Solutions The system of Eqs. (2.2-3) looks deceptively simple due to its elegant mathematical description. But it has to be remembered that it describes rather complex and complicated physical situations and mathematical structures. Most of its explicitly known solutions are therefore distinguished by some symmetry properties of the remaining 3-dimensional Riemannian space $3 and the a-model fields gbi(x) reducing the number of essential variables. The maximal symmetry group for S 3 is the 6-parameter euclidean group of motions, which singles out the trivial "vacuum" solution, 4-dimensional Minkowski space with vanishing vector field strengths and constant scalar fields. A physically more interesting class of solutions are the spherically symmetric solutions with an isometry group S0(3) acting on 2-dimensional orbits. Note that if the NUT-charge is non-zero the action of S0(3) on the 4-dimensional space-time has 3-dimensional orbits. Using polar coordi- nates the metric of X3 can be parametrized as ds 2 =habdxadxb=dr2+f(r)2(dO 2 + sin20drp2). The Eqs. (2.2-3) become under these circumstances 0, (4 a, Rrr = -2f -a d2f dqbi d4J (4.1b) d~-=~'J(~) dr dr' -2 d • The third equation has the general solution The last equation has the general solution f ( r ) 2 = ( r − r 0 ) 2 + c 2 . � ∞ f(r) z = (r-- ro) 2 + c. (4.2) f − 2 ( s ) ds , which is a harmonic • Introducing τ ( r ) := − r function on Σ 3 equipped with the metric γ ij , the first equation Introducing ~(r)=- ~ f-2(s)ds, which is a harmonic function on X3 equipped r above becomes with the metric hab, Eq. (4.1a) becomes dZq~' d~J d~k =0 (4.3) dz 2 + F~((o) dz dz with ˆ φ i ( r ) = ˆ with ~bi(r)=~)i(z(r)). This is the equation for a geodesic φ i ( τ ( r )). in the symmetric space G/H. The decomposition of q~: ~3 ~ G/H into a harmonic map z: S 3 ~R 1 and a geodesic • This is the equation for a geodesic in the symmetric space G / H ∗ = SU (2 , 1) / S ( U (1 , 1) × U (1)), with signature (+ + −− ). The decomposition of φ : Σ 3 → G / H ∗ into a φ : R → G / H ∗ is harmonic map τ : Σ 3 → R and a geodesic ˆ in accordance with a general theorem on harmonic maps according to which the composition of a Eels & Sampson, 1964 harmonic map with a totally geodesic one is again harmonic. • Such factorization into geodesic and harmonic maps is also characteristic of higher-dimensional p -brane supergravity solutions. Neugebauer & Kramer 1964; Clement & Gal’tsov 1996; Gal’tsov & Rychkov 1998 8 / 21
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