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Geometry and Supersymmetry Heterotic Horizons IIB Horizons M-Horizons

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Geometry and Supersymmetry Heterotic Horizons IIB Horizons M-Horizons String special geometry George Papadopoulos King’s College London Geometric Structures in Mathematical Physics Varna, Bulgaria, September 2011 Work in collaboration with Jan Gutowski and Ulf Gran Geometry and supersymmetry Heterotic Horizons IIB Horizons M-Horizons Topology and geometry of black hole horizons The problem that will be addressed in this talk is the classification of topology and geometry of supersymmetric black hole horizon sections I This is a problem in Riemannian geometry despite its Lorentzian origin I Horizon sections are compact manifolds which admit special geometric structures. The classification of horizon sections is closely associated to the understanding of solutions to certain non-linear differential systems I Several explicit solutions will be presented. Questions will also be raised on whether there is a general existence theory of solutions to these differential systems. Geometry and supersymmetry Heterotic Horizons IIB Horizons M-Horizons Classifying supersymmetric gravitational backgrounds Supersymmetric gravitational backgrounds are solutions to the Einstein equations which in addition solve a set of at most first order differential equations, the Killing spinor equations (KSEs). The latter arise naturally in supergravity. The KSEs take the form Dµ ≡ rµ + Σµ(F) = 0 A(F) = 0 where is a spinor and hol(D) ⊆ GL. r Levi-Civita connection, F fluxes, ie forms on the spacetime. I An outstanding problem in Lorentzian geometry is the classification of supersymmetric backgrounds in all supergravity theories. Geometry and supersymmetry Heterotic Horizons IIB Horizons M-Horizons I Much progress has been made the last 10 years to solve this problem and a solution is known in some supergravity theories k;1 n I For special solutions, like R × N , the problem becomes a problem in Riemannian geometry. Typically, Nn admits a special geometric structure which is accompanied with a reduction of the structure group to a subgroup of SO(n). I A large number of such geometric structures have been identified and extensively investigated which include the traditional geometries, like hyper-Kähler, Calabi-Yau, G2 and Spin(7). I They also include some other geometries like those with skew-symmetric torsion, HKT, CYT, and G2 and Spin(7) with torsion. Geometry and supersymmetry Heterotic Horizons IIB Horizons M-Horizons Black holes and near horizon geometries Another classic problem in Lorentzian geometry is the classification of black hole spacetimes. For extreme black holes, one can adapt a coordinate system near the black hole (Killing) horizon such that the metric can be written as 2 I 2 I J ds = 2du[dr + r hIdy − r ∆du] + g~IJdy dy where u; r are light-cone coordinates I h = hI(y)dy ; ∆ = ∆(y) ; g~IJ = g~IJ(y) I Horizon section S is the co-dimension 2 subspace given by u = r = 0 I (S; g~) is assumed to be compact, connected, and without boundary I @u is the stationary time-like Killing vector field of the black hole ∆ > 0 Geometry and supersymmetry Heterotic Horizons IIB Horizons M-Horizons Supersymmetric horizons The main aim is to find the topology and geometry of S. The strategy how to do this is as follows: I Solve the KSEs for the near horizon geometries and identify the geometry on S I The field equations, like Einstein, are solved either by partial integration arguments using the compactness of S, or by the use of the maximum principle. I This is a problem in Riemannian geometry as S is a Riemannian manifold. Geometry and supersymmetry Heterotic Horizons IIB Horizons M-Horizons Heterotic supergravity Heterotic supergravity is a 10-dimensional Lorentzian theory with bosonic field, a metric g, a closed 3-form field strength H, dH = 0, and a real scalar Φ. The Killing spinor equations of Heterotic supergravities are 1 r^ = r − H Γνρ = 0 ; µ µ 8 µνρ 1 Γµ@ Φ − H Γµνρ = 0 ; 2 ∆+ µ 24 µνρ 16 Both have been solved in all cases [Gran, Lohrmann, GP; Gran, Roest, Sloane, GP]. Geometry and supersymmetry Heterotic Horizons IIB Horizons M-Horizons Solution of KSE for dH = 0. Black hole horizons are indicated in red. L Stab(1; : : : ; L) N 8 1 Spin(7) n R 1 8 2 SU(4) n R ; 2 8 3 Sp(2) n R ; −; 3 2 8 4 (× SU(2)) n R −; −; −; 4 8 5 SU(2) n R −; −; −; −; 5 8 6 U(1) n R −; −; −; −; −; 6 8 8 R −; −; −; −; −; −; −; 8 2 G2 1; 2 4 SU(3) 1; 2; −; 4 8 SU(2) −; 2; −; 4; −; 6; −; 8 16 f1g 8; 10; 12; 14; 16 Geometry and supersymmetry Heterotic Horizons IIB Horizons M-Horizons Heterotic horizons The near horizon geometry of a supersymmetric black hole, preserving one supersymmetry, in heterotic supergravity can be written [Gutowski, GP] as 2 − + 2 2 i j ds = 2e e + d~s(8) ; d~s(8) = δije e ; − + ~ ~ H = d(e ^ e ) + H(8) ; H(8) = dW ; − i + i i I Φ = Φ(y) ; e = dr + rhie ; e = du ; e = e Idy : The dilaton field equation is 1 1 r~ 2Φ − 2r~ iΦr~ Φ + (H~ ) (H~ )ijk − h hi = 0 ; i 12 (8) ijk (8) 2 i ^ 8 I The Killing vector @u is null and hol(r) ⊆ Spin(7) n R . Geometry and supersymmetry Heterotic Horizons IIB Horizons M-Horizons Thus I If h = 0, the dilaton field equation and compactness imply that 1;1 M = R × X8, where X8 is a product of Berger manifolds that admit parallel spinors, H = 0 and Φ is constant. Furthermore I If h 6= 0, N = 1 supersymmetry and compactness of S imply that ~^ r(8)h = 0. ^ ^ I The holonomy of r reduces to G2, hol(r) ⊆ G2, and horizons admit at least 2 supersymmetries. Geometry and supersymmetry Heterotic Horizons IIB Horizons M-Horizons Heterotic horizons, h 6= 0, always preserve even number of supersymmetries. In particular, S is an 8-dimensional manifold with skew-symmetric torsion and the holonomy of the associated connection is ^ N hol(r~ ) 2 G2 4 SU(3) 6 SU(2) 8 SU(2) S satisfies additional conditions. Geometry and supersymmetry Heterotic Horizons IIB Horizons M-Horizons N = 2; G2 8 S is a circle fibration over a conformally balanced G2 manifold 7 2 (X ; d~s(7)) with skew-symmetric torsion H(7), ie θ' = 2Φ, ^ (7) 8 hol(r ) ⊆ G2. The metric and torsion on S are 2 −2 2 d~s(8) = k h ⊗ h + d~s(7) ~ −2 2Φ −2Φ H(8) = k h ^ dh + H(7) ; H(7) = k' + e ?7 d e ' where h2 = k2. Given a 7-manifold with a G2 structure, the geometric conditions which have to be solved to find solutions are −2 ~ −2Φ θ' = 2Φ ; k dh ^ dh + dH(7) = 0 ; d[e ?7 '] = 0 ; dh 2 g2 8 ~ Note that S obeys the strong condition, ie dH(8) = 0. Geometry and supersymmetry Heterotic Horizons IIB Horizons M-Horizons N = 4; SU(3) The horizon sections S8 are holomorphic T2 fibrations over a 6-dimensional conformally balanced Hermitian manifold B6 equipped with the connection with skew-symmetric torsion. Such that 2 −2 −2 2 d~s(8) = k h ⊗ h + k ` ⊗ ` + ds(6) ~ −2 −2 H(8) = k h ^ dh + k ` ^ d` + H(6) ; H(6) = −iI(6) d!(6) The relative properties of B6 and S8 are Geometry B6 S8 Hermitian yes yes θ = 2dΦ yes no hol(r^ ) ⊆ SU(3) no yes hol(r^ ) ⊆ U(3) yes no dH(n) = 0 no yes Geometry and supersymmetry Heterotic Horizons IIB Horizons M-Horizons Given a hermitian 6-dimensional manifold B6, the conditions that have be solved to find solutions are k2 dh ^ !2 = 0 ; d` ^ !2 = − !3 ; de−2Φ!2 = 0 ; (6) (6) 3 (6) (6) ^ −2 −2 ρ~(6) − d` = 0 ; k dh ^ dh + k d` ^ d` − diI(6) d!(6) = 0 : where 1 ρ~^ = R~^ i (I )j ek ^ e` = −i@@¯log det g + 4i@@¯Φ ; (6) 4 k`; j (6) i (6) Necessary conditions for the existence of solutions are k2 c (P) ^ [e−2Φ!2 ] = 0 ; c (Q) ^ [e−2Φ!2 ] = − [e−2Φ!3 ] ; 1 (6) 1 (6) 6π (6) 6 c1(B ) − c1(Q) = 0 ; c1(P) ^ c1(P) + c1(Q) ^ c1(Q) = 0 ; Geometry and supersymmetry Heterotic Horizons IIB Horizons M-Horizons Examples Explicit examples of horizon section S8 include 3 3 2 I S × S × T I SU(3) 1 3 I S × S × K3 Geometry and supersymmetry Heterotic Horizons IIB Horizons M-Horizons S as T4-fibrations Suppose that B6 is a holomorphic T2 fibration over a 4-d Kähler manifold X4 with Kähler form κ. Thus S is a holomorphic T4-fibration over X4, the conditions that should be solved to find solutions are k2 dh1 ^ κ = 0 ; dh2 ^ κ = 0 ; dh3 ^ κ = 0 ; d` ^ κ = − e2Φκ2 ; 2 −i@@¯log det(iκ) − d` = 0 ; dκ = 0 ; dh1 ^ dh1 + dh2 ^ dh2 + dh3 ^ dh3 + d` ^ d` + 2k2i@@¯e2Φ ^ κ = 0 ; where h1 = h, and h2 and h3 are along the two new fibre directions. 4 I These are 6 equations for 6 unknown functions on X . I One of the equations is the Monge-Amperé equation but the system cannot be separated. Geometry and supersymmetry Heterotic Horizons IIB Horizons M-Horizons The associated co-homological system is c1(P1) ^ [κ] = 0 ; c1(P2) ^ [κ] = 0 ; c1(P3) ^ [κ] = 0 ; k2 c (Q) ^ [κ] = − [e2Φκ2] ; c (X) − c (Q) = 0 ; 1 4π 1 1 3 X c1(Pr) ^ c1(Pr) + c1(Q) ^ c1(Q) = 0 : r=1 Suppose that there are manifolds X4 such that the cohomological system has solutions. Question: Are these cohomological conditions necessary and sufficient for the existence of heterotic horizons? Geometry and supersymmetry Heterotic Horizons IIB Horizons M-Horizons X4 = S2 × S2 Take as metric and hermitian form on S = S3 × S3 × T2 2 3 2 1 2 2 2 3 2 1 2 2 2 1 2 2 2 ds(8) = (σ ) + (σ ) + (σ ) + (ρ ) + (ρ ) + (ρ ) + (τ ) + (τ ) ; 3 3 1 2 1 2 1 2 !(8) = −σ ^ ρ − σ ^ σ − ρ ^ ρ − τ ^ τ ; S3 × S3 × T2 is not (conformally) balanced.
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