Preprint typeset in JHEP style - HYPER VERSION. Special Geometry. Yang Zhang. Abstract: N = 2 Supergravity. based on hep-th/ , Boris PiolineA

Transcription

1 Preprint typeset in JHEP style - HYPER VERSION Special Geometry Yang Zhang Abstract: N = Supergravity based on hep-th/06077, Boris PiolineA

2 Contents 1. N = Supergravity Supersymmetric multiplets 1 1. Special geometry 1.3 4D action Central charge 4. Calabi-Yau 4 1. N = Supergravity 1.1 Supersymmetric multiplets We consider D = 4, N = supergravity. The supersymmetric multiplets with spin less or equal than are, One gravity multiplet, containing the graviton g µν, two gravitini ψ α µ and one Abelian gauge field A µ known as the graviphoton. (, 3, 1) + (+1, + 3, +) (1.1) n V vector multiplet, each consisting of one Abelian gauge field A µ, two gaugini λ α and one complex scalar z. The complex scalars z take values in a projective special Khler manifold M V of real dimension n V. ( 1, 1, 0) + (0, + 1, +1), n V copies (1.) n H hypermultiplets, each consisting of two complex scalars and two hyperinis ψ, ψ. The scalars take values in a queternionic-khler space M h of real dimension 4n H. ( 1, 0, 1 ) + ( 1, 0, 1 ), n H copies (1.3) We consider the ungauged N = supergravity, i.e., the hypermultiplets is not charged by the vector multiplets. The special geometry describes the vector multiplet. 1

3 1. Special geometry The coupling of the vecot multiplets, including the geometry of the scalar manifold M V, are conveniently described by means of a Sp(n V + ) principal bundle ɛ over M V, and its associated bundle ɛ V in the vector representation of Sp(n V + ). The origin of the symplectic symmetry lies in electric-magnetic duality, which mixes the n V vectors A µ and the graviphoton A µ together with their magnetic duals. Denoting a section Ω by its coordinates (X I, I ), (I = 0,..., n V +1), the antisymmetric product Ω, Ω = ( ) ( ) ( ) X I 0 1 X J J = X I I X I I (1.4) 1 0 The symplectic form is dω, dω = dx I d I. The geometry of M V is completely determined by a choice of a holomorphic section Ω(z) = (X I (z), I (z)) taking value in a Lagrangian cone, i.e. a dilation invariant subspace such that dx I d I = 0. The special geometry constraint is, I J X I I X I J I = 0, (1.5) where J = 1,...n V + 1 and the derivatives are in the projective coordinates of M V. urthermore, we may choose the X I as the projective coordinates, hence (1.5) is simplified to, So we can define the prepotential = 1 XJ J such that, I = X J J X I. (1.6) I = X I. (1.7) The prepotential is an homogeneous function of degree in the X I. The Hessian of the prepotential, τ IJ = I J, is independent of X I. Hence I = τ IJ X J. (1.8) At a generic point on M V, we can choose the special coordiates z i = X i /X 0 (i = 1,..., n V ) as the holomorphic coordinates for M V. Once the holomorphic section Ω(z) is given, the metric on M V is obtained from the Kähler potential, ( ) ( ) K(z i, z i ) = log i Ω, Ω = log i( X I I X I I ). (1.9) The metric g i j = i jk = ie K i Ω, j Ω i K j K (1.10)

4 Under a Kähler transformation, Ω e f(z) Ω, K K f(z) f( z), (1.11) and the metric is invariant. We may define the rescaled holomorphic section as, Ω = e K/ Ω, (1.1) which transforms by a phase, Ω e (f f)/ Ω under the Kähler transformation. The derived section is defined by U i = D i Ω = (f I i, h ii ), where f I i = e K/ D i X I = e K/ ( i X I + i KX I ) (1.13) h ii = e K/ D i h I = e K/ ( i I + i K I ). (1.14) Hence the metric is g i j = i U i, Ū j. (1.15) 1.3 4D action The kinetic term of the n V + 1 Abelian gauge fields (including the graviphoton) is (I = 0,..., n V ), where N IJ is defined to be which satisfies these relations, L Maxwell = (ImN IJ ) I J + (ReN IJ ) I J (1.16) N IJ = τ IJ + i (Imτ X) I(Imτ X) J. (1.17) X Imτ X I = N IJ X J, h ii = N IJ f J i. (1.18) Note that N IJ has X dependence, so the coupling constants of the 4D action depend on the vector multiplets moduli. ImN IJ is a negative definite matrix, as required for the positive definiteness fo the gauge kinetic terms. We may absorb the Yang-Mills angle terms as, where I = ( I i I )/. The dual field of I;µν is, 1 L Maxwell = Im[ N IJ I J ] (1.19) G I = 1 L Maxwell = (ReN ) I IJ J + (ImN ) IJ J (1.0) 1 The functional derivative should be viewed as the formal derivative in I, not I,µν. 3

5 Under the symplectic transformation, ( ) X ( A B C D ) ( ) X (1.1) N transforms as period matrix N (C + DN )(A + BN ) 1, while the field strengths ( I, G I = N IJ µν J ) transform as a symplectic vector, ( ) ( ) ( ) A B. (1.) C D G The 4D action can be rewritten as G which is invariant under the simplectic transformation. 1.4 Central charge The field strength of the graviphoton is, L Maxwell = Im(G I I ), (1.3) T µν = ie K/ X I (ImN ) IJ J = e K/ (X I G I I I ). (1.4) The charges associated to T µν measured at infinity, is the central charge in N = supersymmetry algebra, where i, j = 1,. The BPS bound is, Z = e K/ (q I X I p I I ), (1.5) {Q i α, Q αj } = P µ σ µ α α δi j, {Q i α, Q j β } = Zɛij ɛ αβ, (1.6) M Z. (1.7). Calabi-Yau 4