Happy Birthday Happy Birthday Nigel Nigel The c-map with S. Vandoren, C. Vafa; B. Pioline in progress. Background: B. de Wit, N. Hitchin, U. Lindström math.dg/0603048 What is it? What is a nice way to describe it? An example of how sigma models teach us geometry.
String theory tells us that when we compactify strings on an internal manifold X, the moduli arise as scalar fields in an effective theory in the spacetime M: X M
The nonlinear sigma model these scalar fields parameterize has the geometry of the moduli space. Concretely, we consider compactifying type II strings on a Calabi-Yau 3-fold.
Then the effective theory has N=2 local supersymmetry, and the scalars are components of hypermultiplets or vector multiplets, depending on whether we consider complex structure or Kähler moduli, and on whether we consider type IIA or IIB strings.
Local N=2 supersymmetry restricts the geometry of vector multiplets and hyper- multiplets: The scalars of vector multiplets para- meterize a special Kähler geometry, whereas the scalars of hypermultiplets parameterize a Quaternion Kähler geometry.
How can this be reconciled? The two are very different geometries– Quaternion Kähler manifolds in general aren’t even complex! The answer is the c-map.
To understand it, we need some back- ground material: •Duality •Dimensional reduction •Supersymmetry and superspace •The conformal formalism.
Duality (in D-dimensions): Interchange Bianchi identities and field equations: for F p+1 = dA p , Bianchi: dF p+1 = 0 Field Equation: d(*F) D-(p+1) = 0 Dual: for (*F) D-(p+1) = dB D-(p+2) roles of B.I. and F.E. interchanged.
In particular, in D=4: A 0 <-> B 2 , A 1 <-> B 1 in D=3: A 1 <-> B 0 Questions?
Dimensional Reduction: A p -> (A p ,A p-1 ) D=4 -> D=3: A 0 -> A 0 A 1 -> A 1 ,A 0 A 2 -> A 2 ,A 1 But F 3 = dA 2 is nondynamical in D=3, so really: A 2 -> A 1
Supermultiplets have bosons and fermions that are related by super- symmetry. We are interested in the bosonic content of the supermulti- plets. In D=4, we consider N=1 and N=2 supersymmetry, where N counts the number of spinorial symmetry generators.
D=4 N=1: Bosonic content Scalar multiplet: 1 complex scalar A 0 Vector multiplet: 1 connection 1-form A 1 Tensor multiplet: 1 real scalar C 0 and 1 2-form A 2 The tensor multiplet is dual to the scalar multiplet because A 2 <-> A 0
Comment: Note that the vector multiplet and the tensor multiplet both reduce to the same thing in D=3: A 1 -> A 1 ,A 0 and A 2 ,C 0 -> A 1 ,C 0 Duality? in D=3, A 1 <-> A 0 , so both are dual to the scalar multiplet.
D=4 N=2: Vector Multiplet: N=1 Vector, N=1 Scalar: A 1 ,A 0 ,A 0 * Hypermultiplet: 2(N=1 Scalar): 2(A 0 ,A 0 *) Tensor Multiplet: N=1 Tensor, N=1 Scalar: A 2 ,C 0 ,A 0 ,A 0 *
As for the N=1 case, Vector and Tensor muliplets become the same in D=3, and both are dual to the hypermultiplet in D=3.
The rigid c-map (relates rigid special geometry to hyperkähler geometry): Take N=2 vector multiplet, descend to D=3, and then go back to D=4, reinterpreting the multiplet as an N=2 tensor multiplet. Dualize to a hyper- multiplet either in D=3 or D=4.
Superspace–a flash review: Bosonic and fermionic coordinates. Algebra of derivations; N=1: Chiral superfields
correspond to holomorphic coor- dinates on a Kähler manifold. The superspace Lagrange density is the Kähler potential:
N=2: Write D a , a=1, 2 to make explicit the They obey the algebra: Vector multiplets are described by chiral superfields obeying a reality condition:
Their superspace Lagrange density is given by a holomorphic function F(W) called the prepotential and gives rise to special Kähler geometry with a Kähler potential determined by the pre- potential:
Hypermultiplets and tensor multiplets require projective superspace and twistors (also useful for vector multi- plets); we paramaterize the maximal anticommutative subalgebra of the spinor derivatives by a sphere:
where is the inhomogeneous coordinate on . These derivatives are projectively real with respect a real structure given by complex conjugation composed with the antipodal map. The tensor multiplet is described as a projectively real O(2) section:
Its superspace Lagrange density is a con- tour integral over the sphere of an arbit- rary function:
If we dualize the tensor to a scalar, we get a hyperkähler manifold with Kähler potential given by the Legendre trans- form: End of lightning review–back to more more comprehensible things!
Rigid or classical c-map: F is the prepotential of special geometry; f determines the Kähler potential of the corresponding hyperkähler geometry.
The rigid c-map is not particularly interesting; the interpretation in terms of moduli spaces of Calabi-Yau manifolds only arises in the context of supergravity. Ferrara and Sabharwal gave the explicit form of the metric that follows from the c-map with supergravity–but it is rather messy.
Can we find a description as compact and simple as in the rigid case? Conformal formalism: The Einstein action is not invariant under Weyl rescalings g(x) -> w(x) g(x) Introduce a scalar compensating field:
This is tautologically invariant under and reduces to the Einstein action when we use the Weyl rescalings to set
In the context of supergravity, this is not so trivial. In particular, when we couple hypermultiplets to supergravity, the analog of is a hypermultiplet which combines with the other hypermultiplets to make the Swann space or hyper- kähler cone of the underlying Quater- nion Kähler manifold.
The hyperkähler cone (HKC) of a Quaternion Kähler manifold is a cone above an SU(2) bundle whose connection is the Sp(1) connection of the Quaternion Kähler manifold. Because it is hyperkähler, it can be described in terms of the same superfields as in the rigid case.
The SU(2) rotates the Quaternionic structure and is a superpartner of the Weyl symmetry; to go from the conformal formalism to the usual supergravity formalism, we fix the SU(2) as well as the Weyl symmetry (which scales up and down the cone).
There is also a U(1) symmetry that doesn’t act on the hypermultiplets, but acts on the vector multiplets and is compensated by an extra vector multiplet. The HKC has many uses; in particular it gives the simplest descrip- tion of the Quaternion Kähler quotient.
Hyperkähler Cone U(1) Kähler Quotient 3-Sasakian U(1) Twistor Space SU(2) Quaternion Kähler We now have the tools to give a simple description of the c-map:
In the conformal formalism, the vector multiplet prepotential F is homo- genous of degree 2; one of the multiplets can be interpreted as the U(1) compensator. The f -function describing the HKC is homogeneous of degree 1.
The moduli space has dimension depending on whether we are considering Kähler or complex structure deform- ations. On the special Kähler (vector) side, there are n +1 complex scalars W , where the extra 1 is the compensator.
On the HKC (hypermultiplet) side, there are n +2 tensor multiplets , where 1 of the extra 2 is the compensator, and the other 1 comes from through the c-map from the supergravity fields themselves. The function f has the following simple form:
The hyperkähler potential of the HKC is just the Legendre transform of the resulting Lagrange density L :
If we work out K in terms of F , we find a form that has a remarkable resemblance to the formula for the entropy of a black hole proposed by Ooguri, Strominger, and Vafa (OSV); however they appar- ently had a different Legendre transform. It turns out that it is related to the one we have written down simply by an SU(2) rotation! Thus our conformal formalism
appears to reveal some new and mysterious relation to Black Hole entropy and Quaternion Kähler geometry! This is work still very much in progress. Thank you and once more– a very Happy Birthday, Nigel.
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