Special Kahler Manifold Research Articles

Collapsing hyperkähler manifolds

Given a projective hyperkahler manifold with a holomorphic Lagrangian fibration, we prove that hyperkahler metrics with volume of the torus fibers shrinking to zero collapse in the Gromov-Hausdorff sense (and smoothly away from the singular fibers) to a compact metric space which is a half-dimensional special Kahler manifold outside a singular set of real Hausdorff codimension 2 and is homeomorphic to the base projective space.

Partial breaking in the rigid limit of $\mathcal{N}=2$ gauged supergravity

Using a new manner to rescale fields in $\mathcal{N}=2$ gauged supergravity with n$_{V}$ vector multiplets and n$_{H}$ hypermultiplets, we develop the explicit derivation of the rigid limit of quaternionic isometry Ward identities agreeing with known results. We show that the rigid limit can be achieved, amongst others, by performing two successive transformations on the covariantly holomorphic sections $V^{M}\left( z,\bar{z}\right) $ of the special Kahler manifold: a particular symplectic change followed by a particular Kahler transformation. We also give a geometric interpretation of the $\eta_{i}$ parameters used in arXiv:1508.01474 to deal with the expansion of the holomorphic prepotential $\mathcal{F}\left( z\right) $ of the $\mathcal{N}=2$ theory. We give as well a D- brane realisation of gauged quaternionic isometries and an interpretation of the embedding tensor $\vartheta_{M}^{u}$ in terms of type IIA/IIB mirror symmetry. Moreover, we construct explicit metrics for a new family of $4r$- dimensional quaternionic manifolds $\boldsymbol{M}_{QK}^{\left( n_{H}\right) }$ classified by ADE Lie algebras generalising the $SO\left( 1,4\right) /SO\left( 4\right) $ geometry which corresponds to $A_{1}\sim su\left( 2\right) $. The conditions of the partial breaking of $\mathcal{N}=2$ supersymmetry in the rigid limit are also derived for both the observable and the hidden sectors. Other features are also studied.

Kenmotsu statistical manifolds and warped product

A notion of a Kenmotsu statistical manifold is introduced, which is locally obtained as the warped product of a holomorphic statistical manifold and a line. A statistical manifold is a Riemannian manifold equipped with a torsion-free affine connection satisfying the Codazzi equation. It can be considered as being in information geometry, Hessian geometry and various submanifold theory. On the other hand, a Kenmotsu manifold is in a meaningful class of almost contact metric manifolds. In this paper, we construct a suitable statistical structure on it. Although the notion of the warped product of Riemannian manifolds is well known, the one for statistical manifolds is not established. We consider it in general, and study the statistical sectional curvature of the warped product of two statistical manifolds. We show that a Kenmotsu statistical manifold of constant $$\phi $$ -sectional curvature is constructed from a special Kahler manifold, which is an important example of holomorphic statistical manifold. A Sasakian statistical manifold is also studied from the viewpoint of the warped product of statistical manifolds.

BPS black holes in a non-homogeneous deformation of the stu model of N = 2, D = 4 gauged supergravity

We consider a deformation of the well-known stu model of N = 2, D = 4 supergravity, characterized by a non-homogeneous special Kahler manifold, and by the smallest electric-magnetic duality Lie algebra consistent with its upliftability to five dimensions. We explicitly solve the BPS attractor equations and construct static supersymmetric black holes with radial symmetry, in the context of U(1) dyonic Fayet-Iliopoulos gauging, focussing on axion-free solutions. Due to non-homogeneity of the scalar manifold, the model evades the analysis recently given in the literature. The relevant physical properties of the resulting black hole solution are discussed.

Static BPS black holes in AdS4 with general dyonic charges

We complete the study of static BPS, asymptotically AdS$_4$ black holes within N=2 FI-gauged supergravity and where the scalar manifold is a homogeneous very special Kahler manifold. We find the analytic form for the general solution to the BPS equations, the horizon appears as a double root of a particular quartic polynomial whereas in previous work this quartic polynomial further factored into a pair of double roots. A new and distinguishing feature of our solutions is that the phase of the supersymmetry parameter varies throughout the black hole. The general solution has $2n_v$ independent parameters; there are two algebraic constraints on $2n_v+2$ charges, matching our previous analysis on BPS solutions of the form $AdS_2\times \Sigma_g$. As a consequence we have proved that every BPS geometry of this form can arise as the horizon geometry of a BPS AdS$_4$ black hole. When specialized to the STU-model our solutions uplift to M-theory and describe a stack of M2-branes wrapped on a Riemman surface in a Calabi-Yau fivefold with internal angular momentum.

Superconformal quantum mechanics and the discrete light-cone quantisation of N = 4 $$ \mathcal{N}=4 $$ SUSY Yang-Mills

We study the quantum mechanical sigma model arising in the discrete light-cone quantisation of N=4 supersymmetric Yang-Mills theory. The target space is a certain torus fibration over a scale-invariant special Kahler manifold. We show that the expected SU(1,1|4) light-cone superconformal invariance of the N=4 theory emerges in a limit where the volume of the fibre goes to zero and give an explicit construction of the generators. The construction given here yields a large new family of superconformal quantum mechanical models with SU(1,1|4) invariance.

BPS black hole horizons in N=2 gauged supergravity

We study static BPS black hole horizons in four dimensional N=2 gauged supergravity coupled to $n_v$-vector multiplets and with an arbitrary cubic prepotential. We work in a symplectically covariant formalism which allows for both electric and magnetic gauging parameters as well as dyonic background charges and obtain the general solution to the BPS equations for horizons of the form $AdS_2\times \Sigma_g$. In particular this means we solve for the scalar fields as well as the metric of these black holes as a function of the gauging parameters and background charges. When the special Kahler manifold is a symmetric space, our solution is completely explicit and the entropy is related to the familiar quartic invariant. For more general models our solution is implicit up to a set of holomorphic quadratic equations. For particular models which have known embeddings in M-theory, we derive new horizon geometries with dyonic charges and numerically construct black hole solutions. These correspond to M2-branes wrapped on a Riemann surface in a local Calabi-Yau five-fold with internal spin.

Geometric Construction of the r-Map: From Affine Special Real to Special Kähler Manifolds

We give an intrinsic definition of (affine very) special real manifolds and realise any such manifold M as a domain in affine space equipped with a metric which is the Hessian of a cubic polynomial. We prove that the tangent bundle N = TM carries a canonical structure of (affine) special Kahler manifold. This gives an intrinsic description of the r-map as the map \({M \mapsto N=TM}\) . On the physics side, this map corresponds to the dimensional reduction of rigid vector multiplets from 5 to 4 space-time dimensions. We generalise this construction to the case when M is any Hessian manifold.

A note on tt *-bundles over compact nearly Kähler manifolds

First, we generalize a rigidity result for harmonic maps of Gordon (Gordon (1972) Proc AM Math Soc 33: 433–437) to generalized pluriharmonic maps. We give the construction of generalized pluriharmonic maps from metric tt*-bundles over nearly Kahler manifolds. An application of the last two results is that any metric tt*-bundle over a compact nearly Kahler manifold is trivial (Theorem A). This result we apply to special Kahler manifolds to show that any compact special Kahler manifold is trivial. This is Lu’s theorem (Lu (1999) Math Ann 313: 711–713) for the case of compact special Kahler manifolds. Further we introduce harmonic bundles over nearly Kahler manifolds and study the implications of Theorem A for tt*-bundles coming from harmonic bundles over nearly Kahler manifolds.

Twistors and black holes

Motivated by black hole physics in N=2, D=4 supergravity, we study the geometry of quaternionic-Kahler manifolds M obtained by the c-map construction from projective special Kahler manifolds M_s. Improving on earlier treatments, we compute the Kahler potentials on the twistor space Z and Swann space S in the complex coordinates adapted to the Heisenberg symmetries. The results bear a simple relation to the Hesse potential \Sigma of the special Kahler manifold M_s, and hence to the Bekenstein-Hawking entropy for BPS black holes. We explicitly construct the ``covariant c-map'' and the ``twistor map'', which relate real coordinates on M x CP^1 (resp. M x R^4/Z_2) to complex coordinates on Z (resp. S). As applications, we solve for the general BPS geodesic motion on M, and provide explicit integral formulae for the quaternionic Penrose transform relating elements of H^1(Z,O(-k)) to massless fields on M annihilated by first or second order differential operators. Finally, we compute the exact radial wave function (in the supergravity approximation) for BPS black holes with fixed electric and magnetic charges.

Hypersurfaces with almost complex structures in the real affine space

We study affine hypersurface immersions $f: M \rightarrow {\mathbb R}^{2n+1}$, where $M$ is an almost complex $n$-dimensional manifold. The main purpose is to give a condition for ($M,J$) to be a special Kähler manifold with respect to the Levi-Civita

Tt*-Geometry and Pluriharmonic Maps

In this paper we use the real differential geometric definition of a metric (a unimodular oriented metric) tt*-bundle of Cortes and the author (Topological-anti-topological fusion equations, pluriharmonic maps and special Kahler manifolds) to define a map Φ from the space of metric (unimodular oriented metric) tt*-bundles of rank r over a complex manifold M to the space of pluriharmonic maps from M to {GL}(r)/O(p,q) (respectively {SL}(r)/SO(p,q)), where (p,q) is the signature of the metric. In the sequel the image of the map Φ is characterized. It follows, that in signature (r,0) the image of Φ is the whole space of pluriharmonic maps. This generalizes a result of Dubrovin (Comm. Math. Phys. 152 (1992; S539–S564).

Special geometry of euclidean supersymmetry II. Hypermultiplets and thec-map

We construct two new versions of the c-map which allow us to obtain the target manifolds of hypermultiplets in Euclidean theories with rigid N =2 supersymmetry. While the Minkowskian para-c-map is obtained by dimensional reduction of the Minkowskian vector multiplet lagrangian over time, the Euclidean para-c-map corresponds to the dimensional reduction of the Euclidean vector multiplet lagrangian. In both cases the resulting hypermultiplet target spaces are para-hyper-Kahler manifolds. We review and prove the relevant results of para-complex and para-hypercomplex geometry. In particular, we give a second, purely geometrical construction of both c-maps, by proving that the cotangent bundle N=T^*M of any affine special (para-)Kahler manifold M is para-hyper-Kahler.

Special Geometry of Euclidean Supersymmetry I: Vector Multiplets

We construct the general action for Abelian vector multiplets in rigid 4-dimensional Euclidean (instead of Minkowskian) N=2 supersymmetry, i.e., over space-times with a positive definite instead of a Lorentzian metric. The target manifolds for the scalar fields turn out to be para-complex manifolds endowed with a particular kind of special geometry, which we call affine special para-Kahler geometry. We give a precise definition and develop the mathematical theory of such manifolds. The relation to the affine special Kahler manifolds appearing in Minkowskian N=2 supersymmetry is discussed. Starting from the general 5-dimensional vector multiplet action we consider dimensional reduction over time and space in parallel, providing a dictionary between the resulting Euclidean and Minkowskian theories. Then we reanalyze supersymmetry in four dimensions and find that any (para-)holomorphic prepotential defines a supersymmetric Lagrangian, provided that we add a specific four-fermion term, which cannot be obtained by dimensional reduction. We show that the Euclidean action and supersymmetry transformations, when written in terms of para-holomorphic coordinates, take exactly the same form as their Minkowskian counterparts. The appearance of a para-complex and complex structure in the Euclidean and Minkowskian theory, respectively, is traced back to properties of the underlying R-symmetry groups. Finally, we indicate how our work will be extended to other types of multiplets and to supergravity in the future and explain the relevance of this project for the study of instantons, solitons and cosmological solutions in supergravity and M-theory.

Special K�hler Manifolds

We give an intrinsic definition of the special geometry which arises in global N= 2 supersymmetry in four dimensions. The base of an algebraic integrable system exhibits this geometry, and with an integrality hypothesis any special Kähler manifold is so related to an integrable system. The cotangent bundle of a special Kähler manifold carries a hyperkähler metric. We also define special geometry in supergravity in terms of the special geometry in global supersymmetry.

Special geometry and symplectic transformations

Special Kahler manifolds are defined by coupling of vector multiplets to $N=2$ supergravity. The coupling in rigid supersymmetry exhibits similar features. These models contain $n$ vectors in rigid supersymmetry and $n+1$ in supergravity, and $n$ complex scalars. Apart from exceptional cases they are defined by a holomorphic function of the scalars. For supergravity this function is homogeneous of second degree in an $(n+1)$-dimensional projective space. Another formulation exists which does not start from this function, but from a symplectic $(2n)$- or $(2n+2)$-dimensional complex space. Symplectic transformations lead either to isometries on the manifold or to symplectic reparametrizations. Finally we touch on the connection with special quaternionic and very special real manifolds, and the classification of homogeneous special manifolds.

On isometries relating noncompact sigma -models and supergravity

Some noncompact sigma -models arise in the process of the dimensional reduction of (super)gravity-matter systems. These sigma -models have to be invariant under Abelian shifts of those scalar fields that are produced by duality transformations. The authors observe that such isometries form light-like Abelian subgroups of the full sigma -model isometry groups. This fact will facilitate the study of the correspondence between some quaternionic and special Kahler manifolds (c*-map) arising in superconformal field theories.

(2,2) vacuum configurations for type IIA superstrings: N=2 supergravity Lagrangians and algebraic geometry

The compactification of D=10 type IIA supergravity (the point-field theory limit of the type IIA superstring) on a Calabi-Yau manifold ((2.2) vacuum) is performed. The resulting D=4, N=2 effective supergravity Lagrangian gives rise to an explicit construction of special Kahler manifolds for the (1,1) moduli and dual quaternionic manifolds for the (2.1) moduli. In both cases the scalar manifolds are characterized by a homogeneous holomorphic function of degree two. In the case of the (1.1) moduli, the holomorphic function is given by a cubic polynomial, a result which is valid only in the classical large volume limit (compared to the string scale) of the internal manifold. In contrast, for the (2.1) moduli the holomorphic function, which is related to the 'periods' of a holomorphic three form, is unrestricted and due to non-renormalization theorems, should coincide with the exact (tree-level) string result.